Heirs of Poincaré

It is difficult to evaluate the quality of mathematical work or the particular destiny of mathematics in the 20th century. We have already given some criteria: logical, conceptual and didactic clarity and rigor, clarity of intuition (which, contrary to common myths, is not incompatible with logical rigor),  an orientation towards synthesis and simplification, an orientation towards applications to the sciences embodying novel unifying ideas,  awareness of the deep problems regarding  mathematical models of reality,  awareness and interest in fundamental philosophical problems.

Logicism is irrefutable. Mathematics is in its purest essence is the deployment of computability (and hence logic), but this game cognitively involves intuition (just as chess strategies) as its develops towards perfection. It is a 'game' in the noblest sense: not 'to calculate' in the sense of mere application, but 'to calculate' as in 'be able to calculate' through finding a strategy for the right moves. The very conventionalism is not conventionalist in its a priori logical and computational cognitive presuppositions. Moreover, beyond the genius of Frege, Turing and Church (and Brouwer's intuitionism is just computationalism), there is a presupposition in mathematics of a certain objective-intuitive correspondence and specially a claim to enter into the objective truth of the world in the form of science. Kant saw the tip of this iceberg. The problem of the intuitive correlation of mathematics is similar to that of the efficacy of mathematical models of nature.

In the 20th century (if the number of mathematical publications increased exponentially)  we saw a drastic decrease in logical-conceptual synthesis and clarity and a great lack of philosophical intelligence and awareness, specially regarding the relationship between mathematics and science, regarding the essence and scope of mathematical models themselves (cf. my paper Differential Models, Computability and Beyond).

We saw a proliferation of a kind of junk philosophy and junk science allegedly based on mathematics, specially in the consciousness of the general public, something which can be traced to the reigning ideology of the times (reflected in popular media personalities) and to  questionable core elements of 'general systems theory' and 'game theory'.  We can name a few of these fads: 'chaos', 'fractals', philosophies of vagueness, randomness and uncertainty, 'bifurcations', using the terms 'emergent', 'self-organization',  'self-referential', 'neural',  'non-linear',  'evolutionary' or 'quantum' beyond its narrow legitimate technical sense and at the same time offering no clear and cogent logical and philosophical account of them.

There are not that many people who could be considered the genuine heirs of Poincaré (in geometry, topology and differential equations). These stand out as philosophical-scientific-mathematical giants, towering above others.  We name first of all philosopher-mathematician René Thom (and his collaborators and followers) and Stephen Smale (and the Brazilian school of dynamical systems). For Celestial Mechanics Siegel stands out. Jack Hale has written some excellent textbooks. For geometers we have people like Shing-tung Yau and Milnor.  Ergodic theory (which embodies Lebesgue's measure theory in the mathematical modelling of nature) , originated by Birkhoff,  is also important. And we attach great importance of fluid dynamics (cf. David Ruelle's theory of turbulence - a distant heir to da Vinci).