Explanations: Monstrous Moonshine and the Epistemological Argument for Fictionalism

The Monster Group is well-known to mathematicians as the largest “sporadic group” (that is, a simple group that doesn’t fall into one of the few easily-defined infinite classes of simple groups), and was first generated as a group of rotations in 196,883-dimensional space.

The j function is something that I don’t really know anything about, but it apparently has important connections to the study of elliptic curves, and was studied by Gauss, Hermite, Dedekind, and Klein among others. It turns out that the j function can be represented as:

1/q + 744 + 196,884q + 21,493,760q^2 + …

In 1979, Conway and Norton noticed that 196,884 = 196,883 + 1, suggesting some connection between representations of the Monster Group and coefficients of the j function. This was confirmed when they noticed that the next smallest representation of the Monster Group is in 21,296,876-dimensional space, and that 21,493,760 = 21,296,876 + 196,883 + 1.

Investigation of this connection, and the search for the explanation of this regularity led to the development of the (apparently extremely productive) field of Monstrous Moonshine. (Don’t ever say mathematicians only have boring names for their theories.)

Now let’s consider another mathematical regularity (one I’ve mentioned before) that cries for an explanation. Hartry Field claims (in the introduction to Realism, Mathematics, and Modality) that if we are platonists and believe that most mathematical beliefs held by prominent mathematicians are true, then we should be able to explain this regularity. However, there is no plausible epistemology for platonic mathematics (unlike for the rest of the natural sciences), so we don’t seem to be able to explain this regularity. John Burgess and Gideon Rosen (in A Subject With No Object) give what Øystein Linnebo (in his excellent paper “Epistemological Challenges to Mathematical Platonism”) calls a “boring explanation” of this regularity. That is, they give a historical explanation of how mathematicians came to believe each particular mathematical fact they do, and then are done because the truth of the claims isn’t contingent.

However, note what would happen if we tried this in the Monstrous Moonshine example – instead of explaining why there’s a connection between the coefficients of the j function and the dimensions of representations of the Monster Group, we’d just explain why each particular coefficient is what it is, why each representation has the dimension it does, and then the numerical relations between them are just identities, and so don’t need further explanation, being necessary. If this sort of explanation were sufficient, we’d never have developed Monstrous Moonshine with all its benefits.

All this just goes to support Field against Burgess and Rosen in that we are correct to ask for a deeper explanation of mathematical facts than just the boring one. Linnebo points out why it’s so troublesome to say just what such an explanation would be like, and suggests that it might require a thicker notion of truth than many people want to deal with, but seems neutral about whether such an explanation is actually possible.

In other words, mathematical explanation is still a problem, and has important philosophical consequences for mathematics. But there are clear examples within mathematics itself, and we can look at how these work to give some guidance to a theory of explanation.

Horgan on Common Sense

There’s an interesting discussion between John Horgan (of The End of Science fame – a very interesting set of interviews with scientists and philosophers of scientist, and a bit amateurish in terms of the content, as one might expect – I recall it as being worth reading though) and Leonard Susskind (a relatively important physicist and string theorist, from what I see) and several others (whose points are less interesting) about string theories and the ability of common sense to lead us to reject them. I’m not (yet) a philosopher of science, but it seems that Horgan is onto something – scientific theories can’t be totally unconstrained by some sort of reasonableness. But I think he’s definitely wrong on this particular point. I’m not a believer in Penelope Maddy’s position that “philosophy cannot criticize [nor] can it defend” science. But this is because I think philosophy and science are continuous with one another, and each should be able to shed light on the other. However, it has to be done far more carefully than Horgan is doing it here. Susskind seems to make this point quite eloquently by talking about “uncommon sense”.

At the end, Horgan repeats the common complaint about string theory that it’s not experimentally falsifiable or confirmable. According to Susskind (and I certainly can’t say myself) this just isn’t the case. But even if Horgan is right about it’s non-testability, I assume he means it in the sense that there is no experiment that could confirm or verify string theory as against the contemporary mix of quantum mechanics and relativity. I would be surprised if the theory was totally unable to be falsified or confirmed at all. Even if different settings of parameters make the theory consistent with different numbers of fundamental particles, different sizes of the universe, and so on, I assume that these parameters will have to be set to account for some amount of the observations one has, and once they’re set, they would make further predictions approximately in line with current scientific theory, and not just allow for all the seeming laws to suddenly change at any moment. I certainly hope string theory is verifiable or falsifiable in at least this weak sense – this seems like an important criterion for scientific theories (though the Quine-Duhem problem shows that we need to be somewhat more careful in phrasing this weak requirement).

