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6.2 Finite or Infinite?

There are two principal options open to us. If something is to be actual then we can either maintain that it must be finite, or that can be infinite. Actual things must be determinate, but is not clear whether infinite things can be determinate too. On the face of it, infinite things are unlimited and indefinite, and hence not fully determinate in the required sense.   Mathematics since Cantor, however, has succeeded in giving some kind of determinacy to the notion of infinite sets, and hence it is no longer clear whether actual things are not allowed to be infinite.

    This may seem a rather academic point, but it turns out the the whole difference between classical physics and quantum physics can be made to depend on this decision! If we have actual events, for instance, then the two options are either allowing actual events to succeed each other continuously in time, or requiring events to have non-zero time intervals between them. The purpose of this book to show how this latter choice paves the way for a realistic understanding of quantum mechanics. The fact that quantum theory has proved a good theory therefore provides some kind of evidence to support the idea that all actualities must be finite.

    The hypothesis to be advocated can be framed as the Finiteness Postulate:
*[ 0.3em] Actualities, singly and in aggregates, are necessarily Finite (in the Euclidean sense).
*[ 0.3em]   The qualification `in the Euclidean sense' is included because there is also a problem in mathematics about actual infinities. The mathematical problem, though related to the present options, is to some extent independent of it, because the term `actuality' has a different meaning in mathematics.

Infinity in Mathematics

        The mathematical problem of `actual infinities' is whether certain classes (the classes of whole numbers or reals, for example) are definite sets.   They certainly do not satisfy Euclid's axiom that ``the whole is greater than the parts'', because it is easy to prove, assuming whole numbers to form a set together, that they can be put in a one-to-one correspondence with a proper subset of themselves. There is a simple one-to-one mapping, for example, between all whole numbers and just even whole numbers. This is because we can set up the function

\begin{displaymath}\phi : {\cal N} ~ \rightarrow ~ {\cal N}_e \mbox{ by } \phi (n) = 2n ,

which takes every whole number to just one even number, and vice versa.

  Cantor decided that it is possible for whole numbers and reals to be definite sets, and that if we reject the axiom of Euclidean finiteness, then we are not involved in any contradiction. The mathematical problem of actual infinities is whether this rejection is justified.   It is not accepted in intuitionistic logic, for example, that all whole numbers can be enumerated in any sense sufficiently definite for them to form a set together.   Intuitionistic logicians say, therefore, that they are rejecting the notion of an `actual infinity' (see e.g. Dummett [1977]). They are using the term `actual', however, in a different sense from its use in the philosophy of nature. The mathematical meaning of the term `actual' is practically synonymous with `definite', while its meaning in the philosophy of nature includes `physically existing' along with definiteness. The question of whether whole numbers or reals form definite sets is therefore independent of the question whether infinite things exist actually in nature.



When we are asking in this chapter the question, `Are there any actual infinities?', we want to know whether the particular individuals in the world are themselves in any way infinite, have any infinite or unbounded properties, form infinite sets when aggregated. The Finiteness Postulate asserts that the answers to all these questions are `no'.

There are a number of ways we can justify the acceptance of the Finiteness Postulate. We could argue, as I mentioned above, that it is a point in its favour that it leads to quantum physics. Alternatively, we could follow the line of thought in philosophy which rejects the notion of any `actual infinity' as `a logical self-contradiction [as well as] an empirical falsehood' 6.1 .       This finitist line of thought in the philosophy of nature stretches from Aristotle through Kant to Whitehead. By taking `actuality' in its widest sense, the argument can be applied in the philosophy of mathematics itself to reject Cantor's assumptions, most noticeably by intuitionistic mathematicians after Brouwer. These arguments tie together the problems of natural and mathematical infinities. This connection may or may not be a good thing, so I will present the finitist arguments next, and then discuss what relevance they carry for the philosophy of nature.

Since infinity is what is in-finite, the finitist tradition says, it is greater than any determinations, and an `actual' or `fully determinate' infinity is a logical contradiction. Thus `infinite actuality' can neither be a particular, nor an attribute of a particular thing.     We are not obliged to accept Cantor's construction of definite infinite numbers (he constructs a whole series of what he calls `transfinite numbers' $ \aleph_0 , \aleph_1 , \ldots $), because in his proofs,   Quine [1961] shows, Cantor presupposes that infinities were already definite. In other words, the `axiom of infinity' in set theory is not derivable from the remaining axioms.   Thus, it may be possible to form a set theory without it: see e.g. Mayberry [1988] for some suggestions along these lines. If this could be worked out in sufficient detail, so that calculus (and other branches of mathematics) are still possible, then the Finiteness Postulate would follow automatically because all set would be finite in the Euclidean sense. As yet, however, this working out is not accomplished, so I reserve my judgement as to whether it would be suitable for a philosophy of nature. In the meanwhile, I assume that the calculus of the continuum requires the use of infinite (i.e. `non-Euclidean') sets.  

