Should You Switch
Given that you don't know what X is (but only what is in your envelope), should you switch?
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1 solution
One way of estimating expected outcomes is through computer Monte Carlo simulations, i.e., randomized trials and tally the results. Here, if done this way, it makes no difference, as intuition would tell us. This is one of those times where intuition beats "mathematical reasoning and analysis"----and it should serve to advise or warn us about deficiencies in assumptions being made in such mathematical analysis. The root of the problem, or "paradox", is the exact definition of the problem. When doing this by Monte Carlo simulations, we randomize which envelope has the greater amount, then either stay with the first chosen envelope or switch after choosing, and tally the results. But, as with the example of "breaking a stick randomly into 3 pieces and seeing if they'll form a triangle", we can end up with different results depending exactly what is meant by "random" in setting up a trial.
The expected value in either envelope is
2 1 ( 2 D ) + 2 1 ( D ) = 2 3 D
even after you've picked the first. The flaw in the posted argument is that it assumes that after the first is opened, it reveals something about the value of D . It does not, D is still unknown, we've gained no new information after the first is opened. Contrast with what happens if we're told what the value of D is.
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Yes. It's actually interesting to compare this with the Monty Hall problem and show why for that problem there is new information received anyway.
In fact the observation that the probability depends on the information received after making some choice is key in reasoning about any such sort of problem/puzzles. It's pretty complicated understanding probability by empirical means , like veryfing a large number of cases as in the Monte Carlo simulations. As such that will not be a very convincing argument as it's line of reasoning is pretty fine and subtle. Of course because it will show that there will be no advantage in some long period time running a person who didn't believed at first that the chances are equal will be finally obliged to accept the reasoning as highly possible right but hardly anyway. The only real convincing way is theoretical rigorous understanding of probability and moreover that is the only way of really understanding probability anyway.
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What I'm actually arguing about using Monte Carlo methods is their utility in understanding just what's involved when attempting to understand a problem having probabilities. In other words, an useful guide in avoiding the blind alleys, just by understanding what you are really doing when setting up for a Monte Carlo run.
It's a kind of a personal philosophy of mine----in science, the importance of observational, experimental validation, and replication is stressed. Well, in mathematics, if a thing can't be replicated by computer, there's probably reason to be suspect. In the not-so-far future, we're going to have A.I.s What happens if human mathematicians try to explain a particularly abstruse theory in mathematics, and they can't get any of the A.I.s to understand or replicate it? Then should we worry? I think we should.
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@Michael Mendrin – Interesting , it seems to me that you are elaborating your reasoning based on the reductionist attempt of describing mathematics by means of formally omputable. I think I can see what are the innate principal thinking of your personal philosophy and sense of things implied in this type of thinking but I am not sure anyway.
It may have to do with extending conceptually the now implicitly felt reflex and idea that the world can be understood by means of mathematical modeling this principial idea being extended intuitively and without realizing from the empirical sciences in which it firstly appeared (in the modern sense) to the formal ones , as it actually happened with the evolution of sciences at the beginning and late XX and XIX centuries I think which has as object of study , the more abstract and subtle matter of the forms of thought themselves that can be as well modeled and characterized formally in the same/slightly modified type of mathematical thinking. I pretty much agree that , formally , mathematics as well as our reasoning implied in mathematics is reducible to computations done by machines though this is just a first impression and it may prove not to be completely consistent when properly investigated but it seems something amiss at this idea of reducing mathematics as well as all it's underlying concepts to a model of computation maybe because apart from being rightly characterized formally our thinking has to some extent more than jsut that formal criterion but I'd really like to hear more on your opinions regarding the nature of mathematics anyway.
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@A A – It's already become a common practice to test theories in physics by computer simulations. Partly it's for the practical reasons of not being able to perform actual experiments (such as replicating supernovas, or even nuclear weapons in an age of prohibited nuclear weapon testing), but also partly because such computer simulations have become a new paradigm of testing or confirming hypotheses and the mathematics that go with them. What if a new theory in physics is developed that is in principle cannot ever be tested by such computational means? Then I think we're entering in the realm of philosophy, because that is close to saying it's an untestable hypothesis.
Is mathematics exempt from this? I don't think so. Ultimately, all theories and branches of mathematics should or ought to be repeatably computable. But that's just my personal opinion. If a mathematician can't teach his theories to A.I. to reliably replicate his conclusions, then I think it's his job to show that it's more than just a philosophy. But let it be understood that we are assuming that A.I. one day can match humans in cognitive ability "to do the math". I know that many will deny this possibility.
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@Michael Mendrin – I'd really like to see how such a theory would look like as certainly that is rather a very unfounded and undeserving theoretical construct for the name of theory haha. If however it has it's justification in an a priori solid and rigorous proof which that should be at least in principle enough to assure the thinking of it concerns is right.
I think the part that all theories and branches of mathematics are repeatably computable" should be theoretically justified ,we should firstly understand why it is so theoretically solidly the question being what is actually that makes the structure of mathematics in it's intimate design ultimately reducible to computatin. If this question is deepened correctly or at least partially you can have a better understanding of what makes you assert, and as such become more conscious of the sources that makes you have the opinion, that mathematics is reducible to mechanical computation to enforce or weaken that opinion anyway. Nonetheless what is important is that it enhances that actual understanding of the problem. Therefore in case you just took it as a mere opinion I advice you elaborate on this if you are interested and not let it just at the level of not articulate opinion as besides giving the possibility of new insight in it you may also understand so to say better what you are already thinking and even know yourself better anyway.
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@A A – Well, as a classic example, there's plenty of pseudo mathematics such as "specified complexity", which cannot be reliably replicated by either computers or humans, because the terms involved are so poorly defined. Those are usually roundly ridiculed by the mathematical community, but every once in a while, I do run across a paper published in a mathematical journal that makes me raise an skeptical eyebrow.
The distinction between sound mathematics and pseudo mathematics isn't as clear-cut as much as we'd like to believe, even though the vast number of papers and theories that have already been vetted by the mathematical community are sound.
I'm just making the argument that one measure we can use to judge the validity or meaningfulness of a theory in mathematics is whether or not it makes sense to a computerized A.I.. We're not here yet, we don't have A.I.s that can function at the level of the mathematical community today, but I expect that someday we will.
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@Michael Mendrin – I haven't heard of it yet but , by the way it sounds ,it seems rather to be an improper way of thinking at the object of inquiry of the study and also a superficial claim of such a stuff being a genuine object of study anyway than necessarily related on wrong reasoning. Maybe most such papers are nothing but the result of a poor representation of what is actually rightly defined as true mathematical concepts and if they have some truth in them responsable for rising the eyebrow that truth isn't correctly expressed in authentic terms anyway.
I prefer believing we can distinguish what is true from what is false mathematics and reasoning just by the level of understanding it produces in conceiving it. Almost always an improper representation of things in understanding produces a false note on thinking it's predication which let's something incomplete anyway.
Yes , I understood your argument but I identified that for considering computers as capable of verifying the validity of any mathematical proposition and measuring their degree of accuracy and correctness you have to think that innately mathematics it's reducible to such a formal model of mechanical computation. Just by being reducible to it in truth , and when saying "reducible" I meant ultimately being such a model in it's a priori "content" and as such just by having a vision related to the internal structure and nature of mathematical reasoning as being a form of computation can this point be sustained at the extent at which you propose computers being tools for verifying the validity and value of mathematics I think.
In other words just that is the principle which underlies and , by being a principle, continuously animates at every step of thinking the idea you proposed. By intuitively observing this I thought that , because this reduction claim is the constant heart of the argument proposed you should deepen it to verify and understand it's validity. As such I argued about this (type of) thinking , the thinking in the idea that human thinking is simply mechanical computation and more importanly about the depths which gives reason to this thinking speaking of it's underlying principle saying that once identified it has to be inquired more into what justifies it or in other words into what are the sources that gives content and made you consider that this underlying idea is correct in an articulate way.I understood your argument as meaning that because mathematics and human thinking reducible at the model of a machine of computation ,they are reducible because they ultimately are such machines and therefore that if for advanced computers which are , by this idea anyway , likely to advance to the point of having sort of a thinking as we do , though being more efficient and more accurate in making their computations some mathematical proposition is not understandable it is so because it lacks the true soundness which would so to say otherwise be found by the computer's running program anyway. I suppose this is what you meant by the fact that computer's capacity of understanding is a measure to verify the validity and a such internal coherence or soundness of mathematical statements but may have been wrong. I kind of doubt I misunderstood your statements though anyway.
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@A A – Well, yes, I am saying that ultimately "mathematical truth" should or ought to be "reducible" to verification by machine computation. I know that sounds harsh and maybe even demeaning of the mathematical spirit, but I was very careful to qualify all I have said by saying that it's just a personal opinion of mine. I would never think of publishing such a mere "opinion" in any journal.
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@Michael Mendrin – Great , I though that's what you are saying. You have a great subtility of thought. I understand it's a personal opinion but my point was that , as any personal opinion or most of them it has some reasoning , even if not always acutely seen underlying it and as such that as almost anything in this world there is either a correct or flawed reasoning regarding it. Because what convinces you is the traces of this reasons which are the sources of your convictions I adviced you to make it and identify those traces that make you think like that acutely , in other words to switch from just an unarticualted opinion to understanding explicitly it's reason. Because this traces of reason are responsible for your convictions in their is the heart of the things that actually made you think mathematics like that. Therefore once identified I think you can understand your thinking better and why it is or it isn't so and as such I adviced you to look for that actual reason that makes amthematcis be like that or not which is unclearly conceived by you when you say you have an opinion because that reason it's the actual source of your opinion and therefore that's the actual point , the actual problem to understand.
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@A A – If I could actually prove that mathematics is reducible to verification by machine computation, I think I would have a paper that warrants publication, on par with something like Godel's Theorem. I think I'm a long ways from there.
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@Michael Mendrin – I understand what you mean but I think that thing you are making some steps in completing your understanding of the matter. Even if you will not be able to completely solve it would still significantly improve your understanding of the subject by making it articulate and besides making a progress for your own thinking maybe also you could make some for the mathematical community itself.
I know it sounds very grand anyway. My point is still that because what makes you have this opinion are after all some sort of not articulated or incompletely expressed reasons that ultimately it's still this reasons that convinced you over time and maybe without completely being conscious of it anyway of what you sustain and as such that because of this it's still in those reasons where you should look as it's ultimately reason that is the source of the conviction. It's just for refreshing belief in reason and sense I guess. As such the point was that what you should look is understanding more this conviction/personal philosophy and what justifies it anyway.
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@A A – Okay, the first step is that, like the ideal in scientific methodology, findings in mathematics should be repeatable. That's one of the hallmarks of machine computation. If a computer should compute the same thing several times and get different results (without deliberately introducing randomization), then that's not so useful, and we would suspect that something is broken that somehow introduces variability. Even a flawed computer software will repeatedly give the same flawed results (a broken clock is reliably correct twice a day). Likewise, it's poor mathematics if a theorem cannot be reliably proven to be true (or false). So, that's the connection between good mathematics and machine computation.
I can envision another kind of mathematics where a particular theorem could both be true or false, much like the famous dead-and-alive cat in that Schrodinger thought experiment. What kind of a mathematics would that be, where a theorem could both be true or false in some superposition state, which itself is influenced by some other finding? And there is never any definite answer? From some of the stuff I've seen, maybe we're there already.
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@Michael Mendrin – Isn't rather that we have to search in "repatability" the characteristic or in other words in what makes or is responsible for both the machines and mathematics repeat the same result if it is to be consistent anyway ?
I thought more of the connection between mathematics and mechanical computation as being resulting in seeing that the logic and reasoning on which it is based as sort of a computation which takes based on some axioms, some strings of predication and computes them according to the axioms deriving those initial strings into other strings this being theoretically something which can be reduced as a mechanical model. Now , having this model offered as a preliminary understanding we can ask about the fact that mathematics is repeatable. Therefore we in this way should prove I think that mathematics should be in any of it's predications consistently deriving only 1 result according to axioms and better understand the claim that mathematics is repeatable as such ahving a preliminary theoreticial construct anyway.
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@A A – Okay, there's actually much more to say about that, but there's a parallel between "the art of doing mathematics", and "the art of coding for computers". With the latter, a serious and still unresolved issue is related to Turing's Halting Problem, which is, "can and will this code blow up and cause the computer to go off rails?" It's every programmer's nightmare, and there is still no proven method for determining whether or not a particular code won't fail in that way. So, to combat this problem, restrictive "recommended procedures", much like GAAP (Generally Accepted Accounting Practices) in accounting, are promulgated to limit coding to those that will reliably not fail in that manner. I can see something similar being done for mathematics, but, you know, it would have the result of at least stifling creativity in mathematics.
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@Michael Mendrin – So ,if I understand correctly , you are saying that because there is the risk of mathematics not being able to work at some point we should ellaborate such restrictive recommended procedures that allow mathematics to surely work consistently in the same way it's being done in programming so to say anyway. Yet I do agree that you should elaborate at least a little what you mean that mathematics doesn't work or can't be consistent in the way a computer program doesn't work because this is not clear to me at all so please elaborate that point regarding the analogy you proposed anyway. Besides by elaborating this I think should be gained some insight into mathematics being reducible at computation. It's an important part which has to be said for the argument to be understood and of course I agree that such a procedure would be too restrictive for mathematics anyway.
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@A A – Here's the rub----this kind of thinking, or ambition, has been attempted berfore---back around 1910 or so, Principia Mathematica was published, authors Alfred North Whitehead and Bertrand Russell, an effort to put all of mathematics on a strict axiomatic foundation, in a manner that would or could have been "machine computable" (this was published long before any computers). But a fellow named Kurt Godel came along and rained on their parade, with his undecidability theorems. Mathematics have been in a flux ever since, still groping for "rigor". I think in the field of logic, there have already been efforts much like I talked about with "producing well-behaved coding for computers". The two are closely related.
But, here, let me come around the full circle----let's not forget or lose sight of the original argument or proposal I made at the beginning. What I said was that ONCE a given branch, discipline, or set of theorems in mathematics is made computable, I say it confers it a greater legitimacy over something that isn't expressible yet in computable terms. I'm not talking about trying to create new mathematics "by computer", I'm proposing an analogy with the scientific methodology, which is, "you can hypothesize and theorize, but it's on solid ground only after obervational or experimental verification". Heretofore, mathematics was about proofs, ideally by rigorous means (which people still dispute about whether it's rigorous enough), but now that we are entering the age of computers, that is another objective measure of legitimacy we can and should use in mathematics. In other words, it's a way of "making it real". The original problem posted was about something that seems blazingly obvious intuitively speaking, and yet, "mathematical reasoning" can somehow lead us to different conclusions? I suggested the use of computerized Monte Carlo methods to "make this real", and then by examining the means of setting up such Monte Carlo trials, we're afforded better insight to what's going on with these wayward "mathematical reasonings".
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@Michael Mendrin – Well, this kind of ambition may have been tried before but almost certainly the understanding of what mathematics is and does formally isn't completed. I know of the story with Hilbert's problems and Godel disproving an appealing expectation that mathematics is computable completely by means of his incompleteness proof and of the general accepted vision that mathematics is a bunch of statements derivable from axiomatic system and that can be as such rigorously treated and thought as a model for what we are doing when we do mathematics.
But I was speaking of what makes it that mathematics can be represented in it's formal character by means of mechanical computation and trying to extend or better comprehend at least by making explicit some things in the work which others have done on it as people from the XIX and XX centuries did. In other words I was speaking of exactly that idea and more about the principle underlying it that mathematics can be made computable which is what we should question to better understand the idea of mathematics being computable as it is a principle. this is what I asked you to develop if you please because it's critical for this way of thinking and as such justifies it or not in a very great part of it , the improtant problem being that of if mathematical in truth is actually nothing but computation.
Computers are great for a lot of tedious work in mathematics but they don't offer the theoretical insight and comprehension that logical thinking offers I think. As such they just offer empirical data but not also the understanding of the empirical observations made and in the case of probability working with empirical data already is theoretically doubtful by some. Because of this doubts the problem would still therefore be better solved theoretically. As such the only way for this particular case of probability , not other domains of math ,would be the use of theoretical and analitical powers of thinking than trying to run simulations because utlimately the problem is the lack of understanding of probability and we are searching for that understanding of the data seeking which so to say by me shall never be abandoned anyway.
Your idea in rest for using computers to accumulate empirical data regarding our modelling and predictions in mathematics is right and very good I think. It does indeed shows and speaks about our age of computers and put them to a very good use especially in some new domains of mathematics and modelling systems but I think that their good use can be characterized as simply being able to verify our reasoning as such they being useful because they can compute but this if rightly programmed many verifications that can assure about the truth of our reasoning. Yet still I think you agree that we should base on our thinking even in very complicated matters about the truth of something on our own reasoning. Otherwise all that mathematicians do is to compute something beyond their powers of comprehension which is nothing but an unpassionate image which lacks a lot of the reasons for which (pure) mathematics is being done. We may be mature now to understand that some of our deepest ambitions were wrong but from understanding this if it ever was done rightly so to doing mathematics in which we don't trust for the validity of our claims our own reasoning is a great gap anyway. Ultimately instead of expanding and deepening our understanding we give up understanding in such a way and become completely superficial thinkers as the reason for the validity of a formal something stays in an accessible truth in thinking I think.
Mathematical reasoning , if it is authentic should conclude , for the part in which it can conclude what is right but the problem is that it's not always done right.
As such I can conclude that while the means of verifying by computers is ok because everybody can err sometimes the idea of completely abandoning trying to understand the theoretical means by which something is right or wrong is a superficial think which makes us forget the meaning of truth and mathematics. But that is my opinion anyway.
For my part I'd like to use a computer to verify Cantor's claim of transifnite theory. To me that's completely nonsensical no matter what proof Cantor had and ebcause nobody truly udnerstands the proofs , as it is obvious since nobody was able to make them more explicit and clear and just accepted them without thinking at their meaning I can think we are going on the path of mathematicians lacking the true ideal of understanding which even if it is indeed very very old must still have some truth in it after all with all the experience we have gained anyway.
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@A A – "Computers are great for a lot of tedious work in mathematics but they don't offer the theoretical insight and comprehension that logical thinking offers I think."
Well, you see, I think that's about to change. You give the example of Cantor's transfinite numbers, and that's an excellent example to bring up here. While transfinite numbers themselves are not directly computable, his reasoning nonetheless should be. As a matter of fact, this possibility is already a matter of lively discussion. I'm hardly the first one to propose that we use machine computability to "validate" mathematics, and if you want to see where "the action is" in this matter, go look up axiomatic set theory. There already are mathematicians that share my sentiment about this. This is really a very deep subject, and when you read up on it, it almost feels like it's philosophy that's being discussed. Wasn't it Wittgenstein that had objections with Cantor's theory of transfinite numbers? If axiomatic mathematics is so clear cut, why did controversy dog Cantor's ideas about transfinite numbers? It's interesting that while David Hilbert supported Cantor's ideas about them, Herman Weyl, who worked closely with Hilbert in the same university, did not. You couldn't hardly find such a close pair of such brilliant mathematicians, and yet each have come to different opinions about Cantor's theory of transfinite numbers!
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@Michael Mendrin – Exactly because of that a priori capacity of verification the truth I encouraged you to deepen the problem of mathematics being or not mechanical computation. The idea we can verify the truth of a theory by analyzing how consistent would a computer work according to it has underlying it the idea that the structure of mathematics and ultimately of thinking and logic is reducible to computation.
This principle animates I think also your personal philosophy and sense of things as I said this insight into the "a priori" so to say structure and conception of mathematics itself as it is and I almost imagine that you see mathematics from the point of view of how would it be if it is just mechanical computation. I think also that this idea should be better explained , how and by what the consistency of "truth" itself can be verified by a computer run but this rigorous and articulated with the acute perception of the mapping betwee the two structures anyway. But however I may be wrongt in the idea of your image of the nature of mathematics. Nonetheless I still think we have some kinship in perceiving this conception I just in part agree with such a reductionism from some theoretical concerns anyway.
