Sunday, August 31, 2008

Book Project: Mathematics and Scientific Representation

This year I will be trying to come up with a draft of a book that I have been planning for some time, but that I have been quite unsure how to organize. The general issue is the prevalence of mathematics in science and whether there is a philosophical problem lurking here that can be productively discussed. My current angle of attack is to focus on the contribution of this or that part of mathematics to a particular scientific representation. ("Representation" is meant to include both theories and models.) So, we can ask for a given physical situation, context, and mathematical representation of the situation, (i) what does the mathematics contribute to the representation, (ii) how does it make this contribution and (iii) what must be in place for this contribution to occur?

To avoid devolving into a list of examples, I am also trying to come up with different sorts of representations and different ways that mathematics might contribute. At the moment these are: (i) the math is intrinsic/extrinsic to the content of the representation, (ii) the representation is causal concrete or abstract acausal, (iii) the representation is concrete fixed (i.e. a fixed interpretation) or abstract varying, (iv) the scale of the representation and (v) the global (as in a constitutive framework) or local character of the representation. In future posts I will try to clarify these dimensions and offer examples of different ways in which mathematics contributes to them.

The main point of the book, though, is to argue that the contribution that mathematics makes in all these different kinds of cases can generally be classified as epistemic. That is, mathematics helps us to formulate representations that we can confirm given the data we can actually collect. So, there will be many issues related to confirmation and epistemology that I will try to explore here as well.

Pointers to similar projects or projects pursuing this issue in a different way are welcome!

11 comments:

Anonymous said...

Please discuss if our mathematics if the only possible mathematics. If there were a different sort of intelligent being, could that being develop a different sort of mathematics, and use that to model, in a radically different way, the natural world.

So does math correspond to the natural world so well just because it is the toolbox we have to use to model/represent/analyze the natural world, or there something deeper going on?

Thanks!

Chris Pincock said...

This is a fair suggestion, but I am not so sure how easy it is to tell when a different kind of mathematics would do an equally good job. In many cases, if we have some mathematical representation using mathematical theory M1, there will be an easy way to transform it to another mathematical theory M2. But whether this freedom extends to completely different approaches to mathematics, e.g. mathematics we could never in principle know, is hard to say.

Anonymous said...

Is our mathematics determined by the nature of our cognition, the way our brains work, or is it something more fundamental that any intelligent creature is likely to develop something similar?

Chris Pincock said...

I tend to think of mathematical knowledge as like any other kind of knowledge. Its truth does not depend on features of us. But even if that's the case, I wouldn't know how to argue that any intelligent being would have to agree with most of our mathematics. It may be that our access to objective mathematical truth depends on more than just our rationality or intelligence.

Anonymous said...

"It may be that our access to objective mathematical truth depends on more than just our rationality or intelligence."

What do you mean? That sure is hard to interpret!

Thanks for your patience with all the questions.

Chris Pincock said...

Sorry, I didn't mean to be mysterious. One view of mathematical knowledge is that it depends on a different part of our mind -- some call it "intuition". This is the view often attributed to Kurt Goedel. So, to come to know about mathematical objects, we would need to do more than just be rational. We would have to exercise this faculty of intuition as well. (By the way, this is not my view.)

Anonymous said...

"Sorry, I didn't mean to be mysterious. "

Glad you clarified that, I thought you might have meant we were "reaching out" and accessing some perfect ideas in the mind of God or something.

Kenny said...

Hi Chris,

Sounds like this is a book that could get a marketing boost from a connection to Wigner's "unreasonable effectiveness of mathematics in the natural sciences". You might not be trying to explain the unreasonable effectiveness, but it does look like you're at least trying to get a grip on what this effectiveness consists in!

Chris Pincock said...

Kenny -- thanks, all marketing suggestions are appreciated! I will indeed discuss Wigner, as well as Steiner's more recent arguments. I present these points as claims about discovery, and argue that my prior work in the book makes these discovery problems easier to handle.

Seamus said...

R.W. Hamming has written a "follow-up" to Wigner's piece. I don't know if it adds anything new in relation to this project but I thought it worth mentioning, just in case you'd not seen it.

http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html

Chris Pincock said...

Seamus,
Yes, thanks, I have downloaded this paper, but not yet read it. Any thoughts on it?