This discussion board is a place to discuss our Daily Challenges and the math and science
related to those challenges. Explanations are more than just a solution — they should
explain the steps and thinking strategies that you used to obtain the solution. Comments
should further the discussion of math and science.
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italics
**bold** or __bold__
bold
- bulleted - list
bulleted
list
1. numbered 2. list
numbered
list
Note: you must add a full line of space before and after lists for them to show up correctly
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
Math
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Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3
2×3
2^{34}
234
a_{i-1}
ai−1
\frac{2}{3}
32
\sqrt{2}
2
\sum_{i=1}^3
∑i=13
\sin \theta
sinθ
\boxed{123}
123
Comments
This diagram--this is invention, motivated by some non-rigorous truths? By the way, does "rigorous truths" have a place in this scheme, and if so, where would they be?
This is an approximation of the debate about the discovery or invention of mathematics. Maybe no one will ever be right. the fact that rigorous truths appear in this diagram just explains the ramification of mathematics; when some advance is made, there are plenty of new areas discovered or invented (as you prefer).
Well, it's been long debated whether mathematics is about invention or discovery. A native wood carver, for example, may speak of "discovering" the animal inside the log, and seeks to "free" it. Others will say he is merely creating animal art from an otherwise featureless log.
I am, and many others, are of the strong opinion that mathematics is not all arbitrary invention. For example, given the axioms and definitions of Euclidean geometry, there are only 5 regular polyhedra, no matter how some may wish to creatively make happen many more. As another example, after the discovery of complex numbers and their extremely useful geometrical properties, it was quite natural for mathematicians to wonder and hope for a 3 dimensional analog of it. Sir William Rowan Hamiltonian sought to generalize it, and ended up with quaternions, which have very interesting properties of their own, but nevertheless not quite like complex numbers. Generalization of such algebras is possible through what's called Cayley-Dickson construction, proceeding from complex to quaternions, octonions, sedenions, etc., each with new properties and very little or no leeway for any "creative license in inventing novel algebras." Another famous example is the classification of finite simple groups--which fall into exactly four categories, which are cyclic groups, alternating groups, Lie groups, and "sporadic" groups. And there is only a finite number of such categories, and mathematicians, after more than a century of work, are finally getting around to completing the proof of this, which includes finding all 26 sporadic groups. Again, no creative invention here--that's all there is, folks! However tremendously complicated it may be.
Adam Strange of "Mythbusters" is famous for his line, "I reject your reality and substitute my own!" In mathematics, you really don't have that choice. Creative invention takes you only so far, the rest is discovery of unalterable reality.
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
*italics*or_italics_**bold**or__bold__paragraph 1
paragraph 2
[example link](https://brilliant.org)> This is a quote# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"\(...\)or\[...\]to ensure proper formatting.2 \times 32^{34}a_{i-1}\frac{2}{3}\sqrt{2}\sum_{i=1}^3\sin \theta\boxed{123}Comments
This diagram--this is invention, motivated by some non-rigorous truths? By the way, does "rigorous truths" have a place in this scheme, and if so, where would they be?
Log in to reply
This is an approximation of the debate about the discovery or invention of mathematics. Maybe no one will ever be right. the fact that rigorous truths appear in this diagram just explains the ramification of mathematics; when some advance is made, there are plenty of new areas discovered or invented (as you prefer).
Log in to reply
Well, it's been long debated whether mathematics is about invention or discovery. A native wood carver, for example, may speak of "discovering" the animal inside the log, and seeks to "free" it. Others will say he is merely creating animal art from an otherwise featureless log.
I am, and many others, are of the strong opinion that mathematics is not all arbitrary invention. For example, given the axioms and definitions of Euclidean geometry, there are only 5 regular polyhedra, no matter how some may wish to creatively make happen many more. As another example, after the discovery of complex numbers and their extremely useful geometrical properties, it was quite natural for mathematicians to wonder and hope for a 3 dimensional analog of it. Sir William Rowan Hamiltonian sought to generalize it, and ended up with quaternions, which have very interesting properties of their own, but nevertheless not quite like complex numbers. Generalization of such algebras is possible through what's called Cayley-Dickson construction, proceeding from complex to quaternions, octonions, sedenions, etc., each with new properties and very little or no leeway for any "creative license in inventing novel algebras." Another famous example is the classification of finite simple groups--which fall into exactly four categories, which are cyclic groups, alternating groups, Lie groups, and "sporadic" groups. And there is only a finite number of such categories, and mathematicians, after more than a century of work, are finally getting around to completing the proof of this, which includes finding all 26 sporadic groups. Again, no creative invention here--that's all there is, folks! However tremendously complicated it may be.
Adam Strange of "Mythbusters" is famous for his line, "I reject your reality and substitute my own!" In mathematics, you really don't have that choice. Creative invention takes you only so far, the rest is discovery of unalterable reality.