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* UNIFESP
Some Remarks on Wittgenstein’s Philosophy of Mathematics: The Case of Euler’s Proof
Resumo
Neste artigo discuto as críticas feitas por Wittgenstein, na seção 123 do Big Typescript, a uma demonstração dada por Leonard Euler para a infinitude dos números primos. Wittgenstein propõe ainda uma correção, que é o único exemplo conhecido de trabalho matemático realizado pelo autor. Nas considerações tecidas por Wittgenstein ao redor desse tema, encontram-se algumas das questões centrais de sua filosofia da matemática, tal como desenvolvida a partir do período intermediário. Meu objetivo é separar os diferentes aspectos da crítica wittgensteiniana, ao mesmo tempo em que sugiro um modo de interpretá-la que faça jus a outra posição que Wittgenstein nunca se cansa de repetir: A matemática não tem nada a temer da filosofia, pois a filosofia não pode levar à rejeição de nenhum cálculo matemático. Termino por examinar quais seriam, de acordo com as concepções de Wittgenstein, as verdadeiras tarefas da filosofia em relação à matemática.
Palavras-Chaves:
Filosofia da matemática em Wittgenstein . Big Typescript . Prova de Euler . Construtivismo em matemática
Abstract
In the present paper I discuss the criticisms levelled by Wittgenstein in Big Type-script, section 123 at Leonard Euler’s proof of the infinity of prime numbers. Witt- genstein also offers a correction of that proof, which is his only known piece of math-ematical work. Among the issues at stake here are some of the most central to his philosophy of mathematics, as developed from the middle period on. My purpose is to disentangle the different aspects of his criticisms and suggest a way of interpreting them that does justice to another claim he never ceases to repeat: mathematics has nothing to fear from philosophy, for philosophy cannot lead to the rejection of any mathematical calculus. I conclude by examining what would be, on Wittgenstein’s view, the real tasks of philosophy with respect to mathematics.
T i a g o T r a n j a n *
36
Tiago Tranjan
Keywords:
Wittgenstein’s philosophy of mathematics . Big Typescript . Euler’s proof . Constructivism in mathematicsIn
Big Typescript
, section 123 Wittgenstein discusses a proof given by Leon-ard Euler establishing the infinity of prime numbers. In a recent paper
1
, M. Marion and P. Mancosu analyze this section – as well as a number of pointsrelative to its historical context –, with an acute commentary and many inter-esting results. It is only fair to say that I am very indebted to the authors, and acknowledge that their research provided the main spark for the reflections I will develop in the following. As a matter of fact, around the end of their paper, Marion & Mancosu write: “We are now left with the task of reconstructing this
Standpunkt Witt- gensteins
and the above discussion of his remarks on the constructivization of Euler’s proof should be seen as a contribution to this task”. I took the recom-mendation seriously, and I would like the discussion below to be taken in the very same spirit: as an attempt to further our understanding of Wittgenstein’s standpoint with regard to the philosophy of mathematics.First of all, let me try to summarise some of the points made by Marion & Mancosu in their paper: 1) Section 123 of the
Big Typescript
ends with some awkward calcula-tions, the purpose of which is very difficult to figure out. Marion & Mancosu give the manuscript sources (MS 108) from which the calculations are drawn, and show that they contain a complete version of these calculations, making it possible to fully understand them. 2) Based on this finding, they show that Wittgenstein’s calculations actually amount to a
correction
of Euler’s proof, a proof taken by Wittgenstein to be inadmissible in its srcinal form. 3) Supported by the now complete calculations, interpreted as a cor-rection of Euler’s proof, and by a number of remarks available in Wittgen-stein’s text, they argue for the following position: Wittgenstein is not content merely with purifying the discourse
about
mathematics; he has something to say about the way mathematics – and mathematical proofs – are conducted.
1 [Marion & Mancosu: 2003]
37
Some Remarks on Wittgenstein’s Philosophy of Mathematics: The Case of Euler’s Proof
O que nos faz pensar nº33, junho de 2013
It is with respect to this last claim that I will try to improve upon. I be-lieve some of the issues at stake here are utterly important for a correct un-derstanding of the
Standpunkt Wittgensteins
. Let us then take a look at Euler, Wittgenstein, and Wittgenstein about Euler.
* * * * *
The proof of the infinitude of the prime numbers given by L. Euler in the last decades of the 18th century – a proof I will discuss in detail in the next section – makes use of the “reductio ad absurdum”. Its general structure is as follows.It begins with an equation, taken to be valid
2
:The left side of this equation is an infinite sum (more precisely: it is a di-vergent series, known as the “harmonic series”). The right side is a product, ranging over the prime numbers. Now, if there were a finite amount of prime numbers, the right side would be finite. But that is impossible, by virtue of its being equated to the left side, which is infinite. Therefore, the prime numbers cannot be finite – they are infinite. With respect to this proof, Wittgenstein expresses himself, indeed, in a very critical manner. He puts the situation in his usual colourful way:
“(...) Here once again we have that remarkable phenomenon that we might call circumstantial proof in mathematics – something that is absolutely never permitted. Or perhaps we could call it a proof by symptoms. The result of the summation is (or is understood as) a symptom that there are terms on the left that are missing on the right. The connection of the symp-tom to what we would like to have proved is loose. That is, no bridge has been built, so we settle for seeing the other bank. All the terms on the right side occur on the left, but the sum on the left side yields
∞
and the one on the right only a finite value – so . . . must; but in
2 The reasons why this equation is taken to be valid will not concern us here. Sufﬁce it to say that there is a fairly unanimous agreement upon its validity, from both the point of view of traditional and constructive mathematics.
38
Tiago Tranjan
mathematics the only thing that must be is what is.The bridge has to be built.”
3
(My underlines)
Wittgenstein tries to make us see that there is something very problematic in the way Euler’s proof is presented. But what is, exactly, the target of his criti-cism? Here the complications begin. The examination of section 123 of the
Big Typescript
suggests various candidates, whose mutual connections are far from simple. There are at least four of them:I. In the first paragraph of the above-quoted passage, Wittgenstein seems to criticize a
method of demonstration
which he labels “proof by circumstantial evidence” (
Indizienbeweis)
or “proof by symptoms” (
Beweis durch Symptome
).II. Closely related to the rejection of this method of demonstration is, as it seems, the rejection of the “law of excluded middle” as a general logical law. The law is explicitly mentioned, by the end of the section, in the discussion of an example
4
.III. In the second paragraph of the passage quoted above, there are some critical remarks regarding the use of modality (the use of “must”) in math-ematics. Interpretation here becomes particularly tricky.IV. Finally, we can find an attack directed at the relation between Euler’s demonstration and that which the proof purports to be a demonstration of:
“The connection of the symptom to what we would like to have proved is loose”.
All these are possible targets of Wittgenstein’s criticism, and they maintain complex relations among themselves. I do not doubt the different themes can be adequately related to one another, assuming their place in a coherent interpretation of Wittgenstein’s point of view. In what follows, however, I will not examine all these possibilities, nor discuss their mutual relations. I shall follow a different strategy. As it happens, there is in Wittgenstein’s text what I take to be a particu-larly enlightening formulation of the main problems he is dealing with. The formulation I have in mind is what I shall call the “criticism of the bet”. After presenting Euler’s proof, Wittgenstein asks:
3 Big Typescript, section 123 (p. 434e)4 “What follows from that? (The law of excluded middle.) Nothing follows from that, except that the limiting values of the sums are different; that is, nothing [new].” BT, section 123 (p. 435e)

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