Avril, (01)
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. (02)
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. (03)
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. (04)
In short, I am opposed to *every* proposal that there is or
should be a single ideal foundation for all of mathematics. (05)
> John is a by-stander who does not see the debate very important. (06)
On the contrary, I have a very firm position: (07)
1. All the approaches are interesting as hypotheses, and most
of them are compatible with most practical applications. (08)
2. But *none* of them is a suitable foundation for all
conceivable mathematical structures. (09)
3. There is absolutely *no need* for a single foundation for
all conceivable mathematical structures. (010)
4. Partisan arguments for one approach or another are pointless. (011)
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. (012)
John (013)
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