[FOM] RE: Experimental mathematics

[David Corfield]

Bill Taylor wrote:
 

> it seems 
> to me that the differences have been stated many times 
> before, and are obvious anyway, despite the claims of many 
> like Lakatos (in his amusing but not terribly convincing 
> "Proofs and Refutations"). 

Where does Lakatos claim that there are no differences
between mathematics and science? Do you think he thought 
scientists engage in proof analysis?

Of course, there are problems with Lakatos's account. 
Solomon Feferman likened reading  "Proofs and Refutations"
to listening to a tune from a single instrument when one wanted 
to hear the full orchestra. But let's be grateful that we
got to hear a new instrument, which told us that:

In mathematics we don't start from a definite A and proceed
by secure methods to a B. More commonly there's an interesting-
looking B we'd like to know more about. We attempt to construct 
a path to it from some point A we take ourselves to know. After
much to-ing and fro-ing we may end up with a very different path 
ending at something resembling B, possibly starting from 
something quite unlike A. In the process we arrive at a range of new 
concepts and linkages between them.

This concerns the context of discovery, when what I care about 
is the context of justification, you may say. But the key to getting 
something out of Lakatos is to think of him as explaining how we 
carve out 'good' concepts, those Frege talked about as having 'fruitful
definitions' and being 'organic unities'. Both share the thought that
one can speak of 'getting concepts right'. For Lakatos a considerable
part
of the evidence that one has got them right comes from a reconstruction
of the way they arose. Don't think you've finished the job if there's
something about that initial B which has escaped you. This was enough
for 
him to liken mathematics to science as 'quasi-empirical'.

For my own part, I see both mathematics and science as hugely complex
knowledge producing disciplines. It's easy enough to play the game of
making them out to be very similar or to be very different, just as 
one can when comparing a whale to a mouse. At the present state of play,

however, I think it wouldn't hurt if some philosophers of mathematics 
spent a little more time working on the similarities. I have   
done so, e.g., elaborating Polya's ideas concerning a probabilistic 
rendition of mathematicians' degrees of belief (chap. 5 of the book
mentioned
on the web site below) in light of similar ideas in the philosophy of 
science. It's not a bad way to add a new instrument or two to the
orchestra. 


David Corfield
Faculty of Philosophy
10 Merton St.
Oxford OX1 4JJ
http://users.ox.ac.uk/~sfop0076