Turing, Gödel and the “Bright Abyss” Juliet Floyd ! 2/5/2015 9:25 AM Comment [1]: 1. Check phone number okay to use this ong?; Also need 2. ABSTRACT of 1 Introduction 150-200 words and 3. a CONTRIBUTOR bi- ography of up to 100 words! Xxx, J. They have completely forgotten what is a mathematical creation: a vision that decants little by little over months and years, bringing to light the obvious [evident] thing that no one had seen, taking form in an obvious assertion of which no one had dreamed ... and that the first one to come along can then prove in five minutes, using techniques ready to hand [toutes cuites] • A. Grothendiek, Recoltes et Sémailles, 1986 Mathematics: there is that which is taken on faith and that which is proved; there is that which is buried deep inside the heart of the mathematician1 and that which is wholly public; there is intuition and fact and the “bright abyss” in be- tween; there is the raw, one might say, and the cooked. I hold up my hand and I count five fingers. I take it on faith that the mapping from fingers onto numbers is recursive in the sense of the mathematician’s defi- nition of the informal concept, “human calculability following a fixed routine.” I cannot prove the mapping is recursive—there is nothing to prove! Of course, mathematicians can prove many theorems about recursiveness, moving forward, so to speak, once the definition of the concept “recursive” has been isolated. Moving backwards is more difficult and this is as it should be: for how can one possibly hope to prove that a mathematical definition captures an informal con- cept? This paper is about just that one word, capture, or more precisely the rela- tion “x captures y,” and the question, if y is taken to be computability, definabil- ity or provability, does there exist an adequate and unique choice of x? By which criteria do we shape, commit ourselves to, or otherwise assess standards of adequacy? This is the problem of faithfulness; the problem of what is lost whenever an intuitively given mathematical concept is made exact, or, be- yond that, formalized; the problem, in words, of the adequacy of our mathemati- Juliette Kennedy Department of Mathematics and Statistics Helsinki Collegium for Advanced Studies University of Helsinki, Finland Email: [email protected] Telephone: +358 9 19122866 1 Hadamard writes of nonverbal thought, images that resist statement in words, sudden insight, flashes of inspiration. See (Burgess forthcoming). 2 cal definitions. It is always present in mathematics, but it is a philosophical prob- lem rather than a mathematical one, in our view, as there is nothing in this idea of “fit” that can be subject to mathematical proof.2 Logicians have developed a number of what one might call coping strate- gies. For example, some have raised the question of intensional adequacy in con- nection with the Second Incompleteness Theorem, as establishing the theorem may turn on the meaning of the (formal) consistency statement as read by the rel- evant theory—or so Feferman, Detlefsen, Franks and others have argued.3 In their view, briefly, one should grant the meta-theoretical claim that a theory T cannot prove its own consistency only when there is a sentence which T both “recognizes” as a consistency statement, and which T cannot prove. The criteria for a theory T’s “recognizing” its consistency statement as such are met just in case it can be proved in T that its proof predicate satisfies the three Hilbert- Bernays derivability conditions.4 A consistency statement is nothing but a state- ment to the effect that something is not provable, which means that the adequacy claim in question rests on the further claim that the provability predicate of T is recognised by T as adequately representing “genuine” provability. Another coping strategy involves the use of the word thesis—a word which serves to, in effect, flag an adequacy claim, if not to bracket it. The Church- Turing Thesis, for example, in its present formulation, equates the class of intui- tively computable number-theoretic functions with the class of functions com- putable by a Turing Machine.5 Other conceivable theses in mathematics include Weierstrass’s thesis, as some have called it,6 namely the assertion that the ε − δ definition of continuity correctly and uniquely expresses the informal concept; Dedekind’s thesis, asserting that Dedekind gave the correct definition of the con- cept “line without gaps”7; or, alternatively, asserting that Dedekind gave the cor- rect definition of a natural number; the area thesis, asserting that the Riemann in- tegral correctly captures the idea of the area bounded by a curve. Hilbert’s Thesis, a central though contested claim in foundations of mathematics, refers to 2 Some would claim to the contrary that Turing proved a theorem equating human effective cal- culability with Turing calculability. See below. The faithfulness problem is only a problem for those who take the view that a sharply individuated concept is present, intersubjectively, in the first place. In his (2009) Hodges takes a different view: “Turing’s Thesis is the claim that inde- pendent of Turing we have an intuitive notion of an effectively computable function, and Tu- ring’s analysis exactly captures this class of functions... This kind of claim is impossible to ver- ify. Work like Turing’s has a power of creating intuitions. As soon as we read it, we lose our previous innocence.” 