Calculus without Calculus

NOTE: Full credit goes to N. J. Wildberger for exposing this idea in his DiffGeom playlist, though as pointed out this is in fact a result of Lagrange and (possibly) Euler.

Consider, for example, the polynomial $1 - 3x + x^3$ over the rational number field and the evaluation map $p$ from the rational number field to itself given by $p(\alpha) \equiv 1 - 3\alpha + \alpha^3$.

To evaluate the Taylor polynomial of the example polynomial at a particular value $\alpha = r$ we first compute $p(\alpha + r)$, which, after collecting the terms with respect to powers of $\alpha$, gives us

$\begin{aligned} p(\alpha + r) &= 1 - 3(\alpha + r) + (\alpha + r)^3 \\ &= (1-3r+r^3) + (-3+3r^2)\alpha + (3r)\alpha^2 + \alpha^3 \\ &\equiv q(\alpha). \end{aligned}$

If we define $T_r^{(k)} p(\alpha) \equiv q(\alpha - r)$ to be the $k^{\text{th}}$ Taylor polynomial centred at $\alpha = r$, then we obtain

$T_r^{(0)} p(\alpha) = p(r),$

$\begin{aligned} T_r^{(1)} p(\alpha) &= (1-3r+r^3) + (-3+3r^2)(\alpha - r) \\ &= (1-2r^3) + (-3+3r^2)\alpha, \end{aligned}$

$\begin{aligned} T_r^{(2)} p(\alpha) &= (1-3r+r^3) + (-3+3r^2)(\alpha - r)+ (3r)(\alpha-r)^2 \\ &= (1+r^3) + (-3-3r^2)\alpha + 3r\alpha^2 \end{aligned}$

and

$T_r^{(3)} p(\alpha) = p(\alpha).$

Interpreting the results geometrically (see the picture above, where $A \equiv \left[-\frac{1}{2},\frac{19}{8} \right]$ and $B \equiv \left[\frac{3}{2},-\frac{1}{8}\right]$), we have that $T_r^{(1)} p(\alpha)$ and $T_r^{(2)} p(\alpha)$ correspond respectively to the tangent line and the tangent conic of $p(\alpha)$. This will generalise for any (finite-degree) polynomial. Furthermore, for a polynomial of degree $n$ with evaluation map $p$, we have that $T_r^{(0)}p(\alpha) = p(r)$ and $T_r^{(n)}p(\alpha) = p(\alpha)$.

So, we have here a purely geometric/algebraic framework for differential calculus, which really just boils down to understanding tangents. The framework provided will generalise not only to arbitrary fields but also for algebraic curves, as well as surfaces. In fact, this framework would provide a firm foundation for differential geometry, which is really just the study of curves and surfaces.