Comment by chriswarbo

> If there is no way in our universe to ever know exactly what BB(111) is, can it really be considered a well-defined number?

There's a branch of Maths called constructivism which requires values/proofs/etc. to be "constructable" in principle. For example, the law of the excluded middle ('for all X, either X is true or (not X) is true') is not constructive, since it doesn't give us a value: we don't know whether we're dealing with X or (not X).

http://en.wikipedia.org/wiki/Constructivism_(mathematics)

Constructive mathematics turns out to be very closely related to computation, and is one reason why functional programming gets so much research attention.

Constructivists don't have a problem with infinite objects, like the set of all Natural numbers, if they can be represented in a finite way (eg. as a function). There's another branch of Mathematics knows as finitism, which regards infinities as not existing. What you're describing is ultrafinitism, which regards really big things as not existing: http://en.wikipedia.org/wiki/Ultrafinitism