## Sunday, 3 February 2013

### Mathematics as "Pretense"?

A rather fashionable "nominalistic" view in the last decade in relation to mathematics is a view, or perhaps bundle of views (also sometimes called "fictionalism"), that mathematical reasoning is a kind of "pretense". Rather than thinking it to be the case that 3 + 5 = 8 or that there are infinitely many prime numbers, we somehow merely play a game of pretense with the sentences "3 + 5 = 8" and "there are infinitely many prime numbers". There are echoes here of old-fashioned game formalism, if-thenism and deductivism. And so it is that some philosophers have come to compare mathematics---a fundamental component of human knowledge and science---with Santa Claus.

My response to this kind of pretense theory is usually along the following lines. At the moment, no actual scientist has used the Santa Claus story to compute the Lamb shift, analyse quark confinement or perturbations in binary star systems and so on. So, I would be very interested to know why not!

The basic claims of such pretense theories are in some sense connected to the meanings of mathematical sentences; but they are neither abstract metaphysical claims (e.g., "mathematical facts are necessities which trivially supervene on all contingencies"), or rather abstract epistemological claims (e.g., normative claims, concerning rationality, "oughts", and so on). What is interesting about such claims is that they seem to have some empirical content at the level of cognitive psychology, and therefore may be subjected to empirical investigation.

In an interesting forthcoming article, "Pretense, Mathematics, and Cognitive Neuroscience" (BJPS 2013), Jonathan Tallant argues:
Abstract
A pretense theory of a given discourse is a theory that claims that we do not believe or assert the propositions expressed by the sentences we token (speak, write, and so on) when taking part in that discourse. Instead, according to pretense theory, we are speaking from within a pretense. According to pretense theories of mathematics, we engage with mathematics as we do a pretense. We do not use mathematical language to make claims that express propositions and, thus, we do not use mathematical discourse to make claims that are either true or false. In this paper I make use of recent findings from cognitive neuroscience and developmental science to suggest that pretense theories of mathematics fail.

1. I consider myself a fictionalist. However, that "pretense" theory badly misses the mark on how I see mathematics.

Perhaps I failed to read the fictionalist's bible (is there such a thing?)

2. Hi, do you think the magnetic field is like Santa Claus? Is the gauge group of electromagnetism like Santa Claus?

Jeff

1. I don't see the relevance. There's at least a physical basis for a magnetic field.

3. The point here is connected to Quine and Putnam's arguments against nominalism, based on science. Here, the example is the magnetic field, which is a vector field, a function from spacetime to a vector space (roughly, $\mathbb{R}^3$). Our best scientific theories refer to fields, vector spaces, wavefunctions, Lie groups, dimensionless constants, and so on. But if one is a nominalist (fictionalist), one rejects such things as functions and vector spaces. So, if there are no such things, then scientific theories are false.

Matti Eklund's Stanford Encyclopedia article, "Fictionalism", might be a kind of bible, I guess!

http://plato.stanford.edu/entries/fictionalism/

Fictionalism is broader spectrum of views, than the more recent pretense theories, though, yes. E.g., Field's fictionalism is not primarily a pretense view.

Jeff

1. Okay, perhaps I was unclear. I should have said that I consider myself a mathematical fictionalist, which seems narrower than the fictionalism you appear to be talking about.

I don't actually have a position as to whether magnetic fields should be considered fictional, because I don't see that it matters. Perhaps it doesn't matter for mathematics, either, though mathematical platonists seem to believe that there is a fact of the matter as to whether the continuum hypothesis is true, while I see the work of Gödel and Cohen as reason to doubt that there is any such fact of the matter.

4. Right, fictionalism is a view that there are no numbers, functions, sets, groups, etc., and that talk of such entities is comparable to fictional discourse (e.g., Santa Claus). Consequently, the sentence "3 + 5 = 8" is not true. Or consider the completeness theorem - if a formula A is valid, there is a finite sequence of a certain kind. But if there are no sequences, then this claim also false. The standard results of these parts of mathematics and logic are false.

Since the magnetic field is also a function, taking values in a vector space, fictionalism implies that there is no such thing as the magnetic field, and Maxwell's laws are therefore also false. Similarly, there are no scalar fields, tensor fields, wavefunctions, etc. So, physical laws referring to these are false too.

So, the cost of such a philosophical position is to claim that a very significant part of science is false; and this, as Putnam argued, makes the success of science a miracle.

But this is all a bit tangential to the topic of Tallant's paper, which concerns a narrower kind of fictionalism that has become fashionable in the last decade or so - a pretense theory - and proposed way of empirically evaluating pretense theory.

Jeff