3 This is in contrast to the First Incompleteness Theorem, which has nothing to do with the meaning of the undecidable sentence G—all that matters in that case is that G is undecidable. See (Feferman 1960/61), (Detlefsen 2001), (Franks 2009) and (Pudlak 1996). 4 As given in their (1934/1968). 5 This is “Thesis T” in (Gandy 1988). 6 See (Shapiro 2013). 7 See (Sieg 1997). 3 the claim that “the steps of any mathematical argument can be given in a first or- der language (with identity).”8 To many working mathematicians the correctness of theses such as Weier- strass’s is, in Grothendieck’s sense of the word, obvious—what more is there to say about continuity than what is encapsulated in Weierstrass’s definition of it? The possibility that meaningful concepts may arise in future practice, which may fall under the intuitive notion, while not fitting, e.g. Weierstrass’s definition, is simply not considered. In such a case one speaks of theorems, not theses. Turing, for example, is thought by some not to have formulated a thesis but rather to have proved a theorem relating human effective computability to computability by means of a Turing Machine.9 In sum, there are “theses everywhere,” as Shapiro notes in his (2013); not necessarily provable, but at the same time in most cases no longer subject to doubt.! How does the nonlogician (or nonphilosopher) cope with the problem of ad- equacy? “Things are robust if they are accessible (detectable, measurable, deriv- able, definable, producable, or the like) in a variety of independent ways,” in the words of William Wimsatt.10 But there is also grounding: the idea that, among the class of conceptually distinct precisifications of the given (intuitive) concept, one stands out as being indubitably adequate—as being the right idea. This hap- pened, evidently, with the notion of “finite.” There are a number of (extensional- ly) equivalent definitions,11 but the notion that a set is finite if it can be put into one to one correspondence with a natural number seems to be what “every” mathematician means by the term “finite”—even though the definition is blatant- ly circular, on its face.12 8 (Kripke, 2013), p. 81. See also (Burgess 1992), (Rav 2007)and (Shapiro 2013). 9 Turing’s Theorem is defined by Gandy as follows: Any function which is effectively calcula- ble by an abstract human being following a fixed routine is effectively calculable by a Turing machine—or equivalently, effectively calculable in the sense defined by Church—and con- versely. (Gandy 1988), p. 83. 10 See (Wimsatt 1994). In computability theory the term “confluence” seems to be preferred over “robustness.” See e.g. (Gandy 1988). 11 In set theory. 12 This is because the concept of “natural number” is usually defined in terms of finiteness. D.A. Martin expressed a similar thought in his recent (2012): There are various ways in which we can explain to one another the concept of the sequence of all the natural numbers or, more generally, the concept of an ω-sequence. Often these explana- tions involve metaphors: counting forever; an endless row of telephone poles (or cellphone towers); etc. If we want to avoid metaphor, we can talk of an unending sequence or of an infi- nite sequence. If we wish not to pack to pack so much into the word “sequence,” then we can say that that an ω-sequence consists of some objects ordered so that there is no last one and so that each of them has only finitely many predecessors. This explanation makes the word “fi- nite” do the main work. We can shift the main work from one word to another, but somewhere we will use a word that we do not explicitly define or define only in terms of other words in the circle. One might worry—and in the past many did worry—that all these concepts are incoher- ent or at least vague and perhaps non-objective. 4 And it also happened after 1936 in connection with the notion of computa- bility, when the Turing analysis of human effective computability was taken by logicians to have solved the adequacy problem in that case.! The number theorist Michael Harris uses the word “avatar” to express a con- dition of no grounding, or inadequacy—and within that, the condition of knowing that, but not why: ...I suggested that the goal of mathematics is to convert rigor- ous proofs to heuristics— not to solve a problem, in other words, but rather to reformulate it in a way that makes the solution obvious..