[numpy-thagoras]

Read Wildberger if you want to know what he thinks.

I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation.

The whole point is that as a theory for the foundations of mathematics, you do not need to assume numbers with infinitely long decimal expansions in order to do math.

[birn559]

> I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation.

Could you elaborate? What is the output of that function if not an entity in it's own? Having studied math with philosophiy minor long time ago I am curious.

[numpy-thagoras]

It's part of a dependency relation, the function computes and produces an output that we call sqrt(2).

On the other hand, using the axioms of ZFC, one can say any real number exists without having a function to compute it, or a proof to construct it.

For an ultrafinitist, or any finitist for that matter, we say that you only need the minimum of ingredients to produce math -- you do not need to assume anything over and above that, as it's not even helpful in the verification process.

So assuming only finitely many symbols and finitely many numbers, I can produce what we call sqrt(2). We only ever verify it numerically and finitely anyways. We can never reach decimals at infinite ordinals.

So it makes no sense to say, "Hey I assume transfinitely many entities, and my assumption says these numbers exist even though the proofs and decimal expansions are only ever finite."