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To what does Poincaré refer in this article http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html speaking about the intuition of pure number?

My answer is that he may refer to a sort of intuition related to the knowledge of the properties of numbers.

(For completeness of information, I say that he refers also to 2 other forms of intuition, beside the "intuition of pure number", namely, "analogical intuition" and intuition which presupposes "mathematical induction").

PS. To facilitate you to answer, I copy and paste the passages of the text in which he refers to - without defining it clearly, which in my opinion is a philosophical mistake - the "intuition of the pure number":

"We have then many kinds of intuition; first, the appeal to the senses and the imagination; next, generalization by induction, copied, so to speak, from the procedures of the experimental sciences; finally, we have the intuition of pure number, whence arose the second of the axioms just enunciated, which is able to create the real mathematical reasoning [...] I have shown above by examples that the first two can not give us certainty; but who will seriously doubt the third, who will doubt arithmetic? Now in the analysis of to-day, when one cares to take the trouble to be rigorous, there can be nothing but syllogisms or appeals to this intuition of pure number, the only intuition which can not deceive us. [...] I have said above that there are many kinds of intuition. I have said how much the intuition of pure number, whence comes rigorous mathematical induction, differs from sensible intuition to which the imagination, properly so called, is the principal contributor. [...] Could we recognize with a little attention that this pure intuition itself could not do without the aid of the senses? [...] It is the intuition of pure number, that of pure logical forms, which illumines and directs those we have called analysts. This it is which enables them not alone to demonstrate, but also to invent. By it they perceive at a glance the general plan of a logical edifice, and that too without the senses appearing to intervene. [...] Is there room for a new distinction, for distinguishing among the analysts those who above all use this pure intuition and those who are first of all preoccupied with formal logic?"

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The original French edition was : La Valeur de la Science (Flammarion - 1905).

The context of the discussion is the debate on The Foundations of Mathematics following the so-called arithmetization of analysis (Weierstrass, Dedekind) and the development of the Logicist school : Dedekind, Peano, Frege and Russell.

In a nutshell, arithmetization of analysis succeeded in the project of establishing the calculus on the ground of the theory of real numbers, avoiding unwanted "geometrical intuition".

In turn, real numbers were grounded on natural numbers, through Dedekind and Peano axiomatization, and the "emerging" set theory.

The Logicist school, in turn, try to reduce natural numbers to purely "logical" concepts (including the concept of class).

The discovery of the paradoxes "stopped" the completion of the logicist program.

Henri Poincaré point of view is in opposition to logicism :

Historically, [Poincaré's] theses [concerning logic and foundations of mathematics] are directed broadly against the founders of modern logic and set theory such as Cantor, Peano, Frege, Russell, Zermelo, and Hilbert.

according to Poincaré, mathematics requires intuition [emphasis added], interpreted as an element of understanding, not only in the context of discovery, but equally in the context of justification. As we have already seen, in arithmetic pure intuition is necessary to justify the complete (or mathematical) induction principle. It should be noted that the term “intuition” is quite ambiguous, a fact that is well known and explicitly discussed by Poincaré himself. In The Value of Science, he distinguished three kinds of intuition: an appeal to sense and to imagination, generalization by induction, and intuition of pure number—whence comes the axiom of induction in mathematics. The first two kinds cannot give us certainty, but, he says, “who would seriously doubt the third, who would doubt arithmetic?” (Poincaré, La Valeur de la Science, 1905: page 33).

See The Value of Science, page 19 :

We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue. In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition.

Compare these four axioms: (1) Two quantities equal to a third are equal to one another; (2) if a theorem is true of the number $1$ and if we prove that it is true of $n+1$ if true for $n$, then will it be true of all whole numbers [induction axiom ...]. All four are attributed to intuition, and yet the first is the enunciation of one of the rules of formal logic; the second is a real synthetic a priori judgment [see Kant], it is the foundation of rigorous mathematical induction [...].

In conclusion, according to H.P., we have an intuition of the "endless" succession of natural numbers, based on the unlimited possibility of iterating the basic "arithmetical operation" of adding one.

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When Poincare referred to the intuition of pure number he was speaking of the fact that humans have the concept of the number of things, the cardinality of a set of objects from almost the beginning of life. At the time it would have been difficult to explain this as anything but a Kantian a priori intuition, knowledge given by the mind prior to experience.

With the work of Elizabeth Spelke and others we now have solid evidence that the underpinnings of the concept of number are in fact neurophysiological, i.e., provided by evolved structure of the human brain.

We assume that we have always had the idea of adding more things to increase the number of things in hand. I've got one apple and if you give me another, why then of course I have two then. In fact though it takes more than two years for human children to grasp that the difference between "one thing" and "more than one thing" can be more abstractly understood as the successor function which generates all of the numbers with which we count, i.e., the natural numbers (1, 2, 3...some include 0). A child can proudly perform the counting script they have been taught, naming, say, one toy fish, two toy fish, three and so on. However, typically the 2.5 year old child who has just "counted" the toys in that way will hand you one fish if asked for one, but give you an arbitrary handful if asked for any number other than one!

After additional months of experience the child slowly, in a stepwise fashion, learns to understand "two" then "three." Sometime after this comes the great leap forward, where the child grasps implicitly the induction definition of natural numbers, i.e., that each word in the counting routine actually defines how many things you are considering and that each successive count adds one to the number of things (in your set) and that this can be continued indefinitely, with no upper bound (see Evolutionary and developmental foundations of human knowledge, by Marc Hauser and Elizabeth Spelke).

Although a chimpanzee can laboriously learn to associate a number symbol with a particular number of objects, they never (at least not after 20 years of training on one particular subject) progress to the understanding of the successor function, i.e., chimps cannot learn that a new number symbol means that one has been added to the previous set of items. It appears though (from research done by Spelke and others) that humans and some non-human primates both draw on a core neurophysiological basis for numerosity, i.e., (1) representing the approximate cardinal values (about how many items are present) of large groups of objects or events and (2) representing the exact number of object sets or events when there are only a small number of individual units.

It appears that the uniquely human capability to construct the natural numbers (i.e., use the successor function) relies first on the core perception of one versus many, then mapping other number words to larger numerosities, then noticing that the progression in the language (the words representing numbers) of the counting routine corresponds to increasing the cardinal value of the set, the number of units in hand. This (and other research) suggests that natural language ability is involved in the human leap from those core perceptions shared by some non-human species to the natural number concepts unique to humans.

In his later work (Last Essays) Poincare added that "there is in all of us an intuitive notion of the continuum of any number of dimensions whatever because we possess the capacity to construct a physical and mathematical continuum; and ... this capacity exists in us before any experience.." In this also the work by Spelke and others has demonstrated some validity to this notion, research revealing that infants are born with a host of perceptual capacities, including mechanisms for perceiving depth and using depth information to guide spatially appropriate actions (quoting loosely from the earlier cited source). It is clear that such mechanisms include the perception of continuity, else there would be no contemplation of action requiring, say, the movement of a hand over a distance to grasp an object (since there would be no sense that the object could be reached by extension of limb over a gulf of unkown property).

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