I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here.
Also some time ago I read about Grothendieck's "Denunciation of so-called “general” topology" with interesting comments also made here:
According to Winfried Scharlau's book, Grothendieck described his work
in a letter to Jun-Ichi Yamashita as: "some altogether different
foundations of 'topology', starting with the 'geometrical objects' or
'figures', rather than starting with a set of 'points' and some kind
of notion of 'limit' or equivalently) 'neighbourhoods'. Like the
language of topoi (and unlike 'tame topology'), it is a kind of
topology 'without points' – a direct approach to 'shape'. …
appropriate for dealing with finite spaces…
So I am wondering what progress has been made here and in what directions. Does there currently exist an approach to the foundations of general topology that is not based on a notion of "points", in the spirit of Grothendieck's denunciation?
Best Answer
Reading section 5 in Grothendieck's essay Esquisse d'un programme it becomes clear that with regard to topology Grothendieck was bothered by some artificial foundational problems introduced by the fact that the foundations of topology were created by analysts rather than by geometers and topologists. Specifically he refers to phenomena such as space-filling curves which he thinks should be ruled out at the foundational level by a more careful choice of definitions of the basic objects we work with.
The basic model is Hironaka's semianalytic sets (or what Grothendieck proposes to call piecewise analytic sets) where such phenomena do not occur, and which on the other hand is sufficiently rich to accomodate various constructions in geometry and topology, such as coning, stratification, etc. What Grothendieck seeks to do is provide an axiomatisation that would be more or less satisfied by Hironaka's proposal, but that would be realizable in other models as well. Notes Grothendieck:
This situation, like so often already in the history of our science, simply reveals the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question, thus at the context in which we live, breathe, work – accepting it, rather, as immutable data.
My conclusion is that Grothendieck's proposal in this context does not necessarily amount to a search for a foundation not based on points. Rather the idea is to get away from the continuous category with its odd phenomena that are viewed by Grothendieck as being a function of inadequate foundations rather than intrinsic mathematical merit. Not an uncommon phenomenon I must say.