[FOM] Predicativism and natural numbers
Giovanni Lagnese
lagnese at ngi.it
Mon Jan 9 01:22:55 EST 2006
Nik Weaver wrote:
> I start with the idea that words like "set" and "collection"
> have no literal referents. Actually, it surprises me that
> this is not taken for granted by modern philosophers.
> Ordinary language assertions involving such words
> can always be rephrased in equivalent ways which
> do not use such words.
I think that no word has "literal referents", because in the natural world
there are no objects.
Objects are our categories that we use to modeling the natural world.
Similarly, sets are our categories that we use to modeling the natural
world.
I think that this does not exclude a conceptualistic view about foundation
of mathematics.
In my opinion, the point is not giving sense to words, but giving sense to
statements.
The point is not identifying a structure which plays the role of the set of
natural numbers, but giving sense to quantification in the set of natural
numbers.
Similarly, the point is not identifying a structure which plays the role of
the powerset of natural numbers, but giving sense to quantification in the
powerset of natural numbers.
I think we can give sense to quantification in the powerset of natural
numbers by assuming something like Brouwer's continuity principle.
GL
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