(i) Questioning strictly linear models of evolution:
Progress in mathematics is often portrayed as a strictly linear affair — a process in which old theories or ideas are rendered essentially obsolete, and hence forgotten, as soon as the essential content of those theories or ideas is “suitably extracted/absorbed” and formulated in a more “modern form”, which then becomes known as the “state of the art”. The historical development of mathematics is then envisioned as a sort of towering edifice that is subject to a perpetual appending of higher and higher floors, as new “states of the art” are discovered. On the other hand, it is often overlooked that there is in fact no intrinsic justification for this sort of strictly linear model of evolution. Put another way, there is
no rigorous justification for excluding the possibility that a particular approach to mathematical research that happens to be embraced without doubt by a particular community of mathematicians as the path forward in this sort of strictly linear evolutionary model may in fact be nothing more than a dramatic “wrong turn”, i.e., a sort of unproductive march into a meaningless cul de sac.
Indeed, Grothendieck’s original idea that anabelian geometry could shed light on diophantine geometry [cf. the discussion at the beginning of [IUTchI], §I5] suggests precisely this sort of skepticism concerning the “linear evolutionary model” that arose in the 1960’s to the effect that progress in arithmetic geometry was best understood as a sort of
strictly linear march toward the goal of realizing the theory of motives, i.e., a sort of idealized version of the notion of a Weil cohomology.
In more recent years, another major “linear evolutionary model” that has arisen, partly as a result of the influence of the work of Wiles [cf. [Wiles]] concerning Galois representations, asserts that progress in arithmetic geometry is best understood as a sort of
strictly linear march toward the goal of realizing the representation-theoretic approach to arithmetic geometry constituted by the Langlands program.
(ii) Examples of atavistic development:
An alternative point of view to the sort of strictly linear evolutionary model discussed in (i) is the point of view that progress in mathematics is best understood as a much more complicated family tree, i.e., not as a tree that consists solely of a single trunk without branches that continues to grow upward in a strictly linear manner, but rather as
a much more complicated organism, whose growth is sustained by an intricate mechanism of interaction among a vast multitude of branches, some of which sprout not from branches of relatively recent vintage, but rather from much older, more ancestral branches of the organism that were entirely irrelevant to the recent growth of the organism.
(iii) Escaping from the cage of deterministic models of mathematical development:
The adoption of strictly linear evolutionary models of progress in mathematics of the sort discussed in (i) tends to be highly attractive to many mathematicians in light of the intoxicating simplicity of such strictly linear evolutionary models, by comparison to the more complicated point of view discussed in (ii). This intoxicating simplicity also makes such strictly linear evolutionary models — together with strictly linear numerical evaluation devices such as the “number of papers published”, the “number of citations of published papers”, or other like-minded narrowly defined data formats that have been concocted for measuring progress in mathematics — highly enticing to administrators who are charged with the tasks of evaluating, hiring, or promoting mathematicians. Moreover, this state of affairs that regulates the collection of individuals who are granted the license and resources necessary to actively engage in mathematical research tends to have the effect, over the long term, of stifling efforts by young researchers to conduct long-term mathematical research in directions that substantially diverge from the strictly linear evolutionary models that have been adopted, thus making it exceedingly difficult for new “unanticipated” evolutionary branches in the development of mathematics to sprout. Put another way,
inappropriately narrowly defined strictly linear evolutionary models of progress in mathematics exhibit a strong and unfortunate tendency in the profession of mathematics as it is currently practiced to become something of a self-fulfilling prophecy — a “prophecy” that is often zealously rationalized by dubious bouts of circular reasoning.
In particular, the issue of
escaping from the cage of such narrowly defined deterministic models of mathematical development stands out as an issue of crucial strategic importance from the point of view of charting a sound, sustainable course in the future development of the field of mathematics, i.e., a course that cherishes the priviledge to foster genuinely novel and unforeseen evolutionary branches in its development.