Time Gunk and Zeno

I suppose this is an odd topic to post about immediately after FEW, having nothing to do with what went on there. But I talked to several people about this at lunches and in the evening, and they don’t think anyone has written up this idea before.

It seems that some objects may be “metaphysically simple” in the sense that they have no proper parts. For instance, it was once thought that atoms had this property (hence their name) and it may still be thought that quarks and electrons do. However, some people believe it’s at least possible that some objects are atomless, in that every part of the object can be further subdivided into more parts. Any such object is said to be made up of “atomless gunk”.

Of course, all this is talking about the spatial parts of some object. But as Ted Sider points out (in his chapter on Temporal Parts), it seems natural to consider the temporal parts of some objects as well. Some examples he gives are “Ted when he had long hair”, “Ted when he had short hair”, “the piece of clay while it was shaped into a statue”, and “the current time slice of the Eiffel Tower”.

But the talk of time slices (Sider’s “stage view”, I think) seems to me to make the presumption that he didn’t want to make in the case of spatial parts. If we think that some spatial parts of objects may be atomless, in that they have no part without a proper spatial subpart, then why not think that some temporal parts of some objects might have a similar divisibility property? In the spatial case, it seems possible that some physical objects have atomic parts and some not. In the temporal case this would seem weirder, but there’s obviously no logical contradiction with this idea.

I think Sider wants to allow that space and time both be coordinatized by real numbers. Thus, it could still make sense to talk about points of space even if they are located in the region spanned by some gunky object – there’s no need to suppose that the object and space have any parts in common, especially if one is a relationalist about space rather than a substantivalist. Similarly, an object could be temporally gunky even if it makes sense to talk about instantaneous moments of time. This would allow a temporally gunky object to exist at the same time as some non-gunky one.

If two objects cannot spatiotemporally overlap without sharing some parts, then this would also mean that no matter how you break the universe up into parts, there are only countably many disjoint ones. (Assuming the coordinatization of space-time is Archimedean.)

If all objects are temporally gunky, then this would provide a nice resolution of Zeno’s paradox of the arrow. The paradox says that at any moment, the arrow has a specific location. Thus, there is no moment at which the arrow moves. So the arrow must be stationary. However, if the arrow is temporally gunky, then it doesn’t make sense to talk about the arrow at any particular instant. It may only make sense to talk about the arrow extended over a (perhaps extremely small) interval of time. Any such part of the arrow occupies slightly different spatial regions at each moment, and thus every part of the arrow is moving. The “time slices” don’t move, but they also don’t exist on this picture.

For someone who believes what I imagine Sider to believe, that every collection of objects forms a further object, no matter how the parts are arranged in space and time, the paradox also goes away even if time is atomic. However, the explanation seems to miss something about the intuitive idea of motion. The theory suggests that it doesn’t make sense to talk about the motion of a temporally non-extended object, and I would agree. But this would mean that there are some objects (temporally extended ones) of which it makes sense to ask if they are in motion, and some objects (the stages) of which it doesn’t. Perhaps this is no worse than saying that an object must be spatially extended in order to ask whether it has a direction, but it does seem at least minorly more troubling.

And to say that there are some times at which the arrow is moving and some times at which it isn’t would only make sense (if at all) when talking about the whole arrow, and not its stage, because the stage is also part of many other objects that don’t move at all. But if it has no temporally atomic parts, then in the interval when the arrow moves from the bow to the target, it has no parts which are not moving. In some larger intervals it may have some parts that move and some that don’t. But at any rate, Zeno’s paradox disappears

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Back from FEW

FEW was tons of fun this year. Last year I didn’t get to meet as many people, but this year, since it was a bit longer and there were suggested bars each night, basically everyone knew everyone else by the third day or so. It was also nice to meet various people that I’ve seen online a lot, like Matt Weiner, Jonah Schupbach, Gillian Russell, Neil Sinhababu, and of course Sahotra Sarkar, whom I had met only briefly last year. And of course it was nice to meet the others as well, but I figured I’d link to my fellow bloggers. (If there’s anyone I left out, let me know and I’ll include the link.)