I am one who finds Cantor's ideas also insane and irrational and yes I know Wittgenstein was well against Cantor's ideas.
I also know that Wittgenstein had some problems with Godel's incompelteness of it proof this without truly knowing the content of Godel's proof which I find correct and amazingly clear from what i know of it so I can't take Wittgenstein too seriously to be such a believable/reliable person in matter of such a problem.
The problem with infinity is that it's a subtle concept and predication about it shouldn't be done with the iresponsability of which that old point of view of rigor might easily accuse that Cantor had in speaking of it and moreover in "demonstrating" without actually understanding the theory of transfinitism. For philosophical concerns regarding this concept some other people opposed Cantor's proof like Poincaire and because how you said so many brilliant mathematicians had different opinions on matter of Cantor's proofs and ideas it is clear that proof is not that convincing.
I'd like to see either a proof of it by means of computers , though right now I have no idea how that is being done or a clear theoretical understanding of infinity which Cantor certainly didn't offer despite making maybe significative progress showing either why Cantor's proof is correct and explaining it as Cantor wasn't capable or showing why it is wrong and in what part because of ambiguous claims and predication produced in his mind Cantor went wrong. That would be a good proof of the udnerstanding which I am talking about. Nobody in more than 100 yearshad done that and you might believe that we are more mature in thinking than those in the past to avoid promoting , as in the past was done with the Ptolemeic system a great mistake of thinking. Nonetheless it's clear that if Cantor would have tried to be rigorous in his claims , not just proofs and make explicit everything about infinity and his view of it Set Theory wouldn't be in this situation as it is today anyway and we would understand better the concept.
To me between Cantor's transfinite theory and Zeno's paradoxes is almost no difference as both seem right and nobody understands why they can be wrong excepting the fact that while Zeno's paradox can be easily dismissed on more directly intuitive grasp than reason because it refers to something directly seen in our thinking that happens despite Zeno's reasoning (motion) Cantor's ideas are more about an esoteric idea (infinity) which can't be verified with the same conceptual ease by direct thinking or in a technical sense ,intuition , of it and therefore pretty much tend to be accepted by mathematicians especially because they seem surprising and this "surprising" trait of a "theory" alludes thinking. I find the situation of today's mathematics the same as promoting a ridicolous thing before we truly understood it and the only excuse for promoting something which we don't understand would be that we will never understand it anyway. I'd like to not wait 1000 years or more as with the Ptolomeic system to prove it wrong.
Sorry for talking this much about Cantor. To me he saw something into the notion of transfinity and how can some infinite be different but he didn't actually told what it was and went on the wrong track considering the proved something while actually maybe he just proved something else about infinity anyway.
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@A A – Bringing up the example of Cantor's transfinite numbers is a good one, because this sort of thing crops up regularly "in the pursuit of pure mathematics", and I'm glad we are talking about it. Let's take the field of philosophy. Is it computable? I have major doubts about that, so to me it's mostly speculation more rooted in sentiment than hard reality, but it doesn't mean philosophical discourse is frivolous and a waste of time. It does simulate thought, it can help us think of things in new directions. Likewise, "speculative mathematics", which is what I'd like call such forays as transfinite numbers (trust me, there are many more!) possibly can help direct us towards more concrete mathematics that is, yes I'm saying it, machine computable. Once it is finally machine computable, then that does meet a certain standard of concreteness. For example, General Relativity was rather esoteric in the very early 20th century, but today supercomputers routinely use it to simulate things like black hole mergers. General Relativity has long passed the realm of mere philosophical speculation into something that is resembling practical engineering.
I wouldn't be reading any obituary of Cantor's transfinite numbers anytime soon.
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@Michael Mendrin – You have a view of philosophy somehow similar with Russell's or Kan't sview of metaphysics when he says that metaphysics should be done because it is stimulating intellectually although preferably abandoned in whatever serious concern to understand the world as it's a waste of time for this purpose haha. Nonetheless my belief is that the "structure" of philosophy is very different from just being a speculation towards truth as it was proposed by many people not just Kant or Russel haha under the model of science and as such that has an internal and independent dignity regarding it's concerns being an irreducible construct that replies to a very authentic part of the human spirit not regulated nor being possible to be regulated or reduced at something else.
I do not also consider this part to which philosophy replies having to do with emotions because for that is mostly art nor with a "synthesis" between art and science because by that it is somehow reducible at the two of them and not even to religion or being ethical and guiding your life rationally and being wise as somehow it's clear that it's different from religion by it's very method conducted not on unjustified revelations from gods and mysticism but on a way of life and predication which adreses itself to the reason and which dedicates it's insight to others by reasoning anyway which sufficient in itself to convince. Philosophy actually is just as debusolated by the model of science and by the insufficiency of what it actually is and it's somehow funny because in the past science was the one which was restricted to logic while philosophy was in delirium speculating while in the last centuries , and especially from Kant's proof that the suprasensible is not ever achievable by means of our thinking , serious philosophy seem to have restricted itself to logic and reasoning making theories relating to it while science is in delirium speaking of weird and many times irrational things.
I agree that speculative mathematics and in general any sort of speculation and imagination are useful and producing and more importantly understanding things. Sometimes I think that , and you may not like what I am about to say , any sort of creation and thinking is rooted in the irrational or at least no compeltely rational and as such it's almost unavoidable to use anything we find as making sense for that speculation and irrational and is appealing to understand things better anyway.
I'd really like to understand more about this idea of mathematics being reducible to computations and being as such a computation. I'm not very sure of your way of thinking at the standard of cocnretness of mathematics though, shoudln't the actual standard of cocnretness be found in understanding the concepts and theories rather than in verifying it by the use of a machine ? For example if I will be able to verify the truth of anything with a computer and he would say No or Yes about it then with what my desire of understanding better a problem , for which anyway I actually find that I do the mathematics as a result of the fact I want to understand better soemthing which bothers me and makes "no sense" is anyway going to be be improved other than just knowing it's true or false anyway ?
From this point of view it seems I gain nothing cause my lack is in understanding which is not completed with anything and I'll still feel I do not understand. Therefore if the problem of mathematics is to understand I do not gain to much but anyway just will be able to verify a truth in some possibly more reliable way than thinking itself and as such mathematics. For a way of validating a theorem would be ok. For a great paradigm shift in perceiving mathematics it doesn't seem so anyway.
It's somehow funny that Cantor manage to convince a lot of people he is right. Nonetheless I'm sure those who were convinced didn't understood it anyway.
To tell you the truth I have a great problem with General Relativity too. To me it makes also no sense of speaking of "bending space". So much lack of rigor in such ideas make me not take elabroated physics seriously. To me it's just sort of imagination in the same way (with some more modern ideas) Pytagoreans spoke of understanding the world by the use of numbers and pretty superficial and childish but that's my opinion anyway.
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@A A – So you have a problem of "bending space". While theoretical physics has become nearly indistinguishable from pure mathematics, there is nevertheless at least one distinction, and that is, it does rely on conceptual paradigms. So, for example, General Relativity is based on the paradigm of "curved spacetime", just like Electromagnetic theory is based on the paradigm of "electric charges and electromagnetic fields", and Statistic Thermodynamics is based on paradigm of "randomly colliding atoms". Okay---but you don't actually have to believe any of those paradigms as hard statements of reality. What you can do is say, "okay, it behaves as if spacetime is curved", or "it behaves as if there are electric charges and electromagnetic fields", or "it behaves as if there's a lot of randomly colliding very small atoms." And then we look at the mathematics of it and see where it takes us. One of the most interesting things about theoretical physics is that in spite of seemingly disparate conceptual paradigms, time after time it's been shown that the underlying mathematics of some of them are actually equivalent, or nearly so. In other words, it's probably better to treat such conceptual paradigms as merely a shorthand way of imaging how things work. For example, we know that the "very small atoms" in Statistical Thermodynamics are far more complicated than merely hard little balls---but it's useful to imagine them as if they were nothing but that.
By now General Relativity has become so well established in practice that it's going to be very difficult to knock it off its block. What's mostly likely to happen is that sometime in the future, a more mature version of it will arise, that will mesh better with Quantum Field Theory, so that the laws of physics are not solely or even largely determined by "the geometry of curved spacetime", even though it is a good approximation to assume so.
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@Michael Mendrin – Haha , but what about the description of light ? I can't say with too much logical consistency it behaves as it's both particles and waves in the same time anyway.
I certainly find irrational the idea of bending space to not say of the madness of the more esoteric and religious problems of quantum physics. Your proposal it's nonetheless pretty right when it can be applied but it still is problematic I think. If it just behaves like space can be bounded and it is not then what are the empirical facts that makes the computations right ? This problem can't be trully avoided in physics because it's by it's own domain speaking of the true physical and empirical realities underlying it and , from it seems to me , because it wasn't really corrected in such time I think indeed that empirical reality is "bent" space whatever and however that bent "space-time" can be bent which to me sounds completely weird and just by some complicated reasoning saveable anyway. The case for thermodinamics may be just a case of "idealizing" which is pretty much accepted but the case of saying it behaves like X though the empirical reality which makes it is not X but X1 has a greater conceptual gap as one is to say that some atoms aren't actually in that shape but somewhere close and another to consider that some very different X's are the same anyway. It's such irationalities for which I didn't ever study XX century physics rigorously anyway.
I think we should rather say that because our observations don't seem to fit with our understanding our model is wrong and not that the world is irrational anyway.
Moreover, how would Black holes for example be explained without the bent and empirical space ? Did anyone saw a black hole or the observations are themselves interpreted and there might be no Black Holes though ? A lot of problems. Imagination is useful but reason is critical anyway.
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@A A – You mean you have problems with the wave-particle duality of quantum physics too? Well, what I can say is that once you've examined the body of experimental evidence for it, amassed over a century, starting with Einstein's famous photoelectric experiment, you're going to find it hard to dismiss this strange nature about quantum mechanics. If you can successfully offer a "naturalistic" explanation for quantum physics without resorting to this duality, and correctly predict results by mathematical means, you should be published and you will be one of most famous people in the history of science. If you are interested in work that's already been done in trying to resolve this "madness of the more esoteric and religious problems of quantum physics", you might want to start here
and good luck!
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@Michael Mendrin – Thanks.... but there are more interesting things to do that proving quantum mechanics wrong and more serious I think also anyway.
It's truly a madness , you rather have to believe things are like that than understanding it so it's more like religion. In fact I think a lot of New Age so said spiritualists claim quantum mechanics to be the "scientific" proof of their views.
Nothing is rational in it and as I said , instead of saying the world is irrational because we arrive at irrational results I prefer saying we are irrational and wrong in our interpretation and modelling since there is certainly a lot more proof that the world is rational that were ever quantum experiments done! My skeptical view simply seems to me at least more healthy by having the wisdom of not pretending understanding the not-understandable so to say anyway.
If quantum mechanics considers itself right it just has to continue with it's madness until finally it arrives at complete inconsistency and self-destruction.
Just as we measured the "curvature" of spacetime as it's said it was done we did experiments to sustain the belief in strange stuff related to quantum physics and said it's a great data for believing this incredible and deep nature , even irrational nobody comprehends which is in truth really childish for a serious scientific endaveour. But so to say this stuff is being done by humans , though Newton wouldn't be proud anyway.
If it is that the "naturalistic" view is wrong nonetheless I really have to ask you why everyone's thinking is rooted in it and there is no alternative for understanding ?
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@A A – One of the most incredibly difficult measurements have been completed just recently, the Gravity Probe B test, which was done in orbit of about 400 miles above Earth. It proved that the bizarre General Relativity prediction of "frame dragging" is a fact after all. It's one off the hardest things to even imagine how can there be such a thing, but there it is, the results are in, and Einstein once again has been vindicated. It's about as un-naturalistic as it can get in science, but so far nobody has ANY idea of how to counter the data from Gravity Probe B test "with a better explanation than relativistic frame dragging".
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@Michael Mendrin – This unfitted interpretations together with the necessity and openess toward other interpretations for it make me think I should apply what I adviced you and try to understand better what the fellow physicists say such that they can be ejither disclaimed or proved right but more acutely and clear. The only way for me that the bent space could work would be accepting the philsophical idealism as a matter of fact and as such attesting that the universe itself is a great thought ,maybe in the mind of God such that eventual irrationalities are explainable as thoughts.
@Michael Mendrin – I am ultra-critical of childish stuff and with pseudo-science as you observed anyway.
Quantum mechanics is either a religion or a pseudo-science like alchemy and astrology not a serious discipline rooted in the only understanding naturalistic one and if it demands soemthign special as therefore as not being the "naturalistic" kind it should check besides it's experiments it's principles regarding other understandings possible than "naturalistic" ones which it very superficially dismissed making me think most people doing it aren't able to make even simple logical inferences anyway. This even if it has more than a century though. I really regret the view and immaturity this modern science has and to me it's nothing but speculation related to empirical things we don't understand a speculation which all too often got a form accepted by the community similar somehow with the speculations the neo-platonists alchemists pytaghoreeans medieval thinkers had in the past just that hidden in a more modern and intimidating form or cloth anyway.
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@A A – Quantum Field Theory has yielded some of the most accurate predictions known to science, numerically. That can't be a consequence of luck. It may be possible to reformulate Quantum Field Theory in such a way that it yields the same super-accurate predictions and yet avoids this "pseudo-science of wave-particle duality", but so far nobody's has done it---literally after decades of trying. Trust me, many others have shared your skepticism and frustration with the "childish" and "astrology"-like aspects of Quantum Field Theory (i.e., a positron can be described as an electron going backwards in time?), but we're still waiting for someone to come forward and clear the matter up with a naturalistic alternative explanation.
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@Michael Mendrin – Well , I'm pretty happy to know then that the community of quantum physicists knows the problems of consistency in their domain. From what i knew or what it seemed to me when reading matters of quantum mechanics it's most promoters agreed to the idea that this domain at it is now truly reflects reality so it's really very very very very very very good not everyone is that closed-minded in the field.
Thanks for revealing that to me. It makes me have more respect for it knowing it's able to detect it's irrationalities despite the accuracy in empircial predictions. As such I suppose the general attitude in quantum physics would be something of the sort "we are given a set of empirical predictions which are succesfully described by such a crazy model but we don't understand why and how. This is a healthy attitude and not such a schizofrenia New Age thing which actually may reflect pretty well the state of physics today though that would mean it should be also opened to alternative physics and I'm not sure the community really is so anyway. Though you still didn't say anything about the reason for why you think that the structure of mathematics is reducible ("isomorphic") to mechanical computation.I'd really like to hear and understand your view on that but you may hesitate saying it because it's a lengthy thing to dicuss and while it may be so please state at least a few ideas regarding the reasons that justify this reducibility claim anyway.
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@A A – None other than Richard Feynman himself, the man responsible for the development of Quantum Field Theory, said, "I think I can safely say that nobody understands quantum mechanics". So, you're in good company.
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@Michael Mendrin – And..... about mathematics being reducible to mechanical computation you still didn't say anything haha.
Can you please elaborate that reasoning such that it is more clear what your view on the subject is along with it's specific determinations anyway ?
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@A A – Sure, I can give one example. Until fairly recently (i.e., the time I was born), deriving integrals was often a feat of mathematical reasoning. Now, math software routinely derive indefinite integrals, through symbolic manipulations. Likewise tensor analysis and some aspects of group theory. Use of A.I. techniques is rising exponentially in high end math software. Even synthetic geometry, a classic field of mathematics long requiring human reasoning, is now steadily being tackled by computational means, i.e., computers are now coming up with proofs in synthetic geometry. And there's the case of the 4-Color Theorem , which for over a century resisted efforts by the finest minds at a proof. It was finally proven by computer and results announced in 1976. The proof was so long and so complex, nobody in the mathematical community could verify it, without having to eventually resort to using computers to help! That incident raised a lot of questions and concerns about the role fo computers in arriving at proofs in mathematics.
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@Michael Mendrin – Hmmmm but you gave me some examples rather than sometheoretical justification of why it is so , right ?
It's just those examples , along with maybe a lot more of course, that convinced and inspired you or it's rather that you found something in thinking at the cocnept of mathematics and cocnldued that , considered in large what is done in the mathematics can be characterized inprinciple as mechanical computation ?
I knwo the 4-colors theorem to not have yet a trully reliable theoretical proof.
The proof was just splitting it into cases I think which lead to a lot of possibilities that had to be mechanically computed.
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@A A – Again, I am not and wasn't proposing that all of mathematics can be reduced to machine computation, much less offering a means of doing so, or proving that it can (or should) be. What I am suggesting is that a given topic or theorem is "more grounded in reality" once it reaches the point of being computable. Computing is more than manipulation of numbers, software today are capable of handling symbolic expressions and even abstract thought. This is now an active field of study. How can any part of mathematics be reduced to computability would strongly depend on the nature of that part. Maybe you are asking for a "toy example" of such?
It's an really interesting and daunting challenge to even "rigorously justify" why mathematics should be machine computable. Nevertheless, that was actually the original spirit and intent behind Whitehead and Russell's Principia Mathematica .
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@Michael Mendrin – Of course it's a daunting challenge to rigorously justify that , it would improve the understanding of the nature of mathematics greatly.
Well , but you stated that your opinion is that mathematics can reach that stage of development which opinion I found is so accordingly to a certain view of mathematics , view which assures and has it's reason of principle in the idea that mathematics has to be mechanically computable I think. Why has it to be so ? Because if the character and structure of mathematics isn't mechanically computable then for proving things more than relying on computing verifications like in the 4-color theorem for example being able to understand a complicated theorem of mathematics and verify the "sense" of it by comp won't be possible. Therefore for that age of new mathematical methodology to be possible it should be that mathematics is ultimately structuraly reducible to complex computation. Since I thought you were consciouss of this view of mathematics when proposed that sort of methodology I also thought that maybe unconsciously you also had a "feeling" related to what makes mathematics in general therefore mechanically computable as it should be in this view for the proposal to work and could have some glimpse into why it is so this sort of "glimpse" being of an "a priori" kind since it speaks of the conditions of the possibility of the reduction in cause I think.
I meant to deepen that understanding proposed in the Principia Mathematica. The ambition there was rather maybe to jsut understand better math anyway.....
Note that Godel's result that some propositions will be undecidable given an axiomatic system is no important for proving mathematics machine computable. Mathematics can be very well machine computable whitout being compeltely decidable as Godel proved already anyway. Therefore the problem of mathematics being so it's simply and solely one of the structure of mathematics itself anyway. This therefore will have no effect or imply results related to decidability in mathematics but deepen it's understanding in principle.
I'd like though some toy results about computers thinking or "thinking" rather if you want to show anyway.
In what way can that be done and do you mean they can eventually achieve the same development of thinking as we humans have , that it can be so in theory ? That will enhance that idea that mathematics and thinking and ultimately humans machine computable and therefore reducible to them anyway.
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@A A – Again, I am and was not proposing that computers and A.I. can "eventually achieve the same development of thinking as we humans have". What I keep trying to say is that, in analogy with the scientific methodology where the ideal is to obtain observational or experimental verification, the ideal in cutting-edge mathematical thought is to reach the status of machine computability. Once it becomes a matter of machine computability, then it gets "more grounded in reality". But as I said in numerous times, this is just a personal opinion of mine.
And I am already acutely aware that even the day should come that A.I. "equals or even exceeds human cognitive capabilities", they (A.I.), too, can suffer from ambiguities and undecidability. We humans are not suffering from these problems "because our brains are mostly fat!" Such issues are fundamental to the very nature of abstract thought---whether carried out by humans or machines.
I think probably you're looking for a "toy mathematics" as an example. Let me look around and think about this for bit.
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@Michael Mendrin – Yes , I understand your meaning. But to be possible to compute mathematics shouldn't mathematics be in principle computable in it's structure though ?
