Wittgenstein on Mathematics

Severin Schroeder

• Publisher: Routledge
• Year: 2020
• ISBN:
  9781000318296
• Edition: 1
• Topic: Matemáticas
• Subtopic: Aritmética

This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketches of Wittgenstein's early views of mathematics, in the Tractatus and in the early 1930s. Then (in Part II), Wittgenstein's mature philosophy of mathematics (1937-44) is carefully presented and examined. Schroeder explains that it is based on two key ideas: the calculus view and the grammar view. On the one hand, mathematics is seen as a human activity — calculation — rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters (on mathematics as grammar; rule-following; conventionalism; the empirical basis of mathematics; the role of proof) explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein's provocative views on Hilbert's Formalism and the quest for consistency proofs and on Gödel's incompleteness theorems.

Part I
Background

1 Foundations of Mathematics

Philosophy of mathematics has been to a large extent a discussion of the foundations of mathematics, that is, of attempts to show that all mathematical developments were reliably true, laying to rest any worries about unproven assumptions or hidden inconsistencies. Monographs on the philosophy of mathematics often contain three main sections covering the three major schools offering new and more rigorous foundations to mathematics: logicism, formalism, and intuitionism. At least at first glance, Wittgenstein’s writings appear to be no exception to this trend. Of course, the title of his book Remarks on the Foundations of Mathematics was chosen by the editors, Anscombe, Rhees and von Wright, who are also responsible for the selection of remarks included in this posthumous collection. But that title is not inappropriate in light of some of Wittgenstein’s own characterisations of his work (cf. Mühlhölzer 2010, 1–10). The Introduction to his Philosophical Investigations promises remarks on many subjects, including ‘the foundations of mathematics’ (which in the end he decided to remove with a view to a separate publication). And the first sentence of his 1939 lectures on the philosophy of mathematics is: ‘I am proposing to talk about the foundations of mathematics’ (LFM 13).

On the other hand, one of the main concerns of Wittgenstein’s philosophy of mathematics was the rejection of any foundationalist project:

What does mathematics need a foundation for? It no more needs one, I believe, than propositions about physical objects—or about sense impressions, need an analysis. What mathematical propositions do stand in need of is a clarification of their grammar, just as do those other propositions.

The mathematical problems of what is called foundations are no more the foundation of mathematics for us than the painted rock is the support of a painted tower.

(RFM 378bc)

If, in spite of his hostile attitude towards projects such as logicism, intuitionism, or formalism, Wittgenstein had no qualms describing his own work in the area as concerned with the foundations of mathematics, that is no doubt partly because he thought it important to provide a critical discussion of such foundationalist attempts, especially of logicism, having been a collaborator and friend of Bertrand Russell’s, one of logicism’s leading proponents. However, Wittgenstein often used the expression ‘foundations of mathematics’ in a different sense: for exactly the kind of ‘clarification of the grammar’ of mathematical expressions that would show up the foundational project as misguided.¹ Thus, in The Big Typescript he writes:

1.Wittgenstein displayed the same kind of ambivalence with respect to the term ‘meaning’. Taking the word in a referentialist sense (illustrated by Augustine’s ideas) he rejects questions of meaning as irrelevant: ‘No such thing [as meaning] was in question here, only how the word “five” is used’ (PI §1). But later on he suggests a positive account of meaning as use (PI §43).

what has to be done is to describe the calculus—say—of the cardinal numbers. That is, its rules must be given, and thereby the foundation is laid for arithmetic.

Teach them to us, and then you have laid its foundation.

(BT 540)

Again, in the final remark of the typescript that was published as ‘Part II’ of Philosophical Investigations he explains this different sense of the expression ‘foundations’ (Grundlagen):

An investigation entirely analogous to our investigation of psychology is possible also for mathematics. It is just as little a mathematical investigation as ours is a psychological one. It will not contain calculations, so it is not for example formal logic. It might deserve the name of an investigation of the ‘foundations of mathematics’.