There were also a lot of nice interactions between the topics of the various talks. For instance, Alison Gopnik’s work on the way children learn the causal structure of the world seemed to have a lot of relevance to some of Andy Egan’s discussion of causal decision theory. And Gillian Russell’s inference (implication?) barriers had a lot to contribute to the discussions on several other papers.

FEW

I’m heading off to the second annual Formal Epistemology Workshop tomorrow, so I probably won’t be posting here for a week. (I’m not coming back from Austin until the 31st.) Last year I commented on a paper by Brian Weatherson – this year I’ll be presenting my own paper! Fortunately, I’m up first on the line-up, so I can relax and meet people for the rest of the week. Looks like some other logic-bloggers will be there too, and I’m looking forward to meeting them in person!

Elementary Equivalence and Structuralism

I was just grading an essay question on my students’ logic final asking them to contrast two of the views of philosophy of mathematics that we’ve discussed. In discussing structuralism, a few of them made statements that were ambiguous between talking about isomorphism and talking about elementary equivalence. But now that I think about it, maybe there is a reasonable version of structuralism where all that one wants to preserve is elementary equivalence rather than isomorphism.

Standardly, a structuralist (of one particular sort) says that there is no fact of the matter about what objects we’re talking about when doing mathematics beyond the isomorphism-type of the structure as a whole. That is, maybe the natural numbers really are the von Neumann ordinals (n={0,…,n-1}, so 0 is empty, 1 is {0}, 2 is {0,{0}}, etc.), and maybe they really are the Zermelo ordinals (n={n-1}, so 0 is empty, 1 is {0}, 2 is {{0}}, etc). But they most definitely are not one of the non-standard models of Peano arithmetic that contain “numbers” larger than any actual natural number.

But now what if there is no fact of the matter there either? The standard structuralist runs into a problem when trying to explain why she thinks we’re talking about one isomorphism class rather than the other, unless she brings in second order logic and maybe some other heavy (and questionable) machinery. But as long as we just limit ourselves to questions that can be stated in the language of number theory, these issues just don’t arise. We could try to say that in the standard model, no natural number has infinitely many predecessors – but “infinitely many” isn’t something we can say in the language of (first-order) number theory. If it’s set theory that we’re talking about, there really is an internal statement that says something like “there are uncountably many real numbers” – but this statement is true even in countable models, because it’s just talking about what pairing functions exist within the model. Unless we’ve decided once and for all what model of set theory we’re using, there’s no way to say what cardinality means in any given model – cardinality is only understood by talking about what actual functions exist, not what can be stated in some first-order language.*

This view isn’t exactly structuralism, because it says that we don’t even know what structure the natural numbers instantiate. But it’s more than empiricism, because it accepts all sentences we can have evidence for as equal, rather than prioritizing the observational ones. Once we’ve fixed the complete theory, then we’ve fixed everything there is to know. The other questions are somehow metaphysical only in a vague and problematic way, rather than the questions that can be stated in the internal language. I think this view might be related to the one Carnap proposed in “Empiricism, Semantics, and Ontology”, but of course that paper came out long before structuralism, so it might have been thought of as just a type of formalism.

*I think this issue points out why all those problems in mereology asking how many objects there are are troubling. I don’t know what it might mean to say that there are (or aren’t) inaccessibly many objects in the world, unless you’ve got an antecedent notion of set. And even more so for how many sets there are. Sure ZFC proves that the universe has certain closure properties just like those of V_k for k inaccessible. But this doesn’t mean that there “really are” inaccessibly many ordinals or whatever. That statement seems to me to mean nothing. After all, Skolem’s paradox shows us that there really could be just countably many in some sense, and we wouldn’t know the difference.

Nihilism vs. Universalism

When I was at the APA Pacific Division conference at the end of March, I attended an interesting talk by Matthew Slater opposing mereological moderation – the doctrine that for some things, there is an object made up of them as parts, and for some things, there is no object such that every part has a part that is part of one of those things. However, he suggested that he might be neutral between nihilism (the thought that no collections of things are the parts of some further thing) and universalism (the thought that any collection of things is the parts of some further thing).

