Ingvar Johansson <ingvar.johansson@xxxxxxxxxxxxxxxxxxxxxx> writes: (01)
> Steve Newcomb schrieb:
> >
> > If this is your example of a "fact", then I remain unsatisfied.
> > Mathematics is grounded in itself. So what?
> >
>
> Some questions to Steve N:
> (1) Is it a fact that mathematics is grounded in itself? Or: (02)
I see that I misspoke. As I recall, Russell says that the
correspondence between mathematics and reality can neither be proven
nor disproven, because we lack access to the catalog of all things,
and therefore we cannot know whether that catalog contains a listing
for itself. (03)
Which makes my statement completely wrong, at least according to
Russell. (I'm not aware of any post-*Principia* results that
undermine the force of Russell's argument.) (04)
I should have said that mathematics cannot itself provide its own
grounding, and that such grounding cannot be proven to exist. (In
that light, the fact that mathematics is such a large and imposing
edifice can be seen as a naive kind of evidence that it provides its
own grounding, Gödelian misgivings notwithstanding. Indeed, most
people *do* believe, or at least pretend they believe, or exhibit
behavior that would lead any reasonable observer to conclude that they
believe, that Maths is provably grounded, and this doesn't seem to
cause many problems for them. Rightly or wrongly, that's what I was
really thinking when I said "mathematics is grounded in itself".) (05)
> (2) Is it merely your subjective opinion that mathematics is grounded in
> itself? (06)
Nah. But for the sake of the question you're developing here, let's
stipulate that it is indeed merely my subjective opinion. (07)
> (3) If it is merely your subjective opinion, is then your opinion an
> opinion *about a fact* or an opinion about something else? (08)
I'm having trouble following your question. You seem to be making a
distinction between opinion and fact (which is a useful thing to do),
but you haven't characterized the distinction. We've stipulated that
it's my opinion [which it isn't] that mathematics is grounded in
itself. That's my opinion. Now, what's this "fact" you're talking
about? (09)
> (4) If "something else", what kind of entity is it? (010)
I don't understand the question. The facts, if there are any, are in
another universe. In my universe, there are opinions as to what facts
exist in the universe of facts, but, in fact, I lack access to that
universe, just as I lack access to the catalog of all things. (011)
Indeed, from a higher perspective, the situation is even less
satisfactory. We have my original *statement* that mathematics is
grounded in itself, but I have now told you that that statement does
*not* reflect my opinion. So even as a statement of opinion, my
statement is wrong. And statements are all we have. (012)
The most we can hope for, from statements, is that they accurately
reflect the beliefs of their utterers. And my statement did not meet
that criterion. (013)
Let's pretend that my original statement is what I really believe, so
that it's an accurate reflection of my actual belief as to the
grounding of mathematics. What kind of entity is such a belief?
Well, uh, it's a belief. It's part of the furniture of my mind. I
have to walk around it to get to my mental kitchen, or bathroom, or
whatever; it affects my behavior and my thinking in countless ways.
It's a belief that's different from the one that Bertrand Russell led
us to believe that he believed. (Different people believe different
things, sometimes for reasons, and sometimes not.) A "fact", on the
other hand, is something unspeakable to which I have no direct access,
but which I *believe* exists, I hope for good reasons. I can talk
about my beliefs, but about real things, I have no choice but to
remain silent (to paraphrase early Wittgenstein). At most, a
statement is a statement of a belief. The only other possibility is
that it's noise. (014)
-- Steve (015)
Steven R. Newcomb, Consultant
Coolheads Consulting (016)
Co-editor, Topic Maps International Standard (ISO/IEC 13250)
Co-editor, draft Topic Maps -- Reference Model (ISO/IEC 13250-5) (017)
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irresponsible actions of the pusillanimous and corrupt 109th Congress,
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having to make this difficult choice, but I urge you to make it
wisely.) (021)
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