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@A A – I want to believe that mathematics should be in principle computable, because as we have seen, even David Hilbert and Herman Weyl, some of the greatest mathematicians in history, couldn't even agree on Cantor's theory of transfinite numbers. Because they couldn't quite agree on what it means exactly. But setting up the computer software so that this can be resolved through computation, it would establish at least one "mechanistic interpretation" of transfinite numbers. Then different interpretations of transfinite numbers can be seen how the mechanistic interpretations may differ. But each of them could have some value. A classic analogy would be non-euclidean geometries.
Let me say here that, as with theoretical physics, it is the structure of any particular mathematical thought that matters at the end, i.e., the means of coming to a conclusion. The conclusion may be absurd or sounds illogical, nevertheless, there is value in the manner it was decided. It is this structure that has to be repeatable to have any value at all. If, given a room full of mathematicians tasked with proving a particular point, we continually end up with different conclusions, then what's the value of mathematics? It is as useless as a scientific hypothesis that nobody can repeat or replicate reliably.
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@Michael Mendrin – It's not just a question of wanting I think because as I said , that mathematics is computable is the underlying principle of your personal philosophy and view so it is a question of being necessary to be like that and , in my opinion , for that imagined age of new methodology you spoke about to be possible anyway.
Hmmmm , I just have to made the point that I am not very very sure if we should equivalate "structure" of mathematical thought formally with the means of coming to a conclusion as long as we understand by the "means" the actual processes but with the "way" of arriving at the conclusions. By the "way" of arriving at the conclusions I mean the actual a priori structure of the (heuristic) path it travels for arriving at the conclusion , or ,more formally , with the way a mathematical truth is determined in the structure of the mathematical thought. The difference between means and way of arriving would seem unimportant here but i feel it should be pointed out because it gives a good articulation , or expression of the understanding that can be proved crucial some other time anyway.
Nonetheless , how would you use a computer to decide about the truth or falsity of Cantor's transfinite theory ? Please sketch that because I have no idea anyway.
Now ,you're talking of mathematics more of how it is in and by itself/a priori which find especially pleasing as we're getting to the heart of things. Btw I like that you make that analogy between the possible a priori "interpretations" of Cantor's theory by means of computability (if possible) and the non-euclidian geometries part and say they each have some value. I think for example that the value on information given by the non-euclidian geometries is in their fondatory principle which is very easy to be seen when you realize that the reason for which they are all called geometries or rather the reason for which we still see in them some resemblance or common trait with euclidian geometry , as such , a general principle underlying any geometry is that they are still speaking of "metrical space" but at this point , at this conceiving of the metrical space they depart , some making different postulates and obtaining different "a priori" spaces to speak of. In fact this is very much a way of more advanced idealization very used for solving and also of course describing complex problems as in physics in the way you proposed me to think of some problems in quantum mechanics and relativity. To me it's pretty easy to consider now that the problems with new geometries could have been simply avoided if they would have amde the observation , by a conduct of logical and reflexive reasoning that assures rigor , that the geoemtries are different representations (or if you prefer nterpretations) of the ideatic metric space a thing which is very valid once it is rigurously established and not maddening anyway. Indeed , a computer characterization of Cantor's theory would be verry usefull if it could be made as clear as non-euclidian goeemtries can be made in thinking.
Oh , btw I don't think what actually matters is if it is rough saying we are machines or not but that what matters is truth anyway.
If in truth we are machines , however unpassionate that may seem it's truth and therefore maybe just has to be accepted and not feared anyway.
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@A A – Okay, let me suggest a toy " mathematics". Imagine it's some kind of a puzzle, where some piece has to go from point A to point B, and there are certain rules about how the piece may be moved. As with a lot of puzzles, how that can be accomplished may not be obvious, or may even seem impossible. We can develop software to find a way, if there is a way. It may not reliably find a way, but once it is found, it is a bona-fide solution. Proving a theorem is a lot like that, we're trying to go from point A, which is that we don't know if the theorem is true or false, to point B, which is that we do know, true or false. We employ definitions, axioms, rules and inferences to get from point A to B. Once we get to the point where this exercise is even possible to carry out in practice , then we've reached the point where the theorem (and maybe the associated branch of mathematics) is machine computable.
What's important to note here is that it doesn't matter if the theorem "proven at point B" even makes any intuitive sense. What matters is that a "proof" solution has been found, mechanically following the rules established before the exercise.
It might interest you to know that Whitehead and Russell were able to derive a number of theorems having to do with transfinite numbers. Their work in Principia Mathematica might provide the template for developing software for making the matter of transfinite numbers machine computable. If you have an interest in this, you might want to start with that book. Never mind the reality about undecidables.
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@Michael Mendrin – Well , I suppose your example is just meant for illustrating the general idea of the mechanistic way of deciding the truth value of a proposition. But , while such an example illustrates that mechanical computation showing and explaining a simple and rigid mechanical process by which starting from a statement A we can move by a series of steps (according to the set of rules) to deducing it being linked with some other proposition it doesn't show too much about the nature of how those deductions are done for specific kinds of problems in other words of the structure and way of inference this structure being very different for very different problems and most likely not havign the same structure of the way of moving A to B anyway.
Therefore I suspect the actual problem in each such "exercise" should be with the way of deductions and the structure can be mechanically modeled. Since this exact way of mechanically modeling some such problem for making it computable hasn't been showed or illustrated in your example which just illustrated the general idea I'm still not understanding how to do that anyway.Can you please explain more or give an actual example you may have in mind for the illustration? Btw , for feeling like it don't hesitate to use formalism as it can clarify many things anyway. I have in mind you can model what you say like speaking of an output making some processes by use of some deterministic model on the account of theinput and ending the program it will ever end with showing the result based on the problem. The part which I find unclear is related to how to mechanically model the proceses for theoretical problems and is critical to understand this anyway.
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@A A – Okay, imagine that we have a chessboard. The rules of how pieces may be move can be encoded in a computer program. Let's say we have a knight in one corner and it has to go to another, and there's pieces already on the board blocking entry by the knight. It actually isn't difficult to code something that can search for a solution. Automatic symbolic derivation of indefinite integrals works pretty much in the same way, except that for efficiency, math software developers have devised a library or toolkit of "moves" in which the computer can try to use to work them out. Most high end math sofltware are able to work at the symbolic level, and already A.I. techniques are being employed to avoid wasting time on brute-force methods. I would dare say that the level of sophistication these software have in deriving indefinite integrals is far higher than most anybody in school, including undergraduate college.
I suspect you're trying to push on the idea of how A.I. can actually compete with humans in insight and reasoning in cutting-edge mathematics, not something as rote as deriving indefinite integrals. I actually do believe that Principia Mathematica is not a bad place to start on how to craft coding for proofs by machine. The symbolism of logic is the ideal medium for conversion into coding for computers. In other words, given a branch in mathematics, first "code" it in terms of logic as explained in that book, and then convert it into computer code. I'm sure that's what Whitehead and Russell would have liked, if they lived to see the kind of computers we have today.
Can you be a little more specific about what it is it that you are looking for?
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@Michael Mendrin – Yes , I want to understand better your view about the actual structure or anyway if you prefer some more "naturalistic" and intuitive terms nature of mathematics and how and by what sort of theoretical support you consider mathematics is computable. You just made me curious about your view and proposal , which I find has some depth and enough denseness to be solid and by being solid also having some value for deepening this sort of mechanicistic view and the way it anyway resembles or not accurately mathematics and thinking in general anyway.
As such my point is not that of sustaining some idea as it might seem like comparing mechanical thinking to human thinking which is a subject for other day. It's just the underlying understanding of what mathematics is in it's strcuture or a priori form that is extremely interesting and in which I find of lot of weight anyway. Your example with the knight and it's analogy with integral derivation too but anyway I suppose not all problems can be treated in that manner , because they are not reducibile structurally to that same structure that helps you solve the knight problem and integral derivation. Or if it isn't the structure , why and by what some problems in mathematics aren't still solvable mechanically in theory ? What would be the problem with Cantor's transfinite theory ?
The point of this questions is to try to understand better the structure of mathematical derivation of logical propositions because , since there are such problems which avoid at least untill now such a comptuable representation anwyay that source should stay I believe in principal reason regarding mathematics. Therefore I take that as a very general and superficial first pointing to some factual reality of importance for understanding that structure that may avoid computing.
I'm still not very sure though I really udnerstand how can a computer prove a theorem because that seems very different at least by intutiive means of representation of what is a theorem than a game of moving a knight over some pieces on a chessboard whose intuitive representation is closer to mechanicism . I suspect it has to do with some sort of mapping in computer terms the theorem but how you do that I am not very sure I understand though. Oh yes. Maybe therefore since you recommended the book so many times , and sorry for not speaking of it but I was always caught in the reasoning of some other matters relkated to what you said and didn't thought important to speak of it , though i did received what you said pretty well I will try reading those volumes of the Principia Mathematica and thanks anyway.
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@A A – I do want to clarify that my use of the word "naturalistic" is only in reference to paradigms in theoretical physics, i.e., e.g. "spacetime really can't be warped---there has to be a commonsense way of looking it that doesn't involved warped worlds!"
The branch of mathematical logic is the ideal gateway between computers and mathematical reasoning, as it does have its "feet" in both worlds. If you had taken a class in Euclidean geometry (today called synthetic or axiomatic geometry), as well a class in propositional logic (see Propositional Calculus ), you should see a striking similarity. Then go study computer languages and programming. After a while, a picture should emerge for you.
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@Michael Mendrin – Ok, the use of "naturlistic" wasn't very suitable sorry.
But you still didn't reply regarding modelling , by means of computers a "proof" for a theorem anyway
It would be very cute if you would sketch how that could be done because I really don't find it is as simple and certainly incomplete no matter the simplciity to say it's being done exactly as using some sort of algorithm for solving a chess game in the same modelling of the "mechanics" way. But most importanly you didn't say about what makes some problems harder provable by means of computer wether they structure or something else , this being a crucial thing for understanding your view I think anyway.
I'll have to go now so unfortunately I will not be able to continue the discussions in case you want continuing it for some time as I need some sleep I think haha.
Nonetheless be sure that I will reply and tell my thinking regarding your examples and thanks for all that advice and your trying to explicit your view anyway.
@Michael Mendrin – Btw , seeing you haven't reply for this long period of time I am starting to think that unless it happened something important you don't want to continue it anyway.
If that is the case and you do not want to explain further how can a theorem be tested by a computer please let me know and if you are also kind to say anyway because if this is the case and you are not willing to discuss any longer I wonder why please also state the reasons for which you don't want to anyway.
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@A A – It does sound like you really want to get going on how we go about implementing "machine computation" of mathematical proofs. If you really want to dive right into it, I can give it a go, even though this won't be the first time anybody's done it. But before I start, how about if you gave some examples of theorems to be "proven by machine computation"? Just to get an idea? Always start with a small project before trying to build Versailles.
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@Michael Mendrin – Well , to speak you the truth I just had a small idea you where testing my interest in the problem too and thanks for trying to offer me an example of what you mean. Hmmmm , I have a very vague idea of what theorems could be proven by mechanical computation and how would that work.
I can think of various types (regarding our intuitive representation) of simple theorems such as geometric theorem (like Pytaghora's) , algebric theorems or formulas like for example proving the form of the solution for quadriatic equations , combinatorics theorems like those in Ramsey theory I suppose , or anyway topological theorems like the simple Konigsberg bridges theorems which therefore I suppose somehow though , of course I don't have any idea how this could be done or in other world how this could be mapped into mechanical computation even if on a least ammount intuitively it can be seen that such a way and structure by the mathematical logic underlying it and a very little familairity of that type of thinking it can be effectively somehow on attent thought be done so said anyway under that sort of mathematical computable philosophy of yours of course though. For my part anyway I think that even if it wouldn't have any importance what type of theorem it is for the mechanical translation of them it would still be advantageous to see a theorem which is more easily translated into such mechanical "language" since that makes and points out more clearly the relation between the way the mapping of the intuitive representation we have of the theorem and the mechanical represented is done and recognized offering a better instructive content therefore making me prefer such theorems as combinatorics ones which have a better structure more appropriate for thinking in computer terms as it's predication is about , msot often , of a more concrete content more appropriate for atomistic and therefore mechanicistic thinking in our way of mroe intuitive thinking and representation anyway.
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@A A – A A, I just got back into my office. During the day, I had suddenly recalled that I already wrote up a wiki for Brilliant, that would be one example of how logical propositions can be found to be true or false by "machine computation" Have a look at this. I'll come back here later, after I've taken care of a few things
Propositional Logic Using Algebra
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@Michael Mendrin – Ok , unfortunately after 2 hours or less I'll be sleeping and nto be able to reply to your illustration of how youc an derive the truth of a formula though.
I know of the propositional logic and the so called "logical calculus" and that's pretty much the vision of which I was talking at a point when saying that there might be some glimpse by which you can understand and have a conception of the mechanical computation and how therefore that can be used by a deep elaboration of such a type of thinking for understanding the possible way of deriving the truth of a theorem just by use of mechanical computation. I'll recheck my knowledge of the matter. It's more than required maybe for understanding the contents of the illustration you are about to propose and good luck at the office anyway.
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@A A – I can actually write the coding for "proving" such propositional statements, the wiki explains how. The "algebra" that I described in the wiki is not the usual variety algebra that people learn in grade school. It has a couple of wrinkles to it which are nonetheless easily implementable by program.
This is a direct example of "theorems" being "proven by machine".
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@Michael Mendrin – Emmmmmm , yea but it's more of proving simple logical statements.
The idea would be therefore for some slightly mroecomplciated problemto describe it if you can do that in propositional logic and then apply your arithmetics but of course certainly just the elementary application of the principle wouldn't be enough right ? There should be some sort of issues regarding such computer implementation for problems which are just a very little bit harder. That is indeed a good toy example to keep the idea going but it's elaboration isn't to be sure of. You can't know if it will work besides those such easy logical statements. Actually I have to say that I suspect for a more serious theorem it already posses problems which require the modification of the way of applying it greatly right ?
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@A A – Fundamentally, there should be no difference between logical propositions and mathematical theorems. I think maybe we need to discuss this matter first? I think that probably explains why we can't quite get on the same frequency on this subject, so to speak.
The whole point of Principia Mathematica WAS to reduce mathematical theorems into logical statements. Which then, with the computers we have today, we can use mechanical computation to attempt proving them.
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@Michael Mendrin – Very well pointed indeed. We should discuss this problem of reducing theorems to statements of the atomistic propositional logic to understand better this kind of mechanicist conception because in that theoretical "device" so to say namely propositional logic stays maybe the best expression of this view anyway.
But I just have to tell you once more that I'm just trying to understand the problem and view of mathematics being computable better and do not attack this view though nor do I accept it without letting myself of the implications anyof this views might have which is just another problem and story for after the problem is clear. If I am saying that there are some problems with the applicability of the view is in purpose of understandingand pointing out a reality which by being a reality can't be avoided therefore this method being that of making manifest critical view of what is right and wrong for understanding how things actually are anyway. So before entering into the problem of quantative implemention of qualitative statament let's do as you proposed very very well and try to understand better this mechanical or expresion of mechanical view expresed in propositional atomism. Please note that I name it propositional atomism and not propositional logic because in my opinion atomism and the so to say "mechanicism" articualte in propositional logic may in be considered so much linked that they are innately correlated meaning by that that conceivement of one necessarily implies the conceivement of the other in the same moment of thinking it and havign said this , what do you think are the important things to point out from the way of thinking of propositional logic therefore that you find important to point out for the purpose of this discussion ?
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@A A – Unfortunately I have to be out of town for a few days, so when I get back, I'll pick this up. There's just too much to reply to in the time that I have left now before I have to go.
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@Michael Mendrin – Ok. No problem anyway.
I hope it will not be soemthing which gets complicated for weeks from now though. Good luck with the problems though and btw the last commentaries whicha re the ones to which this message I hope was intended are 4 and also 3 messages upper or for another way to find them 1 and 2 commentaries downwards to your second so to speak last one about homomorphisms anyway.
@Michael Mendrin – Well , I hope you didn't forget of this discussion but nonetheless if you did I'll remind you of it. I guess those few days have passed and you are back home so I suppose we can continue if you agree with continuing the discussion anyway.
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@A A – I'm back now and I've got a few days before I leave again. Can you summarize where we are right now with this discussion? There's at least a few days' worth of discussion here covering nearly everything except statistics of football games.
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@Michael Mendrin – Sure , we are at the points which i made about the formal concept of "structure" which I detailed upper with 5 and 6 messages upper from this one or 1 and 2 comments downwards from your upper comment in which you send me to the links about homomorphisms and other simple structure preserving stuff.
So to summarize our discussion or how we got to this point we tried to understand better the idea of "mathematics being reducible to computation" thesis which lead by a good intuitive and substantial grasp into understanding that we are actually asking when grasping this idea of the "structure" of mathematics. Because the central problem was related therefore to the "structure" of mathematics anyway somehow we ended up trying to make precise what we mean by "structure" and as such arrived at the point of making the concept precise. I argued as you can see in those two comments of which I was speaking that "structure" understood in an a priori way ,or by virtue of it's principial form in thought is intimitely related to the , what i called so to say , "atomistic" understanding of the world. Also anyway in the comments written there I argued ,in particular , about what we actually mean when we speak of the "structure of mathematical thought" showing that we actually mean the structure of a process seen somehow synthetical in what actually this process is if ,described in and by itself or in it's "a priori" form so to speak , consists of at the most principial level in our conception of it so to say . This process or thinking and doing mathematics is interpreted in the legacy of propositional logic to which it should be for the most part reducible as a bunch of "deductions" which is also an atomistic and mechanicitic view concerned with the description of mathematical thought therefore as being a proces of deriving from some components the truth of others in an axiomatic system such that the deductions arelogically coherent with the system ,process whose atoms are the "deductions" , such making us to somehow name the process as one concerned with the "structure" of deductions for which I argued nonetheless that it would maybe be better to name the object of the investigations being done here as rather an "deterministic structure" concerned with understanding formally how things are determined but you can read it upper ,you can also go on slack anyway.
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@A A – Hmm, I do go on Slack from time to time, but most of the time "nobody's home". I know you go there to talk. But what you've just said here seems about right so far---you seem to be going in the right direction. The only thing I'm not sure of is "what is the problem now?" It really does sound like you're prepare to tackle the very same challenge that both Whitehead and Russell attempted to tackle a hundred years ago, which is, "How to put mathematics on an axiomatic basis as to make it machine deterministic?"
On the flip side, maybe I should ask, "Do you believe that there are any branches or theories of mathematics that is proven and understood by mathematicians, and yet is and will stay beyond machine verification?" If so, can you cite some examples of such?
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@Michael Mendrin – Well indeed let's hope we are going in the right direction to clarify and understand better this problem of ours from here anyway. What problem ?
The problem of which i am talking about is different from the one of axiomatically describe the structure of "mathematics" or rather the mathematical construct. From what I know in Principia Mathematica the authors tried to show the logical structure of mathematics to get or at least get around an affirmative answer to the hope mentioned in Hilbert's problem that all statements in mathematics can be deduced from each other in an axiomatic system. This gave a beautiful image of mathematics and an idealistic view of it's beauty and structure. Nonetheless in rather some other way (more abstract and less constructive) Godel showed by his meta-logical and meta-mathematical also theorems this is impossible , therefore that not all mathematical statements can be deduced from each other , given any sort of primitive statements as such making a little bit of a gap between our hopes and reality pretty much in "fashion" for XX century anyway.
Having said this I interpret that the problem in Principia Mathematica was an attempt to show and understand if all the statements in mathematics are derivable from one another by trying to get a view of the structure of mathematics , of the way mathematics so to say "works". My problem is rather what is the structure of mathematics to be isomorphic to mechanical computation or even better an attempt to understand better if the way mathematics "works" by that meaning the way statements are derived (the "deterministic structure") is isomophic to mechanical computation which does uses also a study of structure from this point of view anyway.
