Re: Existence of function
- From: John Bailey <john_bailey@xxxxxxxxxxxxxxxx>
- Date: Thu, 15 Nov 2007 08:56:16 -0500
On Thu, 15 Nov 2007 14:35:32 +0200, "I.N. Galidakis"
<morpheus@xxxxxxxxxxxx> wrote:
This probably belongs to mathematical/philosophical existentialism, but hereThe answer may depend on whether B accepts non-constructive proofs.
goes:
1) Person A claims that a certain real analytic function f having certain
properties exists.
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?
4) To make the case more interesting, assume that the convergence of the series
f_n cannot be proved by algebraic means.
Does f exist?
A proof is nonconstructive if it asserts the existence of some object
without actually constructing or finding that object. Such proofs are
used freely in mainstream ("classical") mathematics. Constructivism
is the practice of avoiding such proofs or at least pointing them out
explicitly. Once a mathematician sees the distinction between
constructive and nonconstructive mathematics, he or she will choose
the former.
http://www.math.vanderbilt.edu/~schectex/papers/difficult.html
After all:
integral from 0 to infinity of (cos(2*pi)*product from x=1 to infinity
of cos(x/n) dx) = pi but only to 41 decimal places.
ref: Future Prospects for Computer-Assisted Mathematics by Bailey and
Borwein http://crd.lbl.gov/~dhbailey/dhbpapers/math-future.pdf
Hello Ioannis,
I note the problem is similar to one from a thread on alti.math
http://groups.google.com/group/alt.math/msg/d161e1ec3da02579
where I raised the issue of the criteria for proving an expression for
pi is valid.
Here are some quotes from that thread:
quoting http://cf.geocities.com/ilanpi/pi-exists.html
Well, the first problem is understanding what the problem is. Indeed,
even the compendium [L. Berggren, J. Borwein, P. Borwein, Pi: A Source
Book, Springer Verlag, New York 1997] fails to provide a proof of pi's
existence! Basically, you need to figure out what the exact definition
of pi is, and then rigorously prove that this defines a unique real
number. In effect, this problem is an exercise in mathematical rigor.
http://www.rzuser.uni-heidelberg.de/~tvogt2/999.pdf.
The fact that you can?t compute the decimal expansion of a sum from
the decimal expansions of its addends is a well known phenomenon
that was noticed by Turing. In a fully constructive treatment of the
real numbers, this is often stated by saying (informally) that not
every positive real number has a decimal expansion. More precisely,
there is no constructive proof that every positive real number has a
decimal expansion (or at least we don?t know of one).
http://www.rbjones.com/rbjpub/cs/cs006.htm
Computable reals are defined in Turing's first classic paper
[Turing36] where he uses (what we now call) universal Turing machines
to show that there exist unsolvable problems in elementary number
theory. In this paper Turing defines computable reals as those whose
decimal expansion is computable by Turing machine. Pretty soon he
spotted that this is not so good (not what we would now call an
admissible representation) and comes out with a correction in
[Turing37], where he goes for convergent sequences of nested intervals
with rational end-points (which is an "admissible" representation).
The discussion this thread followed got lost on the confusion of
whether constructive was a term to be applied to the number or to the
proof. The 42 decimal point approximation to pi illustrates the
difference.
.
- References:
- Existence of function
- From: I.N. Galidakis
- Existence of function
- Prev by Date: Re: Algebra with isomorphism and group.
- Next by Date: Re: JSH: Where are you ?
- Previous by thread: Re: Existence of function
- Next by thread: Re: Existence of function
- Index(es):
Relevant Pages
|
