McDowell on Field on Tarski

25 02 2005

Last week I reread Hartry Field’s “Tarski’s Theory of Truth” and John McDowell’s response, “Physicalism and Primitive Denotation: Field on Tarski”, and realized I hadn’t understood the McDowell at all before. Field’s point seems straightforward enough – Tarski gave a materially adequate characterization of truth that was not explanatory of the concept any more than a listing of the net gravitational force on every pair of objects in the universe explains gravity. Field argues that in order to get a proper explanation, we should be able to explain why a word means what it does, which would presumably involve something like Kripke’s causal theory of names, together with various psychophysical explanations of how we acquire meanings for words.

However, McDowell seems to think that such a story is impossible, and that Tarski’s theory is adequate for what actually is possible. Until I read some Davidson, this didn’t make much sense. But now I think I understand Davidson’s basic idea – that mental events are all identical with physical events, but we can never be satisfied with any lawlike explanation of which mental events correspond to which physical events unless we stop regarding people as rational agents and seeking to interpret them with a principle of charity. Thus, explaining truth merely in terms of truth-conditions, and truth-conditions merely extensionally, is all that we can do, and it is in fact a useful project. For McDowell, it seems that the point of the concept “true” is to allow us to interpret other speakers, so that whenever someone says something, I can say “What she said is true iff x”, where x is some set of conditions expressed in my language. My theory of truth will have to be different for each new speaker I encounter, because I have to understand each of them slightly differently. This is clearest when they’re speaking non-English languages, but even if both speak English languages, they’ve almost certainly got subtle differences in their usage of particular words.

However, for Field, it seems that the purpose of a theory of truth is not just to understand speakers of other languages, but to understand even my own language. That is, I can understand people speaking my language directly, but if I want to understand how it is that language hooks up with the world, then I need the concept of truth.

And of course, this all links back up to Dummett in “Truth” wondering just what the point of truth is. He suggests that it has something to do with the fact that a speaker “usually seeks to make true statements”, but notes that this picture is far from complete. Davidson and Field are just giving two competing pictures of what more it is we might want.


Dummett on Logicism and Field

11 02 2005

In “What is Mathematics About?” (in The Seas of Language), Michael Dummett puts forward a view of logicism that I think is not quite orthodox. Rather than saying that it is an attempt to derive the truth of mathematics purely from logic, he says that mathematics just is the business of working with deductive proofs to produce theorems that can be plugged in later. I took Frege’s program (and Hale and Wright’s revival of it) to be an attempt to justify from logic along quantification over a domain of actually existing mathematical entities, and not merely an attempt to justify mathematical practice.

However, Dummett’s suggestion strikes me as the most reasonable suggestion I’ve heard in all of these debates. He doesn’t require the existence of mathematical entities: numbers don’t need to exist – we just abbreviate long deductive proofs into theorems “about them” that get applied whenever we count things. I think this is also quite close to what mathematicians feel like they’re doing. A group theorist thinks that what she is doing is just writing down results that hold for whatever things happen to satisfy the group axioms, as a topologist is coming up with results that apply for anything that happens to satisfy the topological axioms assumed.

However, in calling this “logicism”, I think Dummett misses some of the point of Hartry Field’s fictionalist program. Dummett says that Field’s program would require a conservativity result for every mathematical theory applied in every scientific theory, and this proof would have to be carried out for each such pair of theories. I think this is basically right. But then Dummett goes on to say that this could never be finished, because there is always the possibility of the discovery or invention of more theories, both mathematical and scientific, and new conservativity proofs would need to be found. Dummett thinks this is a problem, but I think this is also exactly right.

The conservativity proof is just a statement of the applicability of the mathematical theory that is used. After all, although every particular actual scientific application of real numbers might be conservative, it certainly isn’t the case that every application of the real numbers is. One can use real numbers to define physically meaningless quantities of a system and then state that these quantities are conserved, which would result in many many contradictions. (If the physical theory is complete, then of course any non-conservative extention will be inconsistent, as Prior’s addition of “tonk” was to classical logic.) There may be some general patterns to which these conservativity results conform, so that many different applications of real numbers (say to lengths, volumes, times, forces, masses, etc.) can all be seen to be conservative in similar ways. But this makes sense, because it’s basically just showing that the physical objects satisfy the axioms of the real number system, which means that we can take Dummett’s logicist view, that then the application of real numbers is just to give results that were known to be true because of the fact that the system satisfies the axioms. And this is exactly what Field says about conservative extensions – they just shorten the proofs!

Of course, proving the conservativity of the reals over any of these physical systems will be quite hard. Applying reals to two different types of dimensionality sounds confusing! But once we figure out how to do it (maybe Field already has – I should reread Science Without Numbers) it will become much easier to do for all other theories.

