This is a nice overview of some of the fundamental aspects of modern mathematics and how they fit together. It's definitely targeted at an advanced reader though.<p>A book describing the large-scale structure of modern mathematics and its fundamental concepts at an undergraduate level is Mathematics, Form and Function by Saunders Mac Lane. If you like this paper but feel like it's currently out of your reach, you might like this book.<p>Something deep about mathematics that I feel has only begun to be explored in the last few decades or so is how intimately connected computation/logic is to algebra and geometry. Connes doesn't really touch on that here but research programs like Geometry of Interaction and Geometric Complexity Theory are really exciting to me.
I am not a mathematician but I found that an interesting read. I am not sure how rigorous all this is, and even whether I understood the material completely. It just tickles the imagination. So here is my summary, in the hope it will tickle your imagination too. (If a mathematician can read it, please correct my layman mistakes)<p>TLDR: after applying some special re-normalization to results from quantum field theory, coefficients of different models match and we can even get ratio of integers instead of bizarre numbers.<p>It implies quantum field theory is not so special: we just don't understand some of its features, that seem to come from geometric symmetries.<p>If I'm not too wrong, it also implies everything is discrete in the universe.
If you liked this essay, you might also like the book by Marcel Berger, Geometry Revealed: A Jacob's Ladder to Higher Geometry.<p>A quote from the cover:
"The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs"
The use of Roman numerals for centuries in this paper is obnoxious. I already have to mentally subtract 1... now I have to translate number systems too?!