Don't forget the contribution from Dedekind. Yet that doesn't differ actually so much from what either Newton or Leibniz said, even if they didn't invent the definition of a limit.

And here you might add there as a "case solved" Robinson with his rigorous foundations for infinitesimals. And where I think Robinson succeeds is putting down the infinitesimal to a new set of numbers.

Of course, that is then called non-standard analysis.

I hate thought experiments. If I see a fat man on a bridge I run away. I refuse to listen to bat detectors in case they start sharing their experiences. I sign petitions to free people from rooms in which they have to translate stuff they don't understand. And all in vain, if I'm just a brain in a vat.

Of course, that is then called non-standard analysis — ssu

Not a "simple" "intuitive" with "beautiful reasoning" in my opinion. If it were you would see more of it in college curricula. The Leibniz notion is interesting, admittedly. A colleague of mine tried teaching the subject at the U of Colorado some years ago, and neither he nor his students benefited. :cool:

Not a "simple" "intuitive" with "beautiful reasoning" in my opinion. — jgill

See, that's the problem here. I think math is filled with a lot of things that a) work b) are totally obvious at some level and c) to make a rigorous proof why they are is problematic. For example, just how many different fields of math can you find something similar to the Axiom of Choice? Just look how much it has created discussion in mathematical circles.

A colleague of mine tried teaching the subject at the U of Colorado some years ago, and neither he nor his students benefited. — jgill

Who benefits from the History of Math or the Philosophy of Math? Not many I would say.

Usually students aren't interested in the fascinating history of a debate in mathematics.

Far easier just to learn calculus: Learn this, do it so, it works. Next issue in the course, we have to run here...

A colleague of mine tried teaching the subject at the U of Colorado some years ago, and neither he nor his students benefited. — jgill

Who benefits from the History of Math or the Philosophy of Math? Not many I would say.

Usually students aren't interested in the fascinating history of a debate in mathematics. — ssu

It's not a math history course. It's a sophisticated real analysis course, including calculus, based upon a rigorous concept of infinitesimals.

For example, just how many different fields of math can you find something similar to the Axiom of Choice? — ssu

I'm not up to speed in contemporary abstract math, particularly foundations, but I would guess few, if any. fdrake or fishfry might be able to answer your question.

Nice! Thanks,

It's not a math history course. It's a sophisticated real analysis course, including calculus, based upon a rigorous concept of infinitesimals. — jgill

Then there simply is no time for philosophy. You have to go through all the work done by mathematicians and get to the sophisticated ways mathematicians use them. It's simply a matter of time.

I'm not up to speed in contemporary abstract math, particularly foundations, but I would guess few, if any. — jgill

There's a book by Herman Rubin and Jean E. Rubin called "Equivalents of the Axiom of Choice", which states about 150 statements in mathematics that are equivalent to the axiom of choice.

Wikipedia states some:

Set theory

Well-ordering theorem: Every set can be well-ordered. Consequently, every cardinal has an initial ordinal.
Tarski's theorem about choice: For every infinite set A, there is a bijective map between the sets A and A×A.
Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.
Given two non-empty sets, one has a surjection to the other.
The Cartesian product of any family of nonempty sets is nonempty.
König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals. (The reason for the term "colloquially" is that the sum or product of a "sequence" of cardinals cannot be defined without some aspect of the axiom of choice.)
Every surjective function has a right inverse.

Order theory

Zorn's lemma: Every non-empty partially ordered set in which every chain (i.e., totally ordered subset) has an upper bound contains at least one maximal element.
Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. The restricted principle "Every partially ordered set has a maximal totally ordered subset" is also equivalent to AC over ZF.
Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.
Antichain principle: Every partially ordered set has a maximal antichain.

Abstract algebra

Every vector space has a basis.
Krull's theorem: Every unital ring other than the trivial ring contains a maximal ideal.
For every non-empty set S there is a binary operation defined on S that gives it a group structure. (A cancellative binary operation is enough, see group structure and the axiom of choice.)
Every set is a projective object in the category Set of sets.

Functional analysis

The closed unit ball of the dual of a normed vector space over the reals has an extreme point.

Point-set topology

Tychonoff's theorem: Every product of compact topological spaces is compact.
In the product topology, the closure of a product of subsets is equal to the product of the closures.

Mathematical logic

If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem; see the section "Weaker forms" below.

Graph theory

Every connected graph has a spanning tree.

If you call that few if any, well...

If you call that few if any, well... — ssu

Wow! That's pretty impressive. I didn't think the AOC ventured much beyond set theory. Some time back fishfry mentioned Zorn's lemma (or transfinite math) regarding the proof of the Hahn-Banach theorem in functional analysis, but when I checked my ancient class notes I found that a minor change in the hypotheses eliminated that need. The closest I ever came in the examples you cite from Wiki is the basis of vector spaces, and even there it didn't come into play in the stuff I explored.

Thanks for the info. :cool: