I would recommend that the participants in this debate
read Imre Lakatos's seminal book 'Proofs and
Refutations: The Logic of Mathematical Discovery'.

The debate here is not about mathematics it is about
the philosophy of mathematics. Lakatos directly
addresses what I see as the subject of the debate here
in his book when he makes the  distinction between
formal and informal mathematics. Formal mathematics is
contained in the proofs published in mathematical
journals. Informal mathematics are the strategies that
working mathemeticians use to make their work a useful
exercise in mathematical discovery.

I see the objection to the four color proof here. It
was undoubtedly a valid formal proof. However it did
nothing to advance the cause of mathematics beyond

The reason that Lakatos equates proofs and refutation
in his title is his contention that it is the
refutations that are developed that show
mathematicians the deficiencies in their theories. It
is their attempts to deal with unwanted and unexpected
refutations - to preserve a valuable theory in the
face of imperfect axioms and proof methods  - that
teach mathemeticians the true depths of their
conceptions and to point the way to new and deeper

Lakatos shows this by an account of the historical
development of the concept of proof in mathematics and
by showing in historical detail how certain valuable
'proofs' were preserved in the face of refutation. To
this point Lakatos shows that the 'proofs' of the
truth of Euler's number are no proofs at all. The
great mathemetician Euler noticed that for any regular
polyhedron the formula V-E+F=2 where V is the number
of vertexes, E is the number of edges and F is the
number of faces. Euler's and his successors proofs
fall before any number of counterexamples. Does this
prove that the theorem is 'incorrect?' Or does it mean
as the mathemetician's actions show that they thought
it meant was that their concept of what constituted a
regular polyhedron was deficient. Lakatos shows how
these conceptions were modified over a couple of
hundred years as counterexample after counterexample
was faced. These counterexamples all made mathematics
stronger by deepening the conception of what polyhedra
really are and by discovering new classes of them. In
the end Euler's formula turned out not to have a proof
but to be in effect a tautology. It is true for the
regular polyhedra for which it is true by the
definition of what constittues a polyhedron. It is
true because human mathematicians in order to make
progress need it to be true.

The computer proof of the four color theorem was a
triumph of formal mathematics. Its critics complained
and. if interpreted according to what Lakatos wrote,
they complained because it defeated the progress of
informal mathematics.

Mathematical proofs are useful tools. The tell us what
we need to know. Formal mathematics is about finding
them. Informal mathematics is about making them
useful. Mathematics is not some Platonian ideal
divorced from humanity, painting, poetry ... It is a
human endeavor to meet human needs.

[log in to unmask] wrote:
> In a message dated 3/12/01 6:21:53 PM Eastern
> Standard Time, 
> [log in to unmask] writes:
> > I'm of course exaggerating ... . Theory is nothing
> to gloss over and can be 
> > very very useful, elegant and quick. But it's a
> theory, and theories have a 
> > history of being overturned in practice. People
> are often blinded by the 
> > beauty of an elegant theory, but often the real
> test for a theory is when 
> > we apply them to the world; that's usually where
> things start going wrong. 
> > And therefore I think there is definitely
> something to say for being able 
> > to prove something 'uitputtend' as we say in
> Dutch, exhaustive. It's not 
> > always necessary, it's not always elegant, but
> it's rock solid. You also 
> > often really need it when applying a theory to the
> world, because when you 
> > use a theory in practice you also have an impure
> domain to cover; practical 
> > 
> You better get caught up on Wittgenstein or Benjamin
> Whorf, because you're 
> saying things that don't make any sense. What's
> "rock solid" about examining 
> every person in the world to discover, say, that
> nobody in the world has an 
> s-shaped scar on their left shoulder? What is that
> supposed to "prove" about 
> whether there ever was such a person in the past or
> might be in the future?
> p

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