What I learned in Philosophy Hall
Two hours in the epistemological gladiator pit
On the afternoon of Wednesday March 23 I entered Room 716 Philosophy Hall at the invitation of the philosopher Justin Clarke-Doane (JCD), to converse with his graduate class in the Foundations of Metalogic. The professor and his students were too polite to point out how my jet-lagged brain struggled to complete sentences and jumped unpredictably from one topic to a another in the second, conversational half of the class. Fortunately, for the first 45 minutes I could allow myself to be guided by the slides I had prepared more than a week earlier. The first half hour was copied more or less verbatim from the beginning of the last installment of this newsletter. The class smiled in recognition when I proposed my version of disciplinary norm of philosophy, namely that
(N) the primary professional activity of philosophers is to explain why other philosophers are wrong,
contrasting this with the tendency of mathematicians to converge on a common understanding of what our colleagues have got right.
At the end of the class, thinking about my frustration at not having definitively established the point I was trying to make about the relation between formal proof and “ordinary mathematical practice,” I realized that this contrast between the disciplines conceals a striking paradox. I’ll have more to say about the role of formal proof in mathematics later. Here I just want to emphasize the point I developed in Chapter 7 of Mathematics without Apologies, namely that a theorem and its proof is generally seen by mathematicians as
the point of departure for an open-ended dialogue that is too elusive and alive;1
whereas philosophers of mathematics (and their emulators among mechanizers), to the extent that they are concerned with epistemology, take the proof to be a final destination consisting of whatever they understand as truth. I was affirming, in other words, that math is from Eros, formalization is from Thanatos (insofar as the latter is an expression of the Central Dogma). In real life, however, I was frustrated at my failure to pronounce, if not the last word, at least some sort of QED; while JCD reassured me that
you certainly shouldn't feel frustrated for failing to settle a philosophical question definitively! It's well known in the 'biz' that the worst case scenario when you give a philosophy talk is that nobody objects (maybe you even get some — gasp! — compliments). That means the audience thinks your argument is truly embarrassing, not deep or smart enough to even engage with. … The best case is lots of back-and-forth, rethinking assumptions, frustrations, and headaches.
In other words, while philosophers posit an immutable truth as an ideal destination, the ultimate subject matter of their discipline, they are not only aware that, in keeping with disciplinary norm (N) the destination will never be reached, they positively thrive on the “elusive and alive” experience of failing to get there. In contrast, while mathematicians know well that there will never be a last word, and wouldn’t dream of “dreaming of a final theory,” something deeply embedded in the discipline fills us with frustration, if not despair, if we cannot place an emphatic full stop at the end of each performance in the gladiator pit, whether it’s a publication or a lecture.
Try if you can to untangle that antinomy! Toward the end of the class I opined that the doomed attempt to talk about change in a language whose words stay in place is perhaps the driving paradox of (western) philosophy, going back to the pre-Socratics, but I only realized afterwards how I had embroiled mathematics in that very paradox.
What is essential?
I concluded my discussion of the Central Dogma and disciplinary norms with a quotation illustrating Peter Scholze’s attitude to the kinds of set-theoretic issues that philosophers believe must be addressed:
The blue text is Scholze’s, the black is mine, and the question — what is essential? — was addressed to philosophers. The ground was laid for a possible answer with an
Interlude from Plato’s Republic, Book 7 (featuring the inimitable Glaucon)
Socrates: This at least will not be disputed by those who have even a slight acquaintance with geometry, that this science is in direct contradiction with the language employed in it by its adepts.
Glaucon: How so?
S: Their language is most ludicrous, though they cannot help it, for they speak as if they were doing something and as if all their words were directed towards action. For all their talk is of squaring and applying and adding and the like, whereas in fact [527b] the real object of the entire study is pure knowledge.
G: That is absolutely true.
S: And must we not agree on a further point?
G: What?
S: That it is the knowledge of that which always is and not of a something which at some time comes into being and passes away.
G: That is readily admitted for geometry is the knowledge of the eternally existent.
Today’s geometers, I explained, are still getting on Socrates’ nerves. Formal proofs have no verbs because their tense is that of the “eternally existent,” but mathematical proofs are constructed around verbs. Like Socrates, one can wish them away; or one can also take a hint from naturalist philosophy of mathematics and try to observe what they are doing in the proof.
Here I should quote from the point in my article Do Androids Prove Theorems in Their Sleep, which JCD assigned to the class as reading in preparation for my visit, where I address the eternal existence of formal proofs that corresponds to the eternal existent tense of their grammar. The context is my hypothetical encounter with an android endowed with the power of speech. I think of it as a living embodiment of the Central Dogma, the philosophical outlook that matches its nature; as a philosopher it compels me to justify my own language and my own perspective.
