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The Philosophy of Set Theory
An Historical Introduction to Cantor's Paradise
By Mary Tiles
Dover Publications, Inc.Copyright © 1989 Mary Tiles
All rights reserved.
The Finite Universe
Infinite, or transfinite numbers and transfinite set theory are relative newcomers on the mathematical scene. Cantor's most important papers on the theory of transfinite numbers, the culmination of work begun in 1870, were published in 1895 and 1897 (Cantor, 1915). Thus, if one were to proclaim them to be inventions, figments of mathematical imagination, one would not be casting aside centuries of tradition. Indeed, the weight of tradition is firmly opposed to giving credence to talk of any such things. The infinite only gained acceptance and a degree of mathematical respectability because traditional ways of thinking were being cast aside.
Also the revolution has not been complete. We are still more likely to be suspicious of talk of infinite numbers and infinite sets than of talk of the familiar whole numbers and fractions that are used in counting and in the simple computations which are an essential part of many practical activities and all commercial transactions. These misgivings about any theory of transfinite sets or transfinite numbers are reflected by those philosophers who would accept the label 'finitist'. There are prima facie two possible types of finitism:
Strict Finitism The strict finitist does not recognize any mathematical use of the infinitistic notions or of infinistic methods (summing an infinite series for example). The strict finitist might also want to distinguish between 'small' and 'large' finite numbers, arguing that there is a (finite) upper limit on the numbers with which we can deal intelligibly, although there may be much debate about how any such limit can be set or determined.
One route to such a position is explored in Wright (1980) which elaborates on themes suggested by Wittgenstein (1967). The basic claim here is that we can only know of the existence of those numbers which we could actually write down in some notation and 'take in' all at once, or survey. Similarly, it is suggested, we can only be convinced by a proof which we can survey. It is quite possible for a purported proof to be too long and too complex for us to take in (for example a computer generated 'proof' running to a hundred pages). If such a sequence cannot be taken in, it cannot be a proof, for it cannot perform the function of a proof, which is to convince us that its conclusion is true (given agreed assumptions which form the premises of the proof) by showing us why it must be true. On this basis it may be supposed that there is an upper bound on the natural numbers (it just is not true that every number has a successor even though it would be impossible to specify one which does not, since if we can specify n we can specify n + 1). This bound will be set by our cognitive powers coupled with the efficiency of our system of notation. Models of this situation are afforded by computing systems whose upper limits are imposed by the memory size together with the structure of the software.
It is clear, however, that any precise statement of a strict finitist position will be a delicate matter. The distinction between 'small' (surveyable) finite numbers and 'large' (unsurveyable) finite numbers has much in common with the distinction between men who are bald and men who are not; the distinction is real even though the loss of a single hair cannot effect the transition from not being bald to being bald. Similarly the strict finitist will need to say that the distinction between surveyable and unsurveyable numbers is real even though the addition of 1 is not sufficient to effect the transition to unsurveyability. (Further possible motivations for pursuing this position will emerge in chapter 2, but these are essentially linked to arguments which appear to close off the option of classical finitism.)
(Classical) Finitism The (classical) finitist is quite happy about the mathematical status of the familiar natural numbers, however large, but refuses to accept the need for infinite numbers or sets, and indeed does not regard talk of such things as coherent. However, he does not dismiss all notions of infinity or all mathematical treatments of infinite series. His insistence is that such things are only potentially, not actually, infinite; any actual segment of such a series is always finite, but always incomplete. It is in this incompletability that its potential infinity consists. The mathematician, when dealing with these always incomplete, potentially infinite series, must thus use methods which differ from those used when dealing with completed or completable finite series.
Finitists of both types argue not only that we do not need infinite numbers or a theory of infinite sets, but also that experience affords us no basis on which to give sense to talk of them. We do not need infinite numbers or infinite sets because all applications of mathematics are to finite systems, finite quantities and finite numbers of entities (see Hilbert, 1925). Since any application involving measurement can only ever be approximate, in view of the fact that every measuring instrument, however good, has a built in margin of error, we only ever need a finite number of decimal places when assigning a numerical value to a physical magnitude. Moreover, it may be argued, our experience is that of finite beings and takes the form of finite sequences of impressions of entities which are also finite. There can thus be no empirical meaning given to talk of the actual infinite. Following this line of argument, some empiricists have been led to conclude that there is no sense to be given to such talk. These claims clearly rest on (a) an assumption about the finitude of the universe within which mathematics is applied, (b) an assumption that mathematics is only applied to this universe via processes of measurement, and (c) an assumption that meaning is to be equated with empirical meaning.
