[FOM] Re: Question on the Scope of Mathematics - Enduring Value II

[Robbie Lindauer]

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 
> established
> because of formalization of set theory.
>
> My preference is to define mathematics broadly; and use the word 
> "formal
> mathematics" for the narrow notion of mathematics.  However, some 
> believe that
> all mathematics is formal mathematics, and in any case, to avoid 
> uncertainty
> 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 
(foundational/philosophical/empirical) justifications/reasons.

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, 
religions, politics.

Best,

Robbie Lindauer