[hist-analytic] Quine's Holism
Danny Frederick
danny.frederick at tiscali.co.uk
Sat Feb 7 10:30:02 EST 2009
Hi Roger, Rogério & JL,
I’m just catching up with posts, so please excuse me if I have missed one of
the later ones.
Roger’s question was a very good one, viz.: is there an example of a
proposition of mathematics which has been proven and subsequently held to be
false as a result of some empirical observation
My initial response was similar to Rogério’s: Euclidean geometry. But the
axiom of parallels was rejected by mathematicians before Einstein came
along. I’ll come back to this in a moment after a brief digression.
There seem to be many propositions which were held to be analytic which have
later come to be rejected on empirical grounds. Descartes’ physics was
supposed to be an a priori system based on clear and distinct ideas. Does
that make it analytic? Not necessarily. But some of it might have been
regarded so, if the question had been put and the analytic/synthetic
distinction accepted. Yet it was the empirical success of Newton’s theory
that did for Descartes. Similarly, scientists like Kepler and Galileo
developed their theories largely as a priori mathematical theories. Galileo
often seemed to think that he was simply doing mathematics. No one asked
these scientists which of their principles they regarded as analytic (if
any), so we don’t know what they would have said. But it seems reasonable to
assume that, in at least some cases, they would have claimed to be analytic
something that has since been rejected in the course of the empirical
progress of science. This sort of thing seems to happen regularly in the
development of a scientific theory: some parts of the theory are regarded as
true by definition or by self-evidence; yet these parts, as well as the
rest, get rejected in the onward march of empirical science.
A defender of the analytic/synthetic distinction can of course say that what
this shows is that people are often mistaken about which truths are
analytic, not that analytic truths can actually be refuted empirically. And
the same strategy seems to occur in mathematics, to safeguard its
non-empirical character.
Thus, the fact that Einstein’s theory uses a non-Euclidean geometry does not
show that Euclidean geometry has been rejected on empirical grounds. It
shows only that Euclidean geometry has to be rejected IF it is interpreted
as a theory of physical space. But it can instead be interpreted
non-empirically as a theory of Euclidean space. Or, to consider a very
simple example, suppose that someone says that it is false that 1 + 1 = 2
because one drop of water when added to another drop of water = one drop of
water. The response is to say that what has been refuted is an empirical
interpretation of the mathematical formula, but the maths itself is purely
formal and cannot be refuted in such ways.
The problem with this approach is that it makes empty the claim that
mathematics is non-empirical: whenever an empirical refutation is produced,
it can be dismissed as due to an empirical interpretation of the maths. This
amounts to making maths ‘true by convention’ in Poincare’s sense.
Roger is, of course, right that there has been a massive development of
mathematics over the centuries (not just new theorems and new proofs but new
branches of the subject); and Rogério is right that a good deal of this
growth has been stimulated by the demands of physical science (as with
Newton’s development of the calculus). JL is right to refer to Lakatos, who
shows in his ‘Proofs and Refutations’ how many previously ‘proven’
mathematical truths have been refuted by quasi-empirical counterexamples. By
‘quasi-empirical’ I mean that the counterexamples came from
thought-experiments which showed that what was previously taken for a
mathematical truth was in fact false, so what was previously taken for a
proof could not be a proof at all. The problem then was to locate the fault
in the previously accepted proof. There could be disagreement about where
the fault lay and different lines of research in consequence.
To answer JL’s question, Lakatos did apply his methodology of research
programmes to the history of mathematics. His Ph.D. thesis was called
‘Essays in the Logic of Mathematical Discovery’ and part of it is reproduced
in ‘Proofs and Refutations.’ Lakatos’ methodology of scientific research
programmes is a development or variation of Popper’s approach set out in
‘The Logic of Scientific Discovery’ (plus the ‘Metaphysical Epilogue’ to
Popper’s ‘Quantum Theory and the Schism in Physics,’ where Popper spoke of
‘metaphysical research programmes’).
One final point: there can be no such thing as the justification of
mathematical propositions. That idea was buried with Russell’s discovery of
the logical paradoxes (see his ‘My Philosophical Development’ for an account
of the history). Hilbert tried to revive the corpse with his
meta-mathematics; but Godel put paid to that effort. For a discussion see
Lakatos, ‘Infinite Regress and the Foundations of Mathematics.’
I have seen that JL has just sent a post which covers part of what I have
said about Lakatos. But I must say that some of that wiki stuff sounds
suspect to me.
Best wishes,
Danny
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