This week I read an extremely interesting paper by Kenny Easwaran, ‘Probabilistic proofs and transferability’, which appeared in

*Philosophia Mathematica*in 2009. Kenny had heard me speak at the Formal Epistemology Workshop in Munich a few weeks ago, and thought (correctly!) that there were interesting connections between the concept of transferability that he develops in the paper and my ‘built-in opponent’ conception of logic and deductive proofs; so he kindly drew my attention to his paper. Because I believe Kenny is really on to something deep about mathematics in his paper, I thought it would be a good idea to elaborate a bit on these connections in a blog post, hoping that it will be of interest to a number of people besides the two of us!

Drawing on previous work by Don Fallis, Kenny’s paper addresses the issue of why probabilistic proofs are for the most part not regarded as ‘real proofs’ by mathematicians, even though some of them can be said to have a higher degree of certainty than very long traditional proofs (given that in very long proofs, the probability of a mistake being made somewhere is non-negligible). He discusses in particular the Miller-Rabin primality test. (I recall having heard Michael Rabin speaking on this twice some years ago, and recall being both extremely impressed and extremely puzzled by what he was saying!) But for the full story, you’ll just have to read Kenny’s paper, as here I will focus on his concept of transferability so as to compare it to the built-in opponent conception of proofs that I am currently developing.

Kenny’s main claim is that, even if they offer a very high degree of epistemic certainty (in some sense), these probabilistic proofs lack the feature of

*transferability*, and this is why they are not accepted as ‘proper proofs’ by most mathematicians. Thus, he proposes transferability as the ultimate conceptual core of the conception of proofs that mathematicians

*de facto*entertain. Here is how he presents the concept of transferability:

… the basic idea is that a proof must be such that a relevant expert will become convinced of the truth of the conclusion of the proof just by consideration of each of the steps in the proof. With non-transferable proofs, something extra beyond just the steps in the proof is needed—in the case of probabilistic proofs, this extra component is a knowledge of the process by which the proof was generated, and in particular that the supposedly random steps really were random.As he soon adds, “transferability is a social epistemic virtue, rather than an individual one.” A transferable proof is one which can be checked by any expert mathematician (possibly within a given mathematical subfield). Importantly, when checking a proof, a mathematician adopts what can be described as an adversarial attitude towards the author of the proof: she will scrutinize every step looking for loops in the argumentation, in particular counterexamples to specific inferential steps. Once she runs through the proof and finds no fault in it, she is persuaded of the truth of the conclusion if she has granted the premises. Thus, on this conception, a proof is a

*public discourse*aimed at persuasion; this also explains why mathematicians prefer proofs that are not only correct, but which are also explanatory: their persuasive effect is greater.

As Kenny correctly noted, his notion of transferability is very closely related to my ‘built-in opponent’ (BIO) hypothesis. I recall having mentioned BIO in blog posts before (and see here for a draft paper), but here is a recap: I rely on the historical development of the deductive method (as documented in e.g. Netz’s

*The Shaping of Deduction*) to argue that a deductive argument is originally a discourse aimed at compelling the audience to accept (the truth of) the conclusion, if they accept (the truth of) the premises. It is only in the modern period, in particular with Descartes and Kant, that logic became predominantly associated with inner thinking processes rather than with public situations of dialogical interaction.

Crucially, deductive proofs would correspond to a specific kind of dialogues, namely

*adversarial*dialogues of a very special kind, as the participants have opposite goals: proponent seeks to establish the conclusion; opponent seeks to block the establishment of the conclusion. But deductive proofs are no longer dialogues properly speaking, as they do not correspond to actual dialogical interactions between two or more active participants. In effect, the two main transformations leading from the actual dialogues of the early Academy (which provided the historical background for the emergence of the notion of a deductive argument) to deductive proofs seem to be the move from oral to written contexts, and the fact that the deductive method has

*internalized*the opponent in the sense that it is now built into the framework: every inferential step must be immune to counterexamples, i.e. it must be

*indefeasible*. I refer to this conception as the built-in opponent conception of proofs because the original role of opponent (checking whether the dialogical moves made by proponent are indefeasible) is now played by the method itself, so to speak: the method has become the idealized opponent. Another way of formulating the same point is to say that what started out as a strategic desideratum (the formulation of indefeasible arguments) then became a constitutive feature of the method as such.

