Silvanus' book makes calculus simple by adopting an infinitesimal approach, like Newton and Leibniz did when they invented calculus. But that approach was shunned by mathematicians for a long time, because it was only made rigorous in the 60s. After Silvanus' book, I also recommend Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler, for those interested in this alternative pathway to calculus. It can be freely downloaded at https://people.math.wisc.edu/~hkeisler/calc.html
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))
You 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.
[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.
Yeah, 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.
Most 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.
Absolutely 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.
Lest 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.
I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.
As an example of the conceptual richness, pick up a Calculus book and flip to the problem section for L'Hôpital's rule. Without using any special rules at all, attempt to write them out in o-notation and observe that you generally don't need L'Hôpital's rule to work them out. It is possible to produce examples that can be calculated by L'Hôpital's rule, but not by simply understanding o-notation. But it isn't easy, and you're unlikely to find them in textbooks.
It is probably true that as you go on, limits are more useful in higher mathematics than o-notation. But o-notation is far more useful in most subjects that use mathematics. Given how easy it is to master limits if you know o-notation, why not teach o-notation first?
More importantly for this audience, the idea of limits lies at the heart of numerical analysis. Explicating the quantifiers of the definition of a limit is the first step in obtaining control over any estimator of your data. This, among other things, is why I am perpetually baffled at the people in this audience who say that limits is something "they will never need". Limits, their algebra (which subsumes all of high school algebra and inequalities) and their delicate analysis is what makes a ton of numerical algorithms work; Higham's classic has it all, and is perfectly clear about it all.
> I think I'd also prefer the infinitesimal version of calculus, but the idea of limits is applicable to many other areas of math, not just calculus (evaluating infinite series, for example). So learning limits is probably a better pathway to higher mathematics.
I'd say that limits in the sense that you mean (as opposed to category-theoretic limits) are precisely the domain of calculus or, if one wishes so to call it (because one is proving things!), analysis. For example, many US universities, mine included, regard the computation of infinite series as part of Calculus II.
Relevant wikipedia entry: https://en.m.wikipedia.org/wiki/Nonstandard_analysis