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.

Greg(07:47:30) :Hi Kenny–

Thanks for another enlightening post. I just wanted to ask about the “philosophical moral” you draw, viz.

If this sort of explanation [The Burgess and Rosen one] 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.1. I’m virtually certain Burgess and Rosen don’t think the only explanation of any mathematical fact is just the ‘boring,’ historical one. If I were guessing, claims treated as

axiomswithin a given field would be the explananda of boring explanations, but I don’t see that other mathematical facts would necessary be.2. A prodictive (in some sense) research program can stem from a bogus request for explanation: Kepler, for example, asked “Why are there exactly six [sic] planets, and why are the distances between planets what they are?”. He came up with an answer based on the five perfect (Platonic) solids. Another example is type stuff (where people find names of famous historical people and events by looking at every

nth letter in the Old Testament). Or pyramidology, where certain dimensions of the Egyptian pyramids were taken to code important scientific information (e.g. 2xbase of pyramid/height of pyramid = pi; the length of the base of the pyramid/ the width of the average building block = 365, and many more). So we can generate a research program, even if we’re over-explaining.What I’m wondering about is whether there is a difference between natural science and mathematics in this case. I just don’t know whether there is — or could be — a mathematical analogue of Kepler and the others. So it could be that deeper mathematical explanations won’t fall into this trap — for in each Kepler etc. case (to put the matter roughly), an accidental coincidence is mistaken for a regularity, and that is (at least partly) why these explanations seem bogus. But in the mathematical realm, I’m not sure it still makes sense to talk about accidental coincidences: the necessary/ contingent distinction seems to disappear in mathematical contexts.

3. Lastly (and tangentially), I don’t think of “most mathematical beliefs held by prominent mathematicians are true” as a “mathematical regularity” — at least not of the monster group – j function variety. It seems to me to be more of a epistemological/ sociological regularity.

Peter(10:57:02) :I have always liked Monster Moonshine. What is striking is that these two branches of mathematics are very distant, and it is not at all obvious that they would be connnected, let alone so intimately. Indeed, the connection was noticed only accidentally.

One wonders which other far-apart branches of mathematics are also similarly connected, and what underlying theory would coherently and meaningfully (as distinct from superficially) explain the relationship. Mathematics sure ain’t done yet, by any means!

Greg(16:39:50) :oops — there’s a typo in my post. Under 2. it should say: “Another example is “The Bible Code” type stuff. Sorry.

Kenny(00:43:39) :Greg – I’ll have to look at Burgess and Rosen again, and some of their exchanges with Field. I seem to remember that the lack of contingency in mathematics gives them the idea that we don’t need much more than boring explanations, but I’ll have to double-check. Because that really would be odd to deny that we ever need interesting explanations in mathematics.

You’re also probably right that I need to be more careful about why Monstrous Moonshine counts as a productive (in a more worthwhile sense than your examples) research programme. Now I wonder if giving rise to a useful research programme might be an important criterion for what counts as a good request for explanation. Because there’s all sorts of places at which one might want an explanation, but only in some cases do these desires seem reasonable.

I’ll try to think of examples where red herrings in mathematics led people to do worthless stuff, or cases where red herrings ended up giving worthwhile stuff. (Can you think of any of the latter in the other sciences, to go with your examples of the former?)

logicnazi(23:59:25) :This doesn’t seem to pose any particular problem for theories which view mathematical truth as inherintly linguistic. Just as we can discover that a smart bachelor is an intelligent unmarried man is true as a mere mater of linguistics even if at first we didn’t recognize the correct way to rework the definitions so too can we discover the same more complex relationships in mathematics. Just because the definitions may be much more complicated doesn’t mean it isn’t the same issue.

Of course we still have the problem of answering what the basic mathematical meanings are in the first place. Thus structuralism, if-thenism or many of the other theories which seek to break the viscous circle of mathematical theorems being true in virtue of meaning despite that meaning only being fixed by the axioms of our theories. However, it seems that this problem while totally devastating for claims that mathematics is contingent (in a substantive sense not merely whether or not we correctly do the proofs) it seems to be unrelated to the central issues in non-platonic phil of math.

As an aside I just don’t understand why philosophy of math raises any new philosophical issues at all. I will probably post my own blog entry on this subject but in short it seems that once you can explain philosophical meta-talk about analytic truths math is quite easy. For instance if you can make sense of claims like, ‘if X and Y mean the same thing (e.g. X=bachelor, Y=umarried man) then all smart X’s are smart Y’s’ it seems mathematics comes for free. Or to put this in a less optimistic manner, until we have an acceptable phil of math we don’t know what all the philosophical talk about analytic truths means.

I suspect the problem here is the fad motivated (because it would just be too depressing if Quine’s two dogmas convinced anyone) rejection of the analytic/synthetic distinction has pushed these issues out of normal phil of language only to have them reappear in phil of math.