But there is no important need for this stronger sense of falsifiability and confirmability (that is, the kind that requires there to be some experiment to differentiate it from current theory). Sure, it would be nice to be able to have an experiment to settle which one of the two theories was better, but even if there’s not, that doesn’t mean the new theory isn’t scientific. That would make science too much a matter of historical accident – the Copenhagen Interpretation of quantum mechanics would be science, but the many-worlds theories and Bohm’s theories would be dismissed as unscientific. If they had come in a different order, a different one would have been scientific. And even apart from this problem, it seems there are often benefits to debating two theories that are empirically identical. Oftentimes, one theory will suggest different modifications in the face of recalcitrant evidence. Or one theory will make all the calculations far easier. Or one theory postulates fewer invisible elves pushing electrons around, and otherwise fits together more aesthetically.

This is the sort of debate in which philosophy and common sense (or perhaps better, uncommon sense) are important in science. Horgan has staked an extreme position that seems indefensible on these grounds, but makes approximately the right broad claim. However, this broad claim then undermines his more specific argument against string theory, that it’s untestable. (For an example of a theory that philosophical concerns should drive us against, despite its empirical adequacy, see here.)

In other news, I’m leaving for Sydney airport in a few hours, and then will be on a roadtrip through Arizona and New Mexico for about a week or so before returning to Berkeley, so I’m unlikely to post until the end of the month.

Are Mathematical Posits Ultrathin?

As mentioned in my previous post, Jody Azzouni thinks that ultrathin posits (the kind that have no epistemic burdens at all) aren’t taken to exist. He suggests that mathematical posits are ultrathin, and thus we shouldn’t take them to exist, so we get nominalism the easy way.

[Ultrathin posits] can be found – although not exclusively – in pure mathematics where sheer postulation reigns: A mathematical subject with its accompanying posits can be created ex nihilo by simply writing down a set of axioms; notice that both individuals and collections of individuals can be posited in this way.

Sheer postulation (in practice) is restricted by one other factor: Mathematicians must find the resulting mathematics “interesting”. But nothing else seems required of posits as they arise in pure mathematics; they’re not even required to pay their way via applications to already established mathematical theories or to one or another branch of empirical science. (pgs. 127-8)

In a footnote to this passage, he suggests that he argues for this claim more thoroughly in his 1994 book Metaphysical myths, mathematical practice: the ontology and epistemology of the exact sciences. I haven’t looked at that book yet (though I plan to in the not-too-distant future), but the claim here just doesn’t seem convincing at all.

It’s true that mathematical posits aren’t required to have any applications, even to other areas of mathematics. After all, who thought of the applications of non-Euclidean geometries, aperiodic tilings, or measurable cardinals? Generally not the people that came up with them.

But as any set theorist knows from experience, mathematicians don’t just go along with any postulation that one makes. They require it to be consistent, and often even want it to be known to be consistent – so many mathematicians still object to various large cardinal axioms that postulate the existence of structures that actually even have a lot of the unnecessary virtues, like mathematical interest and application. But mathematicians sometimes seem not to want set theorists to just postulate these things willy-nilly. And on a more down-to-earth level, we can’t just postulate solutions to differential equations, or even algebraic equations – we have to prove existence theorems. In algebra, fortunately, people building on the work of Galois were able to show that for any polynomial with coefficients in some field K, there is some field K’ containing K in which that polynomial has a solution (this required a lot of tricks of algebra and knowledge of how to actually construct fields from scratch). Kurt Gödel actually helped this cause of proving existence theorems greatly – thanks to his completeness theorem, we (those who accept ZFC) now know that any consistent set of axioms defines a non-empty class of mathematical structures (this leads to much shorter, more powerful proofs of the existence of things like algebraically closed fields). Thus, thanks to Gödel (and our antecedent acceptance of at least a sufficient fragment of ZFC), Azzouni is almost right about postulating the existence of structures. But we still need to prove them consistent, or otherwise argue for their existence in the cases where we can’t prove them consistent.

Thus, mathematical posits don’t live down to his ultrathin expectations for them. It’s not clear to me if they are really “thin” in his sense either (and they’re almost certainly not “thick”), so it’s not immediately clear whether his criterion should say they exist or not.

