[FOM] predicative foundations
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Fri Feb 17 02:42:19 EST 2006
On Feb 17, 2006, at 12:52 AM, Eray Ozkural wrote:
> As for your comments about "possible marks" I have difficulty in
> understanding what you mean. Any concept that we use could be
> interpreted in that way, for instance "chair" could be rendered
> meaningless by your argument because we are talking about "possible
> chairs" all the time (by such simple phrases as "a chair").
There is no reason to think that talk about possible chairs renders the
word "chair" meaningless, and anyhow usually by possible chair one
means a physically possible chair. Physically possible marks can't
serve as a justification of N, only possible marks that are, in effect,
just assumed, posited or intuited to have essentially the properties of
the natural numbers. If someone harbors doubts about N there is
absolutely no reason to think that "marks on paper" type talk will take
them away.
Of course, there's nothing in the least doubtful about N in the first
place and the point of my remark was simply that Weaver's description
of N in a "metaphysically uncontroversial way" is just an illusion:
there are perfectly coherent ways of raising controversy about Weaver's
way of describing N and it is very difficult to imagine anyone doubtful
about Weaver's form of predicativism being convinced by reference to
"marks on paper".
It is humanly impossible to create arbitrarily large marks or survey
arbitrarily large finite sets of natural numbers. By abstracting away
limitations of time et cetera we obtain an interesting picture on basis
of which we can justify various sorts of mathematical principles of
e.g. predicativism or intuitionism. It is also humanly impossible to
survey infinite sets of natural numbers, but by abstracting away this
limitation we obtain yet another picture which might motivate or
justify other mathematical principles. The latter principles are in a
sense "more impossible" in the sense that the latter principles are
stronger than the former, but it is difficult for me to understand why
this should have any epistemological significance to us humans. Of
course one is always free to cry out that some abstraction or picture
is "incoherent" or has no "philosophical justification" but currently
whether or not, or when, one chooses to do so seems to be a matter of
personal preference.
All this is just belaboring in a tediously detailed fashion one of
Harvey's points, namely that predicativism is not special in any
immediatedly clear sense among the myriard possible motivations for
various "stopping places"; one can coherently doubt predicativism as
usually understood or in Weaver's form, and one can coherently form
equally compelling or uncompelling pictures and justifications for
stronger system.
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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