I expressed a half-formed thought about Geoff Pullum’s paradoxical claim about appearances, and next thing I knew up pops confirmation that I was right to worry about one direction of the paradoxical argument. Isn’t blogging great!
Now that’s out of the way, I wanted to mention one other aspect of Geoff’s original post that I thought was very interesting, namely this quote from G. K. Chesterton:
The real trouble with this world of ours is that it is nearly reasonable, but not quite. Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait.
As I’ve mentioned a few times here, I’ve been reading one or two philosophy of science job files in the last few weeks. Actually I’ve been a little ill-mannered about it, because really it’s interesting work – it’s just hard to always be sufficiently grateful for that. Hopefully as we get towards the finalists and I get to closely read papers rather than trying to make judgments based on first appearances it will be even more interesting. One of the things that came out of a few of the better files (I hope I’m not giving anything excessive away about individual candidates by this, because this was a theme across a few pieces I read) was that something like Chesterton’s claim might be right.
A few years ago there was a lot of buzz about the Miracle of Applied Mathematics. (The link is to Mark Colyvan’s 2001 Synthese paper of that title. I don’t mean to pick on Mark here – his is quite a good paper I think and it’s a moderately convenient source for what I’m talking about.) The idea was that it was an amazing fact, and something a decent philosophy of mathematics needs to explain, that mathematics is so incredibly useful in science.
At one level the point goes back to Ramsey. He said that purely formalist theories of mathematics couldn’t explain its role in practical life. But Ramsey wasn’t talking about cutting edge science. From memory his example was that we can use the mathematical fact that 2+2=4 in inferring that if it’s 2 miles to the train station, and 2 miles from the train station to the pub, then it’s 4 miles to the pub via the train station. I’m not sure this is much of an argument against any kind of anti-realism, but modern arguments are more sophisticated. (It’s also not as funny as Ramsey’s argument against intuitionism in mathematics, but that’s for another day.)
Here’s the kind of thing that people have taken to be in need of explanation. (The quotes are all from Mark Colyvan’s paper.)
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. (Eugene Wigner)
It is very strange that mathematicians are led by their sense of mathematical beauty to develop formal structures that physicists only later find useful, even where the mathematician had no such goal in mind. . . . Physicists generally find the ability of mathematicians to anticipate the mathematics needed in the theories of physics quite uncanny. It is as if Neil Armstrong in 1969 when he first set foot on the surface of the moon had found in the lunar dust the footsteps of Jules Verne. (Stephen Weinberg)
[H]ow does the mathematician—closer to the artist than the explorer—by turning away from nature, arrive at its most appropriate descriptions? (Mark Steiner)
Now it’s undoubtedly true that modern mathematics is useful to modern science. But to conclude from that that mathematics is especially, surprisingly, useful for understanding the world is arguably to commit two fallacies of sampling error.
The less important point is that mathematics is useful to science precisely because those tools that are expected to be useful to science are the subject of the most dedicated mathematical research. It’s no surprise that the calculus starts to get developed in great detail once a physical theory that needs it comes into vogue. Nor is it surprising that the best work on computability all happened once large-scale computing became a viable engineering prospect. This is important to the philosophical question, I think, but it isn’t what’s most relevant to the Chesterton quote.
The more important point is that mathematics is very useful for developing scientfic models of reality, and scientific models are exceptionally useful in all sorts of ways, but they remain models of reality not 100% accurate descriptions of it. And the models break down, as often as not, at just the point where our mathematical abilities give out. The salient fact is not that the world is in some deep way mathematical, but that we have done a simply wonderful job of capturing the respects in which it is, and not a particularly good job at all of capturing the respects in which it is not. If you just look at the success stories of science, you’ll come away with the impression that there’s a striking resemblence between mathematical theories and what the world is like. But really what’s true is that there’s a striking resemblence between mathematical theories and our best approximation to what the world is like, and the approximation is as good as we can do because it’s as far as the mathematics will take us. As Chesterton said, the world is close enough to our idealised description of it that these descriptions are useful, but there’s still some wildness. And of course the wildness matters – to scientists, to engineers, and (one hopes) to philosophers.
At the level of broad generality I’ve covered here, this is just a point that Nancy Cartwright made a couple of decades ago. And of course young philosophers of science aren’t just going around re-telling Cartwright’s story, but correcting it, providing better examples to illustrate it, and often coming to very different conclusions to those Cartwright drew. And this is all (to my mind at least) incredibly interesting stuff. It’s almost enough to make me say, Sawyeresquely, you should want to go on search committees and read tons of files. It’s certainly reason to not complain about such a task as I have done. And it’s certainly a reason to pay attention to what’s happening in philosophy of science these days.
Of course if this kind of story is even roughly right, the real miracle here is how a poet like Chesterton could have figured this out before the philosophers did.
PS: None of this should be taken to imply, in any way, that there weren’t any good applications on other areas of philosophy of science, because of course there were. And if I can find anything half-way interesting to say about them without violating any rules of confidentiality (which I hope I haven’t done already) I might do so.