In other words the problem here is what makes the deterministic structure of mathematical derivation to be or not isomorphic to mechanical computation. Therefore this problem is the same as the Principia Mathematica's because asks about the so called "deterministic structure" but it's different because it asks some more particular things about this deterministic structure for the problem of understanding mechanical computation which articulation wasn't necessary for the purpose of understanding the structure in "large" for the problem of seeing whether or not the structure of mathematics is complete. Therefore even if for the authors of the so named Principia Mathematica the understanding exposed by them about the structure of mathematics was enough that gained understanding isn't complete I think to answer about all sorts of particular questions relating to the structure of mathematics and as such asks for a deepening of what the authors of that work have done anyway. Therefore we can actually make a statement or a question , in the same way the authors had in mind replying say following "is the "deterministic structure reducible to mechancial calculus ?" and try deepen it and understand it better and actually I just think this question is indeed deserving of all attention from an intersted spirit in the pursue of the reflexive and as such meta-mathematical thinking required and great way of gaining by asking such a question an understanding of your philosophy and a lot of other ways of thinking for which it seems mathematics is computable anyway.
Why do I say of this problem that it may not be true ? I say that not because i have some kind of example which i know is necessarily impossible to prove in this manner but because I think that only a full examination and analysis of the question can let us know the truth , in other words so to say , because untill it's proven right there might be some sort of small detail somewhere which wasn't considered in our general thinking so good at skipping such small details and maybe also for the sake of understanding better ourselves when we do the mathematics anyway.
It's quite hard to say if this view is right or wrong especially following the example of the XX century which maybe shall show us not to venture too much in a speculation and religious thinking enhanced by a deep convinction since anyway we are ignorant and don't know too many things. This is the source of my doubts regarding this view I suppose and also the reason of why I want to understand better this kind of thinking as well as a pretty mcuh generalpoint which I make in thinking anything and mroeover also as I know , also from experience and anyway I suppose you can confirm it that truth however simple isn't all that simple and small details ,at least in solving a problem should be considered details which can change all initial expectations that , if I am afforded an analogy , truth even if maybe simple from what we know of it is not like a drawing from a kindergarten child which ignores small details that give the picture weight but like a picture of Michelangelo or also of course Leonardo(da Vinci) or any other great painter who have a more develoepd thinking so to say anyway. I suppose you agree. But to ask you because I can't be sure do you agree where am I wrong though ?
@Michael Mendrin – Just to add a little about the second part and actually to point out better what is the actual so to speak problem let's try imagine the following situation anyway. Supposing we have some proven mathematical statement like some theorem and we'd like to prove it also on the basis of mechanical computation the problem regarding this isn't that of calculating but of implementing the problem in the language of the computer which will do the verification. If , however much we would like we don't find a way to implement it the reason for that might be that the structure of the problem isn't implementable in the computer's language. The reason for this might be that , however much we would want to , the structure of the mathematical statements involved in the problem isn't reducible by somesort of cause we can't see at this moment but may assume to the machine anyway.
Therefore the problem is about the structure of mathematics and what makes the structure to be or not reducible to mechanical computation. Actually I do anyway suppose this is the reason for why there are still problems implementing some theorems in the computer's language , a problem related to the fact that this strcuture is not well enough understood. If we woudl have had such a general udnerstanding implementation into mechanical language would be easier for maybe any sort of problem , we would know how to implement it and solve the problem in amechanical way. The fact that we don't should point out we don't really understand as well as we might like that structure and case study of how and what makes some problems harder to implement that others if appropriately seen for the general reasons underlying this should give us insight into mathematics anyway.
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@A A – Okay, and now we've come around the full circle. Right at the beginning of this long conversation, I said it is a personal philosophy of mine that once any particular mathematics or theorem is "verifiable by machine", then I consider it to be more "grounded in reality". In other words, it's a goal to strive for, once a new branch of mathematics has been developed. I have a problem with "math" that totally depends on human thought alone, something that can't ever be "understood" or implementable in machine language.
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@Michael Mendrin – I don't think we went nowhere in this discussion. As you can see we understood better some stuff related to your philosophy and we shall continue anyway.
That strive for understanding your philosophy is what we were doing here. Meaning that we wanted to understand if it is right or wrong. We did so by proposing to deepen the so called structure of mathematical thought. I suppose that if you agree we shall continue verifying this claim in a more or less objective way and analyze where it will lead and understand better not just if the claim is right but also your own reasoning for why the claim should be right , meaning your thinking itself and as such understand better you (and also understand better me) in the manner I said pretty much at the beginning of the discussion.
So , to not end in a circle let's try to understand rationally why the structure of mathematics is isomorphic to mechanical language as proposed anyway. Considering things in this way I suppose of course that you admit that so to speak you don't know for sure if your view is right so you can admit it may be wrong.
But before we get there maybe I should ask you why do you have a problem with the view that human thought and math isn't reducible to computation anyway ? Putting this question I think we will understand better as I said our way of thinking by presenting the reasons for why this thinking is correct or wrong anyway.
As i said my point is somehow different. I believe that a machine can maybe implement formally the structure of mathematics anyway.
But , I do not believe that our thinking is reducible to mechanical computation and hence that we are thinking and very advanced automatons or thigns like that way. Of course I also can't be sure if this view is right but I also have some rational things considered for why my point of view is right , I have some other model so to say if you prefer by which I think at human thinking showing it is qualitatively apart from the formal stuff it implies different from mechanical computation anyway. Considering this stuff I'd really like to know more about your view. But also I have to say that I'd like to make a rigurous analysis of it and not opinions sharing say.
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@A A – War Going On...
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@Prince Loomba – What war are you talking of Prince haha? We just try to clear some sort of things regarding reflexive and meta-mathematical understanding anyway.
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@A A – Haha. Dont be angry. I meant big paragraph war
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@Prince Loomba – I'm not angry haha. Ok , I thought a little that you meant btw.
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@A A – If this is a big paragraph war, A A's winning with his bigger armies.
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@Michael Mendrin – Who's A A though ?
@Michael Mendrin – Sorry , I didn't get the joke or I don't know who's A A or both haha.
@A A – There's actually a difference between "human creativity" and "machine verification" Whether or not "machines" can be as creative as humans in coming up with new ideas and directions is another matter, and it could very well be that it will never happen. However, my whole point was that for an "idea" to move from philosophy to well-axiomatically-based mathematics, it should at least be put into a form that is machine verifiable.
What you have to understand is that, for example, the notion of transfinite numbers doesn't necessarily have to make any sense, what counts is that the reasoning and deductions behind such numbers is consistent and follow axiomatic rules of proof. This is analogous to paradigms in theoretical physics which often seems to be nonsensical, such as, "quantum information from the future can impact present events", and yet the underlying mathematics is axiomatically consistent and repeatable. Then I would imagine it's really up to human imagination to try to grasp the meaning or relevance of those strange concepts---and maybe use them to find new directions.
I agree with you that it's not that "we're only going around in circles with this"----I only wanted to point out you the original impetus of this whole conversation.
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@Michael Mendrin – Well , but when I say my model of human thinking is qualitatively different from that of the machines I'm not speaking the different trait is "creativity" anyway. Considering this if we are indeed advanced or ultra-advanced machines then the characteristic of a machine endowed with "creativity" already is proven as theoretically possible so if indeed "human thinking" is completely mechanical or at least reasoning is so then there should be possible to make machines who will have the capacity for creativity and insight (at least in reasoning) we humans have who are nothing but more natural developed machines anyway.
I understood your point but my problem is if this picture of the mathematics and way of thinking is correct and what makes it correct and wrong also , my problem is understanding if the structure of mathematics affords implementation in mechanical language always but this has to do with a rigurous study of problem. You have to consider that maybe , since we don't udnerstand it very well to prove it otherwise not all problems who are provable in an axiomatic system can therefore be implemented in mechanical language because there is soemthing in their deterministic structure that prevents such an implementation anyway. Therefore if at some point in time this philosophy and proposal will be considered right and mathematicians will have in mind applying it I think they will be very itnerested in proving that such a decision is justified because otherwise they will just risk trying to prove in a form of mechancial computation some problems for which there might very well be no way of implementing it so to speak. As you can observe once this idea is taken seriously (this being the first step in the story) everyone is interested in trying to understand it's view better and get into the deep causes underlying it concievable or not by human intelect to understand it better this question being of course put for the sake of the question itself and for the problem of wether it's wise to apply it as I said though. So that's why I too try to understand your point in a better way asking what makes it a problem to be implementable ? In what or by what way can we be assured this implemntation applies generally anyway ?
Hmmmmm , ok I can consider that. But ,you rather seem to speak that what matters for proving something is if the line of reasoning is consistent point to which I agree of course which further means rather that it doesn't matter for the proof if the concepts make any sense nor that they don't have to make sense. For the point in which we should say a "concept" doesn't have to make any sense in the thinking for the scope of truly being an authentic concept and jsut for the simple sake of proving something I'm not too sure though it may be right. If there are such concepts which have no sense in thinking , that we can't grasp intuitively and understand though they are still indeed true and authentic concepts I have to ask myself and you also what makes a thing have sense anyway.
As far as I can think of an idea as having and making "sense" when I see the idea right or better said consistent which maybe simply means either when I see it follows rationally from something either when I see the idea itself has some directly seen structure which follows directly by it's aprehension in thought. Therefore we can distinguish between sensical stuff those which are propositions regarding deductions and which are regarding the thought of a concept or thing. Therefore , if I consider that an idea is endowed with "sense" it means I see how it follows in a consistent way from some other things which means that if I can see and if something follows consistently from something I should "see" the sense the concept has. By a thinking like this I can deduce further that , if indeed to have sense means to see how it follows naturally from something if a thing anyway follows naturally from something it should have sense and I should see that sense once it's pointed clearly by someone and my thinking is strong enough. Considering therefore as a criteria for "sense" consistency it should be that anything anyway consistent has to make sense or in other words has to be understood or that anything consistent is understandable and this is maybe seems indeed a naive view to state that everything sound is understandable. Further giving the example of Cantor's transfinite theory we say tha the directly apprehended concept of infinity can't be concieved and as such understood. Though , it's really not clear how from a continuous and consistent reasoning in which the components all follow from the rest it's imposible for anyone to understand the concept of infinity which is larger that some other infinity because since the parts from which this concept were understandable what makes that the resulting concept is not to be understood though derivable from those parts other words why and how would an "emergent" characteristic appear here anyway? The way of justifying how from understandable things we get un-understandable things seems pretty doubtful and doesn't make sense in this problem being very dubious that ,if transfinite theory the right what it states isn't udnerstandable anyway.
Ok. I wasn't sure what you meant.
I still think it is important to deepen that understanding of the mathematics structure letting Cantor alone btw.
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@A A – I think it's a bad idea to insist that things "make sense to us" before we accept a finding that has been axiomatically derived. It's a form of prejudice, and it could blind us to possibilities we may not be aware of. As a classic example of that, when Paul Dirac forced solutions from his own relativistic Dirac Equation, it suggested "negative energies", which seemed nonsense. Well, it turns out that equation was predicting the existence of anti-particles, which are now routinely being created in the CERN Large Hadron Collider as a matter of business.
When something at first doesn't make sense to us, and yet the reasoning behind such findings are well grounded, then we should be looking at it in a different way so that it does. That's the whole idea of the effort behind "quantum interpretations".
I absolutely agree with you that "understanding the mathematical structure" behind Cantor's transfinite numbers bears closer study---instead of simply dismissing it out of hand became "it makes no sense!"
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@Michael Mendrin – Well , but the anti-particles you speak of still make kind of sense though the sense they make is just for the mathematical justification not for the physical reality necessary and though, this is not the situation regarding what we were concerned with in what was stated upper I think. The problem to be traited was if something does or doesn't make sense "a priori" if there are things which are possible to prove as logically consistent though nonunderstandable for which it seems to me that you actually agree with what i said.
Namely you agree after all that anything consistent should be theoretically understandable or at least so I suppose. At a moment when you said in your second last comment of the use of "imagination" I almost thought you meant by that ,that the sense and everything we see as sensical it's not authentic sense and just a falsifying of reality but I didn't develop on that point also. Otherwise if your point was just about the need of being open minded and try hard to understand things as they are even if the data which we receive seems for our so to say present interpretation crazy indeed I agree with you. But please observe that for quantum mechanics we don't have a compeltely consistent and rigurous understanding as you would like because the "interpretation" is completely from the point of view of logic inconsistent and superficial.
I once heard of imaginary particles for explaining physical stuff. This is of course or at least so I hope just an mathematical artifice for describing some phenomen but shows a lack of the understanding regarding empirical physical things which so to say are underlying the phenomen to be modeled anyway.
It is indeed very wise to think at some paradoxical stuff and problem more and I never said nor agreed that we should throw to thrash data which doesn't fit our interpretation of the world and seems inconsistent as that would be a sign of immaturity and as you remarked a prejudice. However , not all the solutions especially in math are rigorous despite appearing so and if they don't give you complete understanding once you think at them deeply and attentively and of course are capable of such a thing their correctness is at least doubtful in rigor. This is the case with Cantor's proof which has to be understood and for which therefore I dare say nobody not even Cantor truly understood. But I have to ask you whether you meant in your second last comment that there are possible thigns which are correct but we can't in principle udnerstand , that are un-understandable despite being a proof or rigorous reasoning that sustain them or if there are things which just seem un-understandale to us at first sight but in truth they are so to say consistent and right anyway ?
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@A A – The mathematics of quantum physics is solid and makes accurate predictions with great reliability---and with such tools, physicists are even able to make "crystal lattices made out of light" refract "matter waves"---a strange reversal of roles! In other words, while a lot of it seems nonsensical, it's very hard to argue with the mathematics or the results. What is happening is that mathematics---when it's done correctly---is pulling us to a lot of strange and unexpected worlds. Remember, it was only a few centuries ago that mathematicians thought the idea of "negative numbers" was ridiculous. Now, theoretical physicists speak of "negative probabilities" (which I guarantee you has never appeared as a problem in Brilliant!)---see Negative Probability , by Richard Feynman, a Nobel Laureate in Physics.
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@Michael Mendrin – You still didn't answer the question though.
It seems I upset you a little because of my view of quantum physics but you need to acknowledge though that I have a point when hearing that almsot everything said in quantum physics regarding the interpretation is completely wild imagination and not just one thing or another but almost everything. As you can see all the points which we talked about sound crazy. The results are ok maybe but while speaking of a mathematical result which we have from physics may not be doubted too much the interpretation of that result required for dealing with the empirical nature of physical stuff which should explain it are doubtful anyway.
I have no problem with a new mathematical concept like negative probability as long as it is rigorously shown right and even more if it proves useful. Moreover so to say I don't even find that "negative probability" is a surprising concept but the meanings associated with negative probability regarding particles for example may be completely senseless and superficial.
Concepts like negative or irrational numbers also aren't very surprising though they may contain a deep understanding once attentively considered and exposed. In case of negative numbers even if there is no sense at first we can understand that they refer and show an abstract idea ,that I have -x stuff which can be a debt for example or a loss in anything. Therefore the concept of negative number while admiting it's senseless to speak of negative quantities (let's say so at least) considers this numbers as expressing an abstraction and representing soemthing else which nonetheless happens in the world (we all have debts or most) and therefore is useful as a modeling of the things which happen in the world and understanding of those things more abstract and less concrete in content. Interesting enough if it is asked what made mathematciians have problems with the concept of negative numbers we can say that's exactly because they confounded that this concept like any other speaks of an abstract reality and doesn't happen to be so much imeresed in the concrete reality of thigns as it happened for natural nubmers of counting or ratios , confusion which because therefore not well seen made a controversy regarding the right use of the concept. How can we speak considering this phenomenon of abstarction further anyway? Well go to a next great problem for mathematics , a new conceptual extension but this oen even harder to see and justify because is even more abstract anyway. Imaginary numbers themselves may be said to be a good example of something which makes no sense but can be used so let's take it's case to illustrate. Some physicist , some modern physicist rather may say something like he uses imaginary numbers which are the square root of negative numbers and say that what he does is incredible because defies all logic and reason but works. Someone may wonder why and how can imaginary numbers work since they defy reasoning and ask what "does it mean that a^2 is negative?" in mathematics how can a square be negative but his surprise will fall if he understands the nubmers work because they are a convention to use in the cartesian plane in physics. Therefore someone may say that because it is "conventional" it's not a true concept but a false concept used as an artifice for some abstract consideration regarding thinking at dimensions or such problems ,some way of modelling a thing and I think he would be right. The confusion between a true cocnept and a cocnept used just because it works in a "frame" of thinking , an artifice illustrating something may be the source of the surprise at first but we can see that the way physics and mathematics work because are in more and more abstract understanding and therefore they need more and more seemingly inconsistent tools raported to cocnrete reality to describe their otherwise sound reasoning raported though not to reality itself but to their abstarct thinking while this thing may be said at the elvel of abstarction can't be easily seen anwyay.
This is how I see the problem with seemingly senseless concepts But anyway considering the problem with quantum mechanics , it is inconsistent not because of the tools but because of the absurd interpretations of which it is full and where none of them stays in tools or subtle and sometimes beautiful concepts though.
Btw thanks for the work of Feynman , maybe I'll read about that idea anyway. Indeed I agree with you that deepening the udnerstanding it makes us wonder about paradoxal things and worlds and I share this feeling and fascination which you have seeing how and why we arrive at a lot of incredible things but we still have to be healthy mentally and distinguish by reason what is indeed and what just seems right by our reasoning because otherwise we are not makign a responsible judgement and thinking I think. In this discussion i like to see what fascinates and makes you like physics though and I admire you for that anyway. Before moving again at the topic of mathematical structure and trying to understand it in a more deep level , because I spoke of Feynman's work which you shared with me can you pelase tell me if you ever read Bertrand Russell's work named Mysticism and logic or something like that ? You seem to have shared his view about mysticism when you said philosophy is like an emotional feeling. And also I'm not very sure if I should read that book so I ask you for advice anyway.
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@A A – I haven't read Russell's "Mysticism and Logic", I just now looked it up (it's available for reading online). Well, that's a whole other subject, and it does seem to me that Russell is trying to "think his way through physics by starting with philosophical principles". But, ultimately, if he wants real physics to come out of his notions, he's going to have to set it down in mathematical form, which then needs to be computable to deliver results which can be checked. That's the hard part, which separates philosophers from theoretical physicists.
Now, I'd like to refer to your comment, "But anyway, considering the problem with quantum mechanics, it is inconsistent not because of the tools but because of the absurd interpretations of which it is full and where none of them stays in tools or subtle and sometimes beautiful concepts though." Once again, in order to make good use of quantum physics and the mathematics of it, one actually doesn't have to believe in any particular paradigm---as long as you just do the math and see what results. In other words, we run into problems of "nonsense" and "absurdity" precisely because we are struggling to imagine an intuitively understandable paradigm---and there's very good reason to believe that reality doesn't work that way. It is by faith that we say that once we have a good and correct interpretation and mathematics of physics, then it will make beautiful sense easily explainable to children in grade school. Or laymen.
Let me now refer to you the Banach-Tarski Paradox This is a totally worked about and carefully studied theorem in set-theoretic geometry. The proof doesn't have flaws, it is consistent, and most likely will one day be machine-verifiable. And yet, look what it is proposing. It says that given a solid unit sphere, it can be divided into a finite number of pieces, which can then be re-assembled to make TWO solid unit spheres. Are you suggesting that this "makes more sense" than, say, the wave-particle duality of quantum physics? How do you decide what makes sense and what does not? How do we test for that?