Is there a moment in history that separates the time before the Thomason-Trobaugh theorem was proved from the time it became a theorem? … the android may … defend the position that the proof has always existed (as potentia or dynamis), that its precise incursion upon history is a detail of no importance, and my persistence in presenting the question in these terms is a symptom of a perceptual defect traceable to my communicative dependence on the narrative form.
After the interlude from the Republic I turned to my “close reading” of the proof of Lemma 5.5.1 of the Thomason-Trobaugh paper2, in the form of a quest narrative (on pp. 30-32 of the article reproduced here), emphasizing the verbs and how they are “directed towards action.” And then I turned over the session to JCD and the class.
Rescuing the 'formalist' view
Less than an hour before class JCD had sent me a list of questions he hoped I might have time to address. As it happened I was too sleep-deprived to make sense of them when the list appeared in my in-box, but JCD made sure at least the first one was discussed, in abbreviated form, as soon as I finished my presentation.
(1) The ‘formalist’ view is that a proof is valid if and only if it somehow ‘indicates’ that there exists a (first-order) formal proof of a formalized version of the theorem from some suitable axioms (maybe ZFC + Large Cardinals). Do you think one or both directions of this claim might fail?
(a) Do you think that the Four Color Theorem has been proved? (If not, what do you mean? Do you simply mean that we don’t understand why the theorem is true (or that the demonstration is otherwise non-ideal), or that the argument is, for all we know, invalid?)
(b) Do you think that the independence results suffice to show that no ordinary (informally) valid proof of CH or ~CH will be forthcoming (if standard math is consistent)? If not, is that because informal mathematics is incommensurable with formalized mathematics, or because the (formal) system that does justice to our informal practice transcends the likes of ZFC + LCs?
The question looks innocent enough but if I were a philosopher I would have immediately realized that it’s riddled with booby-traps. I tried to evade responsibility by reminding the class that no one really cares, or should care, what I think. Is the “you” to whom the question is nominally addressed a stand-in for a consensus position among living (human) mathematicians? Am I supposed to have a position of my own before I can critique any established position among philosophers?
My fallback position is that if the question doesn’t arise in this form in mathematical practice, then one should probably be asking different questions (I later mentioned some options, see below). This is how I dismiss the Central Dogma: it conforms neither to mathematical practice nor to what mathematicians say about what we do. A proof is considered valid when it has been validated by the reigning disciplinary norms. JCD worried that this sounded like sociology, making the perfectly valid but also perfectly annoying point that, as far as the philosophical quest for truth is concerned, all mathematicians can be wrong about what we do, though we do know how to prove theorems; with the implication that he was prepared to demonstrate this principle on a live mathematician. As I write now I’m tempted to say what I didn’t find the opportunity to say during the class, namely that the kind of truth philosophers are seeking in mathematics may be a chimera. What, apart from mathematics, looks like truth to philosophers?3
I punted on parts (a) and (b), in different ways. I’m not called upon to have a position on the Four Color Theorem, which is too remote from my mathematical interests, but I do understand in a very general sense how the proof is supposed to work. I didn’t say this in class but I have no objection to checking huge numbers of cases by machine. It is generally accepted that Harald Helfgott proved the Ternary Goldbach Conjecture — every odd number greater than 5 can be written as the sum of three primes — for numbers that are very large numbers, but not so large that the remaining cases can’t be checked by existing machines;4 and I haven’t seen any objection to this last mechanical step.
Part (b) is much more subtle. I can only read it as a question of how I would react to an ordinary mathematical proof of either the Continuum Hypothesis or its negation. My first reaction was that I would assume it was based on a different set theory than the one used to prove the independence result. But that was to fall into the trap: I had to explain why my answer wasn’t an (inadvertent) acknowledgment of the Central Dogma! The option of incommensurability is out of bounds, because I don’t have a general framework in which it fits naturally. But I don’t like the alternative either, because it presupposes that there is one formal system that does justice to our informal practice, and why would I suppose that? I was not especially proud of my answer, which is that CH is a special case because it is a statement about ZFC; JCD was too polite to retort that he could just as well have asked about the undecidability of Hilbert’s 10th problem, which is (at least superficially) a question about number theory.