The Thick, the Thin, and the Ultra-Thin

In his book Deflating Existential Consequence (which I’ve been discussing in the last several posts), Jody Azzouni argues that ontological commitments of a theory aren’t necessarily signaled by the presence of existential quantifiers when the theory is put into a regimented language. Instead, he thinks there should be an existence predicate that applies only to some posits of the theory. To decide which posits to apply the predicate to, he separates the posits of a theory into thick, thin, and ultra-thin posits, and applies the existence predicate only to the first two (though he suggests that a community other than our own could theoretically have a different practice of making existential commitments – I want to argue that even our own community doesn’t follow his practices).

Thick posits are ones to which we can establish robust, grounded epistemic access, in ways that can be refined, and that allow us to monitor the posit in question. (He goes into much greater detail in the book about each of these four criteria.) Thin posits are ones that we postulate for reasons of theoretical simplicity and explanatory power, subject to the theory continuing to have all the Quinean virtues, and provided there is an explanation for why the posit hasn’t been epistemically detected (if in fact it hasn’t). Ultrathin posits on the other hand are ones that have no requirement on their postulation – characters in fiction are perhaps the paradigmatic example of them. They don’t need to pay their “Quinean rent”, as he calls the requirements on thin posits.

It seems clear to me that we might not want to be committed to ultrathin posits, and also that we should be committed to thick and thin posits. However, it’s less clear that there are any truly ultrathin posits in his sense, or that these three categories exhaust the posits of any theory. Azzouni himself seems to suggest at several points that posits can be thick but not thin (as he says most postulation of individuals is – individuals rarely play any important role in the effectiveness of a theory), thin but not thick (like all the stars outside our light-cone, rabbits that have never been seen by humans, viruses that we haven’t yet detected), or both thick and thin (like the Sun, planets in our solar system, and various other things). But he also suggests that there are posits that are neither thick, thin, nor ultrathin.

It’s a subtle matter whether the theoretical links between what’s implied to exist on the basis of theory and what we’ve forged thick epistemic access to is tight enough to justify the conclusion that what’s theoretically posited is actually thin. Imagine, for example, that a particularly well-attested theory implies the existence of a certain subatomic particle, but that it also follows from that theory that the energies needed to actually forge thick epistemic access to such an object are (forever) vastly beyond our capacities. Physicists, as a community (so I claim), would not commit themselves to the existence of that particle on purely theoretical grounds – no matter how much empirical support in other respects such a theory had. (pg. 147)

Theologians, or some of them, anyway, are perhaps different in this respect. Proofs of God, if any could be found, would amount to the acceptance of a rather unique posit on the basis of purely theoretical virtues. … But theologicans, after all, are a desperate lot. (pg. 148)

However, these examples don’t seem to exhaust the sorts of posits that would appear in this leftover category. If the analogy in the physics case is supposed to be with the top quark (which he discusses in a footnote later on), then my understanding is that physicists in that case thought that they couldn’t even exist without the very high energies, not just that we couldn’t see them. Thus, the top quarks weren’t even a posit of the theory until the accelerator experiments were done – until then they were merely a possibility of the system, not an actual part of it. A better example in this case (and closer to home, for me) is the supposed Super Barn supermarket in Canberra. Many people have independently mentioned it to me – someone suggested I should shop there, someone else used it as a landmark to give directions, and someone else just mentioned it as a major store in the downtown area. Thus, for my best theory of the world, it makes sense to postulate this store, because it would explain the similar discussions of different people who may never even have met one another. However, I have wandered around the downtown area several times, at least once or twice even looking for this store, and haven’t had any epistemic access to it, much less the thick sort Azzouni requires. There is no explanation for why I haven’t seen it, so it seems that he would say that it’s not even a thin posit in my theory. Thus, he suggests that I should say that Super Barn doesn’t exist! This seems to me to be a reductio of his view.

Mark Colyvan makes an objection like this in his review of the book, calling these posits “very thin”. Some of his examples of such entities include dinosaurs and Immanuel Kant (it turns out that the requirements on thick epistemic access are rather stronger than one might expect).

At any rate, I think this is a very fun book to read, even though I disagree pretty strongly with most of the arguments in it (I agree with his nominalism about mathematics, but for quite different reasons).

