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

Jlsperanza at aol.com Jlsperanza at aol.com
Fri Feb 6 20:27:30 EST 2009


In a message dated 2/6/2009 4:54:04 P.M. Eastern  Standard Time, 
rbj at rbjones.com writes:
There is no question that changes take  place in mathematics as a result
of developments in empirical  science.
However, these are the development of new kinds of mathematics, not  the
discovery that accepted mathematical propositions are in fact  false.


-----


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).

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

It seems the only people who progress are  "scientists" (or factual 
scientists) and not even _them_, but "Science", with a  capital S.

Philosophers had been discussing same thingos for ages! And  ditto 
mathematicians!

I would not know if 'research programmes', as  coined by Lakatos, would apply 
-- but I do think someone must have thought about  the 'development', 
'progress' or what have you, of Mathematics -- or  "Mathematical Science".

Lakatos thought, I think, that research  programmes either progenerate, or 
'degenerate'. In Factual or Empirical Sciences  (Mario Bunge speaks of formal 
vs. factical sciences), Lakatos viewed, it was  more of the nature of the 
research programme you were engaged (rather than  matters of 'raw' empirical 
evidence) that determined a 'paradigm-switch' as it  were.

As you see, I think nothing has changed much, mathematically, since  Thales 
-- or at least since Thomas edited those two volumes.

Of course,  if you delve (if that's the word) deeper, you'll find a reference 
to those  Babylonians, and how ashamed the Greeks were to be reminded that 
_geometry_  predates arithmetics and that it had to do with the measurement of 
the floods on  the Nile! So, I would not be surprised if some mathematician 
(who will _not_  have a lot of philosophy) would say that Experience _is_ 
Relevant to  Mathematics.

Mathematicians tend to be Kantians, to boot. And so it's  even harder to get 
to admit that a programme of research they are involved in  (notably if it's 
some state-funded university!) is degenerate already! They tend  to think of 
themselves as "transcendental egos" on the bounds of sense, and  dealing with 
possibilities of experience, maybe.

While 'time' and 'space'  are perhaps _empirical_ notions (but vide Kant), 
one should wonder about the  very idea of _number_ -- since Roger has notably 
listed branches of _number_  theories.

Has there been progress via, if not conjectures, 'refutations',  in 'number' 
theory at all? God knows. The same God, incidentally who this German  
mathematician said, created the "natural numbers"! (the rest of the numbers  being the 
offspring of human, evil, wicked minds!)

Excellent question,  Roger. 

Cheers,

JL  

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