John, (01)
I agree with all you said below: mathematics does not
need foundations. Finitism, as I see it, is not a
foundationalist approach. It only aims to prune off
transfinity from mathematics. (02)
Finitism = mathematics minus transfinity (03)
If there were no transfinity, there would be no reason
for finitism either. (04)
> 5. Adopting one or another as the *only* approach would be
> a disaster because it would rule out mathematics that might
> become very important for some future applications. (05)
I believe that pruning off transfinitism will not lead
into a disaster of any sort, because there is no real need for
any constructions above the finite, or potential infinite
in any case, that can in one sense be reduced to finitism. (06)
That is not to say that a lot of very important things have
been reached by holding transfinitist foundations. It is just
that these important things could be reached without the
transfinitist foundations as well, at least the part that
has a real application. And probably much more, since
finitism is a simpler mathematics, constructivist, and more
practical in nature. There is no need for an extra layer of
foundations, or anything above common sense. (07)
What -ism has been critisized in analytical philosophy more than
transfinitism? Perhaps idealism, or nominalism, hard to say. (08)
Why hold on to it, when it is unnecessary, and simultaneously
perhaps the most critisized thing in the history? (09)
Avril (010)
Quoting "John F. Sowa" <sowa@xxxxxxxxxxx>: (011)
> Avril,
>
> As I said in my response to Rob (and in many, many notes to
> this forum and others), I do not believe that there is any
> one-size-fits-all ontology that is adequate for all possible
> problems and perspectives.
>
> My only complaint about Cantor's theories is that many
> mathematicians and logicians consider them the preferred
> foundation for all of mathematics. But the mathematicians
> who prefer category theory believe that category theory
> is more fundamental than set theory. I agree with them.
>
> But I have even more sympathy with those mathematicians who
> believe that there is no reason to have *any* universal
> foundation for all of mathematics. In other words, there is
> no reason why there should be *one* single set of axioms in
> terms of which all other mathematical structures are defined.
>
> In short, I am opposed to *every* proposal that there is or
> should be a single ideal foundation for all of mathematics.
>
> > John is a by-stander who does not see the debate very important.
>
> On the contrary, I have a very firm position:
>
> 1. All the approaches are interesting as hypotheses, and most
> of them are compatible with most practical applications.
>
> 2. But *none* of them is a suitable foundation for all
> conceivable mathematical structures.
>
> 3. There is absolutely *no need* for a single foundation for
> all conceivable mathematical structures.
>
> 4. Partisan arguments for one approach or another are pointless.
>
> 5. Adopting one or another as the *only* approach would be
> a disaster because it would rule out mathematics that might
> become very important for some future applications.
>
> John
>
>
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