It seems to me that if these really are the choices, then Hartry Field’s nominalist reconstruction of physics pushes us most definitely towards the latter. He has to quantify over non-atomic entities in order to do physics, so at least some entities must have non-trivial parts. But this just means that nihilism is not an option. To embrace this nihilism would mean a definite rejection of Field’s program, and thus an embrace of (perhaps more troubling?) abstract objects, rather than composite physical objects.

Not knowing the literature, I don’t know if this argument has already been made (or if the naturalistic prejudice it has is considered legitimate there), but it’s what I was thinking when I was at that talk.

Epistemic “Problems” With Platonism

The more I think about it, the more I think Benacerraf-style epistemic challenges to mathematical platonism aren’t very strong as stated. The point is that it’s supposed to be unclear how we could come to have knowledge of mathematical entities if they are supposed to be acausal and non-spatio-temporal. But to really put the argument forth like this, it seems that you have to have a fully worked-out epistemological theory that requires causation or location or something else that mathematical objects are supposed to lack. Now I’m no expert in epistemology, but I think if you really had a compelling positive position like this about what knowledge requires, then that in itself would be quite an impressive achievement. Sure, causal theories have been proposed, but from what I understand, the arguments in favor of them are primarily negative and don’t obviously apply to objects that are supposed to be non-physical anyway. And more importantly for me, I don’t really understand causation anyway, and I’m not convinced that it really is a part of a mind-independent reality, so I’m hesitant to require it in any of my philosophical positions.

Burgess and Rosen, in their book A Subject with no Object, point out that in fact it seems plausible that we might be able to have non-causal knowledge even of physical things. For instance, physics might predict that in the last five minutes before the end of the universe, particles called “eschatons” will appear, as part of a unified theory of the cosmos. Perhaps in a more down-to-earth example, it seems that physicists were in a position to claim knowledge about top quarks even before one was ever produced in a position to trigger a causal chain ending with a physicist. This is because the theory as a whole predicted them, and the theory was supported by its variety of other predictions that were causally verified.

I take it that a supporter of a Quine-Putnam argument in favor of mathematical realism could say that our knowledge of mathematical objects is similar, in that they play a role in the scientific theories that predict the phenomena we can observe. (I think that she even might be able to say that they do causally interact with us – is it really clear that the sun interacts causally through Newton’s law of gravitation, but that the gravitational constant and the squaring function don’t?) However, I take it that Hartry Field’s version of physics without mathematical entities would block this sort of argument. If we can construct a nice alternate theory with the same predictive power as Newton’s that doesn’t involve mathematical objects, then the knowledge we have of our theories could at best limit us to the two cases – one with the objects and one without. This doesn’t seem to be knowledge of the objects, even if they do exist.

Burgess and Rosen ask why the nominalist has to both give epistemic arguments against platonism and reconstruct a nominalistic science – if the epistemic arguments were compelling then the reconstruction would be at most a later task for scientists, and if the reconstruction is better than the original then the epistemic argument is unnecessary. However, it seems to me that on the reading I give in this post, the epistemic argument is incomplete without the reconstruction (as long as one admits the possibility of a non-causal theory of knowledge due to holistic support for a theory), and the reconstruction doesn’t obviously give one grounds to prefer the nominalistic theory without the epistemic arguments. Once you’ve got the nominalistic reconstruction, it’s not clear whether the (somewhat more complicated) nominalistic theory is to be preferred over the (more ontologically committed) platonist one, until reflection on the epistemic argument (suitably bolstered by the nominalist reconstruction) shows that even if the objects do exist, we can’t have knowledge of them. Thus, the two arguments work together to point towards nominalism, even though neither alone is sufficient.

New Logic(ish) Blog

Andrej Bauer of Carnegie Mellon has started a new blog called mathematics and computation. So far he has interesting posts about typesetting math on the web and the law of the excluded middle. Looks like there’s a lot of blogs sprouting up in these areas this year!

