Kevin Buzzard and Grothendieck's metaphysics of identity
Kevin Buzzard and Grothendieck's metaphysics of identity
Why you can't step in the same ring of functions twice
Throughout history, identity thinking has been something deathly, something that devours everything. Identity is always virtually out for totality… (Adorno, The Jargon of Authenticity)
Kevin Buzzard’s May 24 presentation at Chapman University1, entitled “Grothendieck’s approach to equality,” at a conference “honoring and exploring the contributions of Alexander Grothendieck to the field of Mathematics,” argues that the “MULTIFARIOUS GIANT” of the conference’s title falls short of the minimum requirements for axiomatization when it comes to his treatment of identity — the word I will use for the more abstract notion, of which equality as used by mathematicians is a special case.
With this, Buzzard joins the long list of distinguished philosophers who have identified but failed to resolve the problem of what one of Heidegger’s translators called “Identity and Difference.” The pre-Socratics found the persistence of identity in the face of incessant change so baffling that they proposed increasingly radical solutions to the problem, culminating in Parmenides’s denial of the reality of change. Here we recognize a precocious surrender to Thanatos, the computer scientist’s friend. A chunk of code had better be as unchangeable as the platonic idea of a toaster, and as lifeless; otherwise the program that contains it won’t be fit for purpose.
Failing to resolve outstanding problems is what I might have called the “Principle of Identity” of the philosophical vocation. From this perspective, Buzzard’s effort is no less honorable than any of the other attempts of (at least) the past two and a half millenia. Identity, indeed, has been the basso ostinato of most of western metaphysics’s greatest hits. And not only western, as I recalled in the brief review of the topic in Chapter 7 of Mathematics without Apologies, where I attempted to situate Voevodsky’s homotopy type theory in a plausible historical context:
The question of identity would seem to have been settled long ago in mathematics by the adoption of the "=" sign as a standard item in the lexicon used to construct meaningful mathematical propositions. But there is a rich philosophical literature on this question. Leibniz's principle of the identity of indiscernables states roughly that A = B if everything that is true of (or can be predicated of) A is true of B and vice versa — if A and B have the same attributes in any sense that can be given to this term. Cantor's Absolute, impervious to meaningful predication, resembles the God of the church fathers on the grounds of Leibniz's principle…
By insisting that "The essence of entities is not present in the conditions, etc." and that "Nirvāṇa is uncompounded. Both existents and nonexistents are compounded," the MMK2 excluded Nirvāṇa from Leibniz's world system. Nāgārjuna's commentator Candrakirti considered "Nirvāṇa ... cognitive nonsense... a 'scandal' to logic. In order for Nirvāṇa to be cognitively affirmed or denied, it must be reduced to Saṃsāra, to existence in causes and conditions." Nāgārjuna's equation of Saṃsāra and Nirvāṇa has been translated as the identity of "the Phenomenal" and the "Absolute."
Identity in Leibniz's sense is the property captured by the = sign in set theory. But since A is called A and B is called B this does not quite suffice to unravel A = B. In his 1892 article Über Sinn und Bedeutung, Frege introduced the distinction between sense and reference, giving the example of the morning star and the evening star, both of which turn out upon inspection to be the planet Venus: two senses for a single reference. Frege's attempt to solve the problem of identity remains influential among philosophers but is not nearly subtle enough to account for the gradations of identity that inevitably arise in mathematical practice.
Heidegger, meanwhile, was able to spin the tautological equation A = A — which we were just claiming was unproblematic for set theory — into an essay on fundamental ontology, starting with the claim that the proper formulation of "A = A" is that "every A is itself the same with itself."3 Heidegger's doubts about the transparency of identity were anticipated (by nearly 2000 years) by Nāgārjuna, who wrote that "I do not think those who teach the identity or difference of self and things are wise in the meaning of the teaching."
Difficulties arise when philosophers get going on personal identity, as evidenced in the “transcendental unity of apperception,” not to mention in identity politics, but you already get the idea. Identity is complicated. Mathematics produces models of identity that are either (like Parmenides’s and Frege’s) too brittle to be convenient for the purposes of mathematics or (like Nāgārjuna’s and Heidegger’s) too waffly to satisfy the robots.
Kevin Buzzard’s slides show that identity in Grothendieck’s work fails to escape this dilemma. While one might suppose Buzzard leans toward the brittle pole, his investigation of Grothendieck’s approach expresses his discomfort with both alternatives. His specific objection is that Grothendieck’s
use of equality does not conform to the set-theoretic language.… I discovered this fact the hard way – when trying to apply the principle of substitution to two rings which Grothendieck was claiming were equal, in Lean, a computer theorem prover.
