In which, without lapse into solipsism, your correspondent defends mathematical formalism in the face of deep connections between mathematics and the physical universe.

In response to a comment of mine at Ars Math the unapologetic John Armstrong challenges me thus:

So, Matt, you’re a formalist? You seem to have a similar underlying belief that mathematics is a formal system, and a product of the activities of human minds (brains).

Not to claim a Platonic position here, but I challenge you with the same response as I’d give to a hardcore formalist: how do you explain the “unreasonable effectiveness” of mathematics in the physical sciences? Why should the output of human brains have anything to do with physical law, and how is it that truly well-formed sciences are invariably expressed in mathematics? Escapes into radical solipsism will be discarded as the jokes they are.

My response is after the jump.

OK, first some preliminaries. IANA philosopher, so don’t expect to much; I am (I hope) better at getting proofs done than shedding any great light on the nature of proof. Also I want to say from the start that independent of any question of formalism vs Platonism (vs. fictionalism vs…) I would still assert that mathematics doesn’t offer transcendently certain knowledge in any metaphysical sense. This is a general matter of falliblism. If mathematical objects do exist in a Platonic realm outside the physical universe, our reasoning about them is still only as certain as our fallible brains will allow. We can’t (strictly logically) rule out the possibility that everyone, including ourselves, who read the (apparent) proof of given theorem went temporarily insane and had an overwhelming sensation that they had read a valid proof when they hadn’t.

And now to the business at hand. Yeah, I’d say I’m a formalist. I am not, though, what I believe Penelope Maddy terms a “glib formalist” (as mentioned in this document I’ve linked to before); i.e. I think there are good “real world” reasons to pick one formal system to study over another. Indeed I think there is good reason to say that mathematics has always been reasoning about systems which are abstractions of things observed in the physical world. At the most basic level counting numbers are an abstraction of multiplicities of objects of the same kind; most early maths was an attempted to produce a model of the night sky and so on ). So while I assert that mathematics is the study of formal systems it is (usually at least) the study formal systems that have their roots in modelling aspects of the real world.

The second important thing is the flexibility of mathematics. Let’s here fix “mathematics” as “ordinary mathematics done in ZFC without particular thought to the foundations”. Mathematicians can use it to model the physical universe but we can also use it to build structures far from the physical universe. The vast majority of valid mathematical statements have no way of being interpreted in terms of physics at all. The vast majority of remainder describe physics that is wrong (at least in this universe ;)). That the few mathematical structures which can provide useful models of the physical universe get a lot of attention is a (non-arbitrary) human decision.

So, we have a hugely flexible formal abstract game, developed from the start in a way dependent on out ancestors understanding of the physical universe and which people deliberately (and successfully) use to model that universe. A relevant analogy now is natural language.

The well-developed study of linguistics means that nearly nobody would assert that natural languages refer to immaculate, Platonic essences of things. That languages develop through the action of human minds and the interactions of people with each other and with the world is too well documented and supported. Still, we find that natural languages are very good at describing the physical world, especially on familiar scales but I think if we try hard enough it can be used to accurate describe models of the very large and very small. It does this despite the fact that it’s only connections to the real world are those that people (consciously or otherwise) put there by shared convention.

In light of this, I think it is clear that we don’t need a metaphysical connection between the rules of game and a real-world phenomenon to be able to model the latter in the former, if we are prepared to put the work in.

Now I guess a discussion of how the universe **can** be described well using mathematics is not quite an answer to the question of how “truly well-formed sciences are **invariably** expressed in mathematics?”. Well, firstly I’d deny the literal truth of this claim. Even physics is expressed in a mixture of natural language and mathematics and as you move away from physics, into sciences that work at more familiar scales, the balance always shifts towards natural language. All the same it is clear that maths is better than natural language for expressing many kinds of science (I don’t think there is any other obvious third contender). I think this is really a matter of “design features” of mathematics. Natural languages are full of short-cuts that work well in familiar situations but bewitch us in others. Mathematics is built for precision and with care to keep definitions well-formed, and as I mentioned above is hugely flexible; this is clearly good for exploring unfamiliar territory without being baffled by expecting concepts from everyday life to fit. All we can infer from maths being the language of physics is that it is better for the purpose than any other language that humans have developed; it doesn’t require a metaphysical connection between the physical world and our mathematics.

Tags: Formalism, philosophy of maths, platonism, unreasonable effectiveness

December 12, 2008 at 8:42 pm

Matt,

It seems you are a quasi formalist! Todd, as far as I know, also holds a similar view about mathematics. It would be interesting to hear his views from him on this issue.

I am currently reading Maddy’s ‘Realism in Mathematics’, and so far, I am not quite there with her even though she puts forward strong arguments

forrealism in mathematics.December 12, 2008 at 9:00 pm

“Quasi” as opposed to “subtle”? :( The latter makes me sound so much more sophisticated ;). Does quasi formalist imply not a formalist? I really don’t know this stuff too well; I only knew that “glib formalist” was Maddy’s term because I read Garth Dales paper that I linked (he as my supervisor’s supervisor) and I’ve only really read stuff available online.

I’d certainly be happy with saying the maths done in other systems (consistent or relatively consistent with the Peano arithmetic) is as

trueas that in ZFC; I assumed that made me a formalist.December 12, 2008 at 9:49 pm

Actually, it is hard to find a definition for “quasi formalism”, “quasi empiricism” being heard or talked about all the time. My guess is that Todd may have come up with that term himself! This is what he says,

My philosophy of mathematics could be described as quasi-formalist. Except that the marks on paper have lots of meanings, as opposed to none.over here.

I take that to mean that he is a certainly a formalist but (clearly) the formalism isn’t just a game of meaningless formulas!

December 13, 2008 at 12:00 am

Yes, “quasi-formalist” is my own silly term. It’s meant to dissuade people from thinking I succumb to a straw-man formalism, that maths is nothing but a game of symbolic manipulation without meaning. (But I would hold, for instance, that neither the continuum hypothesis nor its negation can be said to be “true”. This type of truism has been with us since the discovery of non-Euclidean geometries, and it’s strange to me that people still argue about it.)

Well done, Matt! Have you read by any chance Saunders Mac Lane’s book Mathematics: Form and Function? I perceive a lot in common between the point of view you espouse here and his. I particularly like your emphasis that the formal systems studied in practice are far from arbitrary; ultimately they are rooted in and abstracted from human activities [counting, measuring, comparing, estimating, etc.], some going back millenia.

December 13, 2008 at 11:47 am

Todd: Thanks for the kind comments. I haven’t read that book of Mac Lane’s, no.

I agree with everything you say. Actually I think “going back millennia” is even an understatement. I think it’s reasonable to assume that simple concepts of number and space existed way back into our pre-human evolution.

January 6, 2009 at 5:14 pm

[…] of Gödel’s theorems that talk about true, unprovable statements. I’ve said before that I’m a formalist of sorts. Talk of undecidable statements in a system being true gives me headaches. And I’m an analyst […]