When I was an undergraduate, I remember being very struck by some of the early results in the class I was taking on abstract algebra. Of course, I was eventually very struck by the results from Galois theory when we got there, but in the early parts of the class I was struck by the results proving the existence of the algebraic closure of a field, and proving the existence of a field of fractions for every integral domain. In particular, these seemed to me to validate the use of the complex numbers (once the reals were given) and the rational numbers (once the integers were given). I was still vaguely dissatisfied that we hadn’t yet had a proof of the existence of the integers, but I became happier when I saw the definition of the natural numbers as the smallest set containing 0 and closed under the successor operation, especially because this let proof by induction be a theorem rather than an axiom.
However, earlier this week (in conversation with Zach Weber, while visiting Sydney), I started realizing what I should have realized long ago, which is that these theorems really can’t be doing as much work in justifying our use of the various number concepts as I had thought when I was younger. Of course, these theorems are quite useful when talking about abstract fields or rings, but when we’re talking about the familiar complex, real, rational, and integer numbers, it’s no longer clear to me that these theorems add anything whatsoever. After all, what these theorems show is just that, by using some fancy set-theoretic machinery of ordered pairs and equivalence classes, we can create a structure that has all the properties of a structure that we already basically understood. Perhaps in the case of the complex numbers this mathematical assurance is useful (though even there we already had the simple assurance in the form of thinking of complex numbers as ordered pairs of reals, rather than as polynomials over R modulo the ideal [x2+1]), but for the rationals and negative numbers, our understanding of them as integers with fractional remainder, or as formal inverses of positive numbers, is already sophisticated enough to see that they’re perfectly well-defined structures, even before we get the construction as equivalence classes of ordered pairs of integers or naturals.
But this is all a sort of prelude to thinking about the two more famous set-theoretic reductions, that of the reals to Dedekind cuts (or Cauchy sequences) of rationals, and that of the naturals to the finite von Neumann ordinals. Unlike the others, I think the Cauchy and Dedekind constructions of the reals are quite useful – before their work, I think the notion of real number was quite vague. We knew that every continuous function that achieves positive and negative values should have a zero, but it wasn’t quite clear why this should be so. Also, I think intuitively there remained worries about whether there could be a distinct real number named by “.99999…” as opposed to the one named by “1”, not to mention the worries about whether certain non-convergent series could be summed, like 1-1+1-1+1….
But for the reduction of the naturals to the von Neumann ordinals, I think it’s clear that this should do no work in explicating the notion at hand. To prove that enough von Neumann ordinals exist to do this work, you already need a decent amount of set theory. (John Burgess’ excellent book Fixing Frege does a good job investigating just how much set theory is needed for this and various other reductions.) And while some of the notions involved are basic, like membership and union, I think the concept of sets of mixed rank (for instance, sets that have as members both sets, and sets of sets) already strains our concept of set much more than any of this can help clarify basic notions like successor, addition, and multiplication. One might even be able to make a case that to understand the relevant formal set theory one must already have the concept of an ordered string of symbols, which requires the concept of finite ordering, which is basically already the concept of natural numbers!
In some sense, this was one project that Frege was engaged in, and his greatest failure (the fact that his set theory was inconsistent) shows in a sense just how unnecessary this project was. At least to some extent, Frege’s set theory was motivated by an extent to show the consistency of Peano arithmetic, and clarify the concept of natural number. However, when his explanation failed, this didn’t undermine our confidence in the correctness of Peano arithmetic. The same thing would be the case if someone today were to discover that ZFC was inconsistent – most of the mathematics that we today justify by appeal to ZFC would still stand. We wouldn’t abandon Peano arithmetic, and I think we wouldn’t even abandon most abstract algebra, geometry, analysis, and the like, except perhaps in some cases where we make strong appeals to the Axiom of Choice and strange set-theoretic constructions.
Of course, Frege’s actual attempted reduction of the number concepts to set theory would have been a very nice one, and could help explain what we mean by number, because he reduced each number to the collection of all sets with that many elements. However, modern set theory suggests that no such collections exist (except in the case of the number 0), and the proposed reductions are much less illuminating.