This route to making out the finitist case makes it rest heavily on empiricist doctrines about meaning, doctrines which have been seriously challenged, as a result of the failures of logical positivism, by work which follows in the wake of Quine's 'Two Dogmas of Empiricism' (1953). As so formulated it is therefore unlikely to be taken seriously by philosophers advocating realism as the general position to be adopted in the theory of meaning. If we take it that a necessary (but not sufficient) condition of realism with respect to statements of a given kind is that it is held that all statements of this kind are determinately true or false independent of our ability to know which is the case (cf. Dummett, 1963), it will be clear that the realist will not be impressed by arguments against the infinite which appeal to restrictions on our cognitive capacities imposed by either the finiteness of our intellects or the finite character of all experience. In general he will be prepared to allow our ability to conceptualize and entertain possibilities to outrun our capacity (even to know in principle) which, if any, of these possibilities are ever actualized. But what case can the realist make which might persuade the finitist (an anti-realist about the infinite), motivated by empiricism, of the error of his ways?
There are two challenges to which the finitist position, as outlined above, is open. It may be conceded that in our measuring of features of the physical world finite numbers and finite strings of decimals will always serve, but if the finitist thinks of his measurements and observations as measurements and observations of features of a physical world, then he is making two assumptions which require him to think both beyond the finite and beyond the bounds of experience conceived as a sequence of impressions. First he presumes that what he encounters are items extended in space and existing for some, possibly very short, period of time. In doing so he makes at least implicit use of the notion of continuity, or of continuous extension, for space and time are presumed to be continuous. And the notion of continuity brings with it that of infinite divisibility. Secondly he presumes that the things he observes and measures are all parts of a single physical world and can all be located in a single spatio-temporal framework. This all-embracing character of space, time and the universe suggests not only that they must be thought of, even though they are in no sense observable, but also that they must be thought to be infinite since neither space nor time can coherently be thought to have a boundary. So, it would seem, talk of space and time already takes us beyond what can be given content by reference to immediate experience and already threatens to introduce the infinite.
We shall see that, in order to respond to these challenges, the finitist (unless he is prepared to reject the continuity of space and time or to deny the possibility of giving any empirically significant theoretical account of it), must abandon strict finitism and must admit that some sense can be given to talk of the infinite, but without allowing that this legitimates talk of infinite numbers. He will insist that the only possible sense of 'infinity' which can be grounded in experience, more liberally construed as including a reflective awareness of rules and principles, is that of the potentially infinite. It will be argued below that there is a contradiction involved in thinking that a number can be assigned to a potential infinity or in thinking that it forms the sort of determinate collection that a set is required to be. Thus what the finitist has to do is to show that the challenges arising out of the continuity and the unity of space and time can be handled by invoking only the notion of potential infinity. He has to argue that he is not, in virtue of his presuppositions about space and time, committed to the supposition that there are actually, in the physical universe, undetectable infinitely small and infinitely large quantities, or any actually infinite sets of points. Moreover, the route taken by the argument to be considered will reveal grounds other than those tied to forms of empiricism, verificationism or global anti-realism for advocating a finitist position. It will thus suggest that realism about the infinite is no automatic consequence of a generally realist stance elsewhere.
2 Continuity and Infinity
Whereas the question 'How many points are there in a line?' could at least have been asked prior to Cantor, his answer, 'There are 2[??] and 2[??] = [??]1' could not even have been understood or recognized as an answer. It was Cantor who introduced and defined the symbols '[??]0' and '[??]1' as symbols for infinite (cardinal) numbers. We cannot begin to understand his answer without knowing how these symbols are defined, and to understand their definitions it is necessary to know something about the theoretical background which legitimates them as definitions. For we first have to be convinced that there are, or at least might be, such things as infinite numbers for which we can introduce names. Thus it is only within the framework provided by transfinite set theory that it becomes possible to contemplate giving a numerical answer to our question; this framework provides the form of an answer, if not an actual answer. To this extent Cantor's work gives sense to a question which previously lacked any precise sense.
