Poincare says we are born geometric or arithmetic thinkers. Which was Grothendieck and why?

Question

Poincare proclaims that the mathematical continuum originates from the sensible intuition and that intuition by pure number or logic alone could not have given us this notion.

Source for the claim: https://plato.stanford.edu/entries/poincare/#LogFouIntPre

Intuition by pure number meaning induction. Indefinite iteration. Logic is of course the natural language of the "analyst".

In light of this, I immediately thought of algebraic geometry and whether or not the continuum is really something that can only come from the geometric/intuitive (excluding induction) side of things.

Grothendieck's viewpoints are interpreted rather differently by different people. He himself says geometry has both the discrete magnitudes of arithmetic and continuous magnitudes of analysis. It is the only science that has this. It can infer both.

Frequently when reading his own descriptions of how he invents math in his books you see he speaks of visions. These are clearly mental images and therefore ... "Geometric"? But then again is this necessary to his discoveries?

I can hardly see how Poincare can claim the continuum and geometry have such a close relationship. Didn't people just will into existence the square root of negative one???

Why then can we not say "here we will denote with this symbol an object which which is undifferentiated in it's multiplicity and has no concrete numerical identity. you can however set every other number less than to it?"

How does Poincare go from numbers, which are abstract concepts, to the source of essentially the concept of the infinite in math being from pictures?

Puzzles like Grothendieck's make me think he has it all wrong. For if we are born either or, and have a much harder time with the other than our natural inclinations, how come Grothendieck does not seem to lack in any. Considering he is considered the greatest algebraic geometer of all time arguably. Has his work been analyzed from the dichotomy Poincare proposes?

Answer 1

IMO one should not play off one against the other, i.e. geometry against arithmetic or vice versa.

At the time of Poincaré both disciplines were separated. At the base of Grothendieck‘s revolutionary view onto Algebraic Geometry lies the concept of the spectrum: Introducing the spectrum Spec R of a commutative ring R means to consider the algebraic object R as a topological space Spec R.

In the most simple case of the ring Z of integers, the prime numbers - more precisely the prime ideals - become the points of the topological space Spec Z. The integers from Z become functions on the Spec Z, and their prime factors become the zeros of these functions. Grothendiecks concept of the spectrum clarifies the topological properties of algebraic objects.

By the subsequent use of sheaves and the introduction of schemes Grothendieck succeeded to incorporate all of classical arithmetic into this topological setting - not only field theory but also number theory over the corresponding rings of algebraic integers. The result is called Arithmetic Geometry.

In the light of Grothendieck‘s achievement one can ask whether or not Poincarès dichotomy is a bit outdated.

Answer 2

There's a lot to unpack in your post!

Poincare proclaims that the mathematical continuum originates from the sensible intuition and that intuition by pure number or logic alone could not have given us this notion.

Intuition by pure number meaning induction. Indefinite iteration. Logic is of course the natural language of the "analyst".

In light of this, I immediately thought of algebraic geometry and whether or not the continuum is really something that can only come from the geometric/intuitive (excluding induction) side of things.

I think the question is an excellent question is rooted in the nature of the exploration of the primacy of the senses, and ultimately has to be approached with an empirical bent. That being said, human beings are far more predisposed to using their visual systems to their logical ones. It's a fact that even statisticians struggle to use their statistical knowledge well. (See Thinking, Fast and Slow.)

The historical fact is that the developments of continuity from the direction of intuition to rigorous analysis and moving from Euclidian axioms to non-Euclidean axioms both show a bias of the brain as ideas are generally introduced with intuition and then later formalized with logic. In the former case, the idea of continuity started essentially as a visual-kinesthetic intuition that the smoothly curved line was special in a way that other lines were not. Ultimately, analytic continuity was introduced once closure of the reals was put forth. (See Ch. 4, Numbering the Continuum of The Philosophy of Set Theory). As far as why for so long did mathematicians accept uncritically alternatives to Euclid's parallel postulate? Again, I would suggest it's a bias in how our visual system dominates in our cognition and language. There's something inherently off presuming lines that don't look parallel can be considered parallel lines.

I can hardly see how Poincare can claim the continuum and geometry have such a close relationship. Didn't people just will into existence the square root of negative one???

As for the relationship between the continuum and geometry, they are inextricably intertwined. The correspondence between points on a line and the reals is a necessity not only in understanding analysis, but also sets, and this is because what is happening in the brain when moving from the geometrical to verbal domain is a cross-domain mapping that requires development. Lakoff and Nuñez wrote a book offering a theory on what happens based on an idea called the conceptual metaphor, which is related to a scientific study of meaning itself, cognitive semantics, a field in cognitive science.

How does Poincare go from numbers, which are abstract concepts, to the source of essentially the concept of the infinite in math being from pictures?

To answer this, try an exercise. Draw an isoceles triangle with a height greater than a base, and then draw a segment from the two midpoints of the legs. Note the segment from the midpoints will be shorter than the base. Now, put some points on the base of the traingle and draw a segment through the midpoint segments to the vertex between the legs. Then another. Then another. In fact, there are an infinte number of points and segments that can be drawn showing there is a 1-1 correspondence between the points of the midpoint segment and base. Yes, despite having different lengths, they have the same number of points. And yet wouldn't the two differing lengths represent the same continuum? Hence, questions of infinity and the reals arise from a simple Euclidian construction, in this case introducing an analytic proposition which is counter-(geometrical-)intuitive

I am apt to agree with those who have commented on your question. Different people have these mutliple intelligences, and some have lots of all of them, and some struggle with most of them. A square after all can be spoken of, drawn, measured, or represented with abstract, unknown lengths. Despite the origins of geometry and arithmetic being distinct, it is easy to see given the historical developments as well as the development of children that geometry often helps lead us to analytical propositions.

Answer 3

I have to oppose a couple points made by the OP.

Firstly, I believe Grothendieck was, as the OP said, using images or visions to create maths, but he was not a strong algebraist. That would be Weil. Grothendieck's strengths, from what I gather, is extreme abstraction to the point where one could think mathematics has nothing to do with numbers.

He clearly favored shapes; numbers and algebra were simply things he got from geometry. He asserted, as you say, that geometry is the only science which can give both discrete and continuous magnitudes. If you look at his gribouilles you see he clearly thought in diagrams. Equations seemed more like abbreviated sentences, certainly not in the style of Weil or Weierstrass.

He repeatedly considers the possibility of space being made not of points, but of discrete shapes. Shapes are not numbers. And we all know how important spaces are to Grothendiecks work. You can’t transmute a number into a circle. You can describe there points by equations.

But space itself is made of discrete shapes, according to Grothendieck. The ideal positions which algebra and calculus can only approximate is the basic shape at the base of space, not points which were favored traditionally by your so called analysts. Geometry is beyond our equations and numbers. It is not in a hierarchy of dependence, but at the least horizontal in their priority.