From Mathematics to Philosophy (Routledge Revivals)

Hao Wang

• Éditeur: Routledge
• Année: 2016
• ISBN:
  9781134884407
• Édition: 1
• Sujet: Philosophy
• Sous-sujet: Philosophy History & Theory

First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method of approach called substantial factualism which the author asserts allows for the development of a more comprehensive philosophical position by not trivialising or distorting substantial facts of human knowledge.

II
Characterization of General Mathematical Concepts

1 Natural Numbers

We successfully learn the use of the natural numbers 1, 2, 3, etc. at an early age. It does not seem possible to reduce our knowledge of the sequence of natural numbers to anything intuitively more basic. It is hard to imagine how a normal person can fail to use small natural numbers correctly.

A natural number can be used both as an ordinal and a cardinal number. Thus, take the process of counting

We point to one figure and say 1, another and say 2, a third and say 3; then we conclude that there are 3 figures. The square is not 2 but rather the second in our counting. Only the last, 3, is used as a cardinal number. We can count in different orders and yet get the same final result: this seems so obvious that it is hard to imagine how it could be different. It also seems evident that the last ordinal used in counting gives the cardinal number of the collection. We can use different symbols for cardinals and ordinals. We can also, as we often do with natural numbers, use the same set of symbols to stand for both ordinals and cardinals. Such a practice is no longer feasible when we come to infinite cardinals and ordinals. Thus, if we view cardinal numbers as obtained by counting, the concept of ordinal number seems more basic.

Often we can compare the cardinal numbers of two collections without counting; for example, to determine whether there are enough chairs in a room, for people present, or whether there are more males or females at a dance. In such cases, it is true that the acts of matching off must be carried out one by one in temporal succession, even if it is simply the process of looking to see that each chair is occupied. Nonetheless, the ordinal numbers are not involved because we do not have to keep track of the relative order of which of two chairs was first checked: we need only distinguish checked from non-checked. The concept of more or equal is prior to both cardinals and ordinals. Matching off is sufficient for arranging different heaps of pebbles in order of more and less. But in order to get representations of the natural numbers, we need the concepts of one and one more. Given these concepts, both cardinals and ordinals can be gotten. It seems reasonable to say that both ordinals and cardinals presuppose the concept of a unit and the concept of adding one more, but neither is more primary than the other. We should distinguish all sets with 3 members from a particularly simple set with 3 members. A model set is enough to give us the concept 3 even though once given it is to apply to all sets with 3 members. When we are fortunate, we may be able, as a matter of fact, to get representative groups of physical objects for all natural numbers up to 1000, say, even though, without the notion of one more, we are not able to know that this is actually so.

If we represent natural numbers by strings of strokes, the role of natural numbers can be played by such strings of strokes, actual or imagined. ‘2 plus 2 is 4’ means, if we write down two strokes, and two strokes again, we get four strokes. This cannot mean that actually in writing, such a state of affairs must always happen, but rather that if it does not happen, we are inclined to search for other explanations in order to uphold the conclusion that without interference, such would happen. We can avoid all such imaginary difficulties if we avoid them in going to the next stroke. It seems there are still questions in conceiving of arithmetic as concerned with concrete numerical symbols such as strings of strokes whose shape is recognizable by us independently of time and space, and of minor differences in their occurrences. They are, however, useful because we know that as a matter of fact we can use their concrete occurrences, and in theory we can depend on their abstract form. There is no possibility of error with type, and no actual error even with token. To talk about strings of strokes is one way of communicating our thoughts on the justification of arithmetic. It is useful even though inadequate.

The continuing philosophical interest in natural numbers and the continuum may be contrasted with the situation of negative and complex numbers. Historically these numbers appeared to be puzzling at first and there is some resistance to absorbing them when first exposed to them in schools. But they are ‘reducible’ to simpler concepts and do not create serious conceptual problems. When one was accustomed to think only of real numbers, one was puzzled by the question: how could there be any number which, when mutiplied by itself, yields minus 1? what sort of things are the complex numbers? or, more simply, what are complex numbers? There were, therefore, speculative discourses on the true metaphysic of complex numbers. Around 1800, Gauss and C. Wessel supplied interesting and definite answers to this question which are still preserved in mathematics textbooks. Nobody would now regard their representations of complex numbers by pairs of real numbers and by diagrams as philosophy. In fact, as their replies gradually got accepted by everyone, the original question ceased to be of philosophical interest, and speculative doctrines on the topic have been quickly forgotten. Thus, we have a clear case where a philosophical question got solved and immigrated into mathematics.