So I wonder, what role do these proofs play, that demonstrate the existence of structures that behave like the familiar natural numbers, integers, rationals, reals and complex numbers? I’ve suggested that in the case of the reals it may actually do important work, but I’m starting to be skeptical of most of the other cases.
Antimeta 
The whole thing is a trick philosopher’s played on mathematicians. Philosophers convinced mathematicians that logi” was somehow more basic than arithmetic. By making themselves the masters of the “foundation” of mathematics, philosophers hoped to heap mathematics’ well-deserved glories onto themselves. Largely, they were looking for a way to cover up philosophy’s abject failure to progress as a science.
To summarize, philosophy tried to remake itself in the image mathematics, in order to show some progess. That adventure failed, so intsead they decided to remake mathematics in philosophy’s image, or at least, under philosophy’s guise. Since the are good at making convincing arguments, they succeeded to a large degree. Clever fellows.
Why do you think that the concept of sets of mixed rank strains our conception of sets? I think I agree with your point about the ordinals requiring too much set theoretic machinery to justify our familiar use of the natural numbers.
I’m not particularly familiar with how these existence proofs come up outside of set theory contexts. Do they seem motivated? Their role in a foundational debate seems fairly clear. The set theorist wants to show that any mathematical structure is realized in the world of sets. If not then set theory is less attractive for a foundational role. The set theoretic reduction of the reals is useful, then, because, as you point out, it resolves some foundational questions that were left open.
The question that you end on seems to me to be most happily fleshed out as “what role do these results play in which areas?” The representation of the natural numbers doesn’t play much of a role except, maybe, in some foundational debates, if at all. The result for the reals cleared up some uncertainty. I just came across something that might be an example of the representation of the complex numbers. It is in Gray’s book Worlds Out of Nothing. There is a passage describing the unease of geometers in the mid-19th century with extending geometrical ideas to the complex plane because there wasn’t a formal theory of it, or some such. I’m not sure if that is an unease mathematicians today feel before learning the stuff, but it is a possible answer.
KenF – I think that’s a rather uncharitable way to put it. I think the mathematicians managed to confuse themselves, because it’s hard to know what you’re searching for when you’re searching for rigor. The Cauchy/Dedekind analysis of the reals was certainly a real advance. Then the existence theorems for fields of fractions and for algebraic closures are also generally advances, and look like they might help us with our reduction of arithmetical notions. And then Frege looked like he was going to reduce everything to basic concept application, but when his project failed, mathematicians who were interested in this sort of thing tried to replace it, perhaps without thinking about what replacing it would really do for them.
Shawn – I think it takes most people a little practice to get used to the idea of sets of mixed rank. People start out used to sets of urelements, and then get used to the idea of sets of sets. I suppose it’s really functions with domains of mixed rank that might seem weirder to most mathematicians, because variables usually have some particular type or other that they must range over.
Anyway, I think that’s a good point – these results can help clarify and confirm set theory, even if they don’t add anything to the arithmetical structures they appear to be intended to support.
I didn’t mean to be uncharitable, but the longing for mathematical certainty is so striking in so many philosophers. Of course so many were also mathematicians.
I ask the following not to argue, but out of legitimate interest, and naivete.
“The Cauchy/Dedekind analysis of the reals was certainly a real advance.”
Do you think a different analysis of the reals is possible, and might have made an equal or greater advance, perhaps in a different direction? Or was their analysis uniquely determined somehow, the “correct” analysis?
I think that later developments suggest that no other analysis that wasn’t equivalent (as the Cauchy and Dedekind ones eventually turned out to be) would have been as great an advance. There is another analysis that has been generally less important, but still useful, which is the notion of a “real-closed field”. That is, a field extending the rationals, with a linear ordering satisfying the same properties as the linear ordering over the rationals (including interaction with addition and multiplication), such that every polynomial of odd degree has a root. (I believe this suffices to guarantee that it includes every algebraic real number, though it’s not obvious off the top of my head how to show that the square root of 2 is included.)
I think the notion of “real closed” I’ve seen is that of an ordered field such that every odd degree polynomial has a root, and that every positive element has a square root.
Ah, that sounds familiar, and it would explain why I couldn’t figure out the answer to my previous worry.