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@Michael Mendrin – Well , but then know that from I read in passing from that book Russell seems to share your thinking of philosophy to mysticism. The books is about what use and importance may mysticism have as a method of knowing the world , it's rather about the way of thinking mysticism proposes and in what way that may still have something to say in our world from my point of view. He suggests somehow that mysticism or the mysticial way of thinking is still authentic and important for guiding the understanding of things and in my view he is pretty right because any kind of knowledge is in truth esoteric as in the first moment speaks of a mystery. I'd like to remind you Newton for example was as well interested in esoteric knowledge and while it may seem it's contrary to his beliefs and empirical view considering that what he did was penetrating a mystery it's very much understandable why he was mystically inclined and manteined that character. Of course results ultimately come from sort of a cold reasoning but there will always be in any scientific research , as this one type of striving for knowledge , in the experience of gaining knowledge more that just a platitude of deductive reasoning as it will always imply a penetrating of things by thought and and expanding of thinking which are sort of a spiritual experience of understanding anyway. From my point of view anyone who used his thinking knows this and that it's character may be said esoteric is undoubted whoever the final knowledge gained seems to not have any sort of esoterism (I don't agree with this but it's another story) which pretty much is sort of a mystical experience , and alsoa good way of finding from such an experience a way to truth by gaining insight and ideas. Therefore the problem regarding mysticism is that it helps us controling our experience of gaining knowledge and understanding better also what "knowledge" is meaning therefore in my view more than just justifying physics by mysticism as you said. That mysticism is vague and unclear is right , that Cantor had mystical experiences and believed God to be "absolute infinity" greater than all other infinites though he didn't knew very well what he was thinking and therefore that this thing is complete craziness and made him crazy also is right in my view haha. But there is a value in mystical thinking which as Russell says in the book , is not obtainable from other sources anyway.
It's more to say about this and the path to understanding though it's a good thing to debate as I'd like to know your picture regarding how we gain knowledge though. But let's stop with the mysticism if it is a stopping from it and get back to the problem regarding the present state of quantum physics anyway.
The problem with quantum physics is that it should be physics not just formal mathematics , therefore it should serve as a model regarding empircal problems of the material/physical world and model that physical world by the use of mathematics as it was proposed from pytaghoreans and classical physics period. That's the whole spirit of physics or mathematical physics which separates it from simple math and simple formal characteristic ,it's empirical character. Physics therefore about so to speak a modeling of the material world , a trial to udnerstand physical phenomena by using mathematics and this is what makes physicists sort of philosophers as you said because they try to understand the empircial world.Therefore it will never be enough to say that "to make good use fo physics" we jsut need a consistent and well done axiomatic foundation because that violates the very character of physics grounded in mentally mastering by use of formal mathematics empirical things and as such a "good"physics never forgets that. For good physics to be done we need to see how the physical world corresponds to our mathematical description of it in an intuitive way , it's not that the demand of seeing intuitively something is arbitrary it's innately necessary in physics. It's innately necessary because ultimately we speek of physical things which should be there and work , things which can be intuited anyway.
Therefore quantum mechanics which gives such an excuse as so to say for example "we don't care of interpretation" simply violates what physics is about and what made a lot of great physicists liked in physics , the idea that we can grasp and udnerstand this material world and it's beauty compeltely clear maybe also with sort of a mystical feeling very common to the pythagoreans which is felt by anyone who understands this rightly. Therefore if quantum physics has the arrogance of saying such a thing as interpretation is not important and our physics is good is a form of superficiality and a forgetting of what physics is about and it's ideals and just a reduction to compelte formal and useless construct anyway. About your example I have to ask you , is that what it actually says or something close which is rather simplified in those terms ? I don't know of this theorem and have to study it a bit to understand what it says but please answer the question. Noentheles from what i see it is named a paradox or at least a "paradox" and therefore not simply a theorem. It's certainly makes at most as least sense as the wave duality as it contradicts the principle of identity , and doesn't seem by this to follow in a consistent way if it is derived from an axiomatic system which contains the principle of identity though.
It seems you are asking about a criteria of verifying what makes sense. Well , in a naive form we say that something makes sense when we admit it is possible to understand it when it's possible therefore to intuit )immediately grasp in thinking the sense it underlies , naming a simple direct grasping of the "sense". Consider that sense is itself a very subjective concept. Nonetheless to answer of this problem more in depth you also have to answer my question from upper. But considering this subject is very much to talk. How we define sense anyway ?
Do you consider that something which is consistently is and anyway therefore will always be understandable or do you consider that despite deriving from understandable stuuf it is not anyway ? This question is very important and I don't find that because of ome thigns which use subtle reasoning and which are not well understood by anyone disguising themselves in technicalities most often we have a sufficient reason for dismising the idea that anything consistent is understandable by thought anyway.
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@A A – Right now the conversation seems to be about the matter of "things making good sense", we've already moved from the original subject of "putting mathematics on an axiomatic basis which then can be machine verifiable". The best I can understand you, you seem to be making the argument for the need for things to "make good sense", as suggested by your remark, "Therefore quantum mechanics which gives such an excuse as to say for example 'we don't care [of] interpretation' simply violates what physics is about ..." (emphasis added). Additionally, you seem to suggest that I and Russell "share thinking " about the value and role of "mysticism" in physics, and you cited Newton's history of interest in mysticism. I think you have the perception that commonly theoretical physicists first cook up imaginative wild ideas, and then find mathematics to fit. In reality, the other way occurs---the mathematics itself forces theoretical physicists to accept the implications, whether they like it or not. Let me give you a powerful example. Albert Einstein was not the first man to be aware of the famous Lorentz Transform equations of Special Relativity, from which the famous E = m c 2 equation is derived. Other mathematicians before his publication of his paper in 1905 were well aware of these transform equations, and how it does neatly resolves the Michelson-Morley experiment. But all they knew what the implications were, which meant that time and space was not uniform everywhere, and suggested a spacetime that was simply too bizarre for them to accept. Henri Poincaire even developed a entire mathematics on the study of such transforms, looking for alternatives and trying to come up with alternative "interpretations" that would "sound better" to "common sense". What Albert Einstein did was to make the call, and declare that space and time wasn't uniform everywhere. Even after he published his 1905 paper, people for decades still refused to accept his declaration, and some today in 2016 still continue to write crank papers "disproving" Einstein's contention.
The history of "good common sense" guiding the development of mathematics and physics has not been a good one.
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@Michael Mendrin – My problem is not at all about such a matter though. I;m not the kind of person who emphasises in any way "common sense" , that is done rather by philosophers like Russel what I argue is not that quantummechanics doesn't make common sense but that it doesn't really make sense , it's not rational. It's a big difference between being rational and having no sense as seemingly thigns may not have any sort of sense but still be rational but in quantum physics it happens that it's not rational in any way whatsoever anyway.
It's a rational thing for physics , not just a common sense one , to be about the empirical world and speak of it I think. The conversation is and still remains about what is rational and what can be justified as remaining rational. therefore it's nothing about lettin ourselves fooled by seemingly rational but about trying to be rigorous about what we are thinking analyzing seemingly rational things which have the pretension of being so "hyper" abstract that they can't be understood with the seriosity they demand and see if this pretension is actually justified or in fact we are not really knowing what we are talking about too well.
Therefore my argument is rather related to being skeptical and trying to understand better what is being told in such uber abstract things and discerning whether they really deserve their name or they are just an impressive but flawed reasoning because just this emphasized understanding , which in my view wasn't ever obtained such that it is completely clear and direct it's mathematically correct can say if the presented claims are right or wrong hence some who comment rightly. Like i said at the beginning at in the conversation I want to understand it better such that there is no doubt whether the reasoning presented is right or wrong , maybe exactly in the way Godel was able which shall serve as an example of truly coherent work in mathematics. If this same depth and logical clarity of reasoning and argument is achieved such that there will be no doubt , seeing why the reasoning in each step is right just then we can say that we truly understand the most complicated theories we suppose to be rational like "transfinity" anyway. Therefore in such a view , of achieving complete clarity and of truly understanding formulations which are very far from being complete and in my view to fast taken as right by the superficiality and lack of patience which unfortunately is so to say usually manifested in important things I put my hopes. it's very important to remark that since Cantor's theory is not well understood it wasn't ever completely well formulated in it's proof and the important steps of derivation weren't therefore made clear enough for anyone to be undoubtable. In other words the proof of transfinity lacks complete clarity of what are the things which make the theory right steps on which afterwards some persons got to criticize it which would have never happen if Cantor would have had a more rigorous treatment explaining why every improtant step is consistent and in what view they were done and not just stating some vague things and concluding he is right anyway.
Well , I have to say that I don't have that image of physics and I agree with you it's the other way around. I just think all too often physicists lack responsability and even maturity in making claims and all too often we think we understand things which in fact we don't truly understand as they really are or simply that they aren't truly rigorous even if their axiomatic and mathematical reasoning seems correct. As I told you the XX and late XIX centuries launched the fashion that everything regarding common sense is wrong , but this fashion is simply going to superficial claims once it enters into attacking axioms like that of identity it simply turn out to make things irrational and not as it may be the case with some of Einstein's work a marvelous still rational world which violates some rules which we took for granted as right making a poor image for what true understanding should be anyway. Therefore it's not guiding physics by prejudices but by reason , trying to understand everything which seems not well understood , trying to compelte and justify our present understanding and eventually if it is the case that we obtain soemthign which after all our strives still remains not undertstandable try at least to udnerstand how can some things be not udnerstandable but this implies as said maturity and rigor and true deep contemplation. It so to say guiding physics by what animates it and everything , namely by reason and to me this doesn't seem a bad prospect since it's the original motivation of anything we do in knowledge related things with a method which let's nothing not clarified and incomplete and especially the big problems with very unstable answers obtained like transfinity. Therefore I suppose you agree this is a right way to go and that's what I try to indeed do here also. In short the motivation is that we still don't truly udnerstand nor the coherence nor the intuition behind great abstract ideas. It's been a great belief of mien that anything understandable or almost anything can be pointed so clear such that even a child will be able to understand at least the reasoning if not the concept as being right and that true understanding is such crystal like in it's structure anyway.
I don't say that you and Russel share the same views of mysticism. Russel said of mysticism that is sort of a "emotional thing mixed with reason" or anyway something close to that which you said about philosophy characterizing it as an emotional thing and that's the simalarity I spoke of between your thoughts. That idea is somewhere in the first pages of the book. I'll search it if see the book online and showed it at what part I refer. You want me to though ?
About your example we can say and I think you agree with me that if others proposed versions of space time and didn't seem believable that might be anyway a result of the fact that they didn't really understand as well as Einstein what the problem was about in considering space time I think. Therefore , it's again about understanding which was incomplete and superficial while in the case of what Einstein proposed was more close to how things are in our intuitive grasp of reasoning and that's why they seemed and were felt more right I think. Considering this it may be said they were felt as being closer maybe jsut because they just expressed reason itself better and in some sort emphasisez the idea that whatever is right is already authentically felt. there is also the possibility that we have the impression we feel something as right but that doesn't make it to be that authentical thing but the important thing to notice is that reason provides the understanding and feeling of something being correct anyway. But , you still didn't answer the question I put. I still want to continue the discussion about mathematical structure or the deterministic structure of mathematics though therefore please say if you want to continue with our inquiry or do you think maybe there are still some things to say about the topic of method in physics though ?
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@A A – Okay, this is a really long paragraph this time, and I'll have to figure out how to reduce this to the essential point you're trying to get across. Then I'll answer it. Give me some time.
One mistake I made was to make a reference to "common sense", when I should have just stuck with "sense".
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@Michael Mendrin – Well , sure that I'll let you think at what I told. If it helps I state a little bit more detailed that my view is that of trying to understand better and make explicit what does it mean to me to understand better and whata re some of the traits of the model of understanding which I have in mind anyway.
I argue also that I'm not proposing common sense and I'm aware of it'slimtiations but that my motivation stays in a demand for reason and in considering thigns of such importance and high abstarction in a rigorous display of reason which very well may be said from me is lacking in the necessary amount needed so to say. Also I speak of other stuff such that I'm aware a little of the history of physics and the way the understanding and development of a subject goes in physics but know that if you will have time you are very welcome to come slack and ask me about things in which I seemed inconsistent or wasn't pretty clear about anyway.
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@A A – If you could state your "essential point" that you were trying to make in that long paragraph, I would appreciate it.
Regarding the original subject of "discussion about mathematical structure or the deterministic structure of mathematics"---what was the original question? It sounded like to me that you were asking how we may proceed, which is a pretty open-ended question, like, "what can we do to help guarantee that there aren't contradictions in mathematics?"
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@Michael Mendrin – Hmmmm ,let me try to do that. Supposing is the 4th paragraph in the order of reading my point is that we should guide our method and way by a more attentive and deeper reasoning regarding what we can validate as right or wrong especially in things which require highly abstract and complicated thinking. Today it my point of view this is not being done as there are a lot of things in the proof of theories and crazy deep abstract theorems like Cantor's transfinity where the improtant steps aren't all that well argumented to be completely clear. For example if we think of a proof as having four improtant steps and speaking of one important concept it will never be enough , if the reasoning is complicated , to jsut state the steps and not show by the strongest reasoning they are right.
Cantor for example didn't do that and hence his theorem and steps leave place for rightly done criticism because they aren't well enough justified. the proof may seem right but sicne we don't know what we're speaking of we can't trully know if it's right or wrong , we don't know how to trully udnerstand that proof and interpret it things which with more rigor from cantor would have been avoided. Nonetheless remark that maybe Cantor wasn't able to be so rigorous. Because he didn't in a true way udnerstood what he grasped I think anyway. This applies to physics the same way it applies to pure mathematics. We don't really have a deep understanding of things and hence a lot of confusion. If we try to be compelte in our udnerstanding then we would have. This is what I try to state. It's clearer though ?
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@A A – Actually, I do appreciate that you've come to this point, which is a crucial one, central to the entire conversation we've been having. You are criticizing Cantor's lack of rigor in his proof of transfinite numbers. Let me now restate my previous question to this, "what can we do to improve rigor in mathematical proofs?" Mathematicians initially couldn't agree on whether Cantor's proof(s) had the necessary rigor, because too many of the terms lacked precision in definition---so people couldn't agree on exactly what he was talking about. That was one of the impetuses behind Principia Mathematica , which was to introduce "uniform standards" of mathematical proof, using axioms and rules of inference from symbolic logic as the foundation. It just so happens that's the ideal medium for machine verification!
We wandered quite a bit through this entire conversation, but we actually do share a common aversion to "lack of rigor" in mathematics, which is often the problem with unreviewed crank papers. Fortunately, that's rarely a problem in well-reviewed papers, even though there still may be controversies involved.
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@Michael Mendrin – If mathematicians who are so smart couldn't agree on Cantor's proof and if anyway it's agreed that understanding something authentically gives palce for a proof which a child can understand then we can simply conclude that Cantor's proof in no way is a completely clear or rightly done proof and eventually is wrap in mysticism. Moreover since such great mathematicians who should be good at abstarct thinking couldn't agree it's clear Cantor's proof of transfinity is not clear.
I'm glad you come to the point of asking what can be improved at Cantor's work. There are maybe a lot more things but I will state just those to which I now think.
Cantor didn't do that and I doubt anyone really understands the proofs so if you don't mind I'd prefer asking what can we do now to understand better Cantor's theory of transfinite numbers and verify it's correctness or use our imagination in a rational way and try to see how would a real understanding of what Cantor spoke of would look like , how would a completely clear theory of transfinite theory or rather a trully deep and authentic udnerstanding of the infinite. Firstly if we pout the problem in this terms I'd say it's good to see what Cantor speaks of since every theorem is a form of predication who speaks based on some reasoning which it presents more or less clear about something and shows a conception.
The object of his predication is infinity so in his theory he speaks of the concept of infinity trying to show a reasoning by which he claims to prove mathematically some infinities are larger than others but this without speaking too much of what "infinity" is or means. By that I mean he doesn't shows what is his conception , his view of infinity in a rigorous way such that we know how to itnerpret and think of the reasoning he proposes and hence verify by such an interpretation it's correctness and hence we already have problems trully understanding what was in his mind when he spoke of infinity and showed his theory because he doesn't tell it to us in the rigorous way it demanded , he lets this job to us the readers. This all the more so when the concept of infinity is very subtle , in fact so subtle that had contributions to making Cantor crazy that deserves at least some sort of trying to see how rigurous and authentic predication of infinity would look like. Therefore the first step which he should have done for making a truly consistent theory of the infinite is in my opinion realising what he is doing , that he speaks of infinity and try to understand and show slowly how a predication about infinity is so to say legitimate in such a way that nobody goes crazy as we articualte what we are speaking and having in mind thigns which avoids the confusion and contradictions seen when correctly intelectualy mastered. After we realise how to speak of infinity we can go to the proof and make predication about infinity. We shoudl also avoid defining infinity in pretty much mystical ways as that whose part is jsut as large as the whole because that is still confusing or , at least , if we can't do otherwise we should try to explain why and how can infinity be "that whose part is large as the whole" therefore make a substantial predication about an observed trait in an articualte way showing exactly what we have in mind when we see infinity has such a characteristic. For the proof we should really see if the predication we are speaking of is really inherent deduction and show that rigorous. Show that indeed it's rightly speaken of parts who belong to infinity and do that and that ,also show that the traits taken into account and construction proposed trully is a priori and not an artifice to show the proof right or relating to external traits. After doing such a thing and we understand better infinity let's try to understand what it means after all that some infinity is large than other , how can I intuit that thought in my mind and what is the logical/deterministic structure of the reasoning that anyway assures that as therefore a right reasoning. There are a lot of things to say. But maybe it's very important to say I haven't read the original proofs of Cantor though I'm quite familiar with their sketched ideas and I'm pretty sure that rather set theory in general our udnerstanding of infinity is very poor. But do you agree therefore sot to speak that such a thinking should underlie Cantor's proof ? I;m familair with the fact he spoke of and argued what others from past said about infinity and how right or wrong the were , defending his view with a more or less good reasoning of infinity but he never developed a complete and deep conception of infinity apart from his mathematical proofs regarding counting and set theory anyway.
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@A A – Okay, we're at a point where I do have something to chew on. Let me review Cantor's work on transfinite numbers and maybe I can try to propose something. This will take me some time, and when I get back, maybe I'll have some kind of a proposal. I'm leaving late tonight, and I'll be back after the Labor Day weekend.
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@Michael Mendrin – Hmmmm , actually you just made a proposal. Do you know any sourc where I can find Cantor's original works on the net anyway?
I think I'm going to look at them too though there are some other improtant thigns to do but since I just spoke so much of it I'd like to see his personal works. As I have said the solution he proposed needs to be rigorously justified. Therefore there's not problem in trying to contribute or try to understand that better as long as I anyway will be careful enough and rigorous enouhh of course not to get crazy thinking at his works. It will be a very intersting to make after all. Maybe the technical terms will be a problem but I can cope with them I hope. Btw , I know who's DD now haha. And have a good and well time in your trip anyway.
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@A A – Cantor's theory of transfinite numbers is one of those controversial subjects in mathematics, in which people are still arguing about it. There's no lack of resources on the internet, so let me round up a bunch and post them here later.
Remember, this is already Labor Day Weekend for me.
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@Michael Mendrin – Haha. Ok , then have a nice Labor Day Weekend anyway.
I can look for Cantor's work myself though if you guarantee I'll find them. Therefore you don't have to bother searching them as i'll do it myself and thanks anyway.
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@A A
–
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http://ls.poly.edu/~jbain/philmath/philmathlectures/M06.Transfinite.Math.pdf
http://www.cogsci.ucsd.edu/~nunez/web/TransfinitePrgmtcs.pdf
https://en.wikisource.org/wiki/1911 Encyclop%C3%A6dia Britannica/Number/Transfinite_Numbers
This one raises objections to transfinite numbers
https://arxiv.org/ftp/math/papers/0306/0306200.pdf
This one is interesting as how it could relate to physics
https://arxiv.org/pdf/math-ph/9909033v1.pdf
This is an interesting “philosophy” paper on the subject and related matters http://etd.library.vanderbilt.edu/available/etd-04172014-220702/unrestricted/SHammontree.pdf
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@Michael Mendrin – Thanks. I'll search for the original papers of Cantor and translations too not jsut commentaries about his theories as the original is very important.
