[FOM] predicative foundations

[Aatu Koskensilta]

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