Kenny(15:13:28) :The issue I’m trying to get at here is that there is an interesting notion of explanation in mathematics that isn’t encompassed by “boring explanations” – and perhaps this notion is at work in metamathematical explanations as well. The mere linguistic reworkings you suggest are going to convince us of the truth of all these claims, but it doesn’t tell us that we should look for more than just the boring explanation.

logicnazi(03:47:16) :Perhaps I didn’t understand your point. At least as far as I understood your argument it rested essentially on the lack of any reasonable epistemology of platonistic mathematics and went something like this.

The platonists claim that mathematical statements are about real platonic objects but it couldn’t be these platonic objects which are responsible for our beliefs about these statements. Since they believe these statements are not only about platonic objects but true about them we must have some epistemic justification of their truth. One proposed solution is to give a historical account of how mathematicians came to believe each particular mathematical truth. However, this account fails to explain these greater regularities in our mathematical beliefs.

Actually now that I write it out I am utterly confused about how the argument is supposed to work. I mean either it is the case that their is some universal relation between the two branches of mathematics meaning it can be formulated as a true universal claim in the appropriate language and explained like any other mathematical statement or indeed it just ‘happened’ that these two are the same and no additional explanation beyond the individual explanations can be given.

In other words what makes this case any different than say the observation that between every number and it’s double lies a prime? At one point we did not know this was true and merely had many examples but suspected the general statement was true. In this case the general statement turned out to be proveable thus the explanation for the mathematical truth of each instance could be offered in the truth of the universal mathematical statement. However, if it had turned out that it was only true for a finite number of numbers then all we could hope for would be an explanation of each individual number it is true for.

In any case it seems that both your argument and the one in the paper you linked against the boring argument make essential use of the supposed platonic nature of mathematical truth. In the paper his objection relies on mathematicians accepting a theory because it is true while your objection relies on the lack of any epistemic connection (and he may as well). My point was that while this may be an interesting rebutal of platonism it doesn’t seem to be applicable to non-platonic explanations. In particular if mathematical truth is the same as meta-statements about analytic truth there seems no particular problem offering an epistemic account which can explain these regularities you offer.

Regardless, I am not arguing that there isn’t something interesting or hard going on in mathematical truth. Instead I am arguing that there is nothing of special difficulty in mathematical truth that does not show up in philosophy of language.

That is we have statements in philosophy of language where we note that if bachelor means unmarried man then smart bachelor means smart unmarried man as well as substituting other synonyms for bachelor and unmarried man. Yet we also have the meta-statement that for all words X, Y if X means the same as Y then smart X means the same as smart Y. How is this not a recreation of your issue with mathematical explanation as a pure question about analytic truth?

I think the issue is that I really just don’t understand how this example is supposed to differ from any other universal mathematical claim whose truth we first suspect by observing the truth of many instances. Furthermore, if we can explain the truth of a large number of instances can’t we just use induction to offer an epistemic explanation for our faith in the universal statement (and if you reject this it seems you are just demanding an explanation of induction)

logicnazi(04:13:57) :A more succinct version of my confusion might be the following. There are two regularities at issue here. The first is the regularity that facts mathematicians believe are true are actually true (here is where platonism is assumed). The second is the regular relation which holds between the j function and moonshine, i.e., the regularity that for each n some fixed relation occurs between the n-th j thingy and the n-th moonshine thingy. Call the proposition this relation holds for n R(n).

It seems Platonists have a perfectly good explanation of why R(n) in fact holds for many n. It is just the way the platonic objects happen to be and seemingly just as good an answer as physicists can give about why the laws of physics are true. So the problem must be about explaining why mathematicians beliefs that R(n) is true give us good reason to believe R(n) is true.

Now if in fact mathematicians just believe R(n) is true for many n then there is no difficulty if we explain why the statement is true for each n and nothing more is needed. After all it ‘might’ only hold for finetly many n in this case and thus be totally inappropriate to ask for anything beyond an explanation of each case. In other words if we regard these just as a bunch of mathematical facts then what is more problematic about this collection than any other collection that mathematicians believe.

However, this question seems to call out for more of an explanation preciscely because we don’t just believe R(n) is true for many different values of n. What we really want is an explanation of why our belief that R(n) holds for all n is a good guide to mathematical truth. Yet this statement is a perfectly ordinary mathematical claim of it’s own so if the ‘boring’ explanations are valid at all then we should expect a perfectly boring explanation of this particular case.

More generally I don’t see what this sort of example can possibly get you. Since the language of number theory is strong enough to encode any recursive collection of mathematical statements and their truth/untruth there is no difference between situations with a pattern of many mathematical truths we can recognize and the situation with exactly one mathematical truth. The ability to encode the meta-theory in the theory itself seems to erase any important difference between truth in the two situations.

I still may be confused but I think you really need to make it crystal clear which things you are demanding explanations for.