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The Determinacy of Arithmetic: the Puzzle and Its Solution

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The Determinacy of Arithmetic: the Puzzle and Its Solution The Determinacy of Arithmetic: The Puzzle and Its Solution Jared Warren Abstract It seems that our arithmetical notions are determinate — that there is a fact of the matter about every single arithmetical question whether or not we can discover it. But the determinacy of arithmetic is extremely puzzling given the algorithmic nature of human cognition and famous limiting results in mathematical logic, such as Gödel’s incompleteness theorems, which seem to show that no theory of arith- metic learnable by algorithmic creatures like us could possibly be determinate. I call this the puzzle of arithmetical determinacy. This paper has three central goals: (i) to vividly present this puzzle and impress upon the reader that it isn’t a mere idle curiosity (section 1); (ii) to argue that extant attempts to solve the puzzle, in- cluding those using the open-endedness of our arithmetical notions, have thus far failed (sections 2 and 4); and (iii) to clearly explain the notion of open-endedness in a philosophically illuminating fashion and show how the notion can be used to solve the puzzle when coupled with the externalist lessons of contemporary metasemantics (sections 3, 5, and 6). Keywords: Arithmetic, Indeterminacy, Open-Endedness, Incompleteness, Metaseman- tics, Externalism 1 The Puzzle Consider the following three claims: 1. ARITHMETIC: we have a determinate conception of arithmetic 2. COGNITION: human cognition is algorithmic 3. FAILURE: no determinate theory of arithmetic is algorithmic 1 1 THE PUZZLE Here I’m going to briefly explain each of these claims and argue that they’re individ- ually plausible before showing that jointly they give rise to an important puzzle about our grasp of arithmetic. I call this The Puzzle of Arithmetical Determinacy. The puzzle is perhaps best classified as metasemantic or metaconceptual—it resides in the border- lands of the philosophies of mind, language, and mathematics. As such, this puzzle should be of interest to a wide assortment of philosophers, but it isn’t as widely appre- ciated as it should be. (1) ARITHMETIC: we have a determinate conception of arithmetic. Roughly, this means that there is a fact of the matter about any given arithmetical question that can be asked. To illustrate: consider Goldbach’s conjecture, a famous, unproven arithmetical conjecture saying that every even natural number greater than two is the sum of two, possibly repeated, prime numbers. Despite centuries of effort by mathematicians, this conjecture has been neither proven nor refuted. According to ARITHMETIC, there is a fact about Goldbach’s conjecture even though we haven’t found it. What’s more: even if we never prove or refute Goldbach’s conjecture, according to ARITHMETIC, the conjecture is currently either true or false, not both, not neither. ARITHMETIC encodes the plausible idea that arithmetical truth is bivalent: every single arithmetical sentence f is either true or false, not both, not neither.1 This is intimately related to, and often taken to be explained by, the fact that every single interpretation of our arithmetical language that isn’t somehow ruled out by our practice has the very same structure. Say that an interpretation I of our arithmetical language is “admissible” if it isn’t ruled out by our practice, “inadmissible” otherwise. If every admissible interpretation of arithmetic has the very same structure, then any sentence f will be true in an admissible interpretation of arithmetic I just in case it is true in every admissible interpretation of arithmetic. Against this background, ARITHMETIC claims that in every admissible interpreta- tion of our arithmetical language, the “numbers” form an w-sequence. This is a claim that platonists and structuralists alike can endorse.2 In addition, even nominalists— those who reject the existence of abstract objects like numbers—will need to provide 1Given this, it may be wondered I don’t simply characterize determinacy using truth and falsity predi- cates (“T”, “F”) and a name-forming device (“ , ”) as follows: T f F f (or perhaps this conjoined p q p q _ p q with “ (T f F f )” and “ ( T f F f )”)? I don’t because there are subtle issues concerning ¬ p q ^ p q ¬ ¬ p q ^¬ p q the relationship between claims like these “internal” to our formal truth theory and “external” claims like bivalence. In particular, I’m skeptical that this disjunction states bivalence in the sense we want unless cer- tain assumptions about the background logic and proof system are also made. In addition, note that the “not neither” conjunct of the more involved statement of bivalence is simply equivalent to the simpler, dis- junctive statement if the truth theory is formulated in a logic with the DeMorgan laws and double negation elimination. 2Platonism is endorsed in, e.g., Gödel (1964), Maddy (1990), and Woodin (2004); structuralism is en- dorsed in, e.g., Benacerraf (1965), Resnik (1997), and Shapiro (1997). 