Intrinsic and Extrinsic Justifications of Axioms

6 02 2005

In Does Mathematics Need New Axioms? (pdf), Solomon Feferman discusses the kinds of foundational axioms available so far for mathematics. (Foundational axioms are primarily taken to be the axioms of logic, Peano arithmetic, and ZF set theory, together with Choice – these are axioms mathematicians use to justify their mathematical reasoning in general, while other sets of axioms are just used to define particular structures.) The Peano axioms (including induction) are justified by immediate reflection, as are the axioms of logic. Most of the axioms of ZFC are also similar justified (such as the empty set, powerset, pairing, and union axioms), while the others are accepted because of their necessity for rigorizing various mathematical arguments that had already been accepted (in particular Cantor’s arguments about the existence of ever-increasing cardinal numbers, and Zermelo’s argument that the Axiom of Choice implies the Well-Ordering Principle). These last few axioms (primarily separation, replacement, infinity, and choice) are thus justified in a somewhat extrinsic way, by appeal to their consequences. However, all of these axioms in PA and ZFC are in the end accepted for primarily intrinsic reasons – the reason their necessity for these arguments is decisive is that these arguments had already been accepted, and thus the axioms had been implicitly as well. While Frege’s inconsistent system also had intrinsic justification, it turned out to be inconsistent, and ZFC was a weakening that seemed to have similar intrinsic justification and seemed likely to be consistent as well. The extrinsic justification was just to show that these principles had in fact already been used, and thus to point out that the intrinsic justification was stronger than in Frege’s case.

In the case of large cardinal axioms, Feferman has more specific things to say. For those smaller (weaker in consistency strength) than a measurable cardinal, the justification is primarily intrinsic, using a principle Feferman calls “Cantor’s Absolute”. Basically, this principle asserts that whatever closure properties the “cardinality” of the universe has are shared by some actual cardinal. This is analogous to some form of reflection principle, according to which, for any finite true theory T, some ordinal \alpha can be found such that V_\alpha satisfies T. This principle is plausible because of Gödel’s Completeness Theorem, as I suggested in the fourth paragraph of my earlier post, “Consistency and Platonism”. Any true theory must be consistent (on a platonist account), and thus there must be some model satisfying this theory by the completeness theorem. Thus, believing that the axioms of ZFC are true (in a literal platonist sense) gives one strong reason to believe in addition that there are inaccessible, Mahlo, hyper-Mahlo, etc. cardinals.

However, none of this reasoning gives evidence for the existence of measurable (or larger) cardinals. Measurable cardinals were postulated as early as 1930, before Gödel’s completeness or incompleteness theorems, by Stanslaw Ulam, suggesting that they have some inherent mathematical interest beyond their consistency strength. In addition, measurable cardinals (and many other stronger large cardinals) create much more set-theoretic structure that is of interest, and thus have some slight extrinsic justification.

Harvey Friedman has been trying to strengthen this justification by extending the work of Paris and Harrington in providing a statement that is seen to be true by mathematicians but is independent of the given foundational axioms. The Paris-Harrington theorem turns out to be independent of PA, but equivalent to the \omega-consistency of PA (the statement that if P(0), P(1), P(2),… are all provable in PA, then so is \forall x P(x) ), and Harvey Friedman has come up with further theorems that are equivalent to stronger subsystems of second-order arithmetic. However, the results he has come up with that require large cardinal axioms (even just the ones smaller than a measurable cardinal) have no intuitive mathematical justification other than the consistency of these large cardinal axioms, which is certainly not clear for the cardinals beyond measurables. Thus, Friedman’s work on giving better extrinsic justifications (proving theorems that seem to be intuitively true) seems only to have succeeded so far for subsystems of second-order arithmetic, not extensions of ZFC. (Friedman seems to have higher hopes for his Boolean Relation Theory described in his contribution to a panel discussion about new axioms in mathematics in 2000.)

Interestingly, Feferman describes the Quine-Putnam indespensability argument as a kind of extrinsic justification for axioms. I suppose this view makes most justification of physical theories extrinsic, as they are believed to the extent that they predict the observations, not to the extent that they seem plausible. However, I think it is useful to distinguish this sort of extrinsic justification from the kind advanced for large cardinal axioms beyond measurables, which merely postulate interesting additional structure beyond what has already been observed, expected, or intuited. The former kind seems much more justifiable to me than the latter. I suppose I’ll have to read more Maddy to see what she says on the matter. But at any rate, Quine-Putnam style arguments probably won’t support much beyond RCA_0 in second-order arithmetic, much less ZFC or anything larger. Maddy needs to use mathematics (and in particular, set theory) as the scientific practice to respect in order to get this high, where Quine and Putnam just wanted enough to explain our observable physical universe.

Field on Consistency

2 02 2005

In “Realism, Mathematics, and Modality”, pp. 31-32, Hartry Field makes what seems to me to be a muddled argument about the consistency of set theory on platonistic grounds. He starts by suggesting that it is unproblematic that if a theory T is true, then it must be consistent. However, he then suggests that the Gödel and Henkin proofs of the completeness theorem, guaranteeing the existence of a model of T, are a sort of “accident of first order logic”, as if this were an argument for the platonist against identifying logical consistency with the existence of a model.

