Re: Cantorian pseudomathematics

Jean-Claude Arbaut wrote:
> cbrown@xxxxxxxxxxxxxxxxx wrote:

<snip>

> > A formalist might claim "the sqrt(2), like all mathematical objects,
> > has no physical reference; it is an abstraction whose 'existence' is
> > merely a statement about the result of certain operations on certain
> > definitions".
>
> I *really* dislike such an approach ! Even if numbers are abstractions,
> there are geometric verifiable facts about them. There exists some kind
> of limiting process to check that the number - the abstraction - sqrt(2)
> is the diagonal of the square - another abstraction. It suffices to
> measure a physical representation of the diagonal of a square.

What limit process, involving physical measurements, would allow us to
independently confirm "the length of the diagonal of the unit square is
not the ratio of two integers?"

>That's
> the basic way of linking real life measure and mathematics, and I'd be
> very happy not to be forced to reject it - or to let it become an
> insignificant model after the abstract construction has been done.
>
> > A more practically minded mathematician might say "the sqrt(2) is the
> > mathematical abstraction of my intuitions about real things - the real
> > numbers, and distances, and so on".
>
> That's the way I like to see numbers ;-)

"Formalist on weekends, Platonist on workdays" is probably the most
common attitude.:-)

>
> > A finitist might say "Bah, the sqrt(2) does not exist; I only want to
> > consider statements about finite rational numbers".
>
> I tried for myself, just to see where it can lead. With that we loose
> much power, and we also loose the intuition of measure, after having
> noticed the diagonal of a square cannot be a rationnal, hence cannot be
> measurable in this theory. Too awkward.
>

I agree; but it has its own attractions as well.

> > However, because of formalism, we can at least all agree on one thing:
> > by sqrt(2), we all mean a number satisfying sqrt(2)*sqrt(2) = 2; and
> > all agree that it follows from this definition that there are no
> > integers p,q such that sqrt(2) = p/q.
>
> > That, to me, is an improvement!
>
> Well, it was. This has been know for around 2500 years !
>

In particular for this thread, it equally allows us all to agree "under
the standard definition, there is no function which is a uniform
distribution on the naturals".

Cheers - Chas

.

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