2 1 THE PUZZLE some surrogate theory to handle the work that arithmetic does in science and daily life, so nominalists can endorse a version of ARITHMETIC couched in terms of the primi- tive notions of their favored surrogate theory.3 The key point is that ARITHMETIC isn’t a philosophically controversial claim accepted only by those who favor a particular philosophy of mathematics; rather it is a key point of agreement that cuts across most other issues in the philosophy of mathematics. ARITHMETIC is intuitively appealing—it sure seems like there is a fact of the mat- ter about Goldbach’s conjecture that is independent of our ability to discover it. But intuitively appealing claims are sometimes false. Why should we think that ARITH- METIC is true? ARITHMETIC is so fundamental and ingrained in our way of thinking that it’s dif- ficult to argue for it in a non-question begging way. Perhaps the best we can do is to point to the difficulty of conceiving of its falsity. Goldbach’s conjecture is true if there is a natural number that is both even and not the sum of two primes, false otherwise; what possible borderline case could there be? The difficulty of even imagining ARITH- METIC’s falsity is itself some evidence that ARITHMETIC is true. The informal rule of inference being used here is: if D is some state of affairs, from the inconceivability of D’s obtaining, we can conclude that D doesn’t, in fact, obtain: D is inconceivable (Incon) D ¬ It’s plausible that (Incon) is highly reliable, but it’s almost certainly invalid. There are more things on Heaven and Earth than are dreamt of in our philosophy.4 Still, the difficulty of conceiving of the falsity of ARITHMETIC is some reason to accept its truth, however weak. In addition, as far as we can conceive of the falsity of ARITHMETIC, its falsity is implausible.5 Combined with its intuitive plausibility, these considerations give us serious (but perhaps not independent) reason to accept ARITHMETIC. (2) COGNITION: human cognition is algorithmic. An algorithm is a recipe for solving a problem using a sequence of small, mindless steps. To say that some process is algorithmic is to say that it results from the execution of algorithms. The general notion of an algorithm abstracts away from issues of memory, lifespan, and computa- tional speed. This means that a process can be algorithmic without being computation- ally feasible. Logicians have proposed a number of formalized mathematical analyses 3Nominalism is endorsed in, e.g., Azzouni (2005), Chihara (1990), and Field (1980). 4Consider the converse inference, from D to D is conceivable, there is no reason to think this is valid, unless our minds were somehow specially designed for universal fact finding. And in many contexts these two inference rules will be interderivable, e.g., if we start with (Incon) we can derive the validity of its converse using conditional proof and a contrapositive rule ( f y y f). ¬ !¬ ` ! 5Some countervailing considerations are adduced in Field (1994). 3 1 THE PUZZLE of this informal notion, e.g., Turing machines, register machines, recursive functions, etc.6 In perhaps the most remarkable convergence in the history of logic, virtually all formal analyses of the notion of an algorithm that have been seriously proposed turn out to agree—a function is computable by a Turing machine just in case it computable by a register machine just in case it is recursive and so forth and so on. In addition, everything that is intuitively algorithmic is Turing computable, recursive, etc. This alignment of our formal and informal notions gives rise to Church’s Thesis: a func- tion is computable by an algorithm just in case it is computable by a Turing machine just in case it is computable by a register machine just in case it is recursive, etc. Al- though the thesis links an informal notion (algorithmic computability) to formal notions (e.g., Turing computability, recursiveness) and so isn’t open to a rigorous mathematical demonstration, no serious threat to Church’s thesis has ever been proposed.7 So far we’ve just been discussing numerical functions, but as every modern com- puter user knows, algorithms can do much more than crunch numbers when hooked up to action performing input/output machines. In this way, our modern digital computers play movies, process documents, analyze visual data and so on and so forth. This opens up the possibility that anything the human mind can do is done via some algorithmic process. Given Church’s thesis, to say that human cognition is algorithmic is to say that the mind can’t do anything that can’t be done, in principle, by a Turing machine.8 Al- though the human mind isn’t perfectly understood, one central feature of any scientific approach to the mind is that human cognition results from brain activity; the relevant kind of brain activities consist of neurons firing in various complicated patterns and thereby executing various complicated algorithms.
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