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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?

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    For the historical context of the “analysts” vs “synthesists” in mathematics, with its Kantian underpinnings and relation to events of the late 19th century to which Poincare was reacting, see What caused or contributed to Euclid's Elements and Synthetic Geometry falling into disfavor? Already Klein, Poincare's contemporary, noted that the divide was artificial and counterproductive. It is not surprising that Grothendieck, who is extolled by many as the greatest mathematician of the 20th century, could be highly gifted in both compartments. – Conifold Aug 23 at 7:55
  • But this divide does seem to hold weight. Poincare cites examples such as Riemann and Lie who despite appearing to be analysts in their works clearly thought in pictures once you've spoken to them. Weierstrauss on the other hand doesn't call to action a single image. For him, calculus is just a type of prolongation of arithmetic. Does Klein supply any reasons for thinking this is just artificial? Thanks! – Justin Latson Aug 23 at 17:41
  • Bourbaki does not have a single picture on thousands of pages. Does it mean they are all "analysts"? Simplistic speculations about how people think are just that, grand generalizations over paper-thin basis. And just because the extremes exist does not mean that there is no middle, and more on top. Hadamard has an interesting case study in An Essay on the Psychology of Invention in the Mathematical Field, more comprehensive than Poincare-style anecdotes. Although it is still lacking by modern standards. – Conifold Aug 23 at 18:01
  • See also Klein's thoughts on the sort of partisanship this was reduced to at the time in Elementary Mathematics from an Advanced Standpoint: Geometry, p.56:"exaggeration of certain fundamental principles into scientific schools leads to a certain petrifaction". – Conifold Aug 23 at 18:08
  • Poincare does cite Lie as an example to explain your Bourbaki analogy. If you read Lies work you don't see a single image either, yet when you spoke to him. It is unmistakable according to Poincare that he thought in pictures. This cannot be said of Weierstrauss. My basic question to Poincare stems more from his comments on infinity. I can't put myself to believe that the source of the infinity is geometric. – Justin Latson Aug 23 at 20:23
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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.

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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.

  • It almost shocks me that we would not have infinity without the "inner world". Your answer reminds me of a quote from Gauss that goes something like. "If numbers can be understood purely by our minds, then I concede geometry is beyond our minds and therefore can not be ordained apriori as we understand it now." This is coming from someone who says divine thinking is arithmetic. Yet he gives an almost transcendental character to geometry. Is geometry divine? Why can't infinity be analytic as opposed to synthetic in origin? – Justin Latson Aug 23 at 20:20
  • Why must infinity be geometrical in origin? This is extremely counter intuitive to me (no pun intended). Also thank you for your answer! – Justin Latson Aug 23 at 20:31
  • Gauss lived quite a while ago. We've learned somethings about the brains since. I suspect it may boil down to the fact that we 'see' lines and points of geometry and it comes from 'outside' of our mind, where as numbers are created 'inside' the mind, and are applied to objects in the world. As to why would infinity be considered a synthetic truth? I don't entirely believe in the distinction, but if I did, I'd argue that it would be considered a synthetic since questions of infinity tend to be approached from the view they describe characteristics of objects. The universe is infinite, we arent. – J D Aug 23 at 20:34
  • Must infinity be geometrical in origin? I don't think so. In computer science, the infinite loop is an introduction into infinity that has nothing to do with geometry, but I think lines are more comprehensible and easier to understand and therefore easier to intuit from. Children get the idea that lines can go on forever (go in a circle, for instance), but have a harder time wrapping their mind around infinite additions. I'd guess that spatial processing (which is approachable through kinesthetic and visual intelligence) is dominant over temporal processing (which requires an abstraction). – J D Aug 23 at 20:41
  • I think a demonstration of this bias is that we represent time with lines or motion frequently, but we don't represent lines with time. At least nothing in the latter category comes to mind. – J D Aug 23 at 20:44
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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.

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