If S is a sentence that is false in set theory, then set theory shouldn’t imply S. But if no set – no proper part of the set-theoretical reality – could serve as a model for the entire set-theoretical reality, as seems prima facie possible, then set theory would Tarski-imply S. Again, if we are platonists we are saved from this possible extensional divergence between implication and Tarski-implication, by virtue of [the completeness theorem].

I suppose that here by “set theory” he means to refer to the collection of all true first-order statements in the language of set theory (working from a platonistic perspective). What makes this make sense is the fact that “set theory” is not recursively axiomatizable (the way ZFC or any of its standard extentions are), so Gödel’s Incompleteness Theorem doesn’t apply here. Thus, “set theory” can be a theory that implies its own consistency, and thus has models of itself inside any model. I don’t see why the Completeness Theorem is seen as somehow “accidental” and thus not reason enough for the platonist to accept the Tarski definition of implication.

Rather, what seems to me a better argument against the platonist set theorist using the Tarski definition of logical consequence (which relies on the existence of models, which are set-theoretic constructs) is that logical consequence should be prior even to set theory even for the platonist. This is certainly the case for the logicist, who says that the way we know things about the set-theoretic universe is because they are logical truths. Thus, to know that truths about sets are logical truths, we must have some way of recognizing logical truth independent of the existence of set-theoretic entities whose existence has not been justified.

I think even other non-logicist platonists would have to accept the primacy of logical consequence over set theory, because the ideas of model theory are taken to apply to models like “the actual universe of sets” rather than just set-theoretic models. This is an interesting recursivity in foundational mathematics that I have noted, that model theory talks about set-theoretic entities, while set theory assumes that we are working within a model. I think if both are taken literally, then it really is a vicious circle, but we can get out of it by making a move like the one Field argues for in a confused way, by saying that logical consequence is basic, and Tarski merely gave us an extensionally adequate analysis of it.

Consistency and Platonism

1 02 2005

I noticed an odd thing this evening about set theoretic platonists. They seem to accept a proposed axiom as true just in case we can never prove it consistent with the previously accepted axioms, and they reject the ones (like V=L) that are equiconsistent with the previous system. This sounds odd at first, because of course we’d like to believe only things that actually are consistent with the axioms that we already have, so it seems that axioms of the latter sort would be better candidates than the former. But this doesn’t seem to be the practice of most set theorists.

If I understand John Steel right, this is because the equiconsistent axioms add no new interesting mathematical structures, while the others do. After all, if one is interested in studying the structure of ZF+V=L, these all exist inside structures of (say) ZF+”there exists a measurable”, because each of the latter structures gives a structure for the former by the relative consistency proof. However, structures of ZF+V=L don’t give rise to structures that satisfy some of these large cardinal axioms, so we lose that interesting area of mathematical research.

In addition, there seems to be another problem with consistency for platonists, in that whatever system of axioms is “true” (according to the platonists) may turn out not to be consistent. For instance, given a universe satisfying ZFC, either it contains a model of ZFC (in which case we can take the smallest one) or else it contains no model of ZFC. Now, based on Gödel’s Completeness Theorem, there is a model for a theory iff that theory is consistent, so if the universe contains no model of ZFC (as it mustn’t, if we “choose” the universe to be the smallest model of ZFC inside the “actual” universe), then ZFC is inconsistent. If the universe satisfies some large cardinal axioms, then these can be added to the picture as well.

What exactly this inconsistency means is a bit confusing to me. After all, it doesn’t seem like it could mean that there’s “actually” a proof of “0=1” from ZFC, but rather just that the universe “thinks” that some sequence of formulas actually is such a proof. If the model of ZFC that thinks ZFC is inconsistent is a set model within the actual universe, then it’s clear that this proof must “actually” either be infinitely long or use some steps or axioms that the model mistakenly thinks are valid but “actually” aren’t. But for the “actual” universe, I’m not sure what this could mean. Thus, I can see why a platonist who believes that ZFC is “actually true” might be forced to believe as a result that Con(ZFC) is “actually true” as well, and thus Con(ZFC+Con(ZFC)), and so on. This sequence of claims is I believe weaker even than the claim that there exists even a weakly inaccessible cardinal, but I can see how the analogy might hold. The set theorist is forced to accept a sequence of claims, each of which she knows can never be shown to be consistent merely given the (relatively) uncontroversial claims of ZFC. Thus, higher consistency strength, together with a lack of an obvious disproof, can be seen as a good guide to truth, allowing the set theorist to ascend the hierarchy of large cardinals. The fact that the large cardinals are linearly ordered only makes this seem more convincing.

However, I think if one doesn’t take the platonist view about ZFC, then one isn’t forced into this chain of reasoning. For the fictionalist (or formalist, or intuitionist, or whatever), it doesn’t make sense to say that ZFC is “actually” true, and so Gödel’s Completeness Theorem doesn’t force us into any awkward positions for denying Con(ZFC) – especially since we’re withholding judgement on whether Con(ZFC) itself is “actually true” also. If “ZFC is consistent” is interpreted in its natural language sense, we can believe it to be literally true while remaining non-committal about the set-theoretic claim Con(ZFC). Thus, we are never forced to take even one step up the hierarchy of consistency strengths.