On Proof and Progress in Mathematics

How do people understand mathematics? How do mathematicians advance human understanding of mathematics? What are some of the principle tools and skills of mathematical thinking? What is mathematics? What is proof? What motivates people to do mathematics? What are the roles of ideas, speculations, intuitive reasoning, and rigor in mathematics? How is it that progress is made in mathematics?

These questions are explored in an extraordinary 17 page essay by 1982 Field's Medal (https://en.wikipedia.org/wiki/Fields_Medal) winner William P. Thurston (https://en.wikipedia.org/wiki/William_Thurston) (1946-2012) "On Proof and Progress in Mathematics" Bull. Amer. Math. Soc. 30 (1994), 161-177.

A careful reading of Thurston's paper is strongly recommended for this discussion so we can discuss its content in detail. Although he mentions a few advanced topics, the essay is quite readable and it stands on its own without prerequisites.

Download and read Thurston's paper "On Proof and Progress in Mathematics" here (http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf).

This discussion will primarily explore the culture and nature of mathematics rather than knowledge in mathematics as we usually do. So it will be more cultural and philosophical than most of our events.

However, since many of our members like to explore technical mathematical questions, we will dedicate some time during the event to discussing the eight characterizations of the derivative of a function that Thurston gives in the paper. In addition, if participants are interested, we can spend a few minutes (but only a few) trying to apprehend a few of the advanced ideas mentioned in the essay including manifolds (https://en.wikipedia.org/wiki/Manifold), foliation theory (https://en.wikipedia.org/wiki/Foliation), Haken manifolds (https://en.wikipedia.org/wiki/Haken_manifold), Kleinian groups (https://en.wikipedia.org/wiki/Kleinian_group), dynamical systems (https://en.wikipedia.org/wiki/Dynamical_system), geometric topology (https://en.wikipedia.org/wiki/Geometric_topology), diffeomorphisms (https://en.wikipedia.org/wiki/Diffeomorphism), geometric group theory (https://en.wikipedia.org/wiki/Geometric_group_theory), Lie groups (https://en.wikipedia.org/wiki/Lie_group), Teichmüller spaces (https://en.wikipedia.org/wiki/Teichm%C3%BCller_space), pseudo-Anosov diffeomorphisms (https://en.wikipedia.org/wiki/Pseudo-Anosov_map), geometric group theory (https://en.wikipedia.org/wiki/Geometric_group_theory), hyperbolic geometry (https://en.wikipedia.org/wiki/Hyperbolic_geometry), low-dimensional topology (https://en.wikipedia.org/wiki/Low-dimensional_topology), and Thurston's geometrization conjecture (https://en.wikipedia.org/wiki/Geometrization_conjecture).

The Context of Thurston's Paper:

This background may be helpful while reading the Thurston essay "On Proof and Progress in Mathematics".

Thurston's essay is ostensibly a response to the essay "'Theoretical mathematics': toward a cultural synthesis of mathematics and theoretical physics" by Arthur Jaffe and Frank Quinn (http://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00413-0/). I do not recommend reading the Jaffe and Quinn article because it is too willing to adjudicate mathematics history making it challenging to assess let alone discuss (largely because I do not know enough about its historical references, its context nor the multifarious results referenced).

Thurston responds mainly to a paragraph in their essay which basically says that while we [Jaffe and Quinn] believe that speculation and intuitive reasoning is important in mathematics, Thurston's geometrization conjecture is an example of how speculation in mathematics can be done badly. They suggest that some believe the geometrization conjecture has been a roadblock to progress in mathematics.

It should be noted that 16 years after Thurston's paper "On Proof and Progress in Mathematics" was written, in 2010, Arthur Jaffe (https://en.wikipedia.org/wiki/Arthur_Jaffe)'s brainchild, the Clay Mathematics Institute (https://en.wikipedia.org/wiki/Clay_Mathematics_Institute), awarded a $1,000,000 prize to Grigori Perelman (https://en.wikipedia.org/wiki/Grigori_Perelman) (who declined it) for proving the Poincaré conjecture (https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture). Perelman resolved that famous problem by proving Thurston's geometrization conjecture which Jaffe and Quinn had criticized. It might be suggested that Perelman's dramatic result suggests that Thurston's approach to his geometrization conjecture was, if anything, not a roadblock to progress in mathematics?