Yahoo Tuning Groups Ultimate Backup metatuning for those interested in philosophy of mathematics

--- In metatuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In metatuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /metatuning/topicId_3132.html#3132
>
> > i just learned about goodstein's theorem:
> >
> > http://astronomy.swin.edu.au/~pbourke/analysis/goodstein/
> > http://mathworld.wolfram.com/GoodsteinsTheorem.html
> > http://www.maths.bris.ac.uk/~maadb/research/seminars/online/fgfut/
> > http://www.maths.bris.ac.uk/~maadb/research/topics/logic/
>
> ***Isn't this, though, a little like "blind faith?...
>
> JP

this is very deep stuff.

the axiom of mathematical induction is simply:

IF a property holds for the number 1;
AND (if a property holds for the number N, then it holds for N+1 too);
THEN it holds for all whole numbers.

see if you can understand the above and convince your self that it's
true. how could it not be true? try to think of a counterexample.

assuming this ordinary axiom of mathematical induction is true
implies the truth of goodstein's theorem (this is an example of
Godel's Incompleteness Theorem, which you need to understand before
any of this will make any sense to you).

but the truth of goodstein's theorem cannot be proved by using
mathematical induction itself (only in the ordinary variety) in the
proof.

since mathematical induction is so "obviously true", even for
philosophers of mathematics who shun all talk of infinity, we are
forced to believe goodstein's theorem just as strongly.

the fact that the truth of goodstein's theorem is true, and so simple
to express, makes mathematicians wonder if they can understand *why*
it is true.

the explanation as to *why* it is true has to draw on a fancier kind
of induction, called "transfinite" induction -- but the "transfinite"
numbers can simply be thought of as rates of growth of everywhere-
finite functions. the explanations *make sense* to many
mathematicians, but whether they make sense to you or not, the fact
is that the truth of goodstein's theorem is implied by the truth of
ordinary mathematical induction, so if you believe in ordinary
mathematical induction, you *have* to believe that goodstein's
theorem is true -- explanation or no explanation.

--- In metatuning@y..., "Paul Erlich" <paul@s...> wrote:

/metatuning/topicId_3132.html#3153

> --- In metatuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> > --- In metatuning@y..., "Paul Erlich" <paul@s...> wrote:
> >
> > /metatuning/topicId_3132.html#3132
> >
> > > i just learned about goodstein's theorem:
> > >
> > > http://astronomy.swin.edu.au/~pbourke/analysis/goodstein/
> > > http://mathworld.wolfram.com/GoodsteinsTheorem.html
> > >
http://www.maths.bris.ac.uk/~maadb/research/seminars/online/fgfut/
> > > http://www.maths.bris.ac.uk/~maadb/research/topics/logic/
> >
> > ***Isn't this, though, a little like "blind faith?...
> >
> > JP
>
> this is very deep stuff.
>
>
> the axiom of mathematical induction is simply:
>
> IF a property holds for the number 1;
> AND (if a property holds for the number N, then it holds for N+1
too);
> THEN it holds for all whole numbers.
>
> see if you can understand the above and convince your self that
it's
> true. how could it not be true? try to think of a counterexample.
>
>
> assuming this ordinary axiom of mathematical induction is true
> implies the truth of goodstein's theorem (this is an example of
> Godel's Incompleteness Theorem, which you need to understand before
> any of this will make any sense to you).
>
> but the truth of goodstein's theorem cannot be proved by using
> mathematical induction itself (only in the ordinary variety) in the
> proof.
>
> since mathematical induction is so "obviously true", even for
> philosophers of mathematics who shun all talk of infinity, we are
> forced to believe goodstein's theorem just as strongly.
>
> the fact that the truth of goodstein's theorem is true, and so
simple
> to express, makes mathematicians wonder if they can understand
*why*
> it is true.
>
> the explanation as to *why* it is true has to draw on a fancier
kind
> of induction, called "transfinite" induction -- but
the "transfinite"
> numbers can simply be thought of as rates of growth of everywhere-
> finite functions. the explanations *make sense* to many
> mathematicians, but whether they make sense to you or not, the fact
> is that the truth of goodstein's theorem is implied by the truth of
> ordinary mathematical induction, so if you believe in ordinary
> mathematical induction, you *have* to believe that goodstein's
> theorem is true -- explanation or no explanation.

***Thanks, Paul. I'm getting a "glimmer..."

for those interested in philosophy of mathematics

🔗Paul Erlich <paul@...>

🔗John Starrett <jstarret@...>

🔗Joseph Pehrson <jpehrson@...>

🔗Paul Erlich <paul@...>

🔗Joseph Pehrson <jpehrson@...>