The Creating Subject, the Brouwer-Kripke Schema, and infinite proofs Mark van Atten∗ May 2, 2018 . Abstract Kripke’s Schema (better the Brouwer-Kripke Schema) and the Kreisel-Troelstra Theory of the Creating Subject were introduced around the same time for the same purpose, that of analysing Brouwer’s ‘Creating Subject arguments’; other applications have been found since. I first look in detail at a representative choice of Brouwer’s arguments. Then I discuss the original use of the Schema and the Theory, their justifi- cation from a Brouwerian perspective, and instances of the Schema that can in fact be found in Brouwer’s own writings. Finally, I defend the Schema and the Theory against a number of objections that have been made. Indagationes Mathematicae Brouwer’s views may be wrong or crazy (e.g. self-contradictory), but one will never find out without looking at their more dubious aspects. Kreisel [98] Contents Forthcoming in 1 Notation 3 2 Introduction 3 arXiv:1805.00404v1 [math.HO] 30 Apr 2018 3 Three Creating Subject arguments 7 3.1 Ordering relations, testability, judgeability, decidability ........... 7 3.2 Anargumentfrom1948 ............................. 9 3.3 ArefinementbyHeyting............................. 12 3.4 Driftsandcheckingnumbers . 14 ∗Sciences, Normes, Décision (CNRS/Paris 4), 1 rue Victor Cousin, 75005 Paris, France. As of September 1, 2018: Archives Husserl (CNRS/ENS), 45 rue d’Ulm, 75005 Paris, France. [email protected] 1 3.5 Anargumentfrom1949 ............................. 17 3.6 Myhill’s objection to, and repair of, the argument from 1949 ........ 21 3.7 Anargumentfrom1954 ............................. 22 4 BKS and CS 23 4.1 BKS........................................ 24 4.2 CS......................................... 32 4.3 ComparingBKSandCS............................. 35 4.4 Applications beyond Brouwerian counterexamples . ......... 36 5 ABrouwerianjustificationofCSandBKS 37 5.1 AremarkonBrouwerianlogic. 37 5.2 BKSisderivablefromCS ............................ 38 5.3 AjustificationofCS ............................... 39 6 BKS in Brouwer 46 6.1 BKS− ....................................... 47 6.2 BKS+ ....................................... 52 7 Discussion of objections to CS and BKS 53 7.1 BKS− contradicts Church’s Thesis . 54 7.2 BKS+ and MP together imply DNE and PEM . 56 7.3 CSleadstoaparadox .............................. 57 7.4 Proofs are not different at different times . ...... 60 7.5 CS depends on a function that is not unitype . ..... 61 7.6 BKS− contradicts α β-continuity....................... 62 ∀ ∃ 7.7 BKS+ must be restricted to determinate properties . 64 7.8 A palatable substitute for BKS− ........................ 68 7.9 Brouwer does not want to accept BKS− .................... 69 7.10 Brouwer does not appeal to an ideal subject . ....... 75 7.11 BKS− and CS− are incompatible with Brouwer’s notion of infinite proofs . 78 8 Concluding remark 81 A Brouwer’s implicit use of MP in 1918 81 B Brouwer’s proof of the Negative Continuity Theorem 84 2 1 Notation Unless noticed otherwise, the following notation is used: a a sequence of elements each indexed by n h nin i,j,k,m,n,p,v,w variables ranging over the natural numbers f,g,h variables ranging over functions j(x, y) a pairing function N N N × → r, s, t variables ranging over real numbers x, y, z variables whose range depends on the context A, B, C variables ranging over propositions P, Q, R variables ranging over predicates X,Y,Z variables ranging over species α, β, γ, ξ variables ranging over choice sequences αm¯ the initial segment of α α(1),α(2),...,α(m) h i In quotations, notation has been left unchanged. 2 Introduction Can mathematical arguments depend not only on mathematical objects and their proper- ties, but also on the temporal order in which some ideal mathematician establishes theo- rems about them? Brouwer affirmed this, and it led him to devise a number of reasonings now known as ‘Creating Subject arguments’. The first known example occurs in the 1927 Berlin lectures [47].1 Let R be understood as the species of intuitionistic real numbers (convergent choice sequences). Brouwer considers the possibility of an order relation on R based on ‘the naive “before” and “after” ’ according to which r < s if and only if on the intuitive continuum (seen with the mind’s eye), r appears to the left of s.2 Weak counterexample 1 ([47, p.31–32]). Let < be the naive order relation. Then there is no hope of showing that x R y R(x = y x < y ∨ y<x) (1) ∀ ∈ ∀ ∈ 6 → 1Brouwer says in the opening of the 1948 paper ‘Essentially negative properties’ that he had given an example of such reasoning ‘now and then in courses and lectures since 1927’ [34, p.963; trl. 45, p.478]. Known places are the second Vienna lecture (see further on in the text), the 1933 Groningen lectures, and the 1934 Geneva lectures. The latter two have remained unpublished, but Niekus [127, section 7; 128, section 10] makes some comments on them. There is no Creating Subject argument in the 1932 lecture ‘Will, knowledge, and speech’ [30]. 