[FOM] Concerning Ultrafinitism.

Eray Ozkural examachine at gmail.com
Wed Nov 1 17:26:26 EST 2006

On 11/1/06, Bill Taylor <W.Taylor at math.canterbury.ac.nz> wrote:
> So without necessarily making any approbation or disapprobation of either,
> is it fair to regard ultrafinitism as "fuzzy mathematical logic"?

I would be grateful if you would consider my inexpert opinion.

>From what I have read, my impression on finitism has been
that it arises from philosophical preferences.

However, your conclusion is also philosophically motivated.
In particular, you seem to assume the truth of (a particular
form of) realism, and thus you seem to think that there is
something that is _really_ false about ultrafinitism. I beg to
differ. I think we can sort of enjoy a flexible view in which
different philosophical foundations are considered as
mathematics proper.

If you would ask my own view instead, I would say that both
assertions are equally imaginery (I don't say false):
1) there are an infinite number of integers
2) there is a largest integer

Because, in my own view, integers do not exist in any
real sense of the word. On the other hand, both seem
to be equally valid from a flexible philosophical point of
view (as above).

I can also offer a third interpretation, which is computationalism,
this would seem to say something like, a non-halting computation can
generate an ordered list of integers, however, there is a largest
integer that can be generated in any finite *physical* universe.
I take this point of view to be equivalent to what Godel himself
dubbed as Aristotelian Realism. Unfortunately, I haven't heard
any other author use the same phrase.

As an additional claim, I will suggest that there is no
mathematical way of deciding the truth of your claim.


PS: Many cosmologists seem to think that our universe is finite
indeed, therefore the considerations of ultrafinitists actually make
 physical sense, even if they do not designate the only admissible
metaphysics (or mathematics). (I do recognize that
physics itself is beyond the scope of FOM list)

Eray Ozkural, PhD candidate.  Comp. Sci. Dept., Bilkent University, Ankara

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