Tuesday, 12 March 2013

Terry Tao on rigor in mathematics

(Cross-posted at NewAPPS)

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.


  1. Tao's post seems to date from 2009...

    1. Indeed! I don't know why I had gotten the idea that it was recent. Anyway, thanks for the correction, I updated the text of my post.

  2. Terry Tao's theory of mathematical development seems to parallel theories of expertise development in other domains, e.g. Ericsson (1993, 2006, 2009). The pre-rigorous/rigorous/post-rigorous pattern that Tao describes matches what many neo-Piagetian developmental psychologists in the 1980s and 1990s called preformal/formal/postformal cognition. Basseches (1980, 1984) explicitly connected this dialectical pattern of development to dialectical philosophers such as Hegel, Peirce, Dewey, Popper, and others. For a recent example of this kind of theorizing, see the chapter by Michael F. Mascolo and Kurt W. Fischer, "The dynamic development of thinking, feeling, and acting over the life span", in Lerner et al. (2010), pp. 149–194.


    Eeva Kallio (2011). Integrative thinking is the key: an evaluation of current research into the development of adult thinking. Theory & Psychology, 21(6), 785–801.

    Robert J. Mislevy (2010). Some implications of expertise research for educational assessment. Research Papers in Education, 25(3), 253–270.

    Nona Lyons, ed. (2010). Handbook of reflection and reflective inquiry. New York: Springer.

    Richard M. Lerner, Willis F. Overton, Michael E. Lamb, & Alexandra M. Freund, eds. (2010). The handbook of life-span development. Hoboken, NJ: John Wiley & Sons.

    K. Anders Ericsson, ed. (2009). Development of professional expertise: toward measurement of expert performance and design of optimal learning environments. New York: Cambridge University Press.

    K. Anders Ericsson, Kiruthiga Nandagopal, & Roy W. Roring (2009). Toward a science of exceptional achievement. Annals of the New York Academy of Sciences, 1172, 199–217.

    K. Anders Ericsson, Michael J. Prietula, & Edward T. Cokely (2007). The making of an expert. Harvard Business Review, 85(7-8), 114–121, 193.

    Henk G. Schmidt & Remy M. J. P. Rikers (2007). How expertise develops in medicine: knowledge encapsulation and illness script formation. Medical Education, 41(12), 1133–1139.

    Mark E. Quirk (2006). Intuition and metacognition in medical education: keys to developing expertise. New York: Springer.

    K. Anders Ericsson, ed. (2006). The Cambridge handbook of expertise and expert performance. Cambridge, UK; New York: Cambridge University Press.

    Carol Hren Hoare, ed. (2006). Handbook of adult development and learning. Oxford; New York: Oxford University Press.

    Andreas Demetriou & Athanassios Raftopoulos, eds. (2004). Cognitive developmental change: theories, models, and measurement. New York: Cambridge University Press.

    Jack Demick & Carrie Andreoletti, eds. (2003). Handbook of adult development. New York: Kluwer Academic/Plenum.

    Robert J. Sternberg & Elena L. Grigorenko, eds. (2003). The psychology of abilities, competencies, and expertise. Cambridge, UK; New York: Cambridge University Press.

    Jochen Brandtstädter & Richard M. Lerner, eds. (1999). Action & self-development: theory and research through the life span. Thousand Oaks, CA: Sage.

    Jan D. Sinnott (1998). The development of logic in adulthood: postformal thought and its applications. New York: Plenum Press.

    K. Anders Ericsson, ed. (1996). The road to excellence: the acquisition of expert performance in the arts and sciences, sports, and games. Mahwah, NJ: Lawrence Erlbaum Associates.

    K. Anders Ericsson, Ralf T. Krampe, & Clemens Tesch-Römer (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100(3), 363–406.

    Michael Basseches (1984). Dialectical thinking and adult development. Norwood, NJ: Ablex.

    Michael Basseches (1980). Dialectical schemata: a framework for the empirical study of the development of dialectical thinking. Human Development, 23(6), 400–421.

    1. Thanks, these are excellent pointers!

  3. While I know I've seen much less mathematical logic than you have, it seems to me that a great deal contemporary mathematical logic is done in a post-rigorous state of mind very much like the rest of mathematics; model theory and algebraic logic come to mind as particularly sketchy branches of math, but I've also seen this in proof theory: Solovay's completeness proof for modal-provability logic was presented to me at a pretty high level.

    So I'm not sure modern foundations are done in quite the same spirit as Frege and Hilbert. We seem much more comfortable with the formalism these days. And I think that last sentence hints at an answer to your question: the foundation project was not started as a move forward to a post-post-rigorous stage, but rather a statement that the mathematical community didn't have the rigorous understanding of the recently developed structures and techniques required to work post-rigorously. Indeed, formalizing the notion of foundations was something entirely new.

    On the other hand, the more recent development of univalent foundations may provide a glimpse into a post-post-rigorous stage of mathematics. I'm not sure at the moment how to defend this claim, so perhaps I will come back after giving it some more thought.

    1. I agree with pretty much everything you say, in particular that current work on the foundations of mathematics is just one instantiation of the post-rigorous stage described by Tao. About univalent foundations: it is a *very* exciting research program, and I follow it with interest! Not sure yet where it would fit in this classification, so if you have thoughts on why it should count as post-post rigorous, I'd be happy to hear them.

    2. Here are some thoughts. This is far from a complete or convincing argument though; my observations about type theory and homotopy theory could both be very wrong.

      From a purely type theoretic perspective, it seems the only technical novelty is the univalence axiom. However, this isn't to say there's nothing new for type theorists here: the perspective of homotopy leads a whole new insight into the way to think about types. That is, unvialent foundations provide the type theorist with a new perspective, more than a new set of tools. (The fact that it allows the use of tools from homotopy theory is a tremendous bonus!)

      From the homotopy theoretic perspective, it seems the main technical novelty is more a matter of presentation than mode of thought: when working in a type theory with univalence, the homotopy theorist is still working in a model category. The difference is that now the model category is defined in such a way that a more rigorous presentation is possible.

      If those two observations are true, then the (mathematical) contributions of homotopy type theory are primarily perspective and presentation. Perspective is certainly a "post-rigorous" notion. Presentation is a little more difficult. A change in presentation could be a return to rigor, but that suggestion doesn't seem to fully capture what's happening with HoTT. It seems more to be a step forward towards "more rigor" than a step backwards towards rigor.

      But that's a woefully messy argument I just made. Trying to say something coherent has been one more reminder that I'm a minnow trying to swim with the big fish. :)

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