[hist-analytic] Quine's Holism
Roger Bishop Jones
rbj at rbjones.com
Fri Feb 6 16:52:09 EST 2009
Thanks for your response on my question.
I'm afraid I don't see what I was asking for, but you do
present Quine's case without supplying what I consider
essential to Quine's purpose.
We are not really talking about the analytic/synthetic
distinction, but the a priori/a posteriori distinction,
but I think that what I was asking for is particular to
what is needed for disputing the accepted epistemic difference
between mathematics and empirical science.
I shall respond to your arguments and explain why I don't
find them convincing.
On Friday 06 February 2009 19:03:52 Rogério Passos Severo wrote:
> Geometry is traditionally sorted as branch of mathematics. In the
> eighteenth and nineteenth centuries, the axioms of Euclidean geometry were
> thought to be true of physical space.
But surely the belief that physical space is Euclidean is not mathematics
even though Euclidean geometry is a branch of mathematics.
> Nowadays physicists say space is non-Euclidean.
> General relativity is now thought to better explain
> observations than Newtonian physics. So here we have an instance of changes
> in physics coming along hand-in-hand with changes in mathematics. This is
> exactly what Quine's holism says: the sentences of a theory are not
> accepted/rejected one by one, but as "a corporate body".
I don't see that there is any relevant kind of "rejection" of mathematics
here. We still teach Euclidean geometry, we still regard its theorems as
true. It seems to me to be a stretch to say that Euclidean geometry
was rejected in any sense, I doubt that Einstein would have agreed with
that way of describing things.
When physicists change their theories the mathematics they use in
the new theory may not be the same as the mathematics used in the
new theory, but no changes take place to what mathematical propositions
are considered true.
So if this is a prime exemplar of what constitues Quine's holism
then it is hard to see how it is relevant to the epistemological
distinction which is under dispute.
> Here's another example: Descartes and others argued that negative numbers
> were absurd. They thought the idea made no sense at all. And indeed, for a
> long time math had no negative numbers. But then people began to use them
> "instrumentally", so to speak. People began to try them out, and see what
> they could do with them. Later it became clear that math with negative
> numbers was much more useful (for science) than without them. And so
> negative numbers were incorporated. This was a significant change in
> mathematics brought about by empirical considerations (what you could do
> with them in the empirical sciences).
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.
New parts of mathematics stimulated in such ways make no reference
to the stimulus in the justification of mathematical propositions.
Even if a mathematical fallacy was discovered in the course of experimental
research it would not be accepted until flaws had been located in the accepted
proofs of the relevant mathematical propositions, since mathematicians
do not accept empirical claims either in proofs or in refutations.
As far as the development of number systems are concerned, there
remain to this day, as accepted and rigourous parts of mathematics:
1. the theory of natural numbers (which is known as arithmetic)
2. the theory of integers (obtained by adding negative numbers)
3. the theory of rationals (integers and their ratios)
4. the theory of real numbers (analysis)
5. non-standard analysis (reals + infinitesimals)
6. the theory of complex numbers (pairs of reals)
Developments in our number concepts do not involve the discovery
that some previously accepted mathematical propositions are in fact
false, and if such a mathematical discovery were prompted by
empirical considerations, it would not be accepted until a flaw had
been found in previous a priori justifications and a priori justifications
(i.e. mathematical proofs) had been supplied for any new results.
> But I see your point: inside mathematics there is a justification system in
> place which is more or less impervious to what goes on outside (in the
> natural sciences). This is ok by Quine. He acknowledges that in several
> places. His point was a more general and abstract one: no sentences are
> justified in isolation, and mathematics as a whole is not justified
> independently of the rest of science. Changes in the other disciplines may
> have effects on mathematics.
But what world is Quine living in?
Mathematicians have very precise knowledge of what propositions are
involved in the justification of each mathematical theorem, and they
NEVER include proposition of empirical science.
That other disciplines may affect the development of mathematics is
not relevant to the epistemic status of mathematical propositions.
Quine knew this of course! (the first sentence if not the second)
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