[FOM] predicative foundations

[Eray Ozkural]

[in which I point out a philosophical problem]

On 2/16/06, Aatu Koskensilta <aatu.koskensilta at xortec.fi> wrote:
> But these "marks on paper" can't be actual marks on paper, but must be
> something like "possible" marks on paper. The justification for N in these terms
> is circular, because we have to assume that something like "possible marks on paper"
> make sense, which is just as problematic or unproblematic as the natural numbers
> themselves.

I beg to differ. Remembering Turing's original paper in which he described
a computer and general computability in a rigorous fashion, he used the
notion of marks on the paper. There is nothing problematic
about the notion of marks. Indeed, problems arise once you start
interpreting marks as referring to entities that do not exist. Thus, if we
would carefully review what exists when a person is performing
arithmetic, we would find that there are marks in his brain (as
electrochemical events, etc.) and/or on the paper. There is nothing
else that is physically happening.

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").

The real difficulty of course lies in your realist assumption that
a number is a unique and universal entity that has independent
existence from any and all mathematicians. Thus you would tend
to think that using the concept of marks, we would have to think
of a number x as being identical to "all possible representations of
x". I do not believe that this philosophical approach to mathematics
is much progress over Frege. (And one could even argue that such
a "set" of representations is ill-defined)

To sum up, your argument consists in saying that, if we assume
a strong form of mathematical realism, marks do not help us much,
with which I wholeheartedly agree.

Regards,

--
Eray Ozkural (exa), PhD candidate.  Comp. Sci. Dept., Bilkent University, Ankara
http://www.cs.bilkent.edu.tr/~erayo  Malfunct: http://www.malfunct.com
ai-philosophy: http://groups.yahoo.com/group/ai-philosophy
Pardus: www.uludag.org.tr   KDE Project: http://www.kde.org