In my last post, I said the following to defuse the idea that any particular scheme for determining which sets exist, whether the unrestricted comprehension axiom of naive set theory or the more conservative axioms of various later theories, is something we can be particularly confident about:
"The realm of sets, if it exists, is notoriously epistemically inaccessible to us. (In fact, as Benaceraff famously pointed out, if we assume that it exists, it still seems to be the case that, if every set in that realm disappeared tomorrow, we'd never know.) Even if we assume that there's a compelling case for set-theoretic realism--i.e. for the conclusion that at least some sets exist--the question of which sets exist is still very much open."
OK, so, given that, what reason do we have (if any) to suppose that at least some sets exist? Here's what Stephen Yablo says, in his article "A Paradox of Existence":
"About fifty years ago, Quine convinced almost everyone who cared that the argument for abstract objects, if there was going to be one, would have to be a posteriori in nature. And it would have to be an a posteriori argument of a particular sort: an indispensability argument representing numbers, to use that example, as entities that ‘total science’ cannot do without."
If that's one's reason for granting the at least some existence of sets, though, we clearly don't need all the sets you get from the unrestricted comprehension axiom. For example, the sets in the ZFC cumulative hierarchy should be (way, way) more than enough to reconstruct the fragment of mathematics you need for our best current scientific theories.
Remember that the original context of this series of posts is Graham Priest's argument for dialetheism on the basis of the antinomies of naive set theory. It seems to me, though, that we have to start by having a pretty damn good reason to accept unrestricted comprehension if we're going to be willing to sacrifice Non-Contradiction for its sake. Of course, if we're going to engage in the project of arguing with dialetheists (which is, after all, what I'm doing here), "your premises must be wrong because I don't like your conclusion" isn't much of a counter-argument. Fortunately, in this case, we can do better.
After all, to review:
(1) The most plausible argument for thinking that any sets exist at all is the indispensibility argument,
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(2) To set-theoretically reconstruct the fragment of mathematics that's actually indispensible to our best-supported scientific theories, you don't need as many sets as unrestricted comprehension gets you,
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(3) As I argued last time, if we lack evidence one way or the other about what exists in some realm of reality to which we have no direct access, it makes sense to conservatively assume that its logical structure doesn't radically depart from the logical structure of the parts of reality we are familiar with. This seems like a good general principle that everyone can accept--e.g. dialetheists who (like Priest) don't think that there are any counter-examples to the Law of the Excluded Middle have every reason to reject evidentially unmotivated claims that such counter-examples exist in some strange, epistemically inaccessible domain of reality.
Putting (1)-(3) together, it looks like, unless we've already been convinced of dialetheism by some other argument, the paradoxes of naive set theory give us no reason to believe in the existence of sets with inconsistent properties. Considered as an independent argument for dialetheism (which is how Priest seems to view it), we have excellent, non-question-begging reasons to reject one of the crucial premises.
So far, so good. Does this mean, though, that no argument for independent argument for dialetheism could be mounted on the basis of the existence of inconsistent mathematical objects, given our limited epistemic access to such objects, the conservative principle (3) above, and so on?
Actually, no. Indispensibility cuts both ways. Here's what I say about that in my dissertation:
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Now, this suggestion--that our best reason to believe that the total universe of sets fails to contain the Russell Set is that we are only justified in being realists about the sorts of mathematical entities that our best-confirmed scientific theories are ontologically committed to--has a consequence that many of my fellow monaletheists[*] might find extremely unattractive. So as not to look like I’m skirting the issue, it’s worth pausing for a moment to spell it out, and to explain why I regard it not as a bug but as a feature.
While Priest usually likes to portray the intellectual adjustment involved in admitting the possibility of true contradictions as a smaller change than it might appear, one without sweeping consequences for our practices of reasoning--we’ll come back to this point with a vengeance in Chapter Seven, when we consider his “classical re-capture”--in other, more enthusiastically optimistic moods, he likes to speculate about the wide-ranging consequences that the jump to the ‘realm of the transconsistent’ might have.