The Britannica link doesn't work but I searched there about this numbers and found some things about them so there's not problem , I'll look at the wikipedia too haha. I'll post you also the original works of Cantor if I find them for now I found though https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity.pdf this anyway.
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@A A – That's funny, that link still works for me.
By the way, I do own an original 1911 Encyclopedia Britannica set. It's really a classic, and in my opinion, the high water mark in encyclopedias. Today, it's...well, an embarrassment. Not the same any more.
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@Michael Mendrin – Haha that's truly kind of funny. I do not bother reading encyclopedias btw anyway I prefer reading original works on subjects which interest me anyway.
They are much more detailed and rich in content that encyclopedias whicha re good for making a very broad view of things and contribute to general knowledge. I'm not miopic in view necessary but I feel there's so much to explain and see that what an encyclopedia would give , a lot more richness of understanding. But you may be right about superficial encyclopedias and ltierature today. usually when I go on a library I am pretty dissapointed of what I find though I am looking for more humanistic or cultural concerns books than scientific ones. Scientific books or rather Pop science books are as you can guess too superficial in my view anyway. I'm pretty disappointed of the way science and reflection is today , with it's way which to me seems to lack some things but I will not elaborate. Maybe I'm just being superficial but we'll see about that , btw still don't find Cantor's papers. But I'll keep searching. I'll also try to find how a good encyclopedia page was in 1911 haha thought that should be more difficult that those papers anyway.
@Michael Mendrin – https://archive.org/detailsGeorgCantorContributionsToTheFoundingOfTheTheoryOfTransfiniteNumbers
I think this is it. I think it's a collection of articles related to set theory written by Cantor apart from the introduction.
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@A A – Well, that would be getting it from the horse's mouth. I have it up already.
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@Michael Mendrin – Yes , I suppose me studying this work along with what you provided will be a long day in which i will just keep thinking of stuff with components , "agregates" and in an abstract and a priori way at the form of counting trying to see the reasonign and conception the pioneer of set theory Cantor had anyway.
I'll just keep thinking at this as trying to interpret and understand the reasoning exposed in the works ,I suppose it will be an itneresting exegetical work though of course I also hope I will not immerse too much in it to get crazy thinking at infinity and counting but I'll god now though. Tomorow will be an interesting day I think. Once more have a nice labor weekend and thanks anyway.
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@A A – You'll probably go crazy thinking about infinity....many other people have. Good luck!
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@Michael Mendrin – The only way by which that can prevent me going crazy thinking about infinity so to say is in my opinion just rigor and rational thinking at each point.
But in case I'll see it doesn't work I'll most likely stop thinking at it as of course there will be nothing to gain by going crazy and following Cantor's ways anyway.
@Michael Mendrin – Well , I think labor's day has ended and from what remember I suppose anyway you're back home though I may be wrong.
But , in case I am right or when you get home , please tell shall we continue the discussion ?
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@A A – I'm back and I'll get back to you in a bit.
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@Michael Mendrin – Great , since we have a lot of things to discuss by what I think related to Cantor's theory anyway.
But , please say from where shall we continue with the discussion or how to conduct our inquiry anyway.
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@A A – I think we should focus on Cantor's Transfinite numbers for now. I need to examine the literature on it a little more closely.
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@Michael Mendrin – Hmmmmmm , I was thinking that in order to arrive at the problem in a rigorous way of transfinite theory we should try inquiring into the problem of "set theory". But nonetheless , if trying to understand transfinite numbers that inquire into the meaning of set theory will be indeed necessary it will arrive naturally I think therefore let's go along with he study of transfinite numbers as you say anyway.
I read some of the articles you posted and although they were instructive and interesting from time to time I have to say that they seemed superficial though. The objections together with the interpretations of those articles seemed at least to me oversimplified and none seemed to equal the "rigor" of Cantor's work anyway. It appears to me that nobody , nor those who sustain the theory nor those against it truly understand well infinity until this point of time. But what's the literature that you examine though ?
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@A A – I am not a specialist in infinite set theory, and if I'm going to be serious about it, I'd have to start ordering some books on it instead of relying on what's available for free on the internet. But I am already working on other projects, so this will be an undertaking that I don't think I'd take lightly.
Here's one free introductory book on the subject
An Introduction on Transfinite Mathematics
I just now ordered "Sets and Transfinite Numbers", Zuckerman, Martin M, for exactly 1 cent. Plus shipping costs. It'll come to my door in a couple of weeks.
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@Michael Mendrin – Oh , ok no problem. Let me look at your book and after all this more or less serious "packing" where should we start this sailing-journey discussion ?
Btw , I also found some introductory books (and pretty serious seemingly) on thed set theory which I can give you if you want and also some popularising books. Nonetheless I found one which is pretty technical and interesting book too but unfortunately I lack the knowledge for dealing with , perhaps , all that it says.
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@A A – Check this out, The Transfinite Universe , which suggests the depth and complexity of the issues involved with this whole transfinite business. Any field or branch of mathematics is based on agreed-on definitions and axioms that are presumed to be true, even though we know that we often have a choice of making such axioms true or false or undetermined. Here, in the matter of transfinite numbers and the various set theories that go with it, the number of choices simply explodes. There is even an "Axiom of Choice", if I try to be funny about this! We don't have to assume that the Axiom of Choice is true, or is false. That, itself, again, is a choice we have. So, are transfinite numbers "real"? It depends on what we're talking about and which universe we are choosing to be in.
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@Michael Mendrin – Hmmmmm , now you're speaking as a modal logician.
I read until now the few first pages of the book you just sent me and it's simplicity at first made me not like it too much but things start to get a little bit more complicated and interesting as I keep reading it , though it seems to follow the original path of presentation which I find in all books. Start with defining what sets are (eventually speak of the logical paradoxes implied), speak about the one-to-one correspondence , talk about the fact that counting and one to one correspondence helps understanding infinite , try formulate a good and rigurous definition which separates infinite from finite sets , speak of the relations about sets , of ordinals and so on (eventually also speak of axiomatic stuff) such that you establish the conceptual working frame such that you can attack the problem and introduction of transfinity and Cantor's great ideas. The tone in this book though has points in which I am not pleased , as with many others , as looking set theory as a mathematics of the infinite and mainly about infinite which nonethelss is not true in my view , speaking that we develop mathematical ideas such that we can tackle whatever problem when in fact what I think should be the problem here is not of "developing" ideas but of understanding what is the reasoning about sets and how it is correctly done and justified as such having some kind of philoosphical precarity but it's an ok book from what I read. Of course it still seems to me to be oversimplification and I really appreciate I can read Cantor's Contributions to the theory of transfinite because it gives things a little bit more density I think. It's a good book as an introduction for people who don't have maybe the sort of philosophical way of thinking I have so I'll keep reading it to understand better at what stage of udnerstanding is elemetnary set theory so to say.
I think that the problem with our understanding of infinite and infinity results maybe from a precarrity of our ideatical elaboration of the concept. In order to advance predication about it and make derivation we need to understand it better. Therefore in order to really understand it we need to elaborate this concept attentively. Maybe there is no problem none of us has toomuch background knowledge as nonetheless we may have to start from the beginning altogether anyway.
Transfinity firstly of all , from what I read , has some issues related to that improper conceptual understanding of the infinite. The first such problem from what i know is that of justifying or explaining the difference between the actual and the potential infinity which I have no idea if anyway really did so , either showing that actuality or potentiality exclud each other. This step would be crucial for truly advancing a real understand of the infinity. This is maybe the first , but maybe jsut after we establish what is set theory itself that should be of interest.
Btw , I agree with you that in mathematics we are interested with the consistency of things derived from "axioms" , with this axiomatic view and not with the truth or falseness related to the world or our intelectual capacity of grasping the validity of the axioms or not. I have a very intersesting , elementary and simplebook which speaks of axiomatic set theory and I find entertaining for most although of course not completly satisfying as I may never be satisfied with anything on the century in which I leave but that I find it good it's still something. It's called A book of set theory and explains simply and rigorous things. One of them is what you just told me regarding the predication of mathematics starting from axioms anyway.
If you want to I'll upload it on slack or I can give you the link from where I got the book in case you don't already have it. The problem regarding such books which speak of transfinite theory is that they are as it can be seen in their simple introductions based on some not very well developed concepts therefore considerign this offering a not too compelte view of the problem. But nonetheless I'll look at the other book you posted and thanks! I think we should start by trying to understand that distinction about actuals and potentials infinities or maybe the starting point should be what is and what wants "the study of sets" to do and maybe , since they are elementary things we shouldn't worry we don't know too much about the subject maybe this way of considerign the problem being justified more by the simple fact that such a beginning isn't even established rigorously. Therefore shall i ask you if you know of anyone tomake the disticntion between the actual or potential infinity , or of what is considered set theory currently anyway ?
@Michael Mendrin – Emmmmm , I suppose you saw my other reply to this. You seem to avoid expressing your thoughts regarding infinite but why though ?
I hope you don't think we'll get crazy. As long as we're keeping rigorous and know what we are talking about that may not be the case I suppose. Moreover deepening the subject matter might lead to interesting insights. Therefore so to say there's no reason by me to avoid discussing it anyway..
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@A A – A A, even though "infinite set theory" hasn't been one of my favorite subjects in mathematics (or physics) in the past, I am finding it interesting now precisely because it's long been a controversial subject in mathematics, and yet I can see it being amendable via machine analysis, as what we've been talking about. In other words, it seems like an excellent test subject. So now I'm starting to think about this, and, you know, deep matters like this takes time for things to sink in. It's going to require many of my walks for me to think things through.
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@Michael Mendrin – Interestingly enough you seem quite the fitted meditative type of person for elaborating such a problem.
But , more precisely what are the problems that bother you about infinity and machine implementation that require deep attention in your walks anyway ?
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@A A – Okay, let me clear up a few things. The concept of "infinity" is not "what bothers me". It's just a concept, like the concept of 0. But what matters are "rules" that involve infinity. For example, is infinity divided by infinity equal 1 or what? The convention is that it is indeterminate, like how 0 divided by 0 is also indeterminate. Mathematics has settled on a number of such rules, which, by the way, are normally implemented by math software---that usually say both are indeterminate.
The salient point is the cloud of rules that go with such concepts as infinity and 0. This is the "structure" that I keep referred to over and over in this entire conversation. This "structure" can or ought to be implementable by machine, or math software. And for something as simple as 0 or infinity, they [mostly] already are.
Now, getting to Cantor's transfinite numbers, which is actually a world of world of all kinds of infinities, a zoologist's dream of a wealth of "species" of infinities. There's literally now an encyclopedia of "rules" that go with them---and here's the kicker: What those rules are or should be is still an open question, an unsettled matter, there are many possible "set of rules" that can go with Cantor's transfinite numbers. This is a tremendously complicated subject, and specialists have spent careers studying this. What I am suggesting is that perhaps machine analysis can speed up the process and more easily show or illustrate what would be the consequences of one set of "accepted rules" would be as compared to another?
Yeah, I can tell you what disturbs me about this is the depth and amount of work this is going to involve. But kudos to Whitehead and Russell's opus Principia Mathematica for having already gotten an excellent start on this problem. And, indeed, that book should be at the top of my reading list on this subject.
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@Michael Mendrin – I understood your view on the matter as this was a thing which you kept saying more or less subtle , along with others also , through this discussion. You seem to consider that the intuitive "content" of a concept and as such the "sense" which makes it understandable based on our thinking or not isn't important as a criteria for what is validly true or wrong because that part is exhausted by the consistency and derivation of the predication starting from the axiomatic system , in short if we know that they are consistent it doesn't matter if it "makes" sense anyway.
Of course that considering this view we find that it admits as possible that some things are consistent but lack the direct "sense" which makes it understandable or in other words that there can be things which are logically consistent yet despite the entire consistency they have , our mind is not capable to see or understand. This claim nonetheless is a thing which I think should deserve some inquiry itself because it is not at all obvious how can there be things which are logically consistent but not understandable potentially by our mind which can be saw as a system that should have at least the same structure as the consistent predication. The problem of how can we understand at a principal level the relation between understanding and logical consistency of predication is subtle but a valid problem. But here too we get again at the concept of "structure" which is central to solve such problems and , by what we have seen some other related problems anyway.
While I don't agree that there are things which we can't potentially understand and that , supposing I am right and for example Cantor's theory is understandable , trying to understand the concept better is important because it helps us get to that possible understanding I agree that this is not important for the purposes of understanding consistency and mechanical computation and therefore that we can start the discussion from the point of consistent derivation. This moreover so since by understanding such a thing we may get to the problem of also understanding if or not so to say "any consistent predication is understandable or has "sense" " or maybe just for the use of getting into the innate logical structure of things anyway.
I think actually that the problem is finding all such sets of rules to implement in the computer to verify the proofs consistency and of course this problem , as it already is classical in this discussion leads to the problem of structure or rather "deterministic structure". Therefore understanding how the set of rules would look like , what are the actual characteristics for any set of rules implementable and how they relate with each other provides us with the necessary formulation of the problem as it gives us the terms necessary for formally thinking of. But I have to say that at least when I say "structure" I am not thinking just at some defined axiomatic system but I use it referring to the general organization of the derivation process in accordance so to say with such an axiomatic system , since this kind of structure is truly important in my view for settling such problems where the atoms of such a deriving process are simply the "derivation" of a formula from another. It is this sort of derivation process , which we innately find in the mechanical way of working for a system that is responsible for what makes it possible or not that a proposed set of rules works or is fitted for computable testing against others and because this "deterministic" structure is the one which may be thought truly as the a priori or innate cause in it we should look anyway never forgetting that it is about such a structure though.
Regarding division by 0 or with infinity I think that such things are still pretty much accessible in our intuitive grasp and provide a good example of the way the relation between understanding and logical consistency can be though of and at least part of the integrity of a concept is as opposed to a simple artifice of thinking. When i make a true predication about a concept like 0 or infinity that may very well be thought as meaning that my intuitive grasp of the concept , the way so to say I directly see the concept in my mind when I conceive it is consistently kept while making predications of it which in mathematical formal language means that the concept here equivalating with some formal characterization or the "structure" you speak of conserves itself or in the mathematical formulation doesn't imply contradictions.
Btw , I don't think I can let you do all the work alone even without any sort of serious knowledge of the domain of "proof theory" and mathematical logic anyway. Therefore please tell me how or by what though can I start helping (even if it implies books) and of course continuing this discussion which we have been having anyway.
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@A A – "I don't think I can let you do all the work alone..." Why? I might hurt myself? Heh. Well, you know, what's interesting is that even though both of us have been talking about "structure", we don't even agree on exactly what it means. This is one of the great difficulties of doing math----how do we nail down the meaning of words like that? How would you go about defining the word "structure" in this context?
There's nothing wrong with using computers (machines) when "doing math"---it's a force multiplier, like using a backhoe as compared to using a shovel. Not only we know they can do numbers really fast, they are already doing proofs really well too. Think of mathematics as an advancing edge---first it's only in the imagination, then we try to get down the definitions, axioms, then work out a few theorems, etc. Even something as "intuitive" as 0/0 = indeterminate is really not intuitive at all, it took mathematicians some time to get that settled. Once the "new" mathematics is in a form where it can be implement by computer (machine), then it can really take off---and maybe help point new directions for our imaginations to take. Or at least that's how I'd like to picture this. I'm pretty sure it's only a matter of time before the math behind Cantor's transfinite numbers becomes analyzable by software, and then it should it be interesting to see what it has to say about them.
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@Michael Mendrin – Because there is a tremendous amount of work I think not to say that this in a very questionable theoretical context regarding the validity and true consistency of foundations and mathematical logic , context which i dare say can prove pretty overwhelming in more than one way but mainly because it may even require elaborating new and redefining old concepts from scratch , thing which while of course not necessary impossible alone can be done easier in a dialectic way. Well , if I am to be honest , because of this discussion and some other things on which I'm interested I had to consider the problem of defining what "structure" means and did so already so I suppose I should show you how I think of and define what we mean by "structure" .
I firstly should illustarte my thought process so I have to say I thought that the definition related to the conceptual experience of thought which we have when we say and think in a conventional way the meaning of the word "structure" and hence that as it is a concept found in such a thinking experience in which we find it's sense we should inquire. Therefore because we look at such a mental experience when we try to give a definition of the word "structure" and generally when we want to define what a concept is we are trying to get at the a priori content or at the very abstract form of the concept , this meaning at immediate content which is simultaneously perceived with the perception of the concept itself.
Providing a definition means therefore seeking at such a level of abstract thought and in this sense I looked for such a definition anyway.
When we speak of structure , when we say "structure" a first thing to observe is that it provides in our thinking the concievement of an "organization" of certain components meaning by that therefore that whatever implies a structure has the two characteristics of firstly being about some elements which we can perceive as different , some components and therefore has an atomist characteristic and on the other side a characteristic which implies some sort of organization between them whcih doesn't let them to be arbitrary one with each other. Therefore , considering this two characteristics we can say that a structure ,in whatever form it takes always has some sort of "components" and an "organization" by which we think this components as related with each other and correlated with each other the actual ways by which this organization is done between the compoennts and the way the components are themselves being various and not restrcited.
Therefore a structure is as it seems to be nothing but a form of organization between components or more clearly a collection of elements with relations between them by which relation in the experience of thought so to say I associate an element with other elemnts this form of association if , conceived in only one act of thought leading to the way of a complete and integer "structure" from which we can further infer some other rules. An important thing is to understand the emphasis of the elements or atomist characteristic and not consider just the rules because in the cocnept of structure it is analytically implied the cocnept of elements and components which have such rules whether or not this components are empircal or simple elements of thoughts seen as "objects" of predication and abstractly considered together.
This definition applies very well for visual things as well as more abstarct thigns by it's process nature I think.Applyng this definition of structure for mathematical reasoning which is a process that can be articualted as as nothing but a predication resulting in form of consistent derivation related to some axioms the structure of the mathematical reasonign is the structure of deductions in it , or the way this deductions happen in it or by the definition the way such deductions related to each other according to certain rules or characteristic by which the relations considered are defined , thing which to me seems to be quite a correct definition of speaking about the structure of amthematcial reasoning , right ? Nonetheless do you agree with this or you find another one more appropriate? The definition appears catched in my point of view but I haveto know yours anyway.
@Michael Mendrin – Oh ,sorry I wrote so much about structure and our thought experience of the concept that I forgot of the other part of your comment haha.
Yep , I do agree that using comptuers as tools for doing math is a great idea but not only for it's applciation but because makes the problem of what anyway mathematical structure is and looks like even more important and interesting and therefore provides also a theoretical interest which proofs the deep advances have done for such problems despite a lot of superficiality in my view which maybe sometimes can be said to be unavoidable anyway.
@Michael Mendrin – Well , I think you had enough time to think at the meaning of "structure" in the way proposed by me and came with some conclusions regarding it anyway.
Therefore , do you think such a definition is ok and maybe should be chiseled maybe a little or do you think we should replace it with something better anyway?
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@A A – A A, it's a mite early to be thinking about how to define "structure" in prepositional logic form, since in mathematics, the meaning of that word is really broad. Have a look at this Mathematical Structure , and in a more restricted form Structure in Mathematical Logic
Needless to say, the word and notion of "structure" in mathematics is an enormously popular and widely used one.
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@Michael Mendrin – We should think at how to define primitive terms as "structure" to continue the inquiry in an articulate and systematic way or defining at least provisionally.
Without making a definition or at least a provisional definition what we are thinking and advance will be just done heavier and vague in my opinion.
Let's see the definitions from wikipedia. The look pretty close to what I said but they are still very vague.
Also , in the first definition (not from mathematical logic wiki) besides saying that the mathematical object attaches to the set in "some manner" which is extremely vague until said in a very clear way what's the manner it seems that they are actually speaking of "organization" of things rather than structure because they seem to consider the specific set of rules implemented in the abstract set the only thing which makes the "structure". While this is immediately wrong , because the thing which provides the organization between the components of a set , the set of rules heterogeneously applicable to the set , is clearly different from the set of rules plus the components as found in the meaning of "structure" which takes both the rules and the components in the conceiving in thought of the concept but this is not very important though not unimportant either anyway.
