I’ve been reading Akihiro Kanamori’s *The Higher Infinite* to make sure that I know enough about what’s going on in Hugh Woodin’s class on determinacy and woodin cardinals. It’s really an extremely well-written book, giving enough mathematical detail that it’s not quite as dense as Jech’s set theory book, and has more content than Kunen’s. And it also has much more historical detail than any other set theory book that I know of, including citations to the papers where each notion was first introduced, and papers where important use was made of the concepts.

But given that this book is primarily an exploration of axioms extending ZFC, it’s interesting to see the polemical “Appendix (with apologies to Burton Dreben)”. He pushes some Wittgensteinian ideas suggesting that the search for explanations of mathematical practice are misguided, and thus that set theory is best when it is pursued as just another branch of mathematics, rather than attempting to give a “foundation” for the other branches. I think I generally disagree with this point of view, but he tells a lovely story that I heard third-hand reference to coincidentally just the day before from some of the other logic grad students. I’ll quote the entire thing here:

To pursue an analogy, the world of mathematics is like a great cathedral. The thick stone walls along the stately aisles still show the lines of the ancient church that predated the grand edifice. The central dome is supported by high arches of vaulting stone, resilient reminders of the anonymous master masons. Whatever their design, the arches have easily supported the elegant latter-day spires reaching high into the sky. The first adornments can still be seen in the oldest chapels; there in continuing communion with the past steady additions are made, each new age imparting its own distinctive style. In recent memory large new side chapels have been constructed, and new flying buttresses for extra support. Every day the curious enter through the great door of polished wood with the attractive inset figures. Several venture down the long nave seeking instruction, and a few even initiation, quite taken by the order and beauty of the altar. And the work continues: The architects attempt to chart out large parts of the cathedral, some even proposing vast renovations. The craftsman [sic] continue the steady work on the new wood paneling, the restoration of the sculpture, and the mortaring of the cracks that appear with age. And supported by high scaffolding, the artisans continue to work on the fine stained glass. They try to coordinate with their colleagues in the adjoining frames, but sometimes the heady heights inpire them to produce new gems. Those who step back see a larger scheme, but they cannot see across the whole breadth. And they are so high up that they can no longer see their supports. Nevertheless, they are sustained as a community, as part of the ongoing human adventure.

To append an apocryphal tale: A host of industrious spiders started to build an elaborate network of webs in newly excavated vaults beneath the cathedral. It quickly grew so thick and complex that no one could venture across without getting enmeshed. One day, a fearful wind came howling in and blew a gaping hole through the network, and in a desperate response the spiders worked frantically to reestablish the connections. For you see, the spiders had become convinced that their carefully constructed webbing was the foundation without which the entire cathedral would totter. Of course, the craftsmen above hardly raised an eyebrow.

I think he’s right that the large mathematical endeavor has proven so useful and coherent that it is not in any danger of falling. If someone were to prove that ZFC is inconsistent, there would be some scrambling to come up with a new theory (or a move to some alternative old theory, like Quine’s New Foundations, or Russell’s ramified theory of types), but within a decade or so, 95% of mathematics would be shown to be representable in the new foundational picture as well. This happened several times in the early 20th century (though a few of those times didn’t involve a proof of inconsistency, but just a change in fashion about which system to use).

But this doesn’t mean that the cathedral supports itself. I (and others who work in foundations) would like to know how it is in fact supported – there’s no need to think that it is actually supported by our work. A predecessor of this cathedral was the church of analysis built on the work of Newton and Leibniz, and brought to many of its greatest heights by the master craftsman Euler. However, Bishop Berkeley had pointed out to Newton that he was building his edifice directly on a major fault line. Mathematicians responded as Kanamori seems to. In the early 19th century it became clear that Berkeley was right, as Fourier and Cauchy “proved” both that the sum of a series of continuous functions is continuous, and that many discontinuous functions were the sums of Fourier series of continuous functions. [Edited for clarity.] Fortunately, the resulting tremors only damaged a few pieces of work, and Cauchy and Weierstrass were able to restructure the base of the church with epsilons and deltas, finally taking the load off the fault line that Berkeley had pointed out. (Kanamori discusses this entire history in his appendix.)

Just because mathematics has generally withstood the problems of the past doesn’t mean that there are no remaining problems – just that these problems are likely to be only relatively minor. Unlike Kanamori, I think this means we should be continuously vigilant about the foundations. And there are still metaphysical questions. He seems to conflate the desire to do metaphysics with the desire for platonism (which I suppose is something I’ve also done in the title and subtitle of this blog!) but I think many of the foundational programs of Hatry Field, Stewart Shapiro, Michael Resnik, and the like are worth studying, and may result in important foundational gains. Kanamori objects, saying that this drive, “along with the more traditional musings about the starry heavens above or the moral law within, are not in the world but of the mystical, part of the feeling for the unity of experience in the large.” He closes with a couple quotes from the Tractatus and the Tao Te-Ching, saying this is one of those things that must just be shown, not said.

But if this is all right, then the desire for explanation is a sort of mysticism, which would seem to undermine the entire scientific program for understanding the world. Maybe some readers of Wittgenstein would approve of that result, but I think this is an overreaction against workers in the foundations of mathematics, saving mathematics from a lack of foundations in a way that threatens all of science.

Greg F-A(07:37:37) :Hi Kenny —

I want (like you) to reject any view that treats every desire to explain as a merely quasi-religious longing. But I do think

somedesires for explanation can only be satisfied by extra-scientific (~including mystical?) means. Examples from the empirical sciences might be “Why is there something rather than nothing?”; “Why do we have the particular laws of nature that we do, and not some others (or none at all)?”My question is: where do you think

mathematics(or any formal science) draws the line between acceptable/OK explanations and bogus over-explanations?(Have I asked you this question before?)

Peter(13:12:08) :” as Fourier and Cauchy proved both that the limit of a series of continuous functions is continuous”Did you mean to write:

” as Fourier and Cauchy proved both that the limit of a SEQUENCE of continuous functions, IF IT EXISTS, is NOT NECESSARILY continuous”Kenny Easwaran(23:35:51) :Greg – I don’t think you’ve asked me that question before, and I’m not terribly certain that I agree with the premise. I agree that the questions you cite might not have answers, but it may be because they’re not really demands for explanation at all, rather than saying that they’re actual demands for an extra-scientific explanation. I’ll have to think more about it.

Peter – I meant what I wrote, though perhaps I didn’t emphasize enough that the result about sums of series of continuous functions was only “proved” using the non-rigorous techniques that were common at the time. This was an illustration of the problems Berkeley noted long before.

Peter(13:29:26) :Kenny —

Sorry to be difficult, but your statement still troubles me. I don’t know enough history of math to know what Fourier and Cauchy proved, but as an infinite limit of a series of continous functions may neither exist, nor be continuous when it does exist, then they can’t have proved precisely what you claim.

As a counter example, consider the series where:

The first function f_0(x) = x, for x \in [0,1].

And, each n-th function f_n(x) is defined as follows:

For x \in [0, 1/2^n], f_n(x) = x.2^(n-1)

For x \in [1/2^n, 1/2^(n-1)], f_n(x) = 1 – (x.2^(n-1))

For x \in [1/2^(n-1), 1], f_n(x) = 0.

Each of these functions f_n is a little pyramid to the right of zero, and zero elsewhere. The height and width of the pyramids both get smaller and smaller as n gets larger. Each f_n(x) is continuous on the entire interval [0,1].

Let function g_n(x) be the sum of the f_0 + f_1 + . . . + f_n (the first n+1 functions f_n).

Then, as n heads to infinity, this series (the sums of the f_n functions) converges to the function:

g(x) = 1 for x \in (0,1]

and g(0) = 0.

The function g is not continous at zero, even though it is the infinite sum of functions, each of which is continuous there.

Am I missing something?

Peter(13:50:02) :Just re-read your main post, and I see that perhaps you are saying that Fourier and Cauchy “thought they had proved” rather than that they had proved this result.

I think it is tempting to look, as many mathematicians do, smugly back at our “naive” ancestors and criticize their lack of rigour. (I am not accusing you of doing so.) However, all standards of proof, including our very own, are culture- and time- dependent. Ronnie Brown, prominent algebraic topologist, only last week remarked on a category theory list that:

“The situation is more complicated in that what could be classed asspeculation may get published as theorem and proof. For example, in

algebraic topology, sometimes proofs of continuity are omitted as if this was an exercise for the reader, yet the formulation of why the maps are continuous (if they are necessarily so) may contain a key aspect of what should be a complete proof. This difficulty was pointed out to me years ago

by Eldon Dyer in relation to results on local fibration implies global

fibration (for paracompact spaces) where he and Eilenberg felt Dold’s paper

on this contained the first complete proof. I have been unable to complete the proof in Spanier’s book, even the second edition. (I sent a correction to Spanier as the key function in the first edition was not well defined, after Spanier had replied `Isn’t it continuous?’) Eldon speculated (!) that

perhaps 50% of published algebraic topology was seriously wrong!

In homotopy theory, many matters, such as the homotopy addition lemma, had clear proofs only years after they were well used.Surely much early algebraic topology is speculative, in that the language“has not yet been developed to express concepts with rigour so that a clear

proof can be written down. It would be a curious ahistorical assumption

that there is not at this date another future level of concepts which

require a similar speculative approach to reach towards them.

Kenny Easwaran(21:18:35) :I believe Cauchy’s proof, after some investigation (because of Fourier’s counterexamples) was eventually turned into a proof that any series of

uniformly continuousfunctions that convergesuniformly(I don’t remember if both “uniform”s are required) sums to a continuous function.I think it’s probably right, as you and Ronnie Brown point out, that many of our current concepts and results are at least plausibly in the same situation, whether or not they might actually turn out to be incorrect. Some of this goes along with the ideas by Fallis that I posted about in November or so (http://www.antimeta.org/blog/archives/2005/11/what_sorts_of_p.html).

By the way, I read an interesting article by Ronnie Brown, which I think was published in the Mathematical Intelligencer in about ’94 or so, where she basically just lists a whole bunch of philosophical questions that mathematicians might be interested in. Some number of them are the traditional questions of philosophy of mathematics, but there were a bunch more as well, of course.