2In the second Vienna lecture, this is called the ‘natural order’ [27, p.8] and in the Cambridge lectures the ‘intuitive order’ [46, p.43]. 3 Plausibility argument 2. Let P be a unary predicate and e a mathematical object in its domain, such that at present neither ¬P(e) nor ¬¬P(e) have become evident. Define a real number r as a convergent choice sequence of rationals r(n): • As long as, by the choice of r(n), one has obtained evidence neither of P(e) nor of ¬P(e), r(n) is chosen to be 0. • If between the choice of r(m − 1) and r(m), one has obtained evidence of P(e), r(n) for all n > m is chosen to be 2−m. • If between the choice of r(m − 1) and r(m), one has obtained evidence of ¬P(e), r(n) for all n > m is chosen to be −2−m. Then it is true that r = 0 but, as long as neither ¬P(e) nor ¬¬P(e) has become evident, 6 neither r < 0 nor r > 0 is true. As one may always expect to be able to find such P and e,3 there is no hope of showing x R y R(x = y x < y ∨ y<x). ∀ ∈ ∀ ∈ 6 → The justification is a plausibility argument (for the conclusion that the given propo- sition will never be demonstrated), not a proof, because it depends on an expectation, albeit one that is highly likely to be fulfillable. Thus, the justification contains not only mathematical reasoning, but also a value judgement. This is the defining characteristic of weak counterexamples.4 In the Berlin lectures, this weak counterexample serves as a motivation for introducing the notion of virtual order, and it is shown that that is the closest one can come to an order on R; we will come back to that in subsections 3.1 and 3.3. For here, the salient feature of the argument is that the choices in the sequence r depend not only on properties of P and e, but also on the moment at which a certain proof about them becomes available to the mathematician constructing the sequence. It will be clear that this sequence r is therefore not determined by a law, to the extent that one accepts the idea that the mathematician’s activity depends on its free choices in directing its efforts. The role of freedom in intuitionistic mathematics will be discussed further in subsections 7.1 and 7.3. ‘The mathematician’ here is evidently an idealised one, who in particular is always there to make the n-th choice, however large n may become. Brouwer baptised this idealised mathematician ‘het scheppende subject’ in Dutch in 1948 [34, p.963] and ‘the Creating Subject’ in English in 1949 [37, p.1246].5 This idealisation will be further explained in 3See on this point also the remark on the relation between completely open problems and untested propositions on p.70 below. 4As the referee emphasised, Brouwer never calls his weak counterexamples ‘theorems’. 5The basic meaning of ‘scheppend(e)’, the participial adjective of the verb ‘scheppen’, here is ‘bringing something into existence’. According to the historical dictionary Woordenboek der Nederlandsche Taal, the alternative ‘creatief’ had already been introduced in Dutch when Brouwer chose ‘scheppend’, but often 4 subsection 5.3, and an objection to the claim that Brouwer is making it will be discussed in subsection 7.10. Brouwer had given a weak counterexample with a very similar conclusion a few years before, in 1924: Weak counterexample 3 ([20, p.3]). There is no hope of showing that x R y R(x = y ∨ x < y ∨ y<x) (2) ∀ ∈ ∀ ∈ But the way he had argued for it then had been very different. Plausibility argument 4. Let dv be the v-th digit after the decimal point in the decimal expansion of π and m = kn, when in the ongoing decimal expansion of π at dm it happens for the n-th time that the part dmdm+1 ...dm+9 of this decimal expansion forms k1 a sequence 0123456789. Let furthermore cv = (−1/2) if v > k1, otherwise v cv = (−1/2) ; then the infinite sequence c1, c2, c3,... defines a real number r, for which neither r = 0, nor r > 0, nor r < 0 holds. [20, p.3] Note that the choices in the sequence cv are not defined in terms of the moment at which the Creating Subject comes to know the truth of a certain proposition, as they would in Plausibility argument 2. As the decimal expansion of π is lawlike, and for every v it is decidable whether v > k1, r is itself lawlike. When Brouwer wrote this, no value k1 was known, but in the meantime it has been found that k1 = 17,387,594,880 [7]; so r > 0.