Redefining the Finite

There is, however, another way of interpreting the term `infinite'. We can say that Cantor was not really taking the infinite to be definite in any sense. He was not saying that the indefinite was definite, and thereby contradicting himself. Instead, he was merely extending the range of the definite or `finite'.     Sets previously thought infinite, because they did not satisfy Euclid's rule that ``the whole is greater than its parts'', now turn out to be definite and `finite' after all, in an extended sense that includes the Cantorian transfinites. This frees mathematicians to consider sets of integers and reals, etc., as perfectly definite and ordinary sets, susceptible to all their usual axioms and methods of proof.

Those sets which we previous thought to be finite, because they are ``greater than their parts'', are merely a subset of all `finite' sets. They may therefore be called those sets which are `finite in the Euclidean sense'. It is this `Euclidean' sense of finiteness which was employed in the Finiteness Postulate given above.


Potential Infinities

But there is another response, if `actual infinities' are to be logically self-contradictory. This is to take a strict finitist line with respect to actualities, but to allow infinities to pertain to possibilities. This in fact is way all mathematicians regarded infinities before Cantor.     Aristotle, for example, writes6.2
the number of times a magnitude can be bisected is infinite, [but] this infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number. But this number is not separable from the process of bisection, and its infinity is not a permanent actuality but consists in a process of coming to be, like time and the number of time.
  Leibniz believed that the continuous line had an infinity that related to possibilities or potentialities, rather than to actuality. He stated in one of his letters to de Volder6.3
But a continuous quantity is something ideal which pertains to possibles and to actualities only insofar as they are possible. A continuum, that is, involves indeterminate parts, while, on the other hand, there is nothing indefinite in actual things, in which every division is made which can be made. Actual things are compounded as is a number out of unities, ideal things as is a number out of fractions; the parts are actually in the real whole but not in the ideal whole. But we confuse the ideal with the real when we seek for actual parts in the order of possibilities, and indeterminate parts in the aggregate of actual things, and so entangle ourselves in the labyrinth of the continuum and in contradictions that cannot be explained.
  Whitehead takes a view of possibilities which is more realistic than the `ideal continuum' of Leibniz. Whitehead defines his space-time continuum, for example, as the `coordination of all possible standpoints'. Thus, with Leibniz, he draws together the ideas of continuity and possibility, but, against Leibniz, does not see them any the less real for that. As Whitehead puts it6.4,
It cannot be too clearly understood that some chief notions of European thought were framed under the influence of a misapprehension, only partially corrected by the scientific progress of the last century. This mistake consists in the confusion of mere potentiality with actuality.

The Mathematics of Possibility

The second response, allowing only `potential infinities', seems to be directly opposed to Cantor, who asserts for example, that the number of all whole numbers, $ \aleph_0 ,$ is perfectly definite, and certainly not merely the indefinite or non-existent limit of a process of possible additions. I believe, however, that we can effect a reconciliation.

We can argue that, despite its apparent adherence to definite sets of definite objects, modern mathematics is unavoidably concerned with possibilities. Mathematics treats all possibilities as though they were actual, and by considering them all as `objects' in Frege's sense, actualities and possibilities can be all grouped together as `units', and their differences ignored. For example, we will find later in this chapter that purely actual particulars must be unextended, and hence at points in space-time. The question is, `At how many different places can an actuality be put?'. Even in a finite spatiotemporal interval there is a problem, as it appears that actualities can be placed at infinitely many possible points in, say, a 1 metre interval of space.

Mathematics can take all these `possible points of division on the line interval (0,1)', and then, ignoring the fact that these are all really possible divisions, consider them as definite Fregean objects and as forming together definite sets with transfinite cardinality. In this way, we can loosely restate the Finiteness Postulate by saying that transfinite sets (sets which are not finite in the Euclidean sense) can only be sets of natural things if they are sets of possibilities.

Modern mathematicians may not recognise that their activity is so intimately connected with the question of `possibilities'.       This is in part because of the widespread popularity of `extensional semantics' for mathematical logic and set theory. According to the extensional method, the meaning of any mathematical property is precisely the set of all objects which have that property. For example, the meaning of a mathematical function f(x) is precisely the set of all ordered pairs (x,y)  :  f(x) = y ,rather than any idea of a procedure or function that maps its input to its output. The meaning of the property `red' is simply a set of objects, each of which is `red' simply by virtue of being in that set.

In this way, extensional semantics hopes to avoid any reference to the intrinsic or `intentional' meaning of the function or property concerned. The success of extensional semantics depends, however, on the universe of discourse being at least fixed and not varying through time.