Well , a structure is a set endowed with an "organization" where formally an organization is defined as a number of relations between elements that can though be homogenous or heterogeneous in their definition prescribed accordingly with a list of some generally applicable rules for some components from which we can infer the relations between the.The components of a structure can be , as mentioned , in regard with the relevance to the scope intended either homogeneous or heterogenous meaning by that they either have the same characteristics in their definition by which therefore we say that they are homeogenous as the monotoneously apply the characteristic implied regarding their definition or different characteristics therefore we say that they are heterogeneous regarding the conceiving of the formal definition anyway. If when we speak of structure we have in mind this whole of organized relations or take all the relations prescribed together , in only one act or thought by virtue of our faculties of thought we are treating the term "structure" as a whole or entire and doing as such we take it as only one ideatic object of our predication regarding it this being generally necessary for speaking of "the structure" , to speak platonically of the Idea of structure abstractly and generally anyway.
I think this sets things up for what is generally named by structure even if it is a mathematical term or not since if it is to say that the term I described is so sued in natural language but not in the formal language of mathematics then I should say that all thought and conceiving of such terms are initially so related with the use in a natural language and also firstly seen (and consequently also thought) that ultimately they are formed and related with the natural language anyway. We can therefore apply this definition for mathematical logic but in mathematical logic's terms such that it works. Accordingly ,we do this by seeing what a structure is composed in our very general definition which shall apply for the interpretation of mathematical logic and trying to map this terms in the terms of a mathematical logic point of view. Therefore we say that a structure has homegenous/heterogeneous components which have some relations between them prescribed by some rules applicable to the components in a precise or definite way. Therefore considering this sort of definition we can apply it for making a definition in mathematical logic interpreting the homegenous/heterogeneous components , relations and list of rules in the mathematical logic way of thinking and terms which is at it's turn derived from formal mechanical scopes anyway. Maybe it's important to consider that , in mathematical logic we want to make a mechanical sort of definition for the application of computation in it anyway. Considering things in this way , do you agree we should start from such a definition and map it or the terms implied in it in mathematical logic terms anyway?
@Michael Mendrin – So , it's still early to think at structure at the stage were you are or arrived at some preliminary definition though?
And in case you arrived is that different from the "components" showed in my said preliminary definition or contains the notions which I pointed out as i pretty much anticipate any definition of structure will contain and in case you didn't yet and nonetheless still it's early not where are you in this problem ?
@Michael Mendrin – I suppose you are well. You don't reply because you are stuck at something and want to settle that first or are you upset of my proposal anyway ?
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@A A – I really do believe that if I'm going to tackle the problem of "mechanizing Cantor's analysis of transfinite numbers", the best use of my time is to review the book Principia Mathematica and see how far they've gone with this. What I am saying is that attempting to "nail down the definition of structure" in this context is premature. Quite seriously, if this were to be fully worked out, so that computer software can then analyze the matter of transfinite numbers, then this would be worthy of publication in a journal somewhere. In other words, this is a huge project, like building the Hoover Dam, and it will require time.
Edit: I checked on availability of Principia Mathematica . Well, the original 1910 three volume set of books can be had for about $140,000. Okay, I don't have that kind of cash on hand with me. I'll look around for cheaper used reprints.
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@Michael Mendrin – I know of a source of Principia Mathematica with all the volumes available for free on internet. If you will go with your investigation far enough I am quite confident that you will find that even if in terms for which today's knowledge is premature to correctly formulate they will simply follow the sketch I found there as that shows the a priori use of structure which is done in any conceiving of the concept. The only problem nonetheless is maybe with mapping it in a coherent interpretation. But even that translation or interpretation (thinking in terms of) eventually most likely follows the a priori structure (sorry) of the concept if it is authentic anyway. I think I'm going study some mathematical logic for a while as a result of the discussion which we have been having (I pretty much had) as besides the very important mathematical problems themselves it's also fascinating from a philosophical and meta mathematical way of reasoning , things which I think always fascinated or imposed a kind of intimate reflection with me besides a simple sort of intelectual love being of a more spiritual (in the french sense) sort of thing despite in general avoiding entering in any mathematical subject deeply since I have some other very important things to do that do not need too much influence of the thinking and elaborating which I do though. Nonetheless , maybe it's good to study Principia Mathematica too which seems like a pretty kind of treasure for the knowledge and besides the understanding of logic and meta logic with all those more than a hundred pages who's links are the ending of the comment in pdf format which you can also print so in case you didn't know of the work enjoy. But shall we study and discuss this work together though ?
https://ia800804.us.archive.org/23/items/PrincipiaMathematicaVolumeI/WhiteheadRussell-PrincipiaMathematicaVolumeI.pdf
https://ia600307.us.archive.org/18/items/PrincipiaMathematicaVol2/AlfredNorthWhiteheadBertrandRussell-PrincipiaMathematicaVol2.pdf
https://ia800800.us.archive.org/24/items/PrincipiaMathematicaVolumeIii/WhiteheadRussell-PrincipiaMathematicaVolumeIii.pdf
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@A A – The three volume set does delve into the matter of cardinals quite extensively, and it is an interesting read. I think I'll order the paperback version of it, so that I can more easily study the books. My gut feeling remains the same, this is the best place to start if I want to see if Cantor's ideas can be "machine analyzable".
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@Michael Mendrin – More than 18 days in this wor(l)d war
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@Michael Mendrin – Word was my main focus, as you can see 140 long comments on one question, each with average 200 or more than that words
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@Prince Loomba – change the title to when words collide but keep the actors
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@Michael Mendrin – Is that an old movie or soemthing ?? Looks like start trek or star wars , actually I wonder how the world will be after 2000 years. Bet Plato might have thought the same more than 2000 yearsago. Well , i'd say him it's just like you maybe already imagined the humans remained almost as superficial but have more technology.
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@A A – It's a sci-fi novel written by Balmer and first published in 1933. A lot of other later sci-fi stories were inspired by this novel, including Superman , which was first published in 1938.
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@Michael Mendrin – Thanks but how do you know such stuff in detail ? Do you like sci-fi anyway ?
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@A A – I'm old. And, yeah, of course I think the 50s and 60s was the "golden age of sci-fi", but I imagine that many today would dispute that.
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@Michael Mendrin – Haha , well ok then. If will ever want to know anything about (old) sci-fi I guess though I have the right person to ask anyway.
@Michael Mendrin – Btw, at what point are you in the study of Principia Mathematica?
Or haven't you started a serious read yet anyway?
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@A A – A A, it's always been in my habit to have a number of "things floating around in my head" sometimes for months or even years at a time before they come to fruit. Even right now, I've got at least a half dozen really good, difficult problems that I'd like to post on Brilliant---but I'm always being distracted. Anyway, I'm waiting for my 3 volume paperback set Principia Mathematica which should arrive in a week or so. What I am particularly interested in is the question of cardinals and transfinite numbers---once implemented "by machine", will it yield interesting new stuff? Where will this take me? i don't know, and that's why I'm interested in this.
I have no interest in just "proving a point", I just like new directions, places where people haven't been before.
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@Michael Mendrin – Well , I know how it is to think at something and then be distracted by something else which immediately seems as interesting. Actually that happens to me pretty much always so to say. Nonetheless distracted and non concentrate work in a subject while can give more complete results in a long period of time by the fact that it treats a more broad area of problems related to the subject due to the invested time and experience doesn't seem the best thing to do always. Suppose for example that you are having a very hard problem indeed which you would say be able to solve if you invest all your efforts and intelectual capacity in it but would give you no results if you don't. For this case it's pretty clear you better concentrate on the work than do multi-tasking. Moreover solving a problem or a hard problem in a long period of time along with other such problems gives sort of an arbitrarity and luck element to your final results (if they are achieved) which the investment of time and of course attention would make look it more meaningful. Maybe it's jsutme but I never liked solving problems just for the adventure and exploration part of getting to a new point and possibility. For me it always have been a sort of spiritual thing which I do , while also for the sake of arriving at a new view and thing though rather for getting insight into the true sense and causes of a thing fr the sake of understanding. To which in an equal measure I might add of course the simple sort of experience of the way of getting to understand and know itself if that way is done meaningfully which meaningfulness depends mainly on how right the way , the work it's done. But maybe it's jsut me though. Also anyway I'm pretty sure you know all those things. Just wanted to say them for a completing of the picture had regarding research and curiosity which I always seen as a reflex of trying to understand an intuitive penetration into the nature of things which by adequate thinking we inconsciously sometimes get glimpses anyway.
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@A A – See my other reply.
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@Michael Mendrin – Btw I got that it's quite a colossal thing also. If you get to prove by machine Cantor's transfinite number or rather disprove them and also get some further results related to the link you envision between the structure of mathematical reasoning and mechanical computation it may be a starting of some other era for mathematics so maybe you should follow the advice I gave you some comments upper and try to deepen this philosophy of yours along with the innate reasons because either way you find a precise result regarding the soudness of such views it should be pretty great and a great advance in mathematical logic anyway.
@Michael Mendrin – Btw , don't you expect any sort of interesting stuff to happen ? I mean , even if so to say you don't know what are your prelimanry simply expectations though ?
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@A A – I don't think you quite get it. If I tackle this, it's going to be one of the biggest undertakings I've ever done. And almost the only reason why I'm contemplating this is that while the concept of "infinity" in theoretical physics is at best problematic, there are other aspects to physics that makes me wonder if i should re-examine the menagerie of "infinities"---I would at least like to understand the "structure" behind such and maybe see if it parallels with something else in physics. I know that "infinities" has really bedeviled Richard Feynman, who could never feel comfortable with them.
I've already remarked here in Brilliant on a number of times, "who needs infinity?" with regard to physics, but there is a strong possibility that the concept of infinity could be reframed in a different form that doesn't have to involve infinities. In other words, it's a choice if we want to accept the concept of infinity as a paradigm, but we don't actually have to accept that in exclusion of any other alternatives.
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@Michael Mendrin – So the reason for why you want to understand better the concept of infinity though is because it can prove important for various physics related problems about it.
I got it right , right ? Of course Feynman had problems with understanding so very many or infinitely so "infinities" of Cantor's mind! Also Feynman wasn't too good at emphasizing reason since he was interested more in spectacular quantum things. I bet the greatest problem for Feynman would be to speak with the absolute infinity which is greater than all other infinities as Cantor did in his revelations. That very absolute infinity greater than any other from an infinity amount of infinities certainly proves very difficult to understand for anyone and maybe as difficult as the wave-particle duality of light. But he wouldn't have problem with numbers creation maybe. You take1 and add it with 1 and you created the eternal and only 2. Well that sounds more rational. But letting joke aside your idea and track speaks of understanding the structure of infinity. Certainly you had something in mind when you thought of the term structure when you said that so you have a preliminary and a very undeterminate understanding of the word structure so can you get some insight on that initial undetermined sense you saw and name it though ? Can you make that determined ? Also , in what sense do you think the cocnept of infinity can get a formal characterization by which it is excluded ? I mean in what kind of model can we "reframe in a different form" because this can have implications for a stronger thesis and problem related to the use and understanding we have of concepts like "infinity"anyway.
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@A A – Feynman's problem with infinities was more like having to accept as the answer "-1/12" for the infinite sum of numbers 1, 2, 3, 4, 5, ... etc. I wonder if you know about this? His QED was a great success, but he had to resort to some really dodgy work-arounds with infinity issues, and he was never happy about it.
As far as I know, no mathematician has come forth to offer a "rigorous approach" to those issues in QED.
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@Michael Mendrin – You mean if I knew about the infinite sum of a sequence of numbers , i.e a series ? Well , yep i heard of it but I lack all (or almost) technical (calculus) knowledge to cover a rigorous treatment of the problem of infinite series though conceptually such a thing also doesn't make too much sense when you say that the infinite decreasing sequence has a finite answer. Firstly an increasing sequence or divergent series if I remind rightly certainly doesn't have a finite answer for anytime I can add a new number large that the previous which makes the sum increase with a larger quantity , concluding that the sum increases with the iteration. The second case for an decreasing sequence of numbers or a convergent series has quite a pretty good name since we can imagine that the sums converge and the sum increases with a quantity which becomes smaller through the iteration. Because the quantity becomes smaller and smaller it is quite natural to assume there will be reached a limit which the increased sum will never ever pass therefore this fitting very well with the notion of limit in calculus as a thing to which we infinitely approach but never reach conclusion which to me seems right until the point I find that such a covnergent series is equal with some number which maybe has to do of course with the cantorianian doctrine anyway.
Of the fact that the infinite sum of positive numbers is a negative number haven't heard yet and seems contradictory in the context of arithmetic. It might well be right for some other theoretical frame maybe. It should be obviously contradictory as if I have A > 0 and I add A1 to A then A+A1 > 0 and "so on". By using the said and so on though we mean we can iterate the previous rule observed rule in a consistent deterministic way that is consistent because by the deterministic way it preserves such a characteristic. Therefore , unless Cantor didn't say such a thing just for the sake of making the doctrine even more God revealing and paradoxical can you elaborate on what that series which bothered Feynman so to say was and also why are you curious if I knew of series though ?
@Michael Mendrin – Well of course you can elaborate as much as you please but I'd prefer the elaboration on the matter containing words.
Oh and I'mreally curious on the second question of why you wanted to know if nonetheless I'm familiar with that subject, you thought I'm more knowledgeable ?
@Michael Mendrin – So to understand you prefer elaborating with 0 words. Well , if that's your decision very well then anyway.
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@A A – A A, I just got back last night with somebody talking about a book, which is another project I'm now working on, etc. The subject I brought up about infinite sum of numbers is an interesting one, and certainly I'd to talk about it. Infinite sums either converge or diverge (and sometimes that's not so clear cut). The sum 1+2+3+4+...clearly diverges, e.g. becomes infinitely large, and yet some mathematicians are saying it adds up to -1/12, and there is an interesting connection between that and quantum vacuum energy. When I first heard about this, I thought that this was ridiculous---until I read up more about it. After a while, I started to realize that what matters is the way it is "determined* that the sum is -1/12 that matters, not that infinities are involved at all. Let me get some references to you about that.
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@Michael Mendrin – Ok ,very well then too but might you mind if I ask you what book and what's the thing related with it to which to also think though ?
Yes , the way thigns are 'determined" their "deterministic structure" is indeed a principal observation for udnerstanding the formal content implied when making a predication and as such this way of saying is important because in case the thinking simplyspeakls of addition in arithmetics seems quite werid indeed. lease give me so to say those suitable references if you find them. I'm not thinking on the moment we are talking about usual arithmetics but who knows We may veryw ell be on the Cantorian realm again and the sueprficial intepretations it speaks and sorry for not elabroating too much as I would have liekd to say mroe about the deterministic structure and in a better and more articualte way to emphasise what said there but right now Im in a pretty hurry situation and I'll elaborate later on the thigns which I actually tried to say anyway.
@Michael Mendrin – Did you get anything good ? I searched for it and found the explanation for why and while it's not a serious or too serious reasoning I have to say it's a good illustration of that situation which I talked about some messages upper when I told you about the conceptual understanding in mathematics and abstractness in my view related to such issues pretty much also having intuitive content in my opinion when and jsut when thought abstractly and for dealing with abstract "phenomena" anyway.
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@A A – There's actually now a Wikipedia on this 1+2+3+4... Pay attention to the words Zeta Function Regularization Using this method, one can show that 1+1+1+1+....=-1/2. See 1+1+1+1+... In fact, somebody posted a similar problem here about recently. 1+1+2+3+5+8+... See Generating functions to see how this can be analyzed.
Also, look up Ramanujan Summation , which is not something I'm very familiar with, but the point is, Ramanujan was.
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@Michael Mendrin – That's the actual resource I found! Well , I guess that I have to understand that method better yet what i have to say is that it is a very abstract construct and therefore not quite using the usual rules of arithmetic to arrive at those results loosing therefore the real grounding in such a initial referential paradigm anyway.
I found that wikipedia article looking at the Grandi's series article which I stumbled upon while searching of this seemingly curious 1+2.. series which is equaled as -1/2 but that just by a very abstract understanding of a "series" in my point of view since an actual equaling of the series is impossible. When I say a very abstract thing I mean either that it it's concept and results it implies using some other non-primitive and developed abstract notions in the formal sense or that it works as a modelling of a predication regarding "dynamic" concepts. This simply impleis that the used way of thinking and interpretation is changed from what it is initially admitted it is though and therefore that the so called 1+2.. series receives or should suffer a mutation in the meaning which can't be applied consisently for simple quantities as is done in arithmetic desptie the fact that there may be the situation in which the conceptual constructs tends to be sod eveloped that sometiems instead of offering firstly an adequate itnerpretation for the use of those notions and concepts it is applied and gets to seemingly paradoxical results. Actually I think we should name this either false paradoxes or falsec counter-intuitive ideas as they tend making you think they say soemthign extarordinary and eliminate all the initial consistency of a thinkign when , in fact they are speaking of a completely different thing which is right in the conceptual paradigm it speaks of treating the conceptual objects considered in a different way than the way they nonetheless are initially thought and as such at best provide , untill thigns are settled up by understanding the way mathematcis work a curiosity and further once such an understanding of mathematical development is done a good illustration of such points. I'll still have to understand that function to make the point which I want to make articualte. In short if the fucntion changes the interpretation of adding quantities by either changing the itnerpretation of "adding" or of quantities in the cocnept of adding quantities then it's no surprise the change but it may be tricky to get that logical structure underlying it and make it articulate proposing in a simple and explicit way what the predication is talking about in the terms of our conceptual and intuitive understanding anyway.
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@A A – Nobody can actually add up infinitely many numbers like "1+2+3+4+....", so what matters is, "how do we otherwise make the determination what the infinite sum is?" It turns out there's more than one way to make a determination, and they don't necessarily yield the same answers. We make certain presumptions when deciding on a determination to be used, and since we could be making different presumptions about the "how", it's possible, then, to end up with different results. We can't (or shouldn't) insist that just one of them is "correct" just because it "sounds believable", and then reject all the rest out of hand---not if the other methods do not lead to logical contradictions. Contradicting "common sense" is not a true logical contradiction.
There is an analogy in probability theory, where the probability of a certain event critically depends on how the random variable is sampled. Here's a Wikipedia article about that, see Bertrand Paradox
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@Michael Mendrin – To reply that question of how we do determine the answer it is enough so to say to look at the problem in it's principles that is in the prime causes which determine or are respnsable for determing it anyway.
It's not that there is more than one way to make a determination or at least I don't agree with that because that would contradict the principle of identity. Again , this sort of thing said here is not a common sense one just but the axiom which was accepted for any logical valid inference but let me explain a little bit explicit what I mean by contradicting the principle of identity anyway.
If I start with some interpretation of what I mean by the series 1+2+.... then that suff which I mean should have only one result regarding some problem about it which is right regarding it because otherwise my interpretation or my predication of the series would give 2 results for the same interpretation which contradicts the continuity of logical deduction or of the logical form of heuristic determination from the interpretation given for example it would look that steps taken a implies b implies d implies r etc given to results which are not consistent with one another being therefore obvious contradictions in virtue of the principle of identity.Therefore we have to be clear here that for one interpretation I1 of the mathematical equation 1+2+.... we have only one result despite there being possible more interpretations for what the expression 1+2+.... means , like for exmaple an I2 which can give also by virtue of coherent predication about it another result but this result wouldn't have to do anything with I1. So for each interpretation I1 , I2 in virtue of the principle of identity which is not necessarily common sense we have 2 possible different and coherent results regarding a problem which asks for the "same" thign in our two systems of interpretation anyway.