In set theory two sets are equal if and only if they have the same elements. (Don’t try to apply Leibniz’s identity of indiscernables to this version of equality unless your cupboard is fully stocked with logical quantifiers: you will vanish into a vortex of circularity at your first attempt at predication. See below, however.) Buzzard exhibits two sets with different elements, each of which claims to be the ring of functions on an open subset of an affine scheme.
a set theorist would say these rings were not equal. But they are isomorphic.
This looks like it is a serious logical issue. Grothendieck’s “definition” is not well-defined! … this formally breaks the principle of substitution.
Buzzard explains the “substitution property” in terms reminiscent of Leibniz’s principle: A = B provided
if P is any statement about mathematical objects such that P(A) is true, then P(B) must also be true.
I haven’t seen this terminology before4 but I’ll grant that, appropriately qualified (allowing first-order vs. second order quantification, for example), it looks5 faithful to the articulation of set theory with predicate logic.6 Leaving these details aside, Buzzard explains how mathematicians contend with the subject’s inherent propensity to generate paradox:
By restricting what we allow as a valid statement about the real numbers, we can have more leeway in how we treat equality, without violating the principle of substitution.
For example, we can assume that the Cauchy reals and the Dedekind reals are equal.
and how this creates problems for those who wish to make mathematics comprehensible to machines:
“unwritten conventions” … obvious to every algebraic geometer… are subtle to explain to a computer theorem prover written in set theory, simple type theory, or dependent type theory.
Buzzard’s Exhibit A is the word “canonically” in section 1.3 of EGA 1, written by Grothendieck (and Dieudonné).7 His presentation is thoughtful and illuminating but I think it demonstrates the opposite of what I assume he intended. Here, for example, is his critique of the use of the word “canonical,” which is ubiquitous in mathematical exposition:
In my opinion, we have now degenerated into waffle. You cannot type this definition into a computer theorem prover.
My spontaneous and sincere reaction is: tough luck for the computer! It’s hardly controversial to say that Grothendieck’s perspective has been immensely fruitful and transformative. If this perspective doesn’t fit snugly in an a posteriori program, or with the Central Dogma, so much the worse for the program and the Dogma.
The quotation above is Buzzard’s reaction to the Wikipedia page on “Canonical Maps,” which defines the offending term as follows:
In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. In general, it is the map which preserves the widest amount of structure, and it tends to be unique. In the rare cases where latitude in choices remains, the map is either conventionally agreed upon to be the most useful for further analysis, or sometimes the most elegant map known to date.
Buzzard is absolutely right to observe that what is happening on the Wikipedia page is more a failure to define than a legitimate definition, but again, I draw a very different conclusion from this observation. In the first place, it shouldn’t need to be said that while Wikipedia is often useful8, and can often serve as a starting point for scholarly inquiry, it has absolutely no authority over mathematical practice. But leaving Wikipedia aside, you are setting yourself up to be disappointed if you are combing the web in search of the canonical definition of “canonical” as used by mathematicians.
This, I submit, is symptomatic of the so-called mathematical community’s relaxed attitude to internal standards. Although it is common to invoke a hypothetical collective superego as a source of accepted mathematical practice, and although the mathematical community would be able to claim the authority that Wikipedia lacks, if it could speak with a common voice, which it can’t, in reality nothing is wafflier than this community from the standpoint of setting standards.
It’s telling that very little has been written about the phenomenon. Twelve years ago Geist, Löwe, and Van Kerkhove pointed out9, sensibly, that
Given that mathematical correctness of a paper is so important for the decision of whether a paper should be published or not, it might come as a surprise that there have been no studies of the mathematical refereeing process.
A few years after those lines were published I was contacted by sociologist Christian Greiffenhagen, who wished to carry out just such a study.
The main challenge has been to think of a ‘perspicuous setting’, i.e., one where reception becomes visible. Peer review might be such a setting and I thought that I could conduct qualitative interviews with referees just days after they had written a report about the things that they did in coming to a judgement about the refereed paper (Michele Lamont, in How Professors Think… did something very similar for fellowship proposals in the humanities and social sciences). …
do you know whether any of the professional organisations (e.g., AMS) have conducted an overview of mathematics journals’ refereeing practices? Would you know anyone at the AMS whom I should contact about this?
Having more than once seethed at an editorial rejection based on the report of a referee I considered poorly informed, I had an emotional investment in demystifying the process. But Greiffenhagen and I were stymied at the time10 by editors’ insistence on protecting any information that might compromise the anonymity of referees.