"In modern science, the inferentially sophisticated part is nearly always mathematical. An appropriate mathematical theory is found, and its theorems are applied. Hence, a likely way for an inconsistent theory to arise now in science is via the application of an inconsistent mathematical theory. Though the construction of inconsistent mathematical theories (based on adjunctive paraconsistent logics) is relatively new, there are already a number of inconsistent number theories, linear algebras, category theories; and it is clear that there is much more scope in this area. The theories have not been developed with an eye to their applicability in science—just as classical group theory was not. But once the paraconsistent revolution has been digested, it is by no means implausible to suggest that these theories, or ones like them, may find physical application—just as classical group theory did. For example, we might determine that certain physical magnitudes appear to be governed by the laws of some inconsistent arithmetic, where, for example, if n and m are magnitudes no smaller than some physical constant k, n + m = k (as well as its being the case that n+ m ≠k). There are, after all, plenty of episodes in the history of science where we came to accept that certain physical magnitudes had somewhat surprising mathematical properties (being imaginary, non-commuting, etc.). Why not inconsistency?"
The suggestion may seem absurd, but consider the analogy of physical geometry. For thousands of years, pretty much everyone was what we might think of as a “Euclidean monist,” meaning that they took it as obvious that there could only be one shortest path between any two points, and that they understood this not just as a claim about the particular hypothetical realm of one axiomatic system but as a (a priori knowable) claim about physical reality as well. Even when non-Euclidean geometries started to be developed, they were largely regarded as esoteric curiosities that couldn’t possibly model anything in the real world. If a student went to Immanuel Kant’s table at his tavern in Konninsberg and suggested to the great man that future science might plausibly one day falsify Euclidean monism and allow for multiple non-equivalent shortest paths between two physically real points, the young man would have presumably been laughed out of the place. Yet, history has rendered the opposite verdict; Einstein has shown that the true geometry of space-time is non-Euclidean, and so much the worse for our spatial intuitions.
Of course, for the student to demand without special evidence or argument that Herr Professor Kant accept that this was possible (particularly in any richer sense than epistemically possible, where ‘epistemically possible’ is taken to mean something like ‘having a status such that one should keep an open mind about it’) would be to beg the question against Euclidean monism, and the good professor would have been fully rationally entitled to decline to do so. Similarly in this case, if we take Priest’s rhetorical question as a demand that we (monaletheists) take his wild speculation about inconsistent magnitudes seriously as a possibility. The problem, of course, as the Einsteinian case shows, is that the lesson of the history of science seems to be that even intuitively well-grounded claims about impossibility can be falsified, and that we shouldn’t be dogmatically closed off to the (epistemic) possibility that we’re wrong about (logical and mathematical) possibility. Priest lays out a fanciful sort of scenario where monaletheist assumptions about logical and mathematical possibility are in fact falsified by empirical research.
"Let us suppose that come to predict a collision between an enormous star and a huge planet. Using a standard technique, we compute their masses as x1 and y1 respectively. Since masses of this kind are, to within experimental error, the sum of the masses of the baryons (protons and neutrons) in them, it will be convenient to take a unit of measurement according to which a baryon has mass 1. In effect, therefore, these figures measure the number of baryons in the masses. After the collision, we measure the mass of the resulting (fused) body, and obtain the figure z, where z is much less than x1 + y1. Naturally, our results are subject to experimental error. But the difference is so large that it cannot possibly be explained by this. We check our instruments, suspecting a fault, but cannot find one; we check our computations for error, but cannot find one. We have a puzzle. Some days later, we have a chance to record another collision. We record the masses before the collision. This time they are x2 and y2. Again, after the collision, the mass appears to be z (the same as before), less than x2 + y2. The first result was no aberration. We have an anomaly.
"We investigate various ways of solving the anomaly. We might revise the theories on which our measuring devices depend, but there is no obvious way of doing this. We could say that some baryons disappeared in the collision; alternatively, we could suppose that under certain circumstances the mass of a baryon decreases. But either of these options seems to amount to a rejection of the law of conservation of mass (-energy), which would seem to be a rather unattractive course of action.