You seem to say I think rather that because we have 2 different interpretation we are logically inconsistent getting 2 different results. As you see from the upper paragraph that is not what I state. The only thing I said is that we tend to be confused about a result in an interpretation I2 thinking of it in terms of I1 because it's not always clear when the gap between one intepretation and another so to say is being hoped and this as a result of development of technical concepts. Therefore I agree with you that maybe 1+2+.. is -1/12 but this is so in an itnerpretation I2 rather than our starting interpretation I1 of simple arithmetics where a quantity so to sayis added to another this jump from I1 to I2 being mediated by the so called zeta function you just offered me to study. It's not clear though what the Zeta fucntion is to be interpreted in our direct conceptual thinking what does it mean for our direct and intuit thinkingbut it's clear that it provide another itnepretation and hence indeed aanother way of determining the result in accord with that I anyway. The same applies for the analogy in probability I think. It's not that we have to exclude all interpretation besides one which is "correct" , we can have tons of itnerpretation if we prefer but we need to realize that we are speaking of different interpretative systems that are governed by their rules by not confound them. Maybe I wasn't clear in my first comment but now I was ? I think we are saying the same thing and as you can see it's good to formalize things to be precise. From some mathematcal expressions e can given any sort of interpretation I1 , I2 and so on and work in accordance with that interpretation which belong to different realms and as such may give consistently different results , don't you say that though ?
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@A A – One of the strong attributes of "machine computation" is determinism and repeatability. A software engineer knows that even a flawed software will give consistent results, however wrong. Mathematics, if it's based on logic, is or ought to be similarly deterministic and repeatable. That is, given a set of definitions, premises, axioms, theorems, different mathematicians should all arrive at the same conclusion whether or not a proposed theorem is true or false. That is the connection between the two.
So, yes, given a "way of determining the value of an infinite sum", there should only be one unique answer, or, at worse, a family of answers (a problem that sometimes crops up with complex numbers in other situations). Mathematicians shouldn't end up disputing over differing results, because then that would be likely due to a disagreement about the interpretation of a particular definition or axiom being used in making the determination. But there is nothing inherently wrong with having different interpretations, as long as all the parties involved understand and agree that there are separate different interpretations and are carefully identified.
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@Michael Mendrin – Well , I completely agree with what you said and i have quite the same view regarding mathematical reasoning and the content of it's derivation so to say. But what from what I wrote made you think that I say there is anything inherently wrong in considering more possible interpretations and promote a common sense? In my upper comment I just added a little note about the phenomena of jumping from one interpretation to another without completely realizing as a result of the development of ideas and mathematical concepts in a very technical interpretative situation for which it is not immediately clear what their intuit predication is though. I also spoke of the character of such technical concepts by trying to explain in what way they are to be related to immediate conceptual thinking and why sometimes they seem to be irrational or illogical in their own sense while in truth that has to do just with characterization of abstract content objects anyway.
I'm not very sure if we should characterize the relation between mechanical computation and formal mathematical reasoning just by saying that there is some "connection between the two" as that relation may be stronger than just implying a connection between two different things separated besides that connection by other different characteristics which gives them their different objective traits and if not rather they are pretty much the same thing. Therefore you spoke of what makes them alike without considering what would make them different this point having a lot of importance for any sort of view you might have either the equivalence or lack of equivalence of mathematical derivation and mechanical computation as proposed in Principia Mathematica as such a point would offer to both views the density and completeness of any. As such I have to ask you if you consider mathematical reasoning and mechanical computation equivalent though. I'm not very sure if I expressed to well why no matter what view you have it's quite important to understand this but I think you might too incompletely see it is though.
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@A A – "As such I have to ask you if you consider mathematical reasoning and mechanical computation equivalent though." Yes, I do consider them equivalent, ideally speaking. Here's an except from the Wikipedia article about Principia Mathematica
PM was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven
Once this has been achieved, then direct machine implementation is possible.
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@Michael Mendrin – I know of Principia Mathematica's aim and the way of thought it proposes but I'm not sure if I was very clear with what I meant by that question.I didn't ask if the mathematics is implementable in mechanical computation which is pretty problematic as today we don't have from what i think a very good "model" of the logic behind mathematics and we haven't elaborate yet a deep understanding of the structure of mathematics to reply to that question , what I asked was if mathematics and mechanical computation are the same thing.
It's a difference between asking if all mathematics is implementable by mechanical computation and if mathematics is mechanical computation , a pretty objective 1. You might say that no , formally if one thing is implementable by another then that thing is reducible or isomorphic to the other and hence are (formally) considered This because the structure between them is "preserved" in the mapping of the relations which remain the same (invariant) this relations being those which validly attests the fact that two "structures" are similar and any other trait offered is unimportant in justifying the difference between such things when thought by the abstraction of formal thinking which takes away traits that have nothing to do with the a priori or rather formal content of thought , meaning therefore that if mathematics is so implementable in mechanical computation by the valid way of thought of formal thinking it is the same so to say as mechanical computation being naive and missing the point to argue that it can be otherwise. When I asked you though , i was thinking if mathematical reasoning is itself nothing but mechanical computation in disguise if , when I make a valid mathematical reasoning I am making mechancial comptuation and the entire way of thought of mathematics directly linear reducible to mechanical computation. Therefore If you prefer that is that when we look at mathematics in it's true character we should just see the mechanical computation by which it works if it's nothing else but this expression of logic and ultimately has no other heterogeneous trait derivable but ireducible to it anyway. When i say that some something A is reducible to something else B it may mean that A is derivable from B but not that qualitatively they have the same dynamic structure , that they manfiest and are in the same way. For example in physics we have decoherence which ilustartes this thing but I'm not talking just about emergent characteristics. To say it briefly , even if mathematics would be reducible to terms of mechanical computation it's qualitative structure may be more organic in itself that is the structure it have may manifest organicistic cbharacteristics which can't be so to say modeled by the structural form of mechanical computation. In this case I have to ask you again if you think mathematics itself is nothing but mechanical computation or if in the conceiveemnt of mathematics itself lays something more which makes it incompletely fromally reducible to mechanical computation. Characterizing the point that mathematics is rather like an organism explcitly and so on is pretty complicated and would require some maturity of ideas before some more entering in it to establish terms that can make it articulate but , I hope you can udnerstand in what sense I spoke of it here and if not say anyway.
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@A A – Now I think we're discussing philosophy. I know you're trying to address something about the "nature of mathematical [computation] thought", and by the nature of your repeated inquiry into this, it does seem to me that you are suggesting that it is somehow separate and distinct from any "machine process"---which, again, takes us back to the point of beginning of this conversation. And, philosophically speaking (not mathematically rigorously speaking), my answer is the same---I don't think the art of doing mathematics is somehow distinct from "machine thinking or processing", or even should be. But, as I said at the very beginning, that's just my personal opinion. I am not presenting it as a fact. I do envision the day when computerized artificial intelligence can do very well at the "art of doing mathematics" and come up with original and significant discoveries. I do not believe that such an ability is in the unique providence of humans. And, as a matter of fact, that is the power of mathematics itself---its truths are universal, no matter by whom or how they are uncovered.
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@Michael Mendrin – Also , it's truth are universal no matter how they are expressed and inf act that is what universal should actually mean. Yes you got the point somehow but it's not necessarily philosophy it's rather that wasn't very unclear in what I presented about the way mathematics and mechanical computation are different though also so to say not necessarily distinct anyway.
My point wasn't that mathematical formal reasoning is not expressible in mechanical computation or not derivable from mechanical computation this points being to me equally or seemingly equally correct but I'm suggesting we shouldn't stop here with being rigorous about such matters. It may seem i say on one side that mathematical reasoning is different from and on the other side that it anyway so to say is the same stuff and hence I contradict myself ,or that I have a confusion of thought suggesting alternatives which are certainly wrong. But the point which i tried to make there was that formal mathematical reasoning is different from mechanical computation in the sense that while it is derivable from it it is an entire which is by the fact of being an entire displays an emergence trait which is irreducible to the mechanical comptuation from which it is derived althoguht the parts are of course still rightly expressible by it. I see how this say claim of a thing being emergent as a whole displays traits which are irreducible to the stuff from which it is derived seems philsophical as it claims a sort of contradiction or vagueness though the name "philosophy" is unsuitable anyway. If I am to express it in other more technical and apropriate terms whatiw anted to say is that the difference between mathemacial reasoning and the mechanical computation which so to say forms it given is a result of the fact that the manifestation or betetr said the derministic structure it displays is not linear and hence if I am to speak of both deterministic structures I would have to say that one is derivable from other in a not completely linear way. It's this trait of non conmpelte lienarity which i ahd in mind which assures that one of the structure so to say is qualitatively different or heterogeneous from the other as a result of such a not linear itneraction or manifestation by which one is not continuously changed in the other form or by which the expression of the first deterministic structure as aresult of such a trait of "linearity" is not continuous in the structure. Because therefore we see sucha thing we can conclude that the manfiestation or system of determinations between the two concieved structures are different. I hope nonetheless that I am clearer now. If not please say. I may not yet have poitned out completely well what i mean by "not linear derivable from" so to say. Considering also the vague use of deterministic structure here I suppose you didn't quite understood what i meant , but Ihave to ask if you did get it or not and sorry for a lot of grammatical mistakes but now I have to go as I am in a sort of hurry for correcting those grammatical errors anyway.
@Michael Mendrin – I suppose you're pretty busy again since you haven't reply yet. Well , don't fasten to give a reply though and take your time to contemplate whatever you are thinking or do whatever you are doing if it's not mainly thinking now replying when you have time after dealing with that work in which it seems you are so to say pretty much immersed as I don't want distracting you of course and again of course good luck at what are you doing anyway.
@Michael Mendrin – Besides I compeltely agree with your other points regarding common sense which are well said.
Like you say we can't and shouldn't agree to dismiss all other itnepretations just because some sound more believable as a result we are more used with the way of thinking regarding them because that is a subjective thing and not a good way of doing objective reasoning which speaks about what is objectively right.
@Prince Loomba – Really ? So many words haha.
@Michael Mendrin – I looked at the second edition (from the links sent to you) of Principia Mathematica and I have to say you that the notation is awful not to say that the way of thinking and terms used are certainly not the best and today we have better terms to speak and model what we are talking about. Using Sheffer's Stroke is pretty annoying and unnatural so that would be a good resource for understanding the development of ideas and flow of concepts but not the best to initiate or find all the important things regarding mechanical computability. Actually in any topic , unless it's very new and academically elaborated in a a paper/book/volumes/tomes/stuff/etc there always will be advantages of studying introductory and post graduate books in the domain as it contains some advantages. Firstly the notation reflects our current understanding of the topics at hand and innitates the possibility of dialogues mroe easily , secondly such a works presents clearer some ideas then those ideaswere presented by the initial pioneers who by being first had more to give their attention to elaboration then to reflections about elaboration and mostly makign every step crystal clear so to say , thirdly in an introdcutory book you don't find just the improtant ideas of one author as any domain is resulted from ideas of more people which you will not find if you read just one author to the same extent , forth the ideas will be presented mroe in the historical and cultural context which are clearer after some time than they are for the moment of being elaborated , fifth it's easier to avoid some mistakes which the author made and were refined , sixth it shows some new results of other author afterthe publication of a specific work and put the thigns in relation with the entire context and development of ideas.
Such thigns , along with others of which I can't think of at the moment and of which I don't havethe compelte time to list if I would find all favors study of other books than necessary the pioneering material and very firstly infleutnial books making them some sort of second "resource" though. Noneteless for understanding better a historical context and udnerstandign also better the ideas in their genesis and way of thinking it's indeed pretty ok to turn to the original publications which shaped or influenced very much advances of a domain.
@Michael Mendrin – So...... what do you think are the important thigns to point out about the type of thinking found implicit in propositional logic for the purposes of this discussion ?
Btw ,if I am unclear with something , whatever it is tell me without any sort of hesitation because I want my points to be truly udnerstood since I know I'm really speaking things which have sense and also know that I understand the things that are related to the Principia Mathematica are of the past but I think sucha discussion would still be good to articualte things which are actual problems and also that while it is provable not all mathematics is computable by Godel theorems this still doesn't say if the rest of mathematics by it's structure is or not anyway. This is the important problem here , the understanding of this "structure" of mathematics or rather of mathematcial deduction. Or in other words the way therefore in which things are determined ,the structure of such a way of determening propositions in mathematics where this part plays an important part speaking of the underlying structure that makes this way of deductions anyway. Please tell if you want me to elaborate more on what Imeant by "structure" if it is not clear. The problem related to what is actual structure of mathematics therefore isn't the same as that of Principia I have to say though very very much resembles it anyway.
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@A A – If you want to be more familiar with the concept of "structure" in mathematics, you should start in a place like this
or, as an example at the most elementary level,
As a literary analogy, we can have two seemingly different stories that are actually the same---only the names, places, and times have been changed. For example, there are two movies, one called "The Seven Samurai", which is a Japanese movie, the other called, "The Magnificent Seven", which is an American movie. Even though the first involves Japanese samurai in the 16th century and the second involves American gunfighters in the 19th century, the two plots are nearly identical. In mathematics, the two would be said to have "the same structure".
This is really an important concept to understand, not only in mathematics, but in theoretical physics as well.
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@Michael Mendrin – I can tell you I understand this concept pretty well. It may seem non-sensical to speak of the structure of mathematics as I do. I'll try to show you in what sense used the concept of "structure" in this discussion. There are a little bit more ideas to introduce to be clear so please don't skip it where it seems weird anyway. Actually I would appreciate if you would say if you find them sensical or otherwise complete stupidities which have nothing important in content and useless anyway. Considering the way I used "structure" in the discussion though I can say anyway that it may not fit compeltely the mathematical formal sense anyway.
Nonetheless , mathematical or not though it can be maybe formalised mathematically by speaking of isomorphism by it's mapping to computer rather than just any homomorphism to illustrate what I mean the concept of structure to which I refer is that of the structure of the formal logical inferences / deductions. Formally this things can be thought not using the term "deduction" because that is not formal enough and therefore not of the "structure" of deductions but of the way things are determined and as such of the "deterministic structure" of the so to speak logical implementation anyway.
A structure in the sense I conceive it speaks , from an atomistic point of view about the way the components are related in regard to the important relations. Therefore asking and trying to udnerstand how does that or that component relate with others and how they form an entire is a problem of concieving structure in the first case and in the second the "synthetic" structure of the things implied and in that idea I speak of the structure of mathematical inference.
The "preserving" of structure which is from what I think core idea of topology and pelase correct me if I am wrong is a "preserving" of structure exactly because the relations of the components are seen a priori in their "form" in abstract though even for thigns which seem hardly to be thought in atomistic terms as surfaces. In the same sense we can speak of the structure of a mechanical process as it is in the view of propositional logic by considering the way of the derivation of things or in other words their way of being derived from other things in general. This is the same , if you followed my idea until now , sort of atomistic understanding which therefore can be said belongs to the concept of "structure" in general but the components or so to say "atoms" are the elementary derivation of the process. Considering this stuff therefore you can speak and concieve the "structure" of mathematics or rather mathematical way of thinking and understand of course wether or not it is "isomorphic" , if it maps to preserving it's deterministic so to speak content the mechanical process of derivation. Because of this I like to describe such a mathematical structure as part of a "formal topology". Considering that topology is truly speaking of this "form" I think the name is proper anyway. Nonetheless , can you grasp what i mean now ? Can we advance therefore in the discussion or the use of this term here should be ellaborated more anyway ?
@Michael Mendrin – Your movies example is a good example indeed for what I am trying to say too. The structure there is the same because the relation between the components anyway is preserved and is the relation which is important or rather the synthetic so to say concievement of this atomistic components like the characteristics of the plots the story line characters and such other stuff are in the same relation which is of course immediately perceived as almost the same for the relations considered anyway.
Sorry , I wrote thisas a commentary to one of my or yours other commentaries. Nonetheless I wrote it as a reply to the actual reply to which it was intended haha.
@Michael Mendrin – It seems I have not been attentive at the "using algebra" part in the title and therefore spoke I knew something which I didn't though.
Nonetheless the idea of putting logical inference in terms of computation and , as such in terms suitable for the model mechanical computation leads itself by seems to wear a wonderful "revelatory" content at least if it is understood as being not just a tool or trick of implementation but as having sort of an internal character to it demonstrating not an artifice but an actual structural link between qualitative statements and quantitve computation or , in other words in case the propositional logic with it's qualitative content in the process of derivation of the truth value , and not of the propositions themselves , it it's mechanically and formally described logical inference is seen as being a priori eventually such a sort of computation. Saying it in short this seems a legit way for describing the qualitative process of deriving the truth value in computable terms once understood in that conception.Of course there can be a lot of problems as there is still a quantitative character to that qualitative problem and therefore for example you can't choose any numbers at least considering simple and usual arithmetics. There might also be problems in how you describe the equivalence of certain oeprators with operators. Some problem would be therefore why would one describe multiplication for the conjunction of two or more statements , or how would that be dealt with some more difficult semantics which would use some more complcuated operators anyway ?
@Michael Mendrin – Unfortunately I'll be again away for a while. Looking for your message and reply of course.
And yes it's not just a good way to start the mastery of anything with easy things or maybe with the simplest and the most easily understandable concepts , ideas. I believe it's a must for anyone , no matter how gifted if he is from the human species at least such mastery starts with deep and acute simple understanding.
@Michael Mendrin – I suppose you are thinking at it. You have decided what kind of theorem would you like to use to illustrate this type of mechanical proof I think or maybe my many proposals weren't too useful anyway.
@Michael Mendrin – Hmmmmm , having problems with coming with such a proof illustration though ? Maybe then at least describing the problems of or the general way it is done in a more elaborate way to understand better how it could be done if the problem of proving a theorem by means of mechanical computability even for some simple theorem is very hard , though ?
@Michael Mendrin – Btw , I'm not crticiizing you in what I say as I think you may have taken it at little though I for one think thinking is just formally reducible but not completely to mechanicism and as such that the mechanical account of our thinking is right though it's proceses are more than that computations. I just adviced you to take a deeper and more authentic look which you may find refreshing and in some certain way so to say "revelating" regarding what makes you think mathematics is reducible to computation that being an important underlying idea/principle anyway
Probability and that too by AA haha
One way to show there is indeed a perfect simetry between the envelopes and any other reasoning should be fallacious is to draw a table illustrating that perfect simetry anyway.
I believe this argument should be convincing enough to show that there is no new information received based on switching as it points imediately to each particular case and shows that in each of equally likely 2^2 = 4 cases there is a 1/2 chance of being or not being right of whether the next envelope has more or less money anyway.
Yes, we don't know how much money the first envelope has, but we know the second envelope has 2x of the first envelope. In my opinion, I'm not switching.
It doesn't matter. Either way you end up with more money in your pocket than you started with.
Depending on the value of the first envelope, the potential differences could be great. For example, the envelope you open has $50. Logically, the unopened envelope has either $25 or $100 (a $75 difference) If the envelope you open has $10, the unopened envelope will have either $5 of $20 (a difference of only $15) (You'll notice the differences are proportionally the same: the higher possible amount is 4x more than the lower possiblity.)
Emotionally, the gamble is greater with the first, but mathematically the differences are proportionally the same.
If my envelope has got an odd no. of dollars, I will definitely switch with both hands. If not; then reasoning same as others.
You should switch. Although the value of the other envelope is still unknown you will on average still gain more money. Let’s say the envelope you open has $4 in it. If you switch and it’s doubled you have $8. If you switch and it’s halved you have $2. If you stay the average amount of money you will get is $4 but if you switch the average of $2 and $8 is actually $5 so you will get more money if you switch. ALWAYS SWITCH
One argument goes: If you open an envelope and see D dollars, then the expected value in the other envelope is 2 1 ( 2 D ) + 2 1 ( D / 2 ) = 4 5 D > D , so you should switch.
But this doesn't really make sense... clearly there should be perfect symmetry between the two. (And indeed, it doesn't matter whether or not you switch.) To understand the resolution to this "paradox", see two-envelope paradox .