(PPF xiv)

The main part of this monograph will be devoted to a presentation and discussion of Wittgenstein’s own account of the ‘foundations of mathematics’ in the latter sense of the expression: as a clarification of the role and function of mathematical language, the meaning and status of mathematical propositions. First, however, I shall turn to logicism, the foundationalist project that Wittgenstein was most familiar with, briefly characterise its motivation and give a brief outline of its basic ideas, before presenting and discussing Wittgenstein’s objections to it. Wittgenstein’s rejection of logicism, the position of his philosophical father figures, Frege and Russell, can be regarded as clearing the ground for the development of his own ideas. There are two other major contemporary ideas concerned with foundationalist endeavours that Wittgenstein discussed and criticised: the alleged importance of inconsistency proofs and Gödel’s first Incompleteness Theorem. In those cases, however, Wittgenstein’s response is very much a corollary of his positive account of mathematics, so I shall consider them only towards the end of my book.

***

Why should mathematics be thought to need a foundation? The motivation for a foundationalist project may be (i) epistemological or (ii) ontological. I shall briefly consider them in turn.

(i) Epistemological concerns. Morris Kline describes the history of mathematics as a tragedy: after a triumphant beginning of mathematizing the sciences in order to read the book of nature written in the language of mathematics, mathematicians experienced an increasing ‘loss of certainty’. For since the development of analysis in the 17th century,

mathematics had developed illogically. Its illogical development contained not only false proofs, slips in reasoning, and inadvertent mistakes which with more care could have been avoided. Such blunders there were aplenty. The illogical development also involved inadequate understanding of concepts, a failure to recognize all the principles of logic required, and an inadequate rigor of proof; that is, intuition, physical arguments, and appeal to geometrical diagrams had taken the place of logical arguments.

(Kline 1980, 5)

Calculus was a powerful tool in physics, but its concepts were ill understood and apparently illogical. The idea that one could calculate instantaneous velocity, i.e. the speed of an object during an interval of time that is zero, while the object does not actually move, was quite incomprehensible.

Again, in the 18th century there arose the strange problem of infinite series. Consider the much disputed problem of computing the sum of the following infinite series of numbers:

• [S] 1 − 1 + 1 − 1 + 1 − 1 …

On the one hand, it can be argued that S must equal 0, for it can be written thus:

• (1 − 1) + (1 − 1) + (1 − 1) …

I.e. as an infinite sum of bracketed expressions that all equal 0.

On the other hand, it can also be written as follows:

• 1 – (1 – 1) – (1 – 1) – (1 – 1) …

(for − 1 + 1 = − (1 − 1)). So the result must be 1, since one keeps subtracting 0 from 1.

But applying the same arithmetic rule according to which a minus sign in front of a bracket inverts minus and plus inside the brackets, S can also be re-written in this way:

• 1 − (1 − 1 + 1 − 1 + … ).

Hence:

• S = 1 − S.

It follows that S = ½ (Kline 1980, 142; Giaquinto 2002, 6). It would appear from these three contradictory results that important parts of 18th-century mathematics contained inconsistencies.

In response to such conceptual puzzles and flagrant inconsistencies, mathematicians in the latter half of the 19th century made an effort to put mathematics on a clearly defined and rigorous footing. Foundationalist programmes grew out of that 19th-century rigorisation of mathematics. Thus, Russell describes his endeavours to reduce mathematics to logic as motivated by a craving for absolute certainty, of which traditional mathematics often fell short:

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies.

(Russell 1956, 53)

The logicist programme has two parts: (1) showing that all primitive concepts of mathematics can be defined in terms of purely logical terms; (2) showing that all axioms which serve as a basis for mathematics can be derived from a number of purely logical principles by demonstration. However, although a successful reduction of all mathematics to transparent concepts and rules of logic would indeed afford the desired certainty for mathematics, it is not a necessary condition for achieving that aim. To avoid the kind of conceptual confusion and inconsistency illustrated above, mathematicians have to give sufficiently precise and rigorous definitions of their fundamental concepts; but that does not mean that those concepts have to be reduced to purely logical concepts. Indeed, the desired rigorisation of the fundamentals of analysis—the area where conceptual problems and contradictions had appeared so problematic—was completed, mainly through the work of Karl Weierstrass (1815–97), by the end of the 19th century:

Weierstrass’s work finally freed analysis from all dependence upon motion, intuitive understanding, and geometric notions which were certainly suspect by Weierstrass’s time.

(Kline 1980, 177)

Yet such necessary clarification and rigorisation did not require or involve any logicist reductionism. In other words, Russell’s concern for certainty makes it psychologically understandable that the logicist project appealed to him, but it provides no compelling reason to regard that project as necessary.

Still, it would appear that concerns about inconsistency justify the endeavour to ...