Symposium: L.E.J. Brouwer, Fifty Years Later. Free Choice Sequences: A Temporal Interpretation Compatible with Acceptance of Classical Mathematics

Transcription

1 Symposium: L.E.J. Brouwer, Fifty Years Later Amsterdam December 9 th, 2016 Free Choice Sequences: A Temporal Interpretation Compatible with Acceptance of Classical Mathematics Saul A. Kripke CUNY Graduate Center Saul Kripke Center

2 Brouwer s intuitionism had various features and motivations. The most important one was the rejection of nonconstructive arguments, which, according to him, necessitated a new interpretation of the (syntactic) logical constants. Most famously: the interpretations of disjunction and negation, which implied a rejection of the law of excluded middle. Also, of course, the constructive interpretation of existential quantification: denying that an existential statement can be made without in principle being able to provide an instance.

3 Eventually, though this took time, most mathematicians simply ignored the Brouwerian criticisms and developed classical mathematics as before. But there was a time when this was a very live issue. (As everyone here knows, in spite of his own position, Brouwer himself did famous work that was only classically valid.) And interest in intuitionism is alive for many of us here, even if we do not reject classical mathematics.

4 Another feature of Brouwer s work came later. In addition to the determinate or lawlike sequences, he proposed that free choice sequences be allowed. As Heyting says, one does not really need to suppose that an infinitely proceeding sequence be determined by a law: The question how the components of the sequence are successively determined, whether by a law, by free choices, by throwing a die, or by some other means, is entirely irrelevant. (Intuitionism, an Introduction,1956: 32)

5 This newer aspect of his work has a more difficult history. Errett Bishop, a relatively late admirer of Brouwer s constructivism and of his work, categorically rejected the notion of a free choice sequence. He says: In Brouwer s case there seems to have been a nagging suspicion that unless he personally intervened to prevent it the continuum would turn out to be discrete. He therefore introduced the method of free-choice sequences for constructing the continuum, as a consequence of which the continuum cannot be discrete because it is not well enough defined. This makes mathematics so bizarre it becomes unpalatable

6 to mathematicians, and foredooms the whole of Brouwer s program. This is a pity, because Brouwer had a remarkable insight into the defects of classical mathematics, and he made a heroic attempt to set things right. (Foundations of Constructive Analysis,1967: 6) (A similar tendency to sarcasm often appears in Bishop s remarks on foundational issues.) Bishop s real numbers, or sequences of rationals defining them, are given by rules.

7 Martin-Löf s intuitionistic type theory, and the homotopy type theory that developed from it, appear not to use the concept of free choice sequence either. These are major applications of intuitionistic ideas, with connections to theoretical computer science. And it should also be noted that even some authors sympathetic to free choice sequences have tried to show that they can be explained away as a sort of façon de parler, using axioms of open data, or analytic (Σ 11 ) data, to reduce discourse about them to talk of constructive functions.

8 I am not necessarily opposed to such ideas, but I do not regard them as important for getting rid of a fundamental concept. In any event, the idea of free choice sequences I will introduce is completely different from that which motivated these proposals, since it is supposed to supplement classical mathematics, not intuitionistic mathematics, conceived as based on any idea of constructivism.

9 In this talk I shall outline how a concept of free choice sequence could be combined with an acceptance of classical mathematics. I do not wish to identify classical mathematics with any particular axiomatization such as ZFC. Most classical mathematicians other than those working in set theory itself, probably do not think in terms of particular sets of axioms, but apply the set theoretic concepts intuitively.

10 (Category theory is of course an important part of contemporary classical mathematics, and we don t need to go into the issues about its relation or lack of it to set theory. [Intuitionism actually plays some role in category theory, but this is not important here.]) It would be in consonance with Brouwer s own attitude towards intuitionistic mathematics and also its exposition in Heyting s Intuitionism, an Introduction (1956) and Dummett s Elements of Intuitionism (1977) simply to give the theorems intuitively and not rely on an axiomatization. I think this is true of much of classical mathematics also.

11 Now, let s consider a classical mathematician confronted with time. Here we think of time as represented by discrete linearly ordered moments. In other words, with the order type of the positive integers. This was Brouwer s conception as given in his notion of free choice sequence.

12 We assume that the instances of time are given only as a potential totality. There is no end of time from which all the instances of time can be surveyed. Of course, there is a considerable idealization involved here. The creative subject, who chooses the free choice sequences, must be assumed to be immortal. And even when the sequence is determined by a machine flipping a coin or casting a die, it must be assumed to be a perpetual motion machine.

