[hist-analytic] Mathematics and Lakatos's Research Programme Degeneracy

Roger Bishop Jones rbj at rbjones.com
Sat Feb 7 03:50:35 EST 2009


J.L.,

On Saturday 07 February 2009 01:27:30 Jlsperanza at aol.com wrote:

> I don't think alas, Roger, I do have to hand  a case of a mathematical
> 'progress' of any kind -- if that's what we are looking  (Yes, strictly we
> are supposed to be discussing Quine's Holism).

"of any kind"!

I don't actually recall talking about 'progress' but your scepticism about
progress in mathematics is a bit severe.

Here are two different kinds of progress which seem to me conspicuous in
mathematics:

1. There is continuous (possibly even execessive) increase in the number
   of proven mathematical theorems.

   (insofar as discovering and demonstrating mathematical truth may be
   considered the main objective of pure mathematics this is surely
   a kind of progress)

2. There have been considerable (recent) improvements in the rigour with which
   pure mathematics is conducted, and improvements in the rigour with
   which it is possible to conduct mathematical proofs.

   For example, Bishop Berkeley's complaints against the infinitesimal
   calculus have been addressed several times over.  We have even seen
   improvements in the rigour of that paradigm "Euclidean" geometry
   eliminating defects in Euclid's system. 

Of course, you may not think these constitute "real" progress, but I
submit that they are progress of a kind.

> But  reading your original post, Quine's Holism and your reply to Rogerio,
> I was  reminded of Lakatos.

I'm afraid I am not a fan of Lakatos (I'm not really the kind of 
person to be a fan of anyone, but, more to the point, I disagree with
a lot of what he seems to be saying) and it looks to me as if he has
lead you astray, but you are not specific enough for me to understand
what points from Lakatos you are making (and I don't think I have
read what he had to say about "research programs").

If you want to debate Lakatos I would be happy to, if only I could
get a grip on the issue at stake.

Roger




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