Tuesday, December 15, 2009

What the Indispensibility Argument Isn't

When I first heard of Quine's indispensibility argument (We are committed to the existence of abstact objects, since we must quantify over them in order to state our best physical theory), years ago, I misunderstood it.

I thought Quine was trying to draw an analogy between nominalists and, say, people who deny that there are planets, but will admit all the usual observations through telescopes. We find the planet denier's position implausible. We want to say "If there aren't planets, how come -as you admit- everything we see through telescopes behaves just as it would if there were planets? If there are planets, this explains the order and regularity of what we see through telescopes. But if there aren't planets, how come - out of all the mindbogglingly many possible patterns of optical illusions - we happen to have ones that are just like ones that could be produced by seeing persisting objects in space?"

But, even aside from being not what Quine meant (as I was soon told), this is really not a good argument, as I shall now argue for the benefit of anyone else who is tempted by it.

The explanatory inadequacy argument above, crucially turns on our having a notion of the observations we would have if there were planets vs. various other patterns of observation which would *not* be consistent with there being planets. The planet denier's position is unattractive, because on their theory it looks like a miracle that we happened to get a coherent pattern of optical illusions that could have been produced by normal vision of real objects.

But, in the mathematical case, there is no such contrast. There is no pattern in the behavior of physical objects which suggests the existence of numbers. For what would the contrast class to exhibiting this pattern be? It's not like we think: actually cannonballs accelerate towards the earth at 9.8 m/s^2, but if there weren't numbers, they would probably just fly around crazily. On the platonist's own view the objects (numbers and functions) in calculus don't come down and beat on cannonballs to make them behave in ways that are describable by short differential equations. Nor do they prevent cars from going two meters per second for two seconds, in a given direction, but only traveling 1 meter in total (cars don't need to be prevented from doing the metaphysically impossible). So it's not the case that he would expect different behavior if there weren't any numbers (the way we would expect different behavior of telescopes if there weren't any planets). Thus, there's no argument to be made against the nominalist, along the lines of "If there aren't numbers, how can you explain the fact that things happen to look just like they would if there were numbers?"

Instead, the point of the Indispensibility Argument (or the only plausible version of it) is that the Nominalist cannot even *state* his theory of the physical objects he accepts, and how they behave, without quantifying over abstract objects, and hence contradicting himself. To summarize: Quine isn't saying we need numbers to explain observed patterns in the behavior of physical objects. He's saying we need numbers to even state the relevant patterns in the behavior of physical objects.

1. I do sort of like this version of the indispensibility argument, but I'm not totally convinced. How can we rule out the possibility that all the physical facts of the word could be explained by some bizarre "non-numerical" mathematical formalism, i.e. some weird logical system in which you can't even construct the natural numbers, but which somehow can be used to state the laws of gravity and electricity and everything else? I guess that's pretty vague, because I can't imagine a useful mathematical formalism for physics which didn't somehow use or depend on numbers, but maybe I just lack imagination.

2. Yeah, that is indeed the question. In fact, one of the biggest living philosophers of math, wrote his first book trying cook up just such a formalization.

http://www.amazon.com/Science-Without-Numbers-Defence-Nominalism/dp/0691072604

The current state of play seems to be that you CAN state newtonian mechanics in a way that only quantifies over space time points (rather than numbers or sets), but:
- it's not clear whether you can do the same for current physics like QM or GR
- it's not clear whether spacetime points are nominalistically acceptable (I guess it depends why you are attracted to nominalism/denying that there are numbers in the first place)
- nominalistic formulations are much less elegant, and would not be accepted in physics journals