The question about natural numbers could be, what is a natural number? Or how do we characterize the concept of natural number? What do we mean by the term ‘natural number’? At first, we may try to answer the question by giving typical examples such as 25, 61, 1000. Or, we may say that natural numbers are 1, 2, 3, and so on. To explain the phrase ‘and so on’ in this particular context can without exaggeration be said to be one of the central problems of mathematical logic. If we say that a natural number is either 1 or the successor of another natural number, then we have made no sufficient restriction as to what is not a natural number. We can for example also take an arbitrary additional thing ω as a natural number so that we have besides 1, 2, 3, and so on, also ω, ω + 1, ω − 1, ω + 2, ω − 2, and so on; although this sequence contains more things than what we would like to call natural numbers, it satisfies the condition that a natural number is either 1 or a successor. A next attempt is to require further that natural numbers must satisfy the principle of mathematical induction; in other words, every set which contains 1 and contains the successor of every member of it must contain all natural numbers. It was thought that if we take the common part of all such sets, it must contain exactly the desired natural numbers and no more, since otherwise there would be a smaller set which contains 1 and the successor of every member of it. This, however, leads immediately to the question of characterizing the concept of an arbitrary set of natural numbers, and also the question of reducing arithmetic to logic.

There are interesting philosophical views on the nature of arithmetic which have little to do with mathematical logic. A famous example is Kant’s theory in terms of the form of the pure intuition of time. Brouwer argues¹ that the position of intuitionism:

Has recovered by abandoning Kant’s apriority of space but adhering the more resolutely to the apriority of time. This neo-intuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time, as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare two-oneness. This intuition of two-oneness, the basal intuition of mathematics, creates not only the numbers one and two, but also all finite ordinal numbers, inasmuch as one of the elements of the two-oneness may be thought of as a new two-oneness, which process may be repeated indefinitely.

This oft-quoted passage is singularly attractive, capturing or at least hinting at a familiar basic vague thought about the foundation of arithmetic and its place in human thought. It is, unlike additional ideas of Brouwer’s intuitionism, rather neutral to different approaches to the foundation of arithmetic and can even serve to enrich the logic-oriented views we are about to consider.

Leibniz proposes to prove numerical truths such as 2 + 2 = 4 by definitions (of 2 and 4) and axioms only. Frege points out² that the associative law is implicitly assumed in moving from 2 + (1 + 1) to (2 + 1) + 1. The crucial missing step is that in addition to the definitions of each number, 2 = 1 + 1, 3 = 2 + 1, etc., we need also the recursive definition of addition which says in general a + (b + 1) = (a + b) + 1. This is indeed how Grassmann³ corrects the proposed proof of Leibniz. In fact, Grassmann introduces recursive definitions of addition and multiplication, and proves ordinary laws of arithmetic by mathematical induction. He offers a development which amounts to characterizing all integers (positive, negative, and 0) as an ordered integral domain in which each set of positive members contains a smallest element. A formal imperfection of his calculus is the omission of the principle that distinct numerals denote distinct numbers. As a result, his whole system would be satisfied if all integers are taken to be identical with one another.

The nature of recursive definitions presents some complication. Frege feels that they are not proper definitions. Both he and Dedekind prefer to use more complex and essentially set-theoretic explicit definitions which yield the recursive definitions as consequences. Today we generally agree that it is all right to begin with recursive definitions and justify them directly by a sort of combinatorial intuition.

The omission in Grassmann’s calculus gets corrected by Dedekind,⁴ and we arrive at what are commonly known as Peano’s axioms:

• P1 1 is a number.
• P2 The successor of any number is a number.
• P3 No two numbers have the same successor.
• P4 1 is not the successor of any number.
• P5 Any set K contains all numbers, if 1 belongs to K and the successor of any member of K also belongs to K.

In a private letter,⁵ Dedekind gives an account of how he arrived at the above five axioms by analyzing the concept of number. Dedekind poses the question rather exactly:

Which are the mutually independent fundamental properties of the number-sequence N, i.e. those properties which are not deducible from one another and from which all others follow? How should we divest these properties of their specifically arithmetical character so that they are subsumed under more general concepts and such activities of the understanding, which are necessary for all thinking, but at the same time sufficient, to secure reliability and completeness of the proofs, and to permit the construction of consistent concepts and definitions?

According to Dedekind, one is then forced to accept the following facts. 1 N is a set of elements which are called numbers. 2 The elements of N stand in a certain relation to one another, they are in a certain order determined, in the first place, by the fact that to each definite number n, belongs again a definite number n' or ϕ(n), the successor of n. The mapping ϕ maps N into itself and ϕ(N) is a part of N. 3 Distinct numbers have distinct successors, i.e., the mapping ϕ is one-one. 4 Not every number is a successor, i.e. ϕ(N) is a proper part of N. More precisely, 1 is the only number not in ϕ(N). 5 One might think that these facts are sufficient to determine N. But they also apply to every set 5, which, in addition to N, contains also a set T of arbitrary elements t. One can always extend the one-one mapping ϕ so that ϕ(T) = T. In particular, we may use the example mentioned before, where T = {…, ω − 1, ω, ω + 1, …}. According to Dedekind, one of the most difficult points of his analysis is to find what to add to the above facts in order to cleanse our set S from such alien intruders t which disturb every vestige of order, and restrict ourselves to the set N.

If one assumes knowledge of the sequence N of natural numbers to begin with and accordingly permits himself an arithmetic terminology, then he has o...