On 5/29/2012 12:42 PM, Aleksandr Lokshin wrote:
I'll try to explain why choosing an arbitrary element should be interpreted as /a free
will choice in mathematics/.
The difficulty of understanding depends, IMHO, on the fact that in English different
roots of the words are employed in "arbitrary" and "free will". In Russian thre roots
are the same, but my explanation will not base on this fact.
According to phisics, free will choice (if it does exist) is a choice which
1) is not random,
2) is not determined by some law.
Now , consider a *Theorem: *statement A is valid for all x belonging to X.
*Proof. *Let x be an arbitrary element of X. We demonstrate that A(x) is valid.
Since x was chosen arbitrarily, A is valid for all x.
But in this application "arbitrary" just means "without any distinguishing
characteristic"; we specify *only* that x belongs to X and do not specify or use any other
property. That's why "any x" is equivalent to "every x" and the theorem stated in terms
of "every x" doesn't imply any choosing at all.
Brent
*Comment.* As I have pointed out earlier , if x was chosen randomly, the theorem is not
proved.
Analogously, if x was chosen according to some rule, the theorem is not proved.
Therefore, since we are sure that we have proved the theorem by using the arbitrarily
chosen element, we*must inevitably agree* that
the element was chosen by using free will choice.
*
*
*
*
*
*
On Tue, May 29, 2012 at 11:05 PM, Brian Tenneson <tenn...@gmail.com
<mailto:tenn...@gmail.com>> wrote:
It doesn't take free will to prove that every even number is divisible by
2. How to
prove a statement with a universal quantifier is pretty basic.
On Tue, May 29, 2012 at 12:01 PM, Aleksandr Lokshin <aaloks...@gmail.com
<mailto:aaloks...@gmail.com>> wrote:
<</The notion of "choosing" isn't actually important--if a proof says
something
like "pick an arbitrary member of the set X, and you will find it obeys
Y", this
is equivalent to the statement "every member of the set X obeys Y"/>>
No, the logical operator "every" contains the free will choice inside
of it. I
do insist that one cannot consider an infinite set of onjects
simultaneously!
Instead of so doing one considers an arbitraryly chosen object. It is
a very
specific mathematical operation . By using operator "every" we
construct a
formalism which hides the essens of matter - the using of a free will
choice.
On Tue, May 29, 2012 at 10:30 PM, meekerdb <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
On 5/29/2012 10:52 AMOne cannot, John Clark wrote:
On Sun, May 27, 2012 Aleksandr Lokshin <aaloks...@gmail.com
<mailto:aaloks...@gmail.com>> wrote:
> All main mathematical notions ( such as infinity, variable,
integer
number) implicitly
depend on the notion of free will.
Because nobody can explain what the ASCII string "free will" means
the
above statement is of no value.
> A new approach to the Alan Turing problem (how to distinguish
a
person from an android) is also proposed ; this approach is
based on
the idea that an android cannot generate the notion of an
arbitrary object.
But "arbitrary" just means picking something for no reason or
picking
something just because you like it but you like it for no reason;
in other
words it means random. It's true that a pure Turing machine can not
produce
randomness, however this limitation can be easily overcome by
attaching a
very simple and cheap hardware random number generator to it.
Or by computing psuedo-random numbers with a sufficiently long
period that
no one will be able to determine the algorithm.
Brent
Then the android could be as arbitrary as any arbitrary person, if
you
think being arbitrary is a virtue that is.
John K Clark
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