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I have mixed feelings about this. I've been through nonstandard analysis. My response was, "We shouldn't need the axiom of choice to define the derivative." Bu
by btilly 3y ago
I have mixed feelings about this. I've been through nonstandard analysis. My response was, "We shouldn't need the axiom of choice to define the derivative."
But I think that it is extremely important to understand that the infinitesimal notation really MEANS something. Here is some Python to demonstrate.
# d is a functor. It takes a function and returns a second function.
# The second function captures the change in f over a small distance.
# The dx/2 business reduces artefacts of it being a finite distance.
def d (f, dx=0.001):
return lambda t: (f(t + dx/2) - f(t - dx/2))
# d²x / dx²
def second_derivative (f):
return lambda t: d(d(f))(t) / (d(x)(t) * d(x)(t))
def x (t):
return t
def cubed (t):
return t*t*t
print("The second derivative of cubed at 1 is near", second_derivative(cubed)(1))
- btilly 3y agoDang it! I hate re-reading after the time for editing is over and finding a stupid mistake. The Python is correct, but the comment is not. The formula for the second derivative should be, of course: # d²y / dx²
- finite_depth 3y agoMost calculus students don't need the full formal power of rigorous analysis. Calculus, taken alone and with the elementary properties of the real numbers assumed and a few elementary properties of infinitesimals (0 <<< infinitesimal^2 <<< infinitesimal <<< any positive real), can get you a lot of power for very little formal work.
- btilly 3y agoAbsolutely true. However this comes at the cost of having to not think too hard about issues like "what is a function". You generally don't run into trouble with 1, x, 1/x, sin(x) and the like. But when you push past the analytic functions, you wind up having to unlearn a lot of ideas so that you can learn an entirely different foundation.
- ezekiel68 3y agoYou're right. But then again, lots of scaffolding gets discarded when an arch gets constructed also.
- mananaysiempre 3y agoLest you disregard this as completely useless superstructure, note that basically the entirety of the theory of stochastic processes, starting with Brownian motion, is positively infested with continuous everywhere nondifferentiable functions; while the existence of a nonconstant infinitely smooth function with an identically zero Taylor series is what permits the Berezinskii-Kosterlitz-Thouless phase transition to exist. So while the weird animals of elementary real analysis are perhaps not the most important thing in the world, they are far from irrelevant to it.
- krsrhe 3y agoIt’s extremely niche (abstract, “Platonic”) to ever need to care about derivatives of non piecewise-analytic functions.
- btilly 3y agoI think of wavelets and stochastic processes to be a significantly bigger niche than you probably do.
- ZoltanAK2 3y agoYou write, "we shouldn't need the axiom of choice to define the derivative." The good news is that we don't! Only model-theoretic approaches, which justify the infinitesimal methods by constructing a hyperreal field, require (a weak form of) the axiom of choice [2]. However, there are axioms for nonstandard analysis which are conservative over the usual choice-free set theory ZF. The three axioms of Hrbacek and Katz presented in the article "Infinitesimal analysis without the Axiom of Choice" [1] are the best recent example: these axioms allow you to do everything that is done in Keisler's book and more (including defining the derivative), and you never need to invoke the axiom of choice to justify them. [1] https://arxiv.org/abs/2009.04980 https://arxiv.org/abs/2009.04980 [2] Essentially, the set of properties satisfied by a fixed nonstandard hypernatural gives rise to a non-principal ultrafilter over the naturals. The axiom of choice is necessary to prove the existence of non-principal ultrafilters in (choice-free) set theory, but the existence of non-principal ultrafilters is not sufficient to prove the axiom of choice.
- btilly 3y agoYeah, yeah. My comment was my reaction 30 years ago. I find it a mildly interesting intellectual exercise that you can do NSA with weaker axioms than choice. But for all cases I care about, I can already prove it with NSA without ANY additional axioms! How is this possible? From Shoenfield's absoluteness theorem, you can prove that all statements that an be made in the Peano Axioms that can be proven in ZFC, are also true in ZF. (Note, they must be statable in PA, but not necessarily provable there.) But PA can encode any statement we can make about computation. So take any calculation we can talk about that can be approximated on a computer. We can rewrite it in PA. We can prove it using NSA. We then know that it is true in ZF. And we know that it is true without any additional axioms beyond ZF! That which we can actually calculate in any useful way can all be calculated on a computer. And therefore NSA can prove anything about Calculus that I care about without needing any axiom beyond ZF. But in the end this is using a mathematical sledgehammer to drive in a thumb tack. Many approaches to Calculus do not require assertions about the existence of sets that we cannot construct, even in principle. Even though I understand how NSA works, I'd prefer to use any of those.
- deleted 3y ago[deleted]
- krsrhe 3y ago[dead]