Having had a few days to ponder the question I’m now thinking that the best on-the-spot response would have been to extricate the hypothetical contradiction from Glaucon’s “eternally existent” tense and to reinsert it into ordinary mathematical practice. In the latter context “you” — that is “we,” the community of ordinary mathematicians — would look for the mistake. In my advance copy of (New Yorker author) Alec Wilkinson’s A Divine Language, his niece Amie Wilkinson remarks
I can’t tell you how many times Benson5 and I have said to each other over the years, ‘I found a contradiction in math.’
Uncle Alec is puzzled until Amie explains that there aren’t really such things;
It’s what you say when you don’t get the answer you expect.
All it means is that they have reasoned their way to two incompatible statements, one of which is inevitably discarded after they detect the flaw.6
I write “inevitably” because up to now it has always7 happened, but even mathematicians know that just because up to now the sun has risen every morning etc. etc. All mathematics has to offer philosophy on this account is André Weil’s quip:
We know that God exists because mathematics is consistent and we know that the devil exists because we cannot prove the consistency.
And I don’t believe philosophers will ever escape the cycle of “back-and-forth, rethinking assumptions, frustrations, and headaches” in the search for anything better than an empirical answer to question (1)(b). This, however, is where I felt most acutely the (mathematician’s) frustration of not being able to conclude my answer with a decisive QED, achieving what Weil called the “knowledge and indifference” promised by the Bhagavad Gita.
Weil also provides one possible answer to a question that arose during the discussion. One of JCD’s students asked why there are no logicians in Columbia’s mathematics department. Haim Gaifman, the philosophy department’s logician, supposedly told the student that this is because mathematicians see logicians as arrogant, presuming to tell the rest of the department what the subject is really about and how it should be practiced. I didn’t buy this explanation8 and said that Columbia’s relatively small department naturally wanted to expand along its existing strengths. I also suggested that both the philosophy and the mathematics departments could benefit from more consistent interaction around logic; though I mentioned my disappointment when so few of my colleagues attended an excellent Colloquium talk by Hugh Woodin that I had arranged, and worried that so much was already going in inside my department that my colleagues would mainly not be interested in looking to the rest of the university. Weil’s opinion on the subject may explain why logic is of limited interest to most mathematicians:
…si la logique est l’hygiène du mathématicien, ce n’est pas elle qui lui fournit sa
nourriture; le pain quotidien dont il vit, ce sont les grands problèmes.9
Agreement
Here is the rest of JCD’s list of questions.
(2) What explains agreement among mathematicians over whether a proof is valid? Certainly, the answer is not – contra formalists – that, if the above biconditional is true, then this is checkable by a machine, since agreement long predates formalization. Is (part of?) the answer that mathematics is primarily about learning how (like chess) rather than learning that (like history)?
(3) What do you make of heretics in the foundations of mathematics? Do they show that the difference between math and philosophy is just a matter of degree not kind(170)? (There is way more disagreement in ordinary philosophy, but the kind that exists there is the same as the kind that exists vis a vis the Axiom of Choice, impredicative definitions, classical logic, and so forth.)
(4) What is mathematical understanding? How does it relate to physical and philosophical understanding, where narrative seems less important? You suggest that mathematics “is the narrative” (146). But prima facie something external – ‘logic’ – constrains it. Is logic not mathematics?
(5) What is the proper role (if any!) of philosophy of mathematics? Must it be useful to mathematicians in order to be justifiable (compare: must pure mathematics be applicable to be justifiable)? Does the same go for philosophy of physics, biology, art, and so forth?
(6) What do you think that philosophers (qua philosophers) could learn by studying the work of recent leaders in core mathematics (not logic or set theory), like Atiyah, Freyd, Grothendieck, Gromov, Lawvere, Langlands, Lax, Thurston, Connes, Kontsevich, Serre, Simpson, and Zilber?
In class we had time to touch on questions (4) and (2), and only briefly. My immediate answer to (4) was that this is a question that mathematicians should be raising with philosophers, not the other way around; the implication is that mathematicians and philosophers should be spending more time talking to each other, and not mainly (or at all) about set-theoretic or other axiomatic foundations. The only way I could see to begin to answer question (4) is to see how mathematicians use the word “understand” and its cognates, and to apply the method of “close reading,” as in my close reading of the Thomason-Trobaugh proof. Here the problem, for philosophers as well as sociologists or historians, is that it takes so long to acquire the familiarity with the language and concerns of working mathematicians needed for a “close reading” that most philosophers of mathematics limit their attention to the logic they have already learned as part of their professional training.10
The Central Dogma presupposes that agreement about the truth of a mathematical proof, as in JCD’s question (2), must ultimately point to the existence of a corresponding formal proof. Both the Azzouni and Avigad articles quoted in the last installment admit the Central Dogma and offer different answers to question (2), given that in practice agreement is expressed in connection with what they call an “informal proof” rather than its hypothetical formal expression.
