Lecture
15: The Nature
of Mathematics
1.
In the Tractatus,
Wittgenstein produced an original theory of number. It is quite technical, and I shall not discuss it here, but, if
you are interested, try the relevant chapters of Mathieu Marion (1998) Wittgenstein,
Finitism and the Philosophy of Mathematics, Oxford: Clarendon Press or
Michael Potter (2000) Reason’s Nearest Kin: Philosophies of Arithmetic from
Kant to Carnap, Oxford: Oxford University Press.
2.
Between 1929 and 1944, over half of what Wittgenstein
wrote was on the subject of the foundations of mathematics, and one of the
series of lectures he gave in 1939 is now published as a book – Cora Diamond
(ed.), Lectures on the Foundations of Mathematics (LFM). An important source of his late views is Remarks
on the Foundations of Mathematics (RFM).
3. The
Tractatus proposes an answer to the question `What is number?’ But this is a typically philosophical
question, and we know that, in his late writings, Wittgenstein denied that such
questions raised genuine philosophical problems; rather, they create puzzles,
or a challenge to see how we have been led astray by language into thinking
that there are genuine problems. His
later methodology consists of (i) searching for the false analogies that are
liable to distort our thinking and sometimes (ii) to make a joke of the
problem.
4. You
can begin to make a joke of the problem `What is number?’ by asking `What is
the number 4?’. The latter already
begins to seem rather strange. Or
(following PI §1) think of a child with
the slip of paper `five red apples’ following its mother’s instructions to make
sure that she gets exactly what is on the slip of paper. The child asks the grocer `Is that apple
red?’ then `Is that apple five?. These
jokes encourage us to query the assumption that a number is an object or is a
property.
5. But
you might still be inclined to say `OK, the number 4 is not an object and it is
not a property, but what is it?’ But
now recall BB, p.1: `…one of the great sources of philosophical bewilderment: a
substantive makes us look for a thing that corresponds to it?’ Here is where we fall victim to a false
analogy. For, if a child asks us `What
is an apple?’ we can point to one, or to several and, with luck, the child
will, as a result, learn to use the word `apple’ correctly. Or perhaps in a science class, the teacher
might raise the question `What is an apple’ and the task would be to give a
definition, one which listed the essential properties of apples. But we can’t point to the number 4,
and is it sensible to ask `What is the essence of fourness?’, or `What is the
essence of number?’ As we know from PI
§68, Witgenstein thinks that we can use the word `number’ so that the extension
of the concept is not closed by a frontier.
In other words, that there is no essence of number.
6. One
of the things to which Wittgenstein became opposed is the idea that we have to penetrate phenomena (PI § 90), to reach a final analysis
which brings to light something hidden in our usual forms of expression (PI § 91). Wittgenstein deprecates questions such as `What is language ?', `What
is a proposition ?' (PI §
92). His antagonism to such questions
can, perhaps, be explained thus: they commit the fallacy of the complex question.
They are of the form: There is something that being an X consists in;
what exactly is it ? The
unacceptable presupposition is that there is
something that being language, or being a proposition consists in; that there
is some hidden essence to be revealed by analysis. If there is no such essence then the project of logical analysis
is like the hunting of unicorns -- a waste of time, since there is nothing to
be found. But, in another sense, to
analyse something just means to inspect it carefully, to take a good hard look at
it with a view to clearing away some problem.
It is this that Wittgenstein
is constantly doing in his later writings.
The enterprise is explained at PI
§ 90: Our investigation is therefore a
grammatical one. Such an investigation
sheds light on our problem by clearing misunderstandings away. Misunderstandings concerning the use of
words, caused, among other things, by certain analogies between the forms of
expression in different regions of language .
-- Some of them can be removed by substituting one form of expression
for another; this may be called an "analysis" of our forms of
expression, for the process is sometimes like one of taking a thing apart.’
7. What
then, should we say in order to deflate the question `What is number’. The following, taken from a recent paper of
mine `The indefinability of “one”’ provides a Wittgensteinian answer:
Number words typically occur in sentences in noun position or adjective position, and this observation fuels the expectation that numbers are objects or properties (properties not of objects, but of sets of objects). The enquiring mind will then tend to wonder about the nature of these objects or properties, and the project of seeking definitions is underway. But it is worth pausing to reflect that numbers were born not for naming or property-ascribing, but for numbering off (counting), and number words occur in a count neither as nouns nor as adjectives; we have merely a series of distinct sounds in a fixed order. Infants learn to recite the first part of this series and, for them, the sounds initially have no more meaning than do the sounds in a tune. At a later stage, when numbering off a collection of objects, a child, like Wittgenstein’s ponderous grocer, takes any one of the objects, says the first word in the series -- “one” --, puts that object aside and proceeds to another object and to the next word in the series. The useful trick with number words is to take the last-uttered word in a particular count as a measure of the size of the collection. Pleasing results start to flow when we do such things as merging counted collections and re-counting the whole. Of course, when we log such results as equations of arithmetic, we abstract from the particular acts of counting.
8. If numbers are not objects, then one of the most popular philosophies of mathematics – Platonism – is kicked out. For one of the tenets of Platonism is that numbers are abstract (atemporal) objects and that the business of mathematics is to discover truths about the relations between these objects. Other aspects of Platonism that Wittgenstein attacks are the idea that mathematicians discover (according to Wittgenstein, mathematicians invent), and the idea that there are truths in mathematics. In the Tractatus, Wittgenstein said `The logic of the world which is shown in tautologies by the propositions of logic, is shown in equations by mathematics’ (T: 6.22). If it was indeed Wittgenstein’s view that, like tautologies, equations are not propositions (T: 6.2, 6.21) then we can tell a plausible story about the development of the later theory of mathematics as an activity issuing not in truths, but in rules, and of the idea that contradictions, since they are not false propositions, are harmless (WVC: 131, 139, 194-201; RFM, Appendix III: §§11, 12). The claim that mathematicians discover truths is not, of course, unique to Platonism, and Wittgenstein attacks other philosophies of mathematics, including Formalism and Intuitionism. His own position, though, has received a rather hostile reception. Some of the difficulties are discussed, in a fairly non-technical way, in our text, Clear and Queer Thinking, Chapter 4.