Not wanting to leave the sociologist empty-handed, I supplied a few (thoroughly anonymized) snippets from quick opinions I had read as an editor. If you’re looking for a scandalous dependence on a self-perpetuating caste of experts at the dark heart of the mathematical enterprise, you’ll have richer pickings here than in any of the shady dealings that upset the formalizers. Here are a few excerpts from four reports, ranging from clearly negative to unconditionally positive:
…the ideas are not new… a substantial application… is not presented… none of the above mentioned results is really interesting with a slight exception… I do not think that the paper under review meets the novelty and high quality required…
I found the emphasis, and some of the content, of the paper to be misplaced and rather peculiar… I think a shorter and better focused version… might be publishable…
All in all this paper is quite interesting and there are not so many references implementing the… For that reason the paper might well be recommended for publication… Although this is not strictly required, the ideal thing would be a proof of…
the point of this manuscript is to solve a problem… posed… some fifteen years ago… the mathematical import … is clear… and this is the simple reason why I would definitely recommend it.
These are aesthetic judgments in Kant’s sense, with a pretense to “universal satisfaction”: the referee “must believe that he [or she, M.H.] has reason for attributing a similar satisfaction to everyone.”11 For the moment, no one proposes to formalize them.
Buzzard states his conclusion in the penultimate sentence of his presentation:
Axiomatically modelling Grothendieck’s concept of equality seems to me to be an unsolved problem.
I suspect that logicians would argue that in fact there are several solutions; for that matter Buzzard writes down two of them: the one chosen by the proverbial working mathematician
These unwritten conventions … have a definition of equality which is weaker than Grothendieck’s.
As a result, such systems have to work hard to apply Grothendieck’s substitution principle; the inbuilt one is too weak.
and the one proposed by Voevodsky:
Homotopy type theory forgets about “canonical” and decrees that all isomorphisms are equalities.…
However … things can be equal in more than one way; whatever a “canonical” isomorphism really is, homotopy theory type theory also allows equality corresponding to “noncanonical isomorphisms” …
So this approach has gone too far! It also does not capture what Grothendieck wanted.
The unsolved problem, it seems to me, is to formulate a precise characterization of the missing Goldilocks axiomatic model that would make both Grothendieck and Buzzard’s robots happy. Is this a mathematical, philosophical, technical, sociological, or theological problem? Buzzard’s slides don’t contain enough information to answer the question.
You might think the problem of identity is central in mathematics, since so much of mathematical history is driven by discovering that A is also B. The classic instance is Poincaré’s realization on entering the omnibus that
the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry.
(The word in Poincaré’s French is identique.)
It goes without saying that this kind of equality is far less amenable to axiomatization than the set-theoretic version, whose axioms didn’t even exist when Poincaré had his revelation.12 I wrote in Quanta that the obsession with formalizing Wiles’s proof of Fermat’s Last Theorem in misguided because it is moot — there are too many versions of the proof; why choose one rather than another? There I was presuming to declare that the different versions are identical though their manifest contents are very different. What could that possibly mean? What, for that matter, could it possibly mean for Freudenthal, van der Waerden, and Weil to claim that Euclid was doing algebra more than 1000 years before Al-Khwârizmî? Application of Leibniz’s identity of indiscernables to such claims, or even to Poincaré’s sudden illumination, is hopeless. Why, then, can’t mathematicians help using the language of identity in these situations?
I’ll conclude today’s episode with an anecdote from the recent history of number theory. In 1970 and 1971 R. M. Damerell published two articles with the title “L-functions of elliptic curves with complex multiplication” in Acta Arithmetica. Damerell’s theorem, as his results have come to be known, asserts that certain linear combinations of values of Eisenstein series at special points in the upper half plane are equal to certain special values of Dirichlet series (including the “L-functions” of the title). Equality in this case is only moderately problematic: both quantities are the sums of infinite convergent series,13 and the proof proceeds roughly by identifying the index sets on the two sides of the equality in such a way that the corresponding terms are equal. I guess Lean can handle equality at this level.
Although Damerell switched fields to combinatorics after writing these papers, his theorem became famous among number theorists a few years after their publication, because it was the logical starting point for a series of papers on the Birch-Swinnerton-Dyer Conjecture, notably in the 1977 article of Coates and Wiles. Its generalization by Shimura was one of the inspirations for Deligne’s Conjecture on critical values of L-functions and for Katz’s construction of p-adic L-functions of Hecke characters. So it’s fair to say that this particular equality had a marked influence on the development of number theory in the last quarter of the last century.
I have nothing more to say about this (relatively) unproblematic instance of equality. However, in the late 1980s I began wondering about what Damerell’s theorem “really is.”14 To make sense of such a question one must rely on another unformalizable form of identity. Damerell’s theorem is a formula that can be understood in its own terms; as such it made a crucial contribution at a certain stage of the program to solve the Birch-Swinnerton-Dyer Conjecture. I suspect that Damerell’s name is familiar to number theorists, who are generally unaware of his subsequent career as a combinatorialist, for this reason. Special expressions for L-functions tend not to appear in isolation, however. As soon as I asked myself what Damerell’s theorem “really is” I immediately recognized it as a variant of the special case, for the group U(1), of the so-called standard integral representation of L-functions, discovered independently by Garrett and, more systematically, by Piatetski-Shapiro and Rallis.
