1ª Palestra - ‘Números Reais’, ‘Entidades Não-Saturadas’ e ‘Avanços Matemáticos’ 1

• A tese crucial de Wittgenstein: a ideia do “avanço matemático como Mutação Semântica”.

It’s impossible for us to discover rules of a new type that hold for a form with which we are familiar. If they are rules which are new to us, then it isn't the old form. … — we cannot make any discoveries in syntax. — For, only the group of rules defines the sense of our signs, and any alteration (e.g. supplementation) of the rules means an alteration of the sense. Just as we can’t alter the marks of a concept without altering the concept itself. (Frege) (PR, § 154, pg. 182) When I said that a proof introduces a new concept, I meant something like: the proof puts a new paradigm among the paradigms of the language …One would like to say: the proof changes the grammar of our language, changes our concepts. (Wittgenstein, 1983, p. 166) Now how about this – ought I to say that the same sense can only have one proof? Or that when a proof is found the sense alters? Of course some people would oppose this and say: “Then the proof of a proposition cannot ever be found, for, if it has been found, it is no longer the proof of this proposition”. (Wittgenstein, 1983, p. 366)

• O “problema das Regras”: a “determinação”, ou não, da nova casa “nunca antes calculada” de Pi.

• “entidades saturadas” e “entidades insaturadas” podem ambas ser “objetos”?

• O problema da “identidade de uma função matemática” e os

avanços da matemática. … new ideas brought about by the progress of analysis led to the following definition of functions. … Any quantity whose value depends on one or more other quantities is said to be a function from the latter, either we know or we ignore what operations one must undertake to obtain the first from the latter. (Lacroix. 1797)

• Duas noções de “função”: como uma “relação abstrata”, como um “método de obtenção”.

The idea behind 2 is this: we look for a rational number which, multiplied by itself, yields 2. There isn’t one. But there are those which in this way come close to 2 and there are always some which approach 2 more closely still. There is a procedure permitting me to approach 2 indefinitely closely. This procedure is itself something. And I call it a real number. It finds expression in the fact that it yields places of a decimal fraction lying ever further to the right. (PR §183) We [... ] consider an indefinitely proceedable sequence of nested A-intervals λ!! , λ!! , λ!! ,… which have the property that every λ!!!! , lies strictly inside its predecessor λ!! , i = 1,2, . . . .[... ] We call such an indefinitely proceedable sequence of nested λ-intervals a point 𝑷 or a real number 𝑷. We must stress that for us the point P is the sequence … itself, not something like “the limiting point to which according to classical conception the λ-intervals converge and which could according to this conception be defined as the unique accumulation point of midpoints of these intervals”. (Van Dalen, Van Atten & Tieszen - Brouwer and Weyl, The Phenomenology and mathematics of the Intuitive Continuum. Pg. 212)

• Três posições contrárias ao classicismo:

Construtivismo + Tese de Church

As for Church's Thesis, this is not particularly plausible from an intuitionistic standpoint. The assumption that we can effectively recognize a proof of a given statement of some mathematical theory, say elementary number theory, lies at the basis of all intuitionistic mathematics; but to hold that

1ª Palestra - ‘Números Reais’, ‘Entidades Não-Saturadas’ e ‘Avanços Matemáticos’ 2

there is any recursive procedure for recognizing proofs of arithmetical statements would be to run foul of Gödel's Incompleteness Theorem. (Dummett, Elements of Intuitionism. pg 186) Define the function F as follows:

𝐹 𝑥 = 1 𝑖𝑓 𝐺𝑜𝑙𝑑𝑏𝑎𝑐ℎ′𝑠 𝑐𝑜𝑛𝑗𝑒𝑐𝑡𝑢𝑟𝑒 𝑖𝑠 𝑡𝑟𝑢𝑒; 0 𝑖𝑓 𝐺𝑜𝑙𝑑𝑏𝑎𝑐ℎ′𝑠 𝑐𝑜𝑛𝑗𝑒𝑐𝑡𝑢𝑟𝑒 𝑖𝑠 𝑡𝑟𝑢𝑒;

…. This conjecture is still an open problem in mathematics. Is this function F effectively calculable? (Choose your answer before reading the next paragraph.) Observe that 𝐹 is a total constant function. (Classical logic enters here: Either there is an even number that serves as a counterexample or there isn’t.) So as noted in the preceding example, 𝐹 is effectively calculable. What, then, is a procedure for computing 𝐅? I don’t know, but I can give you two procedures and be confident that one of them computes 𝑭. (Enderton. Computability Theory, An introduction to Recursion Theory. Pg 5)

Intuicionismo de Bishop e Sueco

What is a direct proof of a proposition? The answer is that that depends on the proposition, because it is exactly that feature of a proposition which determines it as such, that is, which gives it its meaning as a proposition.

On the other hand, if you ask, what is a possibly indirect proof of a proposition? The answer is that an indirect proof of a proposition is a method of proving it directly, that is, a method which yields a direct proof of the proposition as result. (Martin Löf, Per. Truth of a proposition, evidence of a judgment, validity of a proof. pg. 413)

From the intuitionistic point of view, it is necessary that there exists in the abstract sense calculations of … 100000000000000000000 = 10!"… in order that it should be correct to assert 10!"×10!" = 10!"…; but it is not necessary that these calculations be actually performed or that one of the proofs be constructed. (Prawitz, The conflict between classical and intuitionistic logic. pg. 21-2)

Intuicionismo Brouweriano e Categórico

These examples suggest that a topos may be conceived as a category of variable sets: the familiar category 𝓢𝓮𝓽 is the limiting case in which the variation of the objects has been reduced to zero. For this reason 𝒮ℯ𝓉 is called a topos of constant sets. (Bell. Lectures on the Foundations of Mathematics. Pg. 28)

In topos theory, therefore, a mathematical concept may possess a fixed sense, but a variable reference. The sense of the concept “real number” may be taken as fixed by its definition within a local set theory, but its reference varies with the framework of interpretation. (Bell. Lectures on the Foundations of Mathematics. Pg. 33)

The interpretation of these variable sets was in terms of subjective variation of knowledge; the elements of 𝑃 are called stages of knowledge and 𝐴 < 𝐵 is taken to mean that 𝐴 is a deeper (or later) stage of knowledge than 𝑩; for any set 𝐸 we have, at any given stage 𝑩, constructed certain elements of 𝑬 and proved certain equalities between pairs of elements constructed, giving an abstract set 𝐸(𝐵); if 𝐴 < 𝐵 is a deeper stage of knowledge, the transition map 𝐸 𝐵 → 𝐸 𝐴 reflects that no constructed elements are ever lost and no proven equations are ever disproved, but the map is neither surjective nor injective since new elements may be constructed and new equalities proved at stage A. (Lawvere. Variable Quantifiers and Variable Structures in Topoi, Pg. 105)

… A state of information cannot be considered as determined just by knowing which atomic statements have been verified, not even which are true, since the verification, or the truth, of complex formulas at a node does not depend solely upon which atomic formulas are verified or are true at that node, but on what happens at subsequent nodes. We have, therefore, to think of a state of information as comprising two things: a knowledge of which atomic statements have been verified; and an awareness of the future possibilities of verifying atomic statements, as represented by the subtree determined by the associated node. (Dummett. Elements of Intuitionism. Pg. 282)