13 On the other hand, there may be some scientific models in which there is an end of time, even for a given subject. And there may be general relativistic models in which after an end of time, there starts a new sequence of time. I do not take such things into account here, nor do I worry about absolute simultaneity and special relativity. Everything can be assumed to be the proper time of a single subject.

14 I believe that Brouwer once used the term the classical continuum for the continuum before free choice sequences, and held that the continuum as given by free choice sequences is an extension of it. Here the term classical continuum would mean the continuum as given by lawlike sequences of rationals satisfying the usual convergence conditions (i.e. a constructive continuum). (It might mean the continuum as a whole taken as primitive.)

15 In the conception I have here, the classical continuum simply consists of arbitrary real numbers, defined in one of the usual classical ways (Cauchy sequences or Dedekind cuts). For the mathematics here is simply the usual classical mathematics. The reals determined by free choice sequences are an extension of the classical continuum. However, we are considering sequences of natural numbers, and arbitrary such sequences. (Or, as I will explain later, arbitrary sequences with an upper bound condition.)

16 So, for example, one can consider a coin flipped independently an arbitrary number of times. This might correspond to a lawless sequence. The total number of possibilities is most naturally represented as a tree with the topology of the Cantor set. I am using the term tree rather the intuitionistic spread to emphasize that the mathematics is classical.

17 No doubt this is a simple example of a fan in the intuitionistic sense. However, other finitary trees (spreads) could easily be given, all representing possibilities for what the sequence might be, but never stating which branch corresponds to the real one. Brouwer s proof of the bar theorem, and hence of the fan theorem, has given rise to a fair amount of comment and has some difficulty in intuitionistic mathematics.

18 (The most hostile comment comes, once again, from Bishop, who says, to accept Brouwer s argument as a proof would destroy the character of mathematics. (Foundations of Constructive Analysis, 1967: 70) But he acknowledges that, say, the fan theorem, has never been counterexampled and thinks the situation tantalizing since we will never have a proof, nor will we have a counterexample.) More sympathetic authors have also regarded Brouwer s argument for these theorems as problematic, and may simply take them as axioms.

19 In the present case, there is no difficulty about the fan theorem, which is crucial to the result that a continuous function on a closed interval is uniformly continuous. Since the mathematics of trees with at most finite branching and where every path has finite length (a fan) is classical, König s infinity lemma holds, and hence the fan theorem holds. Similarly, there is no difficulty about the bar theorem.

20 Now, let me give an example of why, even though one accepts classical mathematics, the law of excluded middle will fail for infinitely proceeding sequences (choice sequences, in the present conception). For example, if the terms of the sequence are bounded and are identified with digits of a real number in the unit interval expanded to an appropriate base, we cannot assert at any time that the number is algebraic or transcendental. Remember that there is no end of time, and that probabilistic considerations cannot be used even if the individual terms of the sequence are given by random throws of a coin or a die.

21 In such a case, no doubt there is a probability of 0 that the number will be algebraic, but we cannot use this to assert that the number will be transcendental. (Of course, in the continuum extended by free choice sequences, real numbers need not have a determinate expansion to any base, but this is another matter.)

22 We are really considering here a family of models depending on what restrictions are being placed on the free choice sequence, assumed to be a sequence of natural numbers. If there is a bound on the numbers allowed, this will be reflected in the branching of the tree. It is however assumed that any classical sequence subject to the upper bound restriction, if there is one, is an allowable one in the temporal path. The creating subject is allowed to choose according to any temporally possible sequence.

23 One point needs to be clarified. Ideally, I would like to include all classically admissible sequences in those chosen by the creating subject. But these will be non-denumerable in number. And one might think that this is too much of an idealization. A creating subject can choose any denumerable number of sequences simultaneously simply by alternating his activities from one to another according to some plan.

24 But an uncountable totality of sequences cannot be done in this manner. So one might more weakly postulate that all definable sequences are to be included in those chosen by the creating subject, and these of course could be denumerable in cardinality. Which sequences are definable may depend on the exact language used, a point I have left deliberately vague. But it will include all those sequences that are intuitively thought to be classically definable.

25 It is important to allow all definable sequences since we want all classically definable reals to be included in the model. Note also that on this conception there is no analogue of Troelstra s paradox, other than the known fact that only countably members of the continuum are included. And in an appropriate metalanguage, one could define a diagonal element, given enough extra resources.

26 Also, lawless sequences with any given initial segment are included, as well as sequences with restrictions added by one or more free decisions (which need not be effective). So that α x continuity holds, or at least α!x continuity, but not α β continuity, given Kripke s schema. The definable sequences replace the lawlike sequences in the usual conception of free choice sequences. But here there is no particular effectiveness idea, just definability.

