[hist-analytic] Mathematics and Lakatos's Research Programme Degeneracy
Jlsperanza at aol.com
Jlsperanza at aol.com
Sat Feb 7 08:31:54 EST 2009
Thank you, Roger, for your comments on the two aspects which you'd view as
'progress' in mathematics:
-- number of theorems proved 'truth'
-- rigour in that.
I think it may be connected to your comments on analytic/synthetic, where,
-- and I'd disagree there, but with most philosophers, then -- of defining
'analytic' as 'true in virtue of its meaning'.
You see, I don't _do_ 'true'. When teaching logic, I always had my students
sequencing 0 1 0 1 0 1 0 1 0 1. A student or two would like to challenge my
Excluded Middle, so I'd say, well introduce 0 1 2 ---. I would not be a fan of
introducing 0.5 for the middle value, but perhaps I should. Incidentally,
this leads to the infinity of the continuum, for literally (or potentially,
rather) there's an infinite series that can hold the 'true' predicate.
Sorry to blow his horn, but Grice has a charming section on "Truth" in WOW,
iii. He ends up _translating_ the adjective into: FACTUAL SATISFACTORINESS! If
one reads _Aspects of Reason_ you see the point. Although he does not use
'factual satisfactoriness', he uses 'satisfactoriness' as it applies to
theorems (Oddly the 'theorem' sign in logic equates Frege's assertion sign). So
Grice would say, hoping to bridge the gap between the is and the ought, that
there is another type of satisfactoriness, too, which he calls 'boulemaic', or
'practical', etc -- with "!" as the prefix, rather. These two classes he finds
isomorphic.
But truth has been left behind. In particular, I think he is abiding by
"then _enfant-terrible_ of Oxford philosophy", Ayer (never mind Vienna Circle for
now). In "Language, Truth, and Logic" he is a strict proponent of the view
that mathematics do not speak about the world, so that the word 'true' should
be otiose. It _is_ confusing that 'p v ~p' should not speak about the world
and yet get 1 1 1 1 as a truth table. But if I have to chose a philosophical
description, I'd go with the 'does not talk about the world' rather than it is
true by virtue of its meaning!
---- Now as for Lakatos:
In a message dated 2/7/2009 7:09:15 A.M. Eastern Standard Time,
rbj at rbjones.com writes:
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.
----
Well, here some selections from wiki, which I'll comment as I paste them.
>From the wiki's first sentence on the man (he looks aristocratic -- in a middle
European kind of way)
>most famous today worldwide for his thesis of
>the fallibility of mathematics
which if what I've saying above holds water, is _not_ falsification, because
neither 'false' nor 'true' would apply to a mere calculus of symbols! And if
you say it refers to _space_ or a model of space (geometry, the plane), or
it refers to quantity in its most abstract terms (the idea of 'number') then I
would not know why you don't go the whole hog and call the thing 'empirical',
even if 'two-generations removed' as it were. (The Greeks' idea of number is
funny: '1' was not one, since it was the name of the unity (Wittgenstein
uses 'Unit' in the passages we've been reading re: the Reihe and the Gesatz). Of
course it was not 'zero' -- since Arabic, etc. -- Apparently, it started
with '2'.
>Lakatos received a degree in mathematics (1944)
>In 1960 he was appointed to a position in the London School of Economics,
>where he wrote on the philosophy of mathematics.
And taught I hope. Fancy becoming a teacher and just _writing_!
>It was Agassi who first introduced Lakatos to Popper
>under the rubric of his applying a fallibilist methodology
>of conjectures and refutations to mathematics in his Cambridge PhD thesis.
Should revisit my Agassi.
>Lakatos' philosophy of mathematics was inspired by both Hegel's
>and Marx' dialectic, Popper's theory of knowledge,
>and the work of mathematician Polya.
>"Proofs and Refutations" is based on his doctoral thesis.
>It is largely taken up by a fictional dialogue set in a mathematics class.
>The students are attempting to prove the formula for the Euler
>characteristic in algebraic topology, which is a theorem
>about the properties of polyhedra. The dialogue is meant
>to represent the actual series of attempted proofs which
>mathematicians historically offered for the conjecture,
>only to be
repeatedly refuted by counterexamples.
