Re: what makes it true?
*** T. Winter wrote:
But there is a long way between this proof and
number-crunching evidence. Moreover, number-crunching evidence may
lead you to completely wrong results. Riemann and Gauss asserted that
Li(x) < pi(x) for all x. All number-crunching evidence available now
and ever will tell you that is true. But it is false. The same for
Goldbach's conjecture. All number-crunching evidence shows it is true.
But is it? Nobody knows. You are apparently assuming that
number-crunching can provide evidence for something. That is false.
The only thing that number-crunching can provide is to show that something
is likely. It can not be used as proof of some statement.
You may not expect, but I agree with all of the above. But instead of
"number crunching", let's talk about "idealized number crunching". By
this I don't mean the calculations that actually can be done with the
current machines, but calculations that can be imagined with kind of
an idealized superfast and supersized computer (Turing machines ?).
Now the question reads again: is ( Li(x) < pi(x) for all x ) something
that can been demonstrated false with such an idealized computer? Thus:
does the proof actually "mimick" an idealized calculation? Is the proof
i.e. kind of constructive?
Same questions for Goodstein's Theorem. Because I still don't understand
how the "proof" of _that_ theorem relates to something "realistic".
Han de Bruijn
.
