[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
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.
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