It must be logically possible for new individuals to come into existence as time progresses. (I will be explaining in the next chapter that this is what I believe actually happens.) But whether or not this is part of a correct view of time and change, our logic and mathematics must allow for this possibility. Of course, there has traditionally been considerable tension in philosophy between the claims of logic for immutable truths, and the claims of natural philosophy for the existence of real changes in the world. We must be careful not to adopt a view of mathematical and logical truth that forbids real changes to occur, or that forbids new things to come into existence. The trouble with a purely extensional semantics for set theory is that it falls into this trap.

On the face of it, it seems to us that we can use logic and mathematics to talk truthfully about individuals which have not yet or may never come into existence. For example, if I build five houses tomorrow, and you build four other ones, then we are certain today that tomorrow there will be nine houses altogether. But today, the future individuals do not exist, and hence they cannot be part of any set or an element in any `universe of discourse'. That does not invalidate our addition, however! A purely extensional semantics cannot allow the seemingly valid conclusion as to the total of nine, because as yet there exist neither my nor your houses to be in the extensions of any sets.

The best way for the semantics of logic and set theory to be reformulated to cope with this possibility, is for arguments in set theory to be explicitly concerned with hypothetical sets, and not necessarily with merely those sets which already exist. The argument, for example, that

If X and Y are two disjoint finite sets, then the number of elements in their union, $ Z = X \cup Y $, is the sum of the number of elements in X and in Y,
can be interpreted in two ways. Using extensional semantics, it is traditionally regarded as a statement of the properties of all unions $ Z = X \cup Y $ for all existing sets X and Y.

Alternatively, the argument can be formulated as a single conditional statement:

Whenever X and Y refer to two disjoint finite sets, then the number of elements in their (then existing) union $ Z = X \cup Y $ will be the sum of the number of elements that X and Y will contain.
It is still a valid argument, and will apply truthfully in many circumstances even where now there are no relevant finite sets X and Y, such as the circumstances of our houses above. Mathematical arguments very often have this `if ... then' conditional or modal structure. The fact that this structure cannot be replaced by some extensional semantics except with a severe loss of explanatory power, means that mathematical arguments must be essentially involved with possibilities as well as actualities.

As mathematics is now left free to consider possibilities and non-finite sets as it sees fit, this is congenial with the adoption of the Finiteness Postulate given earlier. It can either be regarded as a statement about actualities, as originally given, or about possibilities, whereby we do allow possibilities (and only possibilities) to form transfinite sets when aggregated.

Semantics of Set Theory

As a digression from the philosophy of nature, it is interesting to consider here some problems in the semantics of set theory. The problem is whether there is a foundation for the conditional or `hypothetical' statements which are seen above to pervade all kinds of set theoretical assertions. We want to know in virtue of what, if anything, these conditional assertions are true. This is an interesting problem here, because it precisely mirrors the problems in chapter 2 concerning the foundations of the conditional or hypothetical statements about dispositions. It seems to me that there are only three ways in which these conditional truths could be founded. They could be based on what actually exists, or based on rules, or based on some kind of real `possibility' or `form'.

If the truth of the conditional statements were dependent on what actually existed, using extensional semantics, then the truth of a statement 4+5=9 would be identical with the truth that there are 9 elements in the union of any of the existing 4-element sets with any of the existing 9-element sets. The difficulty with this option is, as we argued just above, that it fails to allow for new individuals to come into existence. It fails to recognise the truth of the assertion above concerning $ Z = X \cup Y .$
    If the truth of the conditional statements were based on rules of some kind, then we have a position analogous to Ryle's concerning dispositions. He took the conditional statements of dispositions to be `inference tickets' that simply enabled conclusions to be drawn in suitable circumstances. They did not imply or refer to anything real about the world, and do not necessarily have any foundation or basis apart from the fact of their being true. The corresponding position on set-theoretical conditionals is somewhat more plausible, as we cannot point our fingers to real things in the world to which naively we think that they ought to refer. Nevertheless, the status of the mathematical `rules' or `inference tickets' is peculiar.   One might be tempted to follow Wittgenstein and argue that all the `rules' are only `conventional rules', and are only adopted by mathematicians for pragmatic reasons. In this case, however, we would have little incentive to consider them as true, and even less reason to consider them relevant to what really goes on in nature.
  The third option in the present mathematical context follows the `conjectural essentialism' we used in the philosophy of nature. Accordingly, we hypothesise the existence of some kind of basis whose existence would lead to and explain the truth of the relevant conditional statements. In the philosophy of nature, we were concerned with the conditional statements describing dispositional properties, and were led to the notion of `real dispositions' and `real propensities' as existing things whose nature was such that the conditional statements were true. (They do not exist, of course, as `pure actualities': the manner of their existence will be discussed in chapter 7.)

