Wittgenstein’s Philosophy of Mathematics

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Before there was not. To indicate the change of status, Wittgenstein uses several suggestive metaphors. The first is the road building process:. It is like finding the best place to build a road across the moors. We may first send people across, and see which is the most natural way for them to go, and then build the road that way. LFM, p. Here, building the road is a decision, but not an arbitrary one: different crossing paths have been taken over time, but usually one is most often travelled, and thus preferred on a regular basis.

It is this one that gradually emerges as the most suited for crossing, and the one which the lasting road will follow. On the other hand, what is in the archives is protected , withdrawn from circulation — that is, not open to change and dispute. The relations between the archived items are frozen, solidified. Note the normative role of archives as well: it is illegal to tamper with them. A related metaphor we already encountered above is that of the physical process of condensation: just as an amount of vapor enters a qualitatively different form of aggregation turning into a drop of water, so the empirical-behavioral regularities are turned into arithmetical rules.

It is as if we had hardened the empirical proposition into a rule. And now we have, not an hypothesis that gets tested by experience, but a paradigm with which experience is compared and judged. And so a new kind of judgment. RFM VI The elevation to a new status performed because of the robust, natural agreement is indicated by archiving:.

Because they all agree in what they do, we lay it down as a rule, and put it in the archives. Not until we do that have we got to mathematics. Thus, interestingly, there is a sense in which Wittgenstein actually agrees with the line taken by Hardy, Frege and other Platonists insisting on the objectivity of mathematics.

The alternative account he proposes is that arithmetical identities emerge as a special codification of these contingent but extremely robust, objectively verifiable behavioral regularities. Yet, recall that although the arithmetical propositions owe their origin and relevance to the existence of such regularities, they belong to a different order. This situation would only show that pebbles are not good for demonstrating and teaching addition. The mathematical identity is untouchable as it were , and the only option left to us is to suspect that maybe the empirical context is abnormal.

Thus, the next step is to ask whether we face the same anomalies when we use other objects, such as fruits, pencils, books, bottles, fingers, and so forth. So, let us explore, for a moment, for the sake of the argument, the scenario in which a large variety of ordinary items is subject to this strange variation. Under this assumption, the conclusion to draw is a radical one: were this scenario real, the arithmetical practice of summing could not have even gotten off the ground.

This would be the situation defining the supreme pointlessness of arithmetic, which would thus become a mere game of symbols. As Wittgenstein puts it,.

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Wittgenstein On Mathematics - Oxford Handbooks

I want to say: it is essential to mathematics that its signs are also employed in mufti. It is the use outside mathematics, and so the meaning of the signs that make the sign game into mathematics. RFM IV However, the wild variation described above does not exist. It is a contingent, brute natural fact again!


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Moreover, it is precisely the existence of such regularities — together with, as we will see in a moment, regularities of human behaviour — that makes possible the arithmetical practice in the first place. Wittgenstein, however, realized this rather late, as Steiner [] documents.

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A closer look at the contingent regularity relevant in this context — behavioural agreement — is now in order. Its existence is supported by the already discussed facts: i we can be trained to have these reactions, and ii the world itself presents a certain stability, many regular features, including the regularity that people receiving similar training will react similarly in similar situations. There surely is a neuro-physiological basis for this; cats, unlike dogs, cannot be trained to fetch.

Yet, as stressed above, it is crucial to note that speaking in terms of behavioural agreement when it comes to understanding the mathematical enterprise should not lead one to believe that Wittgenstein is in the business of undermining the objectivity of mathematics. According to him, Wittgenstein maintains that at any step in a calculation we could go any way we want — and the only reason that we go the way we usually go is an agreement between us, as the members of the community: in essence, a convention we all accept And which, since it might be changed, is entirely contingent.

Thus, one should say that a mathematical identity is true by convention; that is, it is taken, accepted as true by all calculators because a convention binds them. However, textual evidence can be amassed against this reading. Convincing passages illustrating this point can be found virtually everywhere in his later works, and Gerrard [] collects several of them.

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Given certain principles and laws of deduction, you can say certain things and not others. Is this what I am saying? So, it is simply not the case that the truth-value of a mathematical identity is established by convention. There is no opinion at all; it is not a question of opinion. They are determined by a consensus of action : a consensus of doing the same thing, reacting in the same way. There is a consensus but it is not a consensus of opinion.

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We all act the same way, walk the same way, count the same way p. The specific kind of behavioural agreement in action is a precondition of the existence of the mathematical practice.


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The only degree of freedom is to avoid laying down these rules, not to adopt alternative rules. It is only in this sense that the mathematician is an inventor, not a discoverer. While the behavioural agreement constitutes the background for the arithmetical practice, Wittgenstein takes great care to keep it separated from the content of this practice Gerrard [, ]. As Gerrard observes, this distinction corresponds, roughly, to the one drawn in LFM, p. The very fact of the existence of this background is not amenable to philosophical analysis.

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This is a question that he, qua philosopher, does not take to be his concern. He sees himself as being in the business of only describing this background, with the avowed goal of drawing attention to its existence and overlooked function. But the task of philosophy is, and can be, only to describe, not explain, the fundamental role of the regularity of human mathematical behaviour.

If we recall the black squares example, the guide must play the role of a regress-stopper, constituting the explanation as to how all possible interpretations and distractions are averted. The mind and this guide form an infallible mechanism delivering the result.

Wittgenstein, Mathematics and World

According to P. Thus there is ample room for further discussion of his views, and especially for clarifying how his philosophy of mathematics complements, or even augments, his relatively better understood philosophy of language and mind. The relation between his view of mathematics and that of the professional mathematicians is yet another interesting and potentially controversial matter worth of further study — for, if we are to believe him,.

A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop. He has learned to regard them as something contemptible and… he has acquired a revulsion from them as infantile. That is to say, I trot out all the problems that a child learning arithmetic, and so forth, finds difficult, the problems that education represses without solving. I say to those repressed doubts: you are quite correct, go on asking, demand clarification!

PG , For comprehensive bibliographical sources, see Sluga and Stern [] and Floyd [], and, for material available online, see Rodych [] and Biletzki and Matar []. Sorin Bangu Email: sorin. Ludwig Wittgenstein: Later Philosophy of Mathematics Mathematics was a central and constant preoccupation for Ludwig Wittgenstein — Which 'ism'? The first is the road building process: It is like finding the best place to build a road across the moors. As Wittgenstein puts it, I want to say: it is essential to mathematics that its signs are also employed in mufti.

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The relation between his view of mathematics and that of the professional mathematicians is yet another interesting and potentially controversial matter worth of further study — for, if we are to believe him, A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop.

PG , 7. References and Further Reading For comprehensive bibliographical sources, see Sluga and Stern [] and Floyd [], and, for material available online, see Rodych [] and Biletzki and Matar []. Pears and B. PI Philosophical Investigations , []. The German text, with a revised English translation by G.