soft question - The problem of infinity

This question does not have a mathematical answer.

However, different approaches to formalizing the concept of infinity have been considered in philosophy of mathematics for a very long time (since Aristotle, at least), and have resulted in quite a bit of interesting mathematics proper. Most people here are likely familiar with the modern Cantorian concept of infinity (which is what undergirds modern set theory), and so I will describe a somewhat different mathematical conception infinity.

So, one way of interpreting constructive mathematics is by means of "realizability interpretations". Here, we take the view that a proposition is true when it is possible to give evidence for its truth, and then inductively for each proposition we give conditions for evidence:

• $top$ has the string $()$ for its justifying evidence


• $A land B$ is justified when we can give a string $(p, q)$, where $p$ is evidence of $A$ and $q$ is evidence of $B$


• $bot$ has no evidence


• $A vee B$ is justified when we can give a string $(i, p)$, where $i$ is either 0 or 1. If $i$ is 0, then $p$ is evidence of $A$ and if $i$ is 1 it is evidence of $B$


• $A implies B$ is justified by a computer program $c$, if $c$ computes evidence for $B$ as an output whenever it is given evidence for $A$ as an input.


• $forall x. A(x)$ is justified by a computer program $c$ for evidence, if $c$ computes evidence for $A(n)$ as an output whenever it is given a numeral $n$ as an input.

Now, note that in the cases of implication and quantification, we ask for a computer program which is total on its inputs. So, the question arises, how can we tell whether or not a given program is a legitimate realizer for a proposition or not? The Halting Theorem ensures that we cannot accept arbitrary programs and decide after-the-fact whether or not they are evidence. So we must circumscribe what we will accept to some class of total functions.

So we can decide that certain patterns of recursive program definition (for example, primitive recursion) are acceptable forms of looping, which we believe will always lead to halting programs. These patterns correspond to induction principles. Proof-theoretically stronger theories correspond to logics whose realizability interpretation allows more generous recursion schemes as realizers. In this setup, large infinite sets correspond to very strong induction principles. So this gives a way of understanding Cantorian infinities without having to posit the actual existence of sets with huge numbers of elements.

This should illustrate that how we read mathematical statements shapes what ontological commitments we make regarding them, and so you can't answer physical questions only through pure mathematics. That is to say: physicists can't get out of doing experiments. But funny readings of mathematics may help you interpret those experiments!