It should be clear by now how closely related Kenny’s notion of transferability and my BIO conception are. For starters, we both emphasize the social nature of a deductive proof as a discourse aimed at persuasion, which must thus be ‘transferable’. Indeed, Kenny discusses at length the fact that, in mathematics, testimony is not a legitimate source of information/knowledge (contrasting with how widely testimony is relied upon for practical purposes and even in other scientific domains*): the mathematician “will want to convince herself of everything and avoid trusting testimony.” I believe that the key point to understand the absence of testimony in mathematics is the adversarial nature of the dialogues having given rise to the deductive method: your opponent in such a dialogical interaction is by definition not trustworthy. However, in a probabilistic proof, she who surveys the proof must trust that the author of the proof did not cherry-pick the witnesses, which is at odds with the idea of mathematical proofs as corresponding to adversarial dialogues.

Moreover, the dialogical model explains why, in a mathematical proof, one is allowed to use only information that is explicitly accepted. In Kenny’s terms:

Papers will rely only on premises that the competent reader can be assumed to antecedently believe, and only make inferences that the competent reader would be expected to accept on her own consideration. If every proof is published in a transferable form, then the arguments for any conclusion are always publicly available for the community to check.This is because the premises in a mathematical proof are the propositions that all participants in the dialogue in question (proponent, opponent and audience) have explicitly granted: no recourse to external, contentious information is allowed.

But the upshot is that, while ultimately based on an adversarial dialogical model, it is precisely the constant self-monitoring made possible by the transferability of proofs that makes mathematics a social, collective enterprise as well as an astonishingly fruitful field of inquiry: it allows for a sort of cooperative division of labor (distributed cognition!). Recall when Edward Nelson announced he had a proof of the inconsistency of PA last year: the mathematical community immediately joined forces to scrutinize his (purported) proof, thus adopting the adversarial role of opponent (in my terminology). Before long, Terry Tao found a loop in the proof (see comments in this post, where Tao describes his train of thought); and once he was convinced of the cogency of Tao’s argument, Nelson gracefully withdrew his claim.

I also want to suggest that the social nature of mathematics makes Bayesianism, originally an individualistic framework, ultimately unsuitable to deal with the epistemology of mathematics. But as this post is already much too long, I will not further develop this idea for now. Indeed, these are just some preliminary reflections on the connections between the concepts of transferability and of a ‘built-in opponent’ in a deductive proof; I hope to give all this much more thought in the coming months, but for now I’d be curious to hear what others may have to say on all this.

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* I suspect that transferability in mathematics does have a counterpart in the empirical sciences, namely replication of experimental results, but this will remain a topic for another post.

Thanks for the discussion of the paper! I just want to point out that I accept that all mathematicians do in fact come to know lots of mathematics merely on the basis of testimony. The role of transferability is in making sure that this is not essential for any specific result that is accepted into the body of mathematics.

ReplyDeleteI think I make some minor points near the end of the paper trying to apply this idea to philosophy, to explain what appears to be going wrong in experimental philosophy - they are treating intuitions as evidence for a claim, rather than just as giving premises that the reader happens to accept. I suspect that in philosophy you get more preservation of the adversarial idea, but the same goal of transferability applies.

Hi Kenny, thanks! Yes, there were lots of interesting points in your paper that I had to leave out of the post, which was already too long. I did like a lot your brief discussion on philosophical methodology at the end, for example, but I just couldn't fit it in. Actually, one thing I'd really like to hear your thoughts on is the brief suggestion at the end of my post, whether Bayesianism is a suitable framework for the epistemology of mathematics. It always struck me that Bayesianism is a very individualistic framework, but maybe I'm oversimplifying things here.

DeleteThere are proofs based on testimony. For example, a mathematician may not attempted to oppose the proposed argument for the classification of finite simple groups. Based on testimony, he may still use this classification in his own argument for another result. If conscientious, then he may write "if the classification of finite simple group holds, then my result holds."

ReplyDeleteThis use of testimony happens quite prevalently at extents not as extreme as the above. For example, someone who studies Riemann surfaces may use the topological classification by genus without prior attempt to oppose the topological classification. This happens as a mathematician who studied Riemann surfaces is versed in "analysis" or "algebra" but may not have the ability to intensely find counter examples to the classification by genus, which is in another branch of mathematics, namely "topology".