“What now matters the most is what happens in the middle, with the logic[s] of intermediate values that are much more appropriate to our culture, to our ways of living”
“One way, for example, that catches my attention and that I think is important, is to use the mathematical tools from the last century. Social sciences have, in the sense of the techniques they use for their development, fallen extremely behind with respect to mathematics or the development of mathematics. Mathematics has advanced a lot from the point of view of methodology and logic. On the contrary, sociology and history are still very much binary. They have not been able to use intermediate logics to study the different transformations of historical and social events. The use of alternative logics can be very important and profound. It has not been done. It has to be done. This is long overdue, particularly referring to the most important mathematician of the 20th century, who is little known outside mathematics. I am talking about Alexander Grothendieck (1928-2014), who in the ‘60s invented Grothendieck’s topos, where he conjugated something that seemed to be impossible until that moment, that is, to understand in a universal, superior, and unitary way both the notion of numbers as well as the notion of space. Numbers (arithmetics) and space (geometry) were apparently very far apart. Their union in the concept of space-number is even more sophisticated than Einstein’s unifying of space and time. What Grothendieck proposes—to unite space and number as projections of a superior beam—corresponds to the thinking of a notion of a variable set in the universe, of which some technical applications have already been made, although its full implementation in the “real” will be seen in the future. An interesting fact is that the intrinsic logic of the topos, of these new spaces of thought, are non-classical logics. In fact, it can be proved that they are intuitionistic logic, plastic logic, linked to the deformations of topology. Our natural limitations as human beings are thus overcome, and there are very interesting things which at some point should be applied to history and sociology. It is a matter of time. Mathematics tends to be too adventurous and are sometimes centuries ahead. It is possible that topos start to be used in mathematical biology by the mid 21st century”
“Mathematics must, first of all, create an opening and invent abstract worlds and concepts, which at some point will return to reality. One example is the invention of the complex variable, which was carried out in the 15th century. Its first important implementation was not done until the end of the 19th century when Maxwell invented electromagnetic theory using complex numbers (even worse: quaternions). That is, more than 400 years passed before it was used. With topology, things have been a bit faster because, actually, topology, science and mathematics are advancing at a vertiginous rate. It is amazing, everything that is nowadays going on with science. In art and philosophy as well! There are many fascinating areas where great changes have happened. Topology deeply influenced Einstein’s theory, which was formalized thanks to Riemannian geometry (which come from Riemann, a mathematician of the mid 18th century who invented non-euclidean geometries and introduced brilliant topological techniques to the complex variable).”
https://tripleampersand.org/politics-math-age-post-truth-interview-fernando-zalamea/