[FOM] Re: Question on the Scope of Mathematics - Enduring Value II
robblin at thetip.org
Fri Jul 30 15:29:44 EDT 2004
On Jul 29, 2004, at 9:40 AM, Dmytro Taranovsky wrote:
> I only meant to say that, for example, if you announce that the Riemann
> Hypothesis is true, then unless you have a proof of the Riemann
> Hypothesis, you
> should qualify your announcement by stating that you know the
> truthfulnes of
> the Riemann Hypothesis based on philosophical and empirical reasons
> rather than
> provability in ZFC.
> Also, in the nineteenth century, it was disputed whether set theory is
> mathematics. The agreement that set theory is mathematics was
> because of formalization of set theory.
> My preference is to define mathematics broadly; and use the word
> mathematics" for the narrow notion of mathematics. However, some
> believe that
> all mathematics is formal mathematics, and in any case, to avoid
> and controversy, as much as reasonable of one's mathematical work
> should be
> valid as formal mathematics.
Suppose that someone says "I know the axiom of choice to be true." We
can't then expect them to merely produce a proof of it from ZFC -
wouldn't that be trivial?!
We say, therefore, that that would somehow be a
foundational/philosophical assertion, e.g. not a mathematical question
if knowledge of the truth of a statement is just routine proof in ZFC
(which I suppose it's completely routine to deduce (P&Q&R&S&T&...) -> P
or "ZFC -> Axiom of Choice", just not convincing!)
Would you also conclude that a proof that relied on this further
non-ZFC and therefore (foundational/philosophical/empirical)
non-mathematical question (e.g. one that relied on the truth of ZFC)
was itself therefore non-mathematical too?
I guess my point is - producing a proof from ZFC isn't trivial, but it
is just that, producing a proof from ZFC.
So the statement that "The Reimann Hypothesis is known to be true"
would have to depend on philosophical/empirical reasons rather than
(merely) proof in ZFC, since even a proof in ZFC would depend on these
I guess there is the further question, whether the terms:
"Philosophical, empirical" exhaust the kinds of reasons we use for
accepting non-formal mathematical statements. Perhaps we also accept
them (if we do!) for social, political and religious reasons. I
consider the matter obvious - that we in fact "accept" our formal
systems because of their roles in our languages, societies, cultures,
More information about the FOM