JCD’s formulation points to a different source of agreement. It was at this point in the discussion that I suggested that the attempt to account for the (continuous) phenomenon of change in the (discrete) medium of language is the driving paradox of western philosophy; I pointed out that it took mathematics 2000 years to find a solution acceptable to mathematicians and physical scientists but not necessarily to philosophers. This was said in connection with my claim that in practice mathematicians’ agreement is a process, even an unending process. It’s only the formalizers, their philosophical enablers, and other devotees of Thanatos, who insist on seeing a time stamp before acknowledging the validity of a claimed proof.
Later I sent JCD a quotation from Pascal that expresses my conviction that mathematics, for the mathematician, is a way of being human:11
Il faut acquérir une créance plus facile qui est celle de l’habitude qui, sans violence, sans art, sans argument, nous fait croire les choses et incline toutes nos puissances à cette croyance, en sorte que notre âme y tombe naturellement. Quand on ne croit que par la force de la conviction et que l’automate est incliné à croire le contraire, ce n’est pas assez.12
This is roughly in line with JCD’s contrast between “learning how” and “learning that.” JCD’s home page announces that next fall he will be teaching a graduate course jointly with Haim Gaifman entitled What is Mathematics? Before answering the question one will need to decide what kind of thing mathematics is. Familiar options include “art,” “science,” “technique,” “discipline,” “profession,” “vocation,” “tradition-based practice.” Will “a way of being human” be one of the course’s options?
The discussion of JCD’s question (2) — how to explain agreement among mathematicians absent reference to the possibility of formalization, without reducing it to a matter of aesthetic preference — will have to continue in the future. The answer that it is a “process” that takes place through “dialogue” is absolutely accurate but leaves too much to the imagination to be very helpful. Although the Pascal quotation can be read as behaviorist or even ethological, it can provide a helpful starting point. In the meantime, comments are welcome on this page.
https://www.quantamagazine.org/why-the-proof-of-fermats-last-theorem-doesnt-need-to-be-enhanced-20190603/
Higher algebraic K-theory of schemes and of derived categories, in P. Cartier et
al., eds, The Grothendieck Festschrift Volume III, Boston: Birkhâuser (1990) 247-429.
If we mathematicians really do hold the monopoly on truth then maybe we should form a cartel to improve our negotiating position.
It is taking a long time to appear in print, but not for any metaphysical reasons.
Husband Benson Farb, like Amie a professor of mathematics at the University of Chicago. The quotation is from p. 135.
Whole research programs can arise from the refocusing this requires. More modestly, they can lead to unexpected solutions of outstanding conjectures, as in this example.
Exception made for Mochizuki’s purported proof of the abc conjecture, of course.
My account of Gaifman’s claim is third hand and not to be taken literally. But I did mention that in my former department at the Université Paris 7 the logicians were viewed with suspicion, like a Trotskyist cell always aiming to expand its power. In fact, many of the logicians were supposedly ex-Trotskyists, while many of the non-logicians were ex-Maoists, which was moderately entertaining for someone who arrived in Paris so many years after 1968. The real problem was that the higher authorities in France had created very few research groups for logic, which created strong pressure on the Paris 7 department to hire the department’s own new logic Ph.D’s, and this in turn fueled resentment on the part of the non-logicians.
This is quoted in A. R. D. Mathias’s “The Ignorance of Bourbaki,” a logician’s attempt to formulate an answer to the student’s question rather different from Gaifman’s.
This discussion overlapped with a response to a student’s question about why so much philosophy of mathematics is concerned with logic. I cited the philosopher David Corfield as a notable exception, and mentioned that he knows much more topology than I do, for example. But he really is an exception, possibly the only contemporary philosopher with such a vast knowledge of mathematics.
Updated September 10, 2022: When I met Justin E. H. Smith in Paris last month he mentioned that American philosophers, as far as he knew, were trained in philosophy of mathematics by reading the standard collection of essays edited by Benacerraf and Putnam. This is not surprising but it does help to explain why so many professional philosophers, whether or not they specialize in mathematics, adhere to the Central Dogma.
In fact I used the Wittgensteinian expression “form of life.” But “way of being human” is better when “way of being a machine” is presented as a viable alternative.
Pensees, original text here. Note that the automaton in this quotation is the human being, the soul (âme), shaped by habit, and contrasted with the thinking human whose belief is shaped by the “force of conviction,” presumably instilled by “argument.” This raises the intriguing possibility that mechanical automata can be more human in this specific sense than human mathematicians.