[FOM] Predicativity and Certainty

[Eray Ozkural]

On 2/7/06, Arnon Avron <aa at tau.ac.il> wrote:
> For *me* (again, I am not entitled to speak in the name of others),
> "absolutely certain"
> is indeed the "definition" of "predicative". This does not mean
> identifying it with any current brand of predicative mathematics.

Thank you for the explanations. However, I anticipate that your
definition will raise further questions. You mentioned that a concept
of a natural number is absolutely certain while the concept of
a real number is problematic, and is thus not certain. (I say
"concept of", because there is no evidence that these numbers
have independent existence as the respectable Goedel may
have thought.)

I have not been able to fully understand what you mean by certainty.
Though, I would have thought that you mean a special kind of
conceivability. Both natural numbers and real numbers are
conceivable in one sense, since I can answer questions about
properties of natural and real numbers. On the other hand, while
I can fully represent a natural number in my mind (that is small
enough), I cannot represent some real numbers at all, because they
are not representible. In this sense, the set of real numbers is not
conceivable.

Alternatively, we may be tempted to say that certainty requires
universal agreement by all mathematicians. Via this approach, we
may say that there is no alternative to the definition of a natural
number. On the other hand, alternative definitions of R seem to
exist (for instance we might elect to use the set of computable
reals instead of the whole set of reals), thus R is uncertain.

Could you please elucidate if any of my interpretations of certainty
is close to your thoughts?

Regards,

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