This alternate perspective was indispensable for a series of papers, about unitary groups of different sizes, that I wrote about 30 years ago, all of which were based on an inductive procedure whose initial step was the new interpretation of Damerell’s theorem, as well as for my more recent generalization with Eischen, Li, and Skinner of Katz’s construction. A related but quite different perspective sees part of Damerell’s theorem as a special case of the Ichino-Ikeda conjecture, and thus as a founding moment of the relative Langlands program which is now one of the most active areas of research in number theory.
Seeing a mathematical statement as a special case of some other statement has consequences. It means in practice that one can do things with the first statement — build an inductive procedure around it, for example, as in the case I just mentioned — that one wouldn’t have known how to do without the broader perspective. Seeing the first statement as really a special case of something else — an avatar, in the sense of Chapter 7 of Mathematics without Apologies — has metaphysical consequences. Much of that Chapter 7 was devoted to how the language of avatars entered mathematical use under Grothendieck’s influence. I wrote, however, that
Attempts at greater precision lead straight to the threshold of the abyss of speculative philosophy, where one seeks to explain what it means to take phenomena — as symptomatic of something that remains concealed.
The word réalité appears on 176 of the 929 pages of the pdf version of Grothendieck’s Récoltes et Sémailles. It’s fair to say that Grothendieck’s mathematical obsession was with the réalité des choses, an expression that recurs in his (now-published) manuscript. The “secret identity” he hoped to reveal continues to drive research in all the branches of mathematics with which I am familiar. But it has nothing to do with, and there’s no reason to think it would benefit from, the formalization of the metaphysics of identity.
Whose stated mission is “to provide personalized education of distinction that leads to inquiring, ethical and productive lives as global citizens” but whose home page also points out that it is “15 miles from the beach.”
Mūlamadhyamakakārikā (Fundamental Stanzas on the Middle Way) by Nāgārjuna.
A mathematician would go on to say that A can be identical with itself in many different ways; see Buzzard’s remarks on Voevodsky’s homotopy type theory in what follows.
Frege’s distinction between Sinn and Bedeutung is in part an attempt to resolve puzzles involving the breakdown of a “Principle of Substitution” in logical formulas, but the translation of this into set theory is not straightforward, as far as I can tell.
But I am willing to be corrected.
Buzzard acknowledges that what counts as a “statement about mathematical objects” depends on axiomatic foundations. But this is not really at issue in the example he presents.
In particular, while hiss focus on Grothendieck is to be expected in a conference on the influence of the MULTIFARIOUS GIANT, Buzzard makes no mention of what is arguably Grothendieck’s most influential intervention in the philosophy of equality, namely his systematic use of Yoneda’s lemma in his approach to representable functors. In this view, an object in a category is determined up to isomorphism — that may or may not be canonical — by its relations (morphisms) to all the other objects in the category. Apparently Lean can handle that and quite a lot more category theory.
It is also often just plain wrong, as Buzzard inadvertently revealed to me in 2006.
“Peer review and knowledge by testimony in mathematics,” in Benedikt Löwe, Thomas Müller (eds.). Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. Research Results of the Scientific Network PhiMSAMP. College Publications, London, 2010. Texts in Philosophy.
“Knowledge by testimony” is what Buzzard elsewhere decries as reliance on experts. As Geist et al. write, “how can the epistemic exception of mathematics survive if some of the proofs rely on pointers to the literature?” How indeed? Epistemology is one of western philosophy’s perennial open questions, along with the problem of identity. The authors shed some light on the “how” question but wisely don’t try to solve it.
Checking back with Greiffenhagen six years later, I’ve learned that he persisted in seeking data about the refereeing process. His persistence has paid off: two of his manuscripts, devoted to the two principal aspects of the process — judging significance (quick opinions) and certifying correctness (close reading) — are currently under review (!) for publication, based on no fewer than 95 interviews with journal editors. The papers account for the process comprehensively and in a way consistent with my experience. Moreover, to my mind they together constitute a valuable corrective to the mindset that argues that mathematics is facing a crisis that only formal verification can alleviate, and I look forward to citing them in this site, once they are in final form. (This decision to delay is consistent with one of Greiffenhagen’s own observations: “papers in the social sciences can change dramatically through the peer review process, while papers in mathematics seem to change very little.”)
From J. H. Bernard’s translation of Kant’s Critique of Judgment, § 6.
Now, of course, we would say the groups PSL(2,R) and SO(2,1) are isomorphic, though not canonically …
…in most cases; there are also borderline cases proved by analytic continuation, but we can ignore this subtlety.
I persist in asking such questions as part of my normal routine, even though I know full well that they represent an unforgivably “Whiggish” deviation from historiological standards, as did the speculations on Euclid’s algebra by Freudenthal et al. And I have never regretted asking what a result “really is.”