I have a course in mathematical logic and i heard some argument about intuitionism math. I'am curious about it and i look at some book and i am just confused about phenomenon. But i realized brouwer construct all math and their logic from duality and time intuition. and also he claim that math is a non linguistic concept and any man can do it with have some good memory. I cant find an example of intuitionism proof that show me how duality works please if you have an example or know some book about it, share with me I will be very greatfull.
For Brouwer, mathematics is a mental phenomenon that is intrinsecally "pre-linguistic".
See e.g.: Dirk van Dalen (editor), Brouwer's Cambridge Lectures on Intuitionism (1946-51) (Cambridge UP, 1981), page 4:
First act of intuitionism Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognizing that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics.
Based on his philosophy of mind, on which Kant and Schopenhauer were the main nfluences, Brouwer characterized mathematics primarily as the free activity of exact thinking, an activity which is founded on the pure intuition of (inner) time. No independent realm of objects and no language play a fundamental role.
Brouwer (following the development of non-Euclidean geomteries) rejected Kant's view about pure intuition of space, but followed Kant on the foundations of arithmetics on a pure intuition of time.
The process of generating natural numbers from the iteration of the "+1" operation has its source, according to Brouwer, in "twoity":
"The perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory."
See also Arend Heyting, Intuitionism : An Introduction (North Holland, 3rd ed.1971) page 13: Arithmetic
We start with the notion of the natural numbers $1, 2, 3$, etc. They are so familiar to us, that it is difficult to reduce this notion to simpler ones. Yet I shall try to describe their sense in plain words. In the perception of an object we conceive the notion of an entity by a process of abstracting from the particular qualities of the object. We also recognize the possibility of an indefinite repetition of the conception of entities. In these notions lies the source of the concept of natural numbers [L.E.J. Brouwer 1907 (Over de Grondslagen der Wiskunde, Ph.D. thesis), p. 3].