Re: Existence of function

On 15 Nov., 17:35, "I.N. Galidakis" <morph...@xxxxxxxxxxxx> wrote:
hagman wrote:
On 15 Nov., 13:35, "I.N. Galidakis" <morph...@xxxxxxxxxxxx> wrote:
This probably belongs to mathematical/philosophical existentialism,
but here goes:

1) Person A claims that a certain real analytic function f having
certain properties exists.

Do you have a special property in mind?

yes, I do, but please forgive my reluctance to reveal it. I want to examine the
question from a more "philosophical" perspective. Should an answer become
crystalized from the opinions of the experts, I will consider revealing the
property.

2) Person B claims that f does not exist.
3) Person A then displays a sequence of series
f_n(x)=sum(a_k*x^k,k=0..n), (with a_n calculated numerically) which
appear to approximate the function f to any degree desired, but A
cannot prove convergence of the f_n.

Has A proved the "existence" of f?

No.

Thanks. I suspected that as well.

4) To make the case more interesting, assume that the convergence of
the series f_n cannot be proved by algebraic means.

I might have suggested analytical methods anyway ;)

Yes, sorry, I also meant to write "analytical", but avoided the term in order
not to confuse the purists. In short, no method known so far can prove
convergence of the f_n.

OK, "no method known so far" is of course much
weaker than "cannot be proved".
Thus as of now, neither existence nor non-existence
have been proved nor is the statement "f exists"
known to be independent.
Probably we don't even have something like
"undecidable implies true" (as e.g. if the
Goldbach conjecture were undicidable then
no counterexample [=finite proof of non-Goldbach]
would exist, hence Goldbach would be true).
Thus the whole matter is currently
at the level of conjecture and the weight of
evidence for and against is subject to personal
taste.

--
I.N. Galidakis

hagman
.

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