Epistemic modals and mathematics Craige Roberts and Stewart Shapiro November 19, 2015 Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The cases There is, of course, a large literature in philosophy and linguistics on epistemic modals. It is common, at least in philosophy, to introduce the topic with examples from mathematics, in order to distinguish the targeted epistemic modals from other kinds of modals. John MacFarlane [2011, 144], for example, writes: It is hard to say exactly what makes a word modal, or what makes a use of a modal epistemic, . . . but some examples should get the idea across. If I say “Goldbach’s conjecture might be true, and it might be false”, I am not endorsing the Cartesian view that God could have made the truths of arithmetic come out differently. I make the claim not because I believe in the metaphysical contingency of mathematics, but because I know that Goldbach’s conjecture has not yet been proved or refuted. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The cases In short, this modal is not metaphysical, saying what the world might be like. Similarly, the modal here is not deontic, saying that the conjecture is somehow morally or pragmatically permissible. The statement has something to do with what is known, what is believed, what there is evidence for, or the like. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The current state Despite the controversies, there is broad agreement on some aspects of the semantics of epistemic modals. But those agreed aspects are all incorrect. Virtually all articulated accounts give the wrong truth conditions for these particular epistemic modals. Consider an utterance in the form “It might be that P ” or “It must be that Q ”, both understood epistemically. The embedded proposition, P or Q is called the prejacent of the utterance. In McFarlane’s first example, the prejacent is a statement of the truth of the famous Goldbach conjecture, that every even number greater than two is the sum of two primes; in his second example, the prejacent is that the Goldbach conjecture is false. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The current state On virtually all accounts, the context of utterance will determine an agent, or a group of agents, and a body of information related in a relevant way to the agents. The first utterance, “It might be that P ” is true just in case P is compatible with , or somehow not ruled out , by the indicated body of information The second utterance, “it must be that Q ” is true if Q is entailed by the body of information, or, equivalently, if ¬ Q is ruled out by the information. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The controversies As noted the controversies over epistemic modals are many. First, who are the agents? Is the agent always the speaker only? Or the speaker plus those addressed? Or the speaker plus those who happen to be listening (including eavesdroppers)? Or humanity at large, perhaps at a given time, or at all times? Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The controversies Second, how are the agents determined? Is it from the context of utterance (as with typical indexicals)? Or is it from a separate context of assessment? A matter of pragmatics, or semantics? Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The controversies Third, what goes into the body of information? Is it what the agent or agents know? Or is it what they know that is relevant to the topic at hand? Or is it what they strongly believe? Or what they are supposing in that context, the common ground? Or is there some other contextually supplied body of evidence that is relevant? Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The controversies Fourth, what sort of semantics is appropriate for epistemic modals? Are we in for a variety of contextualism, where the content of the utterance varies from context to context? If so, of what kind? Are epistemic modals like indexicals? Or is there some sort of elided constituent in the utterance? Or a variable in logical form? Or is relativism a better option? Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The problem We will not engage any of those issues here (or not much). Our concern is with what is meant by the indicated body of information being “ compatible with ” the prejacent ( P or Q ) or by the information “ ruling out ”, or “ not ruling out ”, or “ guaranteeing ” the prejacent. Typically, the relevant relation is said to be logical consistency , or entailment . No other relations to serve this purpose have been articulated. Consistency and entailment, however, give the wrong truth-conditions for the mathematical examples–no matter how the other matters are resolved. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The problem Let us modify McFarlane’s example a little. Go back to 1990 (or 1900, or just about any date between 1640 and 1990), and suppose that a mathematician says, then, that Fermat’s Last Theorem might be true and it might be false. The intuitive verdict, apparently in line with McFarlane’s suggestion, is that, in that context (or as assessed from that context), the mathematician’s utterance is true. No proof, of course, and no refutation was known then–by anyone. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The problem However, the standard Dedekind-Peano axioms for the natural numbers were known then, at least tacitly, by the relevant agent or agents–or so we stipulate. One might go so far as to argue that the Dedekind-Peano axioms are somehow implicit in the very notion of being a natural number. They are perhaps believed, and maybe known–tacitly–by anyone who understands the content of Fermat’s last theorem, or any other sufficiently rich statement about the natural numbers. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The problem And the negation of Fermat’s Last Theorem is decidedly not consistent with the Dedekind-Peano axioms (or other things in the epistemic basis). That is what Wiles established (possibly, sort of). So, according to the standard truth-conditions for epistemic modals, it was not true then that Fermat’s Last Theorem might be false. It was not true at any time when the relevant epistemic basis–the Dedekind-Peano axioms–were known, for the simple reason that (thanks to Wiles) the negation of Fermat’s Last Theorem is not consistent with those basic principles–all of which were known at the time. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The problem aside Aside: It is highly likely that Fermat’s Last Theorem has an elementary proof (i.e., from the Dedekind-Peano axioms). See, for example, McLarty [2010]. If it does not, then just expand the epistemic base–what is presumed known by the agent(s) in question–to include the basics of elliptical function theory (the background of Wiles’s proof). Of course, if we have to do that, it is not correct to date our scenario as early as the seventeenth century. The year 1990 will still do. Notice, incidentally, that our above statement that it is likely that Fermat’s Last Theorem has an elementary proof is itself an epistemic modal. One of our (ultimate) goals here is to give truth conditions for it. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The problem It seems that an utterance in the form, “it might be that P ” is false if the negation of the prejacent P can be readily determined by the relevant agents, or at least if it can be easily deduced from what they know. To paraphrase, using our example: “there might be a counterexample to the Goldbach conjecture” is false if the Goldbach conjecture can be readily deduced from what is known (however it is known). Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
The problem Not quite. Consider, for example, a much discussed scenario in Hacking [1967, 148]: Imagine a salvage crew searching for a ship that sank a long time ago. The mate of the salvage ship works from an old log, makes some mistakes in his calculations, and concludes that the wreck may be in a certain bay. It is possible, he says, that the hulk is in these waters. No one knows anything to the contrary. But in fact, as it turns out later, it simply was not possible for the vessel to be in that bay; more careful examination of the log shows that the boat must have gone down at least thirty miles further south. The mate said something false when he said, “It is possible that we shall find the treasure here,” but the falsehood did not arise from what anyone actually knew at the time. Epistemic modals and mathematics Craige Roberts and Stewart Shapiro
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