Melbourne

I’m off to Melbourne this evening for a few days, so I probably won’t post much. I’ll be meeting with Graham Priest and hopefully getting to know some of the other logicians and philosophers of mathematics in the city. (Unfortunately, Greg Restall is in Europe, so I won’t be meeting him.)

Metaphorical Existence

I’ve just read through Steven Yablo’s nice paper “A Paradox of Existence”, which further advances his position that mathematical talk may all be metaphorical, and that metaphorical talk shouldn’t be construed as ontologically committing. His position strikes me in many ways as similar to Jody Azzouni’s, in that both suggest that no rigorization of our language will bring it in line with the Quinean picture where it’s in a first-order language and the ontological commitments can be read off from the quantifiers. The difference is that Azzouni thinks we can get it into that regimented form but the quantifiers aren’t committing, while Yablo suggests that some metaphors may be essential to our descriptions of the world, so any first-order expression of our theories contains some non-literal language, so there is no Quinean rigorization at all, much less one whose commitments can be read off the quantifiers.

I was very tempted by the position when reading this paper – more so than when reading “Does Ontology Rest on a Mistake?”, although the tendency it tempts me towards is more in keeping with the title of the latter.

Yablo points to Davidson’s analysis of adverbial constructions as being properties of events as an example of a theory that is useful and may not be eliminable, but may still be regarded as metaphorical. Davidson’s analysis allows a speaker to learn the language by learning only finitely many symbols, rather than learning every combination of a verb and an adverbial construction separately. But however useful this procedure is, it’s equally useful even if there turn out not to be events and properties thereof, and it’s just a pretense. It can be a purely semantic thesis rather than an ontological one.

But then I analogize this to science. Caloric (a reified notion of heat) is a very useful way of talking about the world for explaining how heat flows from one object to an adjacent one. In fact, it’s useful even if the world doesn’t have any of it. Similarly with electrons and their usefulness for science.

But now, if we’re going to repudiate our ontological commitments just because the world can be described treating this language as purely metaphorical, then it seems we would sink into complete scientific instrumentalism. I assume this is what Yablo means in footnote 53, where he says he wants to “provide a natural brake on the creeping metaphoricalism that might otherwise threaten.” However, I think such a brake isn’t possible. If this is the right account, then we can treat all talk of objects in general as metaphorical (which I think I’m happy with). But as long as we want to talk about objects of any sort, it seems that the constraint on our language is to accept the talk of objects that we can’t paraphrase away – so whether or not we’re committed to mathematical entities will turn again on the indispensability argument that Yablo and Azzouni want to get away from.

Logic in Mathematics, Philosophy, and Computer Science

In discussion with Jon Cohen in the past few weeks, I’ve realized a bit more explicitly the different ways logic seems to be treated in the three different disciplines that it naturally forms a part of. I intended to post this note as a comment on his recent post, but for some reason the combination of old operating system and browser on the computer I’m using here wouldn’t let me do it. So I’m making it my own post.

The thing that has struck me most in my experience with doing logic in both mathematics and philosophy departments is that in mathematics, “logic” is seen as just being classical propositional or first-order logic, while in philosophy a wide range of other logics are discussed. The most notable example is modal logic of various sorts, though intuitionist logic and various relevance logics and paraconsistent logics are also debated in some circles. But in talking to Jon I’ve realized that there are far more logics out there that very few philosophers are even aware of, like linear logics, non-commutative logics, and various logics removing structural rules like weakening, contraction, or exchange (which basically allow one to treat the premises as a set, rather than a multiset or sequence). In his sketch of history, it seems mathematicians are stuck in the 1930’s, and philosophers are stuck in the early 1980’s, in terms of what sorts of systems they admit as a logic. Of course, all three disciplines have developed large amounts of logical material relating to their chosen systems.

The reason for these divisions seems to be a disagreement as to what a logic is. Mathematicians just want to formalize mathematical reasoning in a sense, and so have fixed on classical logic, as it seems to best capture the work that mathematicians find acceptable and necessary. Philosophers on the other hand, have debates about whether classical logic, intuitionism, some sort of relevance, or some other logic is the “one true logic” (or one of the several true logics as advocated by Greg Restall and JC Beall). Although computer scientists study even more types of logic, they don’t seem to argue about which is an appropriate logic for doing their reasoning in – from what I understand, they do all their metareasoning in classical logic (or some approximation thereof). The various systems are studied to gain structural insights, and to model the capacities of various computational systems, but not to talk about truth per se.

Does this sound about right?