>Often the students 'quote' famous mathematicians such as Cauchy.
>What Lakatos tried to establish was that no theorem of
>INFORMAL [my emphasis. JLS] mathematics is final or perfect.
I guess I would _never_ have used that word, 'informal'. It qualifies all
_so_.
>This means that we should not think that a theorem is
>ultimately true, only that no counterexample has yet been found.
and I would not take side with 'true' in the first place. But
'mathematically satisfactory' if you wish (I know you _won't -- wish).
>Once a counterexample, i.e. an entity contradicting/not explained
>by the theorem is found, we adjust the theorem,
>possibly extending the domain of its validity.
validity? or satisfactoriness?
I apply 'validity' to _sequences_ of sentences only, i.e. arguments.
It's true that if
p
p ) q
_____
q
is valid (B. Aune discusses modus ponens at length in ch. iii of his
Empiricism book), then
the theorem:
. (assertion sign)
. ((p & (p ) q)) ) q
will receive a 1 1 1 1 truth-table. But that's stylistically different from
saying an assertion (or meta-assertion? see Hunter, "Meta-logic") is valid.
Wiki continues:
>This is a continuous way our knowledge accumulates,
>through the logic and process of proofs and refutations.
>If axioms are given for a branch of mathematics, however,
>Lakatos claimed that proofs from those axioms were
>tautological, i.e. logically true.)
Or some would say 'analytic', and it's back to 'logical truth'. So perhaps
if Grice did talk about 'factual satisfactoriness', one _could_ talk of
'logical' or 'formal' satisfactoriness here.
Wiki continues:
>Lakatos proposed an account of mathematical knowledge
>based on the idea of heuristics. In Proofs and Refutations
>the concept of 'heuristic' was not well developed, although
>Lakatos gave several basic rules for finding proofs and
>counterexamples to conjectures. He thought that
>mathematical 'thought EXPERIMENTS' [emphasis mine. JLS -- loved that!]
>are a valid way to discover mathematical conjectures and proofs,
>and sometimes called his philosophy
only as a tribute to Quine, I'd add!
>'quasi-empiricism'.
(or "How to Quasi a Meta Quine: Lakatos on two dogmas of mathematics".
Wiki continues:
>However, he also conceived of the mathematical community
>as carrying on a kind of dialectic to decide which
>mathematical proofs are valid and which are not.
at the cocktails, I assume. I'm _never_ invited.
>Therefore he fundamentally disagreed with the 'formalist'
>conception of proof which prevailed in Frege's and Russell's
>logicism, which defines proof simply in terms of formal validity.
Yes, 'formal' seems to be the word we are looking rather than 'logical
truth'. And it's something more 'formal satisfactory in virtue of its _form_
rather than meaning. For 'meaning' is merely an extra manipulation of symbols: 1
and 0" -- they don't have to _look_ at the "extension" of a class (out there)
to _decide_ whether something is 1 or 0. And if they do, it's factual, not
formal!
>On its publication in 1976, Proofs and Refutations became
>highly influential on new work in the philosophy of mathematics,
>although few agreed with Lakatos' strong disapproval of formal proof.
>Before his death he had been planning to return to the philosophy of
>mathematics and apply his theory of research programmes to it.
But then he thought, "Well, Speranza will do it for me" and he left this
world in peace.
Wiki continues:
>One of the major problems perceived by critics is that the pattern of
>mathematical research depicted in Proofs and Refutations does
>not faithfully represent most of the actual activity of contemporary
>mathematicians.
who holiday in Florida, text their friends every other three seconds,
grow big moustaches, speak Russian to non-Russian speakers,
and allow students to use 'calculators'!
References from wiki:
Kampis, Kvaz & Stoltzner (eds)
APPRAISING LAKATOS: Mathematics, Methodology and the Man Vienna Circle
Institute Library, Kluwer 2002 ISBN 1-4020-0226.
Lakatos (1978). Mathematics, Science and Epistemology: Philosophical Papers
Volume 2. Cambridge: Cambridge University Press. ISBN 0-521-21769-52
Teun Koetsier (1991). Lakatos' Philosophy of Mathematics: A Historical
Approach. Amsterdam etc: North Holland. ISBN 0-444-88944-2
Cheers,
J. L.
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