Terry Tao on rigor in mathematics

(Cross-posted at M-Phi)

Fields-medalist Terence Tao (among other feats, he spotted the mistake in Nelson’s purported proof of the inconsistency of arithmetic back in 2011) has a blog post on the meaning of rigor in mathematical practice. He files this post under the heading ‘career advice’, but the post in fact touches upon some key issues in the philosophy of mathematics, such as: What is the role of intuitions for mathematical knowledge? What is the role of formalism and rigor in mathematics? How are ‘formal’ and ‘informal’ mathematics related? 

While Tao’s post is not intended to be a contribution to the philosophy of mathematics as such, and while one may miss some of the depth of the discussions found in the philosophical literature and elsewhere, I find it illuminating to see how a practicing mathematician (and a brilliant one at that) conceptualizes the role of rigor in mathematical practice. (Also, much of what he says fits in nicely with some of the views about formalisms and proofs that I’ve been defending in recent years, as I will argue below -- something that I couldn't let go unnoticed!)

Tao’s take on these matters (at least in the post) is a developmental one. He identifies three phases in the development of mathematical skills upon instruction/education:

•       The ‘pre-rigorous’ stage
•        The ‘rigorous’ stage
•       The ‘post-rigorous’ stage

Ideally, at the end of her mathematical education, say at the end of her graduate studies, a mathematician will have attained the post-rigorous stage. And indeed, as is widely recognized, if one looks at ‘ordinary’ mathematics journals (i.e. not journals specifically for mathematical logic), the proofs contained in the articles are usually very sketchy and not at all ‘rigorous’. The goal seems to be to offer the readers just enough information so that they can reconstruct the proof by themselves if they so wish (something that Kenny Easwaran has described as the ‘transferability’ of mathematical proofs). In other words, Tao is right to point out that most academic mathematicians, i.e. those publishing articles in respected journals, are most certainly not at the ‘rigorous stage’ properly speaking.

However, Tao is not suggesting that at the post-rigorous stage, the rigor of the rigorous stage has been completely abandoned. Instead, he seems to be endorsing a broadly 'Hegelian' picture where a given stage represents a synthesis of the previous ones. In particular, he stresses the role of formality and rigor in order “to avoid many common errors and purge many misconceptions”. “The point of rigor is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.” These observations suggest a conception of formalisms and conventions of rigor in mathematical practice as cognitive scaffolding: external devices which enhance the cognitive performance of the agent in a given task, in particular by offering a corrective to commonly made mistakes. (This is essentially the view on formalisms that I defended in my book Formal Languages in Logic.)

Now, one of the most common mistakes in mathematics, as duly remarked by Frege in the preface of Begriffsschrift, is that of letting presuppositions "sneak in unnoticed":

To prevent anything intuitive from penetrating here unnoticed, I had to bend every effort to keep the chain of inferences free of gaps. … [The] first purpose [of the system presented in the Begriffsschrift], therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed, so that its origin can be investigated. (Frege 1879/1977, 5-6)

However, according to Tao, the post-rigorous stage requires equal emphasis on mathematical intuition – no longer the initial intuitions of the pre-rigorous stage, but the educated intuitions of the mature mathematician, who has thoroughly revisited and revised her pre-rigorous intuitions. It is then the combination of rigor with intuitions that characterizes an accomplished mathematician:

It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture.

So the idea seems to be that formality/rigor and intuition work well together precisely because they compensate for each other’s limitations. Now, according to the dialogical conception of deductive proofs that I have been developing, one might say that intuitions represent proponent’s side of the story – the creative side who formulates and puts forward a hypothesis – whereas rigor and formalisms represent opponent’s side of the story – the corrective side which makes sure that proponent’s ‘confabulations’ stay in check. A significant component of mathematical training would then consist in the process of internalizing this opponent by learning how to comply with standards of mathematical rigor. The role of the two characters, proponent and opponent, must be in balance for mathematical knowledge to come about – again, something like a Hegelian conception of adversariality and synthesis.

To round up, I would like to raise the question of where the research program on the foundations of mathematics – initiated by Frege, continued by Hilbert, and still alive and kicking (albeit by no means the dominant paradigm among professional mathematicians) – fits into Tao’s account, if at all. Does it represent a return to the rigorous stage? Or is it best described as a ‘post-post-rigorous stage’? Does the Frege passage quoted above represent the passage from the pre-rigorous to the rigorous stage, or should it be seen as the passage from the post-rigorous to the post-post-rigorous stage? I’d be curious to hear what others think.