27 Now, let us return to issue of the law of excluded middle and its failure on the present conception. We are assuming a model with branching time, but no one knows which branch is actual. When does the failure set in? According to one reading of Aristotle s De Interpretatione (and my impression is that for a long time it was the received reading), the failure sets in at the very outset.

28 In this reading no one can say that either there will be a sea battle tomorrow or there will not, since the future is indeterminate. One would have to change the example to correspond with free choice sequences. Or make the value of the sequence dependent on whether there will be a sea battle tomorrow. But the idea is clear enough. In this model the idea of the branching time model consists essentially in treating it with the semantics of Kripke models: a disjunction is true at a point iff one disjunct is.

29 But the motivation is quite different from the one I originally gave for such models. The original idea was that the points on a branch represent evidential situations and later points represent situations in which one gets more evidence. One may remain stuck at a single point for an arbitrarily long time, but only if one gets more evidence can one assert a disjunction without being able to assert a given disjunct. This might correspond reasonably well to the intuitionistic idea that proving a disjunction amounts to having a method to prove at least one of the disjuncts.

30 Here, on the Aristotelian model, the idea is different, since the nodes of the tree represent points in time, where later ones represent later points. Then the law of excluded middle fails simply because the future is not determined, though if one waits until the next day, one will know which disjunct is true. Now, this might be a view, but I think it does not so well accord with our intuitions about free choice sequences. If we can wait and see, in some finite time, which of two disjuncts is true here especially for the law of excluded middle then one can assert it in advance.

31 In Aristotle s example, there either will be a sea battle tomorrow or they will not, though we have to wait till tomorrow to see which is true. Putting things a bit more formally: If there is a bar on the branching time tree, so that any path along it will decide between the two disjuncts eventually, then the disjunction can be asserted in advance, though we have to wait a bit to see which one holds.

32 This means that the semantics corresponds to Beth models. Remembering that Konig s lemma and the fan theorem hold for finitary spreads, there will actually be in the case of finite branching an upper bound for how long one has to wait. So here, following Brouwerian intuitionism, we assume the semantics of the connectives for free choice sequences to be given in terms of Beth models. We otherwise continue to take the underlying mathematics to be classical.

33 Thus the idea is that the ordinary classical continuum is to be supplemented by one given in terms of Brouwerian free choice sequences. For these, the usual intuitionistic mathematics is assumed, just as in textbooks such as those of Heyting (1956) and Dummett (1977). I do wish, however, to make some comment on the role of Kripke s schema in this supplementation.

34 I originally proposed it in the weak form: α[( A xαx = 0) ( xαx 0 A)] But there is a stronger and much simpler form: α[a xαx 0] I still prefer the weak form. (And I would hold this preference in ordinary intuitionistic mathematics too, as well as on the present conception.) Moreover models have been found in which the strong form holds. Also, as van Dalen (1977) remarks, the strong schema is trivially true in classical mathematics.

35 What is the reason? Intuitionistically, to prove a conditional A B, one must have a technique so that from any proof of A one can get a proof of B. Now, the idea of the schema, based on Brouwer s own arguments about the creative subject, is that one imagines a sequence in time that is 0 as long as A has not been proved but is 1 as soon as it has been proved.

36 Then one claims that a proof that the sequence is always 0 amounts to a proof that A can never be proved, i.e. that it is absurd. If the sequence gets the value 1, then A has been proved. What would justify the strong form? Well, as Myhill says, if A is ever proved, it is proved at some definite time, and then one gets a value of α that is not equal to 0 simply by looking at your watch (or calendar, as the case may be).

37 However: Is the idea of the intuitionistic conditional (remember that from a proof of A, one can get a proof of B) such that intuitionistic proofs, which indeed must take place at some definite time, have the time of their occurrence as part of the proof, so that the same proof would be different if it happened to occur at a different time? I think not. In that case, the strong form does not appear to be justified, but the weak form is.

38 Remark: Heyting allows his character representing the intuitionist to object even to the weak form: if a law is passed throughout the world prohibiting the making of any mathematical deduction whatever, then the proof of α 0 fails. (1956: 116. In effect, ( A xαx = 0) fails.) But the idea of such a law simply gets rid of the entire intuitionistic interpretation of the conditional, since A B is supposed to mean that from any proof of A, one can get a proof of B, and this becomes a vacuous notion because of the supposed law.)

39 So, what have I argued here? That a Brouwerian theory of free choice sequences could be added to classical mathematics, without any constructive doubts as to its validity. I have tried to be deliberately informal, in the spirit that Brouwer himself would have preferred.

40 Thank you!