Adonis Diaries

Who is towing sciences? (Dec. 6, 2009)

In the previous 2 centuries it was mathematics that was towing sciences and especially physics in its theoretical aspects. Actually, most theories in the sciences were founded on mathematical abstract theorems that were demonstrated many decades ago by mathematicians.

It appears that this century is witnessing a different trend: sciences are offering opportunities for mathematicians to expand their fields of interests, away from the internal problems of solving conjectures (axioms or hypotheses) that were enunciated a century ago.

Cedric Villani, professor of mathematics at the Institute of Henry Poincaré in Paris, thinks that physics still remains the main engine for mathematicians to opening new fields of study.  For example, equations of fluid mechanics are not yet resolved (those related to Navier-Stokes and Euler); compressed fluids; Bose-Einstein condensation; rarefied gas environment. We even cannot explain why water boils.

There is also the study of how the borders that separate two phases of equilibrium among chaotic, random, and unstable physical systems behave.  The mathematician Wendelin Werner (Fields Prize) has been interested in that problem and I will publish a special post on how he resolved these phenomena.

There is an encouraging tendency among a few mathematicians to dealing with new emerging fields in sciences.  The most promising venue is in computer sciences or “informatique”. In computer sciences the problems of verification (for example, theory of verifying proofs) is capturing interest: mathematical tools for validating theorems in the realm of logic or exhuming errors are challenging.

Biology is an exciting field but it didn’t capture the interest of mathematicians: the big illusion that mathematicians will approach biology faded away simply because fields related to the living world is too variable in complexity to attract mathematicians. The number of “variability” is “absurdly” numerous and does not lend simple and clean-cut laws that prove that the world is well structured and “mathematically” ordered.

This has been the case in the 70’s for cognitive sciences and artificial intelligence: scientists in those two fields hoped that mathematicians would get interested and drive them to innovative results, but nothing much happened.

The good trend is this kind of social re-organization within the mathematical community for deeper cooperative undertaking in solving problems.  The web or internet has kind of revolutionized cooperation; it opened up this great highway for sharing ideas instantly and cooperating among several researchers.

For example, Terry Tao and Tim Gowens propose problems on their blog (Polymath project) and then the names of contributors are disseminated after a problem comes to fruition.

Still, individual initiatives are the norm; for example, Tao and Gowens ended up solving the problems most of the time.  Mikhail Gromov (Abel Prize) has given geometry a new life line in mathematics.

It appears that “Significant mathematics” basically decoded how the brain perceives “invariants” in what the senses transmit to it.  I conjecture that since individual experiences are what generate the intuitive concepts, analogies, and various perspectives to viewing a problem. Most of the mathematical theories were essentially founded on the stored vision and auditory perceptions.