EDIT: Andrej Bauer is actually at the University of Ljubljana now, and no longer at CMU.

Gettier Cases in Mathematics?

In the run-up to my qual, I was discussing Benacerraf’s epistemological objections to platonism and the way it used a causal theory of knowledge. John MacFarlane challenged me to come up with a Gettier-type example that could possibly push one to adopt something like a causal theory for mathematical knowledge as well as physical knowledge. At the time the best I could come up with was a lot of the work of Newton and Euler in calculus, which came up with correct theorems, but used a lot of shady dealings with infinitesimals and non-convergent infinite series. By our standards, we shouldn’t be able to say that Newton and Euler knew these results any more than we’d say that a first-year calculus student knows the answer to a homework problem “solved” by plugging in x=0 and then canceling zeros from the numerator and denominator of some fraction. However, I think it’s status as justified true belief may thus also be questionable.

I think I may have come up with a better example recently though. Consider any of Cantor’s results in set theory. He was working with a fairly intuitive theory that may not have been consistent, but managed to establish many important results that were later verified with the machinery of ZFC set theory. Now consider Frege, who may well have formalized some of Cantor’s proofs of these results in his notation, and proved them using unrestricted comprehension. However, if the only instances of comprehension that he needed to use are the ones that appear as the pairing, powerset, union, replacement, separation, and empty set axioms of ZFC, then it seems more plausible to say that he actually was justified in his beliefs. Every step in his proofs was an instance of a schema we now generally recognize as valid. However, he believed these schemas were valid because they were all instances of the larger schema of full comprehension, which is inconsistent, and thus not valid.

So Frege may well have had JTB but not knowledge of various results from ZF (putting aside temporarily the question of whether any mathematical claims are “true”).

Spooky Action at a Distance

I’ve been rereading a bunch of the stuff on Field in the last couple days (my exam is in about 14 hours – I hadn’t planned on publicizing this blog until afterwards, but I guess I ended up doing so early) and I’ve been struck by the thought that a lot of nominalist worries about the causal efficacy of mathematical objects could be put a bit more mildly.

In general I’m a believer in the Quine-Putnam argument, though I think that mathematics actually is dispensible, so it ends up being a nominalist argument. This means that if mathematics had turned out to be indispensible, then I’d have to believe in its entities and confront the epistemological worries. But I think they aren’t really a serious problem, because our theory can be confirmed as a whole. I do think there is a slight issue though, in that these numbers that enter into our physical theories aren’t located near the entities they’re correlated with, or connected to them in any obviously causal way. The appearance of abstract objects in a physical theory seems to me like it’s at least a minor point against the theory, because it’s basically a stronger version of the “action at a distance” that Einstein and the early modern scientists disliked so much.

Of course, if the best theory predicts action at a distance, then I guess we have to live with it, and if the best theory predicts action from nowhere then we have to live with that too. But it doesn’t mean I have to like it any more than I should have to like a theory that is essentially non-deterministic (like quantum mechanics) or has any of a number of other blemishes that I’d like to think an ideal scientific theory wouldn’t have.

Eight Views of Mathematics

It seems to me that the three topics of my qualifying exam relate to three different questions about the objectivity of mathematics. The first is Hartry Field’s question about whether or not mathematical objects exist. The second is the one involved in higher set theory about whether or not there are methods of verifying mathematical statements that go beyond proof from already established axioms. And the third is Dummett’s question about whether or not there is a fact of the matter about mathematical statements that go beyond whatever verification procedures we have. Of course, in the latter two questions, the term “verification” should be taken with a grain of salt, because unless the answer to the first question is affirmative, mathematical statements won’t be literally true. But I think that in whatever sense one approves of standard mathematical theorems as opposed to their negations, one can raise the corresponding two questions.

Now, it seems plausible to me that these three questions are in fact independent, and I’ll try to sketch views that would be characterized as each of these combinations.