Prior to Cantor the natural answer would have been 'Infinitely many', and if the question 'And how many is that?' were further pressed, it would have been taken as showing a lack of understanding of what is meant in this context by 'infinitely many'. This is not to say that finitism was the only possible position prior to Cantor's work; it is just that the non-finitist would not have been able to give an answer couched in terms of transfinite numbers, or any other numbers.
One could take Berkeley as a representative finitist and Leibniz as a representative non- finitist. Berkeley insisted that space can only ever (actually) be infinitely divided and was highly critical of Newton's infinitistic methods (Berkeley, 1734). Leibniz was well aware of the distinction between potential and actual infinites and believed that matter is actually infinitely divided (Leibniz, 1702). He was prepared to admit the existence of infinitesimal magnitudes and himself developed an infinitesimal calculus at much the same time as Newton. But it is clear that Leibniz could have attached no more sense than Berkeley to a question about the number of points in a line. Here there is not only a question about whether 'Infinitely many' is a legitimate answer to a 'How many?' question but also about whether the points on a line form a totality of which one can sensibly ask 'How many are there?'
One might suggest, as Wittgenstein (1967, p. 59) does, that such a question has as much or as little sense as 'How many angels can dance on a needlepoint?' The problem is that one is simply unclear as to how to determine the totality to be numbered. It is not clear that even one angel can dance on a needlepoint, whereas there are points in lines. But a line is a single, continuous whole. How can it be made up of points? If one point is to be continuously adjoined to another, it must be no distance from it and therefore must in fact coincide with it. Alternatively, given any two distinct points there will always be some distance between them, and so also more points between them. This suffices to show that no finite number can be assigned to the points in a line because one will always be able to find more points in between those counted.
This is merely another way of saying that a continuous line is infinitely, i.e. indefinitely, divisible, for points are points of division. The points in a line mark the boundaries of parts (actual or potential) of the line. If a line is potentially infinitely divisible (but only ever actually finitely divided), there must be a corresponding potential infinity of points of division. If the line is actually infinitely divided, there must be an actual infinity of points of division. But neither from the point of view of the finitist with respect to actual division, nor from that of the non-finitist do the points in a line form the sort of totality to which one could, in principle, assign a number. The answer 'Infinitely many' is thus both an assertion that there are more than any given finite number and a refusal of the demand for a numerical answer.
That this is the right response appeared to have been conclusively demonstrated by Aristotle in his treatment of Zeno's paradoxes (Aristotle, Physics, Bk.II). Zeno's paradoxes are important because they seem to show that it is simply impossible to think of a continuum, such as space or time, as made up out of indivisible atoms, whether these be points or minimally extended regions of space and/or time. There is a contradiction between the existence of movement and an atomistic conception of space and time. But we know that movement exists. This is not something we can deny, therefore it is the possibility of an atomistic view of continua such as space and time which must be rejected. This is not just a human impossibility but a logical impossibility, and therefore one which any philosopher, whether of realist or anti-realist persuasions, must accept as indicative of a limitation on what is conceptually possible.
These paradoxes have, naturally, been discussed at length (see, for example, Salmon, 1967). However, the point of introducing them here is not that of presenting a new resolution but of showing how one of their oldest resolutions, that indicated by Aristotle, gave support to a finitist position – the position which rejects all talk of infinite sets, infinite numbers, etc., as unnecessary nonsense; something which could never form part of mathematics conceived as a rational scientific discipline. In later chapters we shall see how the Aristotelian view had to be rejected in order that the case for the infinite could be made out.
3 Zeno's Paradoxes
The strategy of Zeno's paradoxes may be reconstructed in the following way (modified from Owen, 1957).
Thesis I Neither space nor time are pluralities. For if they are pluralities it must be possible to specify the units (atomic parts) of which they are composed. But
Thesis II Any attempt to treat space or time as composed of atomic parts leads to absurd conclusions. For suppose they are composed of atomic parts, then a space or a time must either be divisible without limit or there must exist limits of division.
a. Suppose they are divisible without limit, then
1 a runner cannot complete a racecourse, and
2 Achilles cannot catch the tortoise.
b. Suppose there are limits of division, then either these have size (magnitude) or they do not.
(a) Suppose they have size, then
3 the paradox of the stadium.
(b) Suppose they have no size, then
4 the arrow paradox.
Thus the alternatives lead to absurd conclusions, and neither space nor time are pluralities.
Excerpted from The Philosophy of Set Theory by Mary Tiles. Copyright © 1989 Mary Tiles. Excerpted by permission of Dover Publications, Inc..
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