Vtyjkyuk

The problem of ontology is one much discussed in mathematical philosophy with much categorization into different schools of thought, but the problem of epistemology seems to be less discussed; specifically the question not of how we 'know' mathematical objects in the first place, but of how we can be sure that a particular physical 'process' (whether a calculator or mental arithmetic) gives the 'right answer'.

My question arose after asking this about Chaitin's constant $Omega$, a number defined so that its digits represent the solution to an undecidable problem, and hence (assuming the Church-Turing principle or perhaps one slightly stronger, but not as strong as Church-Turing-Deutsch) we can never work out what its digits are (although we can find some). However, it is possible that a physical constant might have value $Omega$, or $Omega$ might even (as this answer put it) be 'engraved on a monolith buried on the moon alongside the axioms of ZFC'. It is standard to argue that if this were the case we could never know (e.g. see here). David Deutsch's response to this here is as follows:

[It is not] obvious a priori that any of the familiar recursive functions is in physical
reality computable. The reason why we find it possible to construct, say, electronic calculators, and
indeed why we can perform mental arithmetic, cannot be found in mathematics or logic. The reason
is that the laws of physics ‘happen to’ permit the existence of physical models for the operations of
arithmetic such as addition, subtraction and multiplication. If they did not, these familiar operations
would be non-computable functions. We might still know of them and invoke them in mathematical
proofs ... but we could not perform them.... Chaitin (1977) has shown how the truth values of
all ‘interesting’ non-Turing decidable propositions of a given formal system might be tabulated very
efficiently in the first few significant digits of a single physical constant.
But if they were, it might be argued, we could never know because we could not check the accuracy
of the ‘table’ provided by Nature. This is a fallacy. The reason why we are confident that the
machines we call calculators do indeed compute the arithmetic functions they claim to compute is not
that we can ‘check’ their answers, for this is ultimately a futile process of comparing one machine
with another: Quis custodiet ipsos custodes? The real reason is that we believe the detailed physical
theory that was used in their design. That theory, including its assertion that the abstract functions of
arithmetic are realized in Nature, is empirical. [bold emphasis mine]

Thus it appears that we have two different philosophