A Coq and Bull Story

Author: Michael Harris.
Title: Mathematics Without Apologies.
Source: I eventually gave up waiting for a complimentary signed copy to be sent to me in the mail, and so borrowed a copy from the Northwestern University library. I’ve read up to the beginning of chapter 5 so far.

Thoughts before reading the book: Rational thought and continental philosophy have always been non-overlapping magisteria in my mind. Who better, then, to bridge the gap than Michael Harris, the left-bank mathematician who had returned stateside (well, Manhattan-side at least).

In Mathematics Without Apologies, Michael Harris addresses what it is to be a mathematician. There are several questions raised in this book which I hope to consider on this blog. However, before we begin any analysis of the actual content, there is an issue that first has to be discussed. In a book so thick with opinions, it is quite extraordinary that the author believes that he can not only play the simultaneous roles of case study and provocateur, but also, at the same time, present himself as completely impartial observer, one who is merely “offering up” a goulash of narratives and cultural insights for our consideration. Michael Harris is not the naive innocent that he pretends to be; to me, the author is about as impartial as Socrates is when he asks Thrasymachus what makes a “just man.” On the other hand, how can one win an argument with the author on how strongly he holds certain opinions? Rather than take a detour to discuss the relationship between author and text, and argue about the extent to which an author can, by writing supplementary blog posts or otherwise, clarify the intended implications (or lack of implications) of his own words, let me offer the following solution: from this point on, we shall denote the author of “Mathematics Without Apologies” by michael harris (lower case). To be clear, michael harris is a chimera born out of Michael Harris’ words and my interpretation of them. Michael Harris is allowed to disagree with my interpretation of what michael harris says, but he is not the ultimate arbiter of deciding what michael harris thinks; that is up to the reader(s). The only further words I will say on this topic are as follows: for a book claiming to offer no apologies, there are an inordinate number of strongly worded disclaimers, for example: “[one] … should not mistake this book for a work of scholarship.” [OK, duly noted. Of course, these are the words of Michael Harris, not michael harris, the savage cultural critic.] Voloch complains here of his frustration of the author’s apparent inability to commit to any position, to which Michael Harris is sympathetic, but I suggest that both of them read closer between the lines to see that michael harris does indeed hold some strong opinions. With that out of the way, let us begin considering actual issues raised by the text. For now, let me restrict myself to explaining and largely agreeing with a single opinion of michael harris. We begin by recalling a distinction made in the book between the philosophy of Mathematics and the philosophy of mathematics. The philosophy of Mathematics is concerned purely with epistemological questions, and should be thought of as the subject whose intellectual lineage goes back to the crisis of foundations, and which tries to explain the meaning of mathematical truth and its relationship to knowledge. The philosophy of mathematics, on the other hand, is concerned with how actual mathematicians think (about mathematics), and is a topic of primary interest to the author. What becomes clear when reading the book is that michael harris believes that the philosophy of Mathematics — and indeed the subject of foundations more generally — has nothing at all useful to say or contribute to the professional lives of mathematicians, by whom I mean people like myself, or people like Michael Harris. For example, harris quotes Jeremy Gray (from Gray’s account of the Foundations Crisis) as follows:

The logicist enterprise, even if it had succeeded, would only have been an account of part of mathematics — its deductive skeleton, one might say…. mathematics, as it is actually done, would remain to be discussed.

Or, to quote michael harris directly (p.67), “Capital-F Foundations may be needed to protect mathematics from the abyss of structureless reasoning, but they are not the source of mathematical legitimacy.” Nothing controversial so far, I think. But harris goes further. In a book thick with quotations, the key to any reading is to identify the villain. And the villain in chapter 3 is definitely Voevodsky, quoted here as follows:

If one really thinks deeply about … [the possibility that the foundations of mathematics are inconsistent] … this is extremely unsettling for any rational mind.

Certainly, by this measure, I am either not in a possession of a rational mind, or not a deep thinker about these questions. If someone told me today that Voevodsky had discovered an inconsistency in ZFC, I would care slightly less than if someone told me the Collatz problem had been solved, and care much less than if someone (trustworthy) told me that a serious error had been found in the proof of cyclic base change. In the first case, I would presume that what ever fix (large or small) to foundations needed to be made, it wouldn’t make any difference to the mathematics that I think about. Whereas in the third case, it would make quite a lot of difference. Underlying all of this, of course, is some assumption (by me) that there is some formulation of mathematics which is consistent. There seems to be some evidence for this, including a several thousand year history of mathematics which has required barely any modification at all in light of whatever logical issues arose in the 19th and 20th centuries. This is not to say that michael harris completely dismisses any efforts to study foundations. However, to his mind, and to mine, the ultimate judgment about such an enterprise should be on its effect on mathematics qua mathematics (“If and when univalent foundations is adopted as a replacement for today’s … foundations, it will probably be … triggered by a demonstration of the new method’s superiority in addressing old problems”, p.65). This ties together with the attitude that both michael harris and I share towards computer assisted proofs, which is somewhere between “who cares” and a general skepticism that it will have any relevance to mathematics as we practice it (caveat: I believe michael harris discusses these issues in later chapters which I have not yet read). Again, to understand what michael harris thinks, it suffices to judiciously select those whom he quotes. Here is another quote by harris on p.66:

When [Benecerraf] limits the articulation of mathematical truth to logic and then complains that the ability of mathematicians to refer has been lost, it is no wonder; it is also no wonder that number theorists and geometers have not borrowed the language of logic to do their work.

(Confusingly, I can’t quite work out where this quote is from: the footnote refers to a 2009 reprint of writings of Herman Weyl (who died in 1955), but the quote above implicitly (in the context of the book) seems to refer to an article of Benecerraf from 1973.) Here michael harris is drawing the following distinction. The view, coming from the philosophy of Mathematics, is that mathematicians are trying to understand some truth, and then to try to decipher what form of truth mathematics actually consists of; whether it be a chain of modus ponens all the way down, or what have you. After all, if the goal of mathematicians is to seek the “truth” (whatever that is), then surely it follows that it is important to put the notion of truth on some firm philosophical footing. Yet the philosophy of mathematics view is really quite different. If you asked me why I do mathematics, I wouldn’t, if I was honest, say that I was seeking “the truth.” My attraction to mathematics (and even analytic philosophy) is that I find it a lot of fun, and moreover completely addictive. Analytic philosophy (of the flavour I enjoy) is fun, because it is a (logical or linguistic) puzzle. For example, the observation:

“Hesperus is Phosphorus” is not a tautology

can be used to negate certain claims about the meaning of proper names and the relationship of the necessary to the a priori. (Or even better, to go Quine rather than Frege, “9 = the number of planets” is not a tautology, not least of which because it is currently false.) But do these exercises actually tell us anything about epistemology, or are they really just some form of enjoyable intellectual exercise? Those questions are beyond my pay grade. But honestly, they don’t even interest me that much. I simply enjoy those puzzles for what they are, rather than lay any claim that they are revealing deep truths about human thought. (To be honest, I think if I really felt that I wanted to understand what words mean, I would become a computational linguist, not an analytic philosopher.) In mathematics, too, I enjoy playing with objects (Galois representations, automorphic forms) that are my stock in trade; the fun and beauty aspects are more compelling than the “search for absolute truth.” No doubt michael harris will have more to say on these topics in later chapters. These issues also remind of a conflict between mathematicians and historians of Mathematics. A colleague once reported to me a conversation (or argument) with a historian of Mathematics who claimed that Galois did not know that $A_5$ was simple. On the other hand, Serre claims here that Galois not only knew that $A_n$ for $n \ge 5$ was simple, but also that $L_2(p):=\mathrm{PSL}_2(\mathbf{F}_p)$ was simple (for $p \ge 5$), and that (moreover) Galois really understood these groups. For example, Serre claims that Galois knew that $L_2(p)$ for $p > 3$ has a transitive action on $p$ points only for $p = 5,7,11.$ (Exercise for the reader!) Of course, it’s a bit of a running joke in mathematics that everything is really due to Gauss, but I side with Serre here (a bold stance, I know), and moreover think that one of the issues is (or could be) the tendency for a historian of Mathematics to view mathematics through the perspective of a Philosopher of Mathematics rather than of mathematics; to fail to distinguish between a formal proof and a genuine understanding.

Other remarks on the first few chapters: Harris quotes Neil Chriss (p.72 “who chose to forgo a promising future in the Langlands program to work for a … hedge fund”) as saying that “The Glass Bead Game is a favorite novel among my mathematician friends.” (the implication being here that Mathematicians would love to have no responsibility to the outside world). Really? Either those mathematicians completely misunderstood the point of that novel, or I did. It seemed to me that one of the key themes of Hesse’s book was that a complete disconnect between intellectual pursuits and the ultimate responsibilities of humanity towards society was a bad thing; possibly not a surprising message for a book which was published in 1943. On the other hand, the implication of the quote above is that Mathematicians dream of Castalian paradise where they can pursue mathematics unencumbered by the realities of society. Curiously enough, Both Michael Harris and michael harris are entities to whom the theme of Hesse’s book (in my reading) seems to hold some appeal. One certainly gets the sense even from the first few chapters that michael harris is deeply concerned with the interaction between intellectuals and broader society. (I am with harris completely when it comes to his opinions on foundations; I expect to differ on matters related to our role in society.)

(p.123) I think that Tom Stoppard did a better job of demonstrating that the symmetry group of Rosencrantz and Guildenstern is $\mathbf{Z}/2 \mathbf{Z}.$

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4 Responses to A Coq and Bull Story

1. Matthew Emerton says:

Serre is correct about Galois and $\mathrm{PSL}_2(\mathbb F_p)$. Galois studies this group because it is the Galois groups of the $p$-division equation for an elliptic curve. He knew that it was simple (for $p > 3$), and that it had a faithful degree $p +1$ permutation representation (the action on $\mathbb P^1(\mathbb F_p)$), implying that the $p$-division equation can be solved by splitting a degree $p+1$ equation.

On the other hand, Galois’s big theorem on solvability was for polynomials of prime degree: he showed that an irreducible polynomial of prime degree is solvable in radicals iff its splitting field is generated by any two roots. So he was interested in whether he could reduce the degree of the $p$-division equation from $p+1$ to $p$. To this end, he computed when there is a maximal subgroup of index $p$.

Given that Galois knew that $\mathrm{PSL}_2(\mathbb F_5)$ was a simple group of order $60$ that acts faithfully on $5$ elements (and taking into account his overall level of sophistication and insight, which was extreme, to say the least), it’s pretty inconceivable that he didn’t know that it was identified in this way with $A_5$ (and hence also know that this latter group is simple).

• Dick Gross says:

Matt,

I think it is deeper than that. Galois knew that the only primes p where G = PSL(2,p) acted on p letters were p = 2,3,5,7,11. In this case G has a subgroup H of index p. In the first two cases, G is the symmetric group S3 and S4 and the subgroup H is cyclic and dihedral respectively. In the last three cases, the subgroup is A4, S4, and A5. This suggests that Galois may have known the relation between the finite subgroups of PSL(2,C) and the finite subgroups of G. How else would he have discovered H in the latter cases?

Nowadays the finite group theorists are interested in classifying the maximal subgroups of simple groups like G(p), where G is of Lie type. The most difficult ones to identify are reductions (mod p) of groups of the form H = G(Z), where G is a reductive group over Z which is compact over R. For example, G2(Z) = G2(2) occurs as a maximal subgroup of G2(p) for all primes p. Galois’s exceptional subgroups are all of this type, coming from the unit groups of orders in quaternion algebras over Q or Q(sqrt(2)) or Q(sqrt(5)) which are definite at infinity.

Dick

2. DT says:

Hi Dick. There’s another exceptional subgroup of G2(q), a copy of Janko’s J1 inside G2(11). Do you know any similar explanation?