"Then someone, call them Einquine, fixes on the fact that the resultant masses of the two collisions were the same in both cases, z. This is odd. If mass has gone missing, why should this produce the same result in both cases? An idea occurs to Einquine. Maybe our arithmetic for counting baryons is wrong. Maybe the appropriate arithmetic is one where z is the least inconsistent number, and p (the period of the cycle) = 1. For in such an arithmetic x1 + y1 = x2 + y2 = z, and our observations are assumed without having to assume that the mass of baryons has changed, or that any are lost in the collisions! Einquine hypothesizes that z is a fundamental constant of the universe, just like the speed of light, or Planck’s constant."
The story goes on, but the general idea should be clear enough. We have a localized change of the arithmetic assumed by our best science, one that doesn’t require us to change the way we count match-sticks, any more than the non-Euclidean curve of space-time stops us from making Euclidean assumptions about how to make geometrically complicated bank shots when we play pool, but which is just as deadly to monaletheism as Einstein’s discovery was to Euclidean monism. Of course, the details of this story could doubtless be nit-picked in many ways, but I very much doubt that someone in the late eighteenth century trying to imagine a way that science could falsify Euclidean monism could do any better. It’s in the nature of scientific revolutions that, before they come along, they’re not only unpredictable but often almost completely unimaginable. Similarly, at the level of detail that’s Priest’s given us, there’s no way to be sure whether (even in the extreme, strange, counterpossible hypothetical situation being considered) revising the underlying arithmetic of our theory from a classical to a paraconsistent one would be the most rational response. As per our discussion at the end of the introduction, recalcitrant evidence can always be taken as evidence against many different parts of our overall package of beliefs, and any given proposal for belief-revision has to be reasoned out on the specifics of the case, specifics that are unknowable in consideration of these sorts of extreme hypotheticals. So we can’t be too confident in advance that, even in this specific scenario, the best scientific move would be the one Priest hypothesizes.
That said, if Priest’s scenario came true, and it turned out that the best, most reasonable response was to formulate a scientific theory to which inconsistent mathematical entities were indispensible, then, by my lights, we should be realists about such entities, just as we have a good reason to be realists about as much set theory as we need to make sense of our best current science. Thus we have the potentially unattractive consequence mentioned earlier: my solution to Russell’s Paradox leaves the empirical/mathematical back door open for dialetheism. If the empirical results were to go in certain extremely unexpected directions, then inconsistent mathematical results would give us an excellent reason to abandon monaletheism.
As I said, this strikes me as not a bug but a feature of my solution. A belief that no evidence would ever dislodge is a dogma, and one doesn’t need to subscribe to all the details of Karl Popper’s epistemically impoverished theory of justification to agree with him that one of the great lessons of the last few hundred years of scientific progress is that unfalsifiability is a mark of theories that don’t deserve to be taken seriously. If monaletheism is intuitively compelling, there are no good arguments for dialetheism, and the consequences of accepting dialetheism are unappetizing, then we’re within our rational rights to retain the monaletheist structure of our beliefs, but we should always keep the door open for the world to push back against those beliefs and show us that we’re wrong.
Of course, we don’t have a right to even this limited conclusion until we’ve satisfactorily blocked the single strongest argument for dialetheism, which is the argument from the Liar Paradox. It is to this, then, which we now turn.
[*"Monaletheism" is the view that contradictions are never true. When being careful, I like to talk about "monaletheism" rather than "the LNC" as the view being challenged by dialetheists because, technically speaking--if by "the LNC" one just means the logical formula that tells us that for any conjunction of a claim and its negation, the negation of that conjunction is true--then dialetheists don't generally reject the LNC. That formula continues to be a logical truth in systems like Priest's favored logic LP. It's just that dialetheists who accept it also think it has true--and false--counter-examples.]
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But wait! Doesn't this amount to solving Russell's Paradox and the Liar Paradox in different ways, despite their deep structural similarity? And doesn't that violate the Principle Of Uniform Solution?
Stay tuned for next Monday's post to find out!
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