the logic of the infinite

[Armin Rigo]

Hello Jeremy,

Let me give a few words about your quotations about the infinite. I am a
mathematician myself, oriented on the logic side. I am not too fluent in
English, too, so please forgive me for my very inaccurate language.

I must admit that your quotations are of the kind of text that I
personally consider as dangerously confusing. Let me explain. Far from
me the idea of contesting the pertinence or validity of the
philosophical grounds of the arguments you quote. What I am afraid of is
that this might be subject to "playing on the words". The word
"infinite" has a quite precise and mathematical definition. In your
quotations however I see the word "Infinite" used in a quite different
sense (and with a capital I, accordingly): it is closer to the notion of
"what cannot be encompassed", or "ultimate model of the real world"; =

some would say "God". The "logic of the Infinite" is then the way to
express things about this Infinite. With this definition I am ready to
consider philosophical arguments of the kind you give. I would just make
sure that you are aware that the mathematical infinite (which can e.g.
be defined as "more than n, for any integer number n that you can speak
out") is much more modest. There are various sizes of infinites, and we
can compare them, encompass them in models, and so on. There are of
course an infinite number of computer-science-related structures that
can be considered, or programs that can be written in a given language,
or whatever. This is in my opinion just the mathematical infinite (and
in most cases even just the denumerable infinite, the smallest one).
Computers are already good at manipulating *some* such infinite families
-- any compiler can theoretically handle an infinite number of input
programs; and handling arbitrary-sized integer numbers is easy. This
said, having a more general way to handle mathematically infinite
families would be nice, and Arrow is aiming at that if I am not
mistaken.

What I'd like to point out here is that confusing words in different
contexts just because they are the same sequence of letters is not only
misleading but completely off-target in my opinion. This was the
favourite sport of Lacan and I fully reject its pertinence. Now this is
probably not what you are trying to do here. Please do not feel attacked
for this. I might have misunderstood your point here. Your intend might
not have been to make a parallel between infinite and Infinite. Please
also do not consider this e-mail as an attack from a computer scientist
to a non-technically-oriented philosopher :-)  I am personally very
interested in subjects like epistemology, and as my very answering might
show I certainly do not disregard the interest of this kind of
discussion! And finally I have to apologise for the tone (just wanted to
make sure we are not just playing on words) -- you warned it might be
"too unconnected to the hard facts of computer science for some folks"
and a half in me is that folk. I promise I'll let the other half talk
too :-)


A bient=F4t,

Armin.