      To generate and explain mathematical conditionals, we therefore hypothesise that there exist such things as `real possibilities' or `real forms'. In the case of our houses above, we would be postulating something like `possible houses' or `possibilities for houses' or `forms for houses' or `space for houses'. Any of these things would be assumed to exist now, before the houses are built, and their present existence generates and explains the present truth of the conditional statements. Of course, there will have to be considerable discussion as to which (if any) of the above entities is most satisfactory for the purpose. I will be arguing in chapter 8 that the first notion of `possible houses' is not satisfactory, as such things cannot be uniquely individuated, identified and counted. That is, they do not follow the classical laws of identity. Instead, I will argue that the remaining three notions are equivalent to each other, and, at least in the philosophy of nature if not in mathematics, have a true and useful role to play in the analysis of dispositions and changes.

    This interpretation in mathematics may be reminiscent of Platonism, in that it postulates entities (`forms' or whatever) whose nature is responsible for and explains mathematical truths. Here, we are taking these entities as `possibilities', and they `enter in' to all actual things because everything that is actual is at least possible. (One consequence of this interpretation of the semantics of set theory is that impossible things cannot be counted. We may of course still count fallacies, but not the impossible things to which they allegedly refer. Because there is no possibility for an impossible thing, one cannot, therefore, `think six impossible things before breakfast'.)

This third view of set semantics supports the idea that mathematical arguments are essentially connected with possibilities, as the basis of mathematical conditional statements is taken to consist of some kind of possibilities. Mathematicians therefore consider, as a definite (sort of actual) thing, the very possibility for all nine-element sets. It is thus reasonable to consider `all possibilities for say the division of the real interval (0,1)' as if they formed a definite set, for mathematicians can now consider possibilities as if they were actual. It is only when mathematics is applied to the physical world, that we have to pay attention more carefully to whether the elements of sets are actualities or possibilities.  



The problem of the continuity of space is closely bound up with the problems of infinity. One may suppose that provided any `smallest units' were sufficiently small that we could never see them, it would not make much difference as whether the number of subdivisions of a spatial interval were infinite, or merely some (very large) finite number.   It might just be an odd but perfectly true feature of space that, even if we were superhuman, we could only subdivide a spatial interval so many times and no more. It might be that not only actualities are finite, but also space itself. This would definitely be the case if we adopted the finitism described earlier.

These arguments would carry some weight, if everything we imagined existing had to be finite and actual in the full sense. If space were an actual thing, for example, or if space were the only actual thing (as geometrodynamics supposes), then we might be tempted to follow the intuition which lead to the Finiteness Postulate above, and to imagine that space was composed of only a finite number of points which would be like finite `units'. Space would then not be continuous in the traditional sense.

In the present philosophy of nature, however, we have reason to suppose that modal considerations of possibilities have to be taken very seriously. If actualities are ever to `come to be', for example, then they must have been possible or potential in some way, and in fact we will later define space and time in terms of possibilities. If it turns out that space is essentially composed of certain kinds of possibilities, then it cannot be an actual thing. Space is hence not bound by the Finiteness Postulate. If space and possibilities are intimately intertwined, then there will be possibilities in space that are impossible for actual things. In particular, it might be possible for space to be composed as a non-finite set of points.

  Since Aristotle's time, indeed, continuity has been defined in terms of possibilities, in particular the possibilities for endlessly subdividing a finite interval. I refer to the quotations from Aristotle and Leibniz given above. Continuity of the interval (0,1), for example, was taken to mean that there was no limit to the number of possibilities for dividing it into two subregions, and this was accepted without having to accept that the interval was really composed of an actually infinite number of points.

A more recent way of defining continuity does rely on a finite interval being `actually' composed of an infinite number of points. 'Actually' here of course does not mean `actually existing as a natural thing', but, as explained previously, is the mathematician's term for `definitely'.   According to Cantor, the real continuum is really and definitely made up of a transfinite set of points with cardinality $ {\bf C} = 2 ^ {\aleph _ 0} . $  Newton had a similar idea in mind when he conceived of infinitesimals as quantities with definite values although infinitely small. Continuity of a real line is then simply that every adjacent pair of points has not more than an infinitesimal separation.

According to the `mathematics of possibility' outlined above, for the purposes of the philosophy of nature there is not a great difference between these two ways of defining continuity. Space may be composed of possibilities, following Aristotle and Leibniz, but it turns out that we can consider all these possibilities together as forming a set with definite (transfinite) cardinality, following Cantor. No confusion need result provided that we do not forget that the elements of such transfinite sets are possibilities, not actualities.      

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Prof Ian Thompson


Author: I.J. Thompson (except as stated)