Alain Connes and the mathematical world
Alain Connes is currently a professor at the College de France and Vanderbilt University. Connes won the Fields Medal in 1982 and Crafoord Prize in 2001. He has authored a number of books and represented the Platonist point of view in a debate with neuroscientist Jean-Pierre Changeux presented in the Princeton book: Mind, Matter and Mathematics.
Connes is one of the few mathematicians who can speak about his experience of mathematics in an intimate and provocative way. He’s like the guide you were fortunate to get, when you made your trip to an old city, who made your trip memorable because he directed your attention to the things most likely to tell you where you were and what you were looking at. What follows are a few excerpts from some of his writing. I don’t have much to add to what he writes, other than saying that his view contributes significantly to my own, and to the ongoing debate about what mathematics is. The first set of excerpts is from a paper called A View of Mathematics.
The nature and inner workings of this mental activity are often misunderstood or simply ignored even among scientists of other disciplines. They usually only make use of rudimentary mathematical tools that were already known in the XIXth century and miss completely the strength and depth of the constant evolution of our mathematical concepts and tools.
It might be tempting at first to view mathematics as the union of separate parts such as Geometry, Algebra, Analysis, Number theory etc…where the first is dominated by the understanding of the concept of “space,” the second by the art of manipulating “symbols,” the next by the access to “infinity” and the “continuum” etc…it is virtually impossible to isolate any of the above parts from the others without depriving them from their essence. In that way the corpus of mathematics does resemble a biological entity which can only survive as a whole and would perish if separated into disjoint pieces.
Whereas the letters we use to encode numbers are dependent of the sociological and historical accidents that are behind the evolution of any language, the mathematical concept of number and even the specificity of a particular number such as 17 are totally independent of these accidents.
The scientific life of mathematicians can be pictured as a trip inside the geography of the “mathematical reality” which they unveil gradually in their own private mental frame…whatever the origin of one’s itinerary, one day or another if one walks long enough, one is bound to reach a well known town i.e. for instance to meet elliptic functions, modular forms, zeta functions. “All roads lead to Rome” and the mathematical world is “connected”. In other words there is just “one” mathematical world, whose exploration is the task of all mathematicians and they are all in the same boat somehow.
Moreover exactly as the existence of the external material reality seems undeniable but is in fact only justified by the coherence and consensus of our perceptions, the existence of the mathematical reality stems from its coherence and from the consensus of the findings of mathematicians. The fact that proofs are a necessary ingredient of a mathematical theory implies a much more reliable form of “consensus” than in many other intellectual or scientific disciplines. It has so far been strong enough to avoid the formation of large gatherings of researchers around some “religious like” scientific dogma imposed with sociological imperialism.
Most mathematicians adopt a pragmatic attitude and see themselves as the explorers of this “mathematical world” whose existence they don’t have any wish to question, and whose structure they uncover by a mixture of intuition, not so foreign from “poetical desire,” and of a great deal of rationality requiring intense periods of concentration. Each generation builds a “mental picture” of their own understanding of this world and constructs more and more penetrating mental tools to explore previously hidden aspects of that reality.
On the concept of space:
The mental pictures of geometry are easy to create by exploiting the visual areas of the brain. It would be naive however to believe that the concept of “space” i.e. the stage where the geometrical shapes develop, is a straightforward one. In fact as we shall see below this concept of “space” is still undergoing a drastic evolution.
The Cartesian frame allows one to encode a point of the Euclidean plane or space by two (or three) real numbers… This irruption of “numbers” in geometry appears at first as an act of violence undergone by geometry thought of as a synthetic mental construct.
This “act of violence” inaugurates the duality between geometry and algebra, between the eye of the geometor and the computations of the algebraists, which run in time contrasting with the immediate perception of the visual intuition.
Far from being a sterile opposition this duality becomes extremely fecund when geometry and algebra become allies to explore unknown lands as in the new algebraic geometry of the second half of the twentieth century or as in noncommutative geometry, two existing frontiers for the notion of space.
Some of these remarks are restated in a short paper called Advice to the Beginner, along with a number of other interesting observations. This paper can be accessed from his publications list.
For me, algebra unfolds in time: I can see a formula live and turn and exist in time, whereas geometry has something instantaneous about it and I have much more difficulty with it. As far as I go, formulas create mental pictures.
I have often had the impression that there are concentric circles in the mathematical world ; that one begins to work in a totally eccentric part and one tries to get gradually closer to the heart.
What is this heart ? Is it subjective ?
What I mean by the heart of mathematics is that part which is interconnected to essentially all others. A bit like all roads lead to Rome, what I mean is that, when the mental picture you get of a mathematical subject becomes more and more precise, you realize in fact that, whatever the topic you begin with, if you look at it sufficiently precisely, after a while, it converges toward this heart…. if you walk long enough, you are obliged to go towards these domains, you cannot remain outside. If you do, it is a bit out of fear. You can succeed in doing a lot of things by refining techniques in a given topic, but unless you keep moving towards this heart you feel you are left outside.
Perhaps one of the things that discourages so many:
There is always this permanent fear of error which doesn’t improve over the years… And there is this part of the brain which is permanently checking, and emitting warning signals.