Wittgenstein's Conception of Mathematical Possibility Revisited
Kengo Okamoto
Tokyo Metropolitan University,
Graduate School of Humanities, Department of Philosophy
fregewittgenstein@gmail.com
20170617
Why do we accept those mathematical propositions that we do? How can we justify,
or at least make it sufficiently intelligible, the fact that we accept them? ― One might
say this is one of the most central problems (perhaps the problem) in the philosophyof mathematics that Wittgenstein tackled throughout. As a preliminary to my talk,I shall clarify here an important part of what I take to be the intent of the problem.
Some might maintain that the problem is in fact illusory since as evidenced by e.g.
the opposition between classical mathematicians and intuitionists, there would
hardly be any one definite collection of mathematical propositions shared by every
one of us (or even by almost all of us). It is true there is a vast body of mathematical
propositions (sentences or formulae) to which virtually everyone finds no difficultyin assenting. But it should be noted, one would say, those seemingly sharedpropositions are in fact homonymous: e.g. classical mathematicians understandthem in one way and intuitionists do in quite another － probably incompatible －way. Hence it is illegitimate or at least too optimistic to presuppose from the very
start that there could be any plausible collection of mathematical propositions thatwe (or at least almost all of us) do accept.
Let me articulate the idea underlying this line of argument a bit further. For
simplicity, we shall introduce the term the “content” of a (mathematical) propositionand provisionally characterize it as follows: the content of a proposition is just what
we should understand of the proposition so that we are entitled to a knower of it (i.e.
to a person who understands it). Although this formulation needs to be elaborated
much further, the term will serve our present purpose well.
Now, I think the characteristic conception of the content of a mathematical
proposition common to those who advocate the line of argument above － i.e. those
who are skeptical about starting from the supposition that there is a more or lessstable body of non-homonymous mathematical propositions shared by (almost) all ofus － could be described as follows: they do believe that the content of amathematical proposition should consist in the way in which the proposition can bejustified. In other words, one understands a mathematical proposition if and only if
she understands the way in which the proposition can be justified (i.e. thejustification condition of the proposition).

Let us call this conception of the content of a proposition the justificatory

conception (JC). It is true the notion of justification alluded to here also needs to beelaborated still further. (As will be clear from our exposition, the term “justification”
is used here in a rather loose sense that permits non-constructive justifications aswell). But most would probably agree that we can explain the reason why those whoendorse JC are doomed to be skeptical about the supposition of the body of nonhomonymous common mathematical propositions as follows.
On the one hand, some of us (e.g. classical mathematicians) find a certain type ofjustifications (classical proofs) as acceptable and others find other types so while theformer refuses the latter types and the latter do the former type: we never accept
one and the same means of justifications uniformly. On the other hand, those whoendorse JC surely adopt as the criterion of the content of a mathematical proposition
the following: the proposition occurring here (i.e. used in one context of ourmathematical practice) and the one occurring there (i.e. used in another context) should
be counted as identical if and only if one and the same justification condition is associatedwith both of them. Those who endorse JC are thus inevitably led to the conclusion thatthe propositions seemingly shared by us are in fact homonymous ones and that thesupposition mentioned above is simply false or at least thoroughly without foundations.
Now, the main claim that I should like to make in the talk is the following: it is justfreeing us from JC and instead providing with us a different, hitherto not wellappreciated alternative conception of the content of a mathematical proposition thatWittgenstein aimed at throughout. But if so, what is that alternative?
As will be expected, we could and should in the first place make the following remark:according to the alternative conception, the main ingredient of the content of aproposition should be the use to which the proposition can be put, or a bit more
specifically, the role played by it in the process of executing some action by an agent. We
would like, however, to put much more emphasis on the modal aspect of the problem: thecontent of a mathematical proposition should be characterized in terms of the (space of)possibilities it brings in. I shall expound this aspect of Wittgenstein’s conception of thecontent of a mathematical proposition more fully in the talk.