Y,Y,Y – I would think that something like this is the view of most set theorists working on large cardinals, and in particular was probably Gödel’s view. He took a naively platonistic view of sets, thought that there were means to discover new axioms beyond ZFC, and that every question about sets had a resolution in this platonic universe. Contemporary set theorists like Woodin and Martin retreat a little bit and call themselves “conditional platonists” or something of the sort, and also have a more sophisticated view of what counts as confirmation for an unprovable axiom. For instance, for them the axiom of projective determinacy (and thus the existence of infinitely many Woodin cardinals) is confirmed not just by the fact that each of its stateable instances is provable, but also that it predicted claims about sets of Turing degrees and continuous functions that not only have themselves been individually confirmed, but have also since become important topics in the study of their respective structures. Thus, the axiom makes many important and divergent explanatory predictions, and thus should be accepted as true.

Y,Y,N – Any position that states that mathematical objects exist, but that the facts of the matter don’t extend beyond our means of verification is going to be at least somewhat awkward. But I may be able to say that a position like this is what Steel suggests when he says that the continuum hypothesis might be ambiguous. That is, there might be two equally good set-theoretic universes, each isomorphic to a generic extension of the other, one of which satisfies CH and one of which doesn’t. Every statement a set theorist working in one structure makes can be translated into a statement about a generic extension of the universe by a set theorist working in the other. There is no fact of the matter as to which one of the two is talking about the “real” universe of sets, because each can be seen as being contained in the other. Most interesting mathematical statements will be decidable as above, but some (like CH) will just be considered ambiguous because they vary between the two models.

Y,N,Y – This is perhaps the naive view one takes after reading about Gödel’s incompleteness results. One believes that the universe of sets exists, and there are many facts of the matter about it, but that we have no access to any of these facts beyond ZFC. Never mind how we have knowledge about the axioms of ZFC. At any rate, there are strict limits to how much we can know.

Y,N,N – This is perhaps the slightly more sophisticated version of the above view. There are many universes of sets, and each consistent extension of ZFC is true in one of them. However, there is no fact of the matter as to which consistent extension is the “right” one, and we have no principled way to adjudicate between them.

N,Y,Y – This position also seems slightly awkward, but I can imagine someone like Hartry Field holding it. Mathematical entities don’t exist, but to the extent that we pretend they do, we might as well pretend they have a complete theory. Presumably the methods of set theorists will alow us to determine what statements should be satisfied beyond the axioms of ZFC. Or maybe we’re just allowed to make it up freely?

N,Y,N – I think this is the position that I am closest to. I agree with Field that mathematical objects don’t exist, and I agree with Dummett that there’s no point in saying there’s a fact of the matter about things we can’t assert (perhaps even more so when the “fact” of the matter is about a fiction). However, I am also convinced by Martin and others about the fact that certain extensions of ZFC are far more natural than others. If an author leaves enough hints in a novel, it seems that there can be a fact of the matter as to who the murderer was in the story, even though it was never explicitly stated. Similarly, if there is enough evidence in the theory of ZFC, it seems that projective determinacy must be true in the fiction as well, even though it was never explicitly stipulated.

N,N,Y – I’m not sure what would persuade one to adopt this position. However, there may be something like this in an instrumentalist view of science like van Fraassen’s. An agnosticism about the facts of the theory might allow one to believe that they could be secretly true, even though we have no way of knowing them to be true, and there aren’t objects to make them true. But enough agnosticism may allow one to consider this position while considering the next one as well.

N,N,N – This seems to be closest to the one that Dummett describes. If mathematical truth is identified with provability from the axioms, then negative answers to the second and third question follow immediately from Gödel’s incompleteness results. And a negative answer to the first question seems to be presupposed in identifying mathematical truth with provability rather than facts about the mathematical objects.

Now of course, all these views can probably be further multiplied by considering their restrictions to particular domains of mathematical discourse, like Peano arithmetic, ZFC, small large cardinals, large large cardinals, and statements like CH and beyond. So someone like Sol Feferman might take the views “Y,Y,Y” about number theory, but “N,N,N” about statements beyond ZFC. But this is just a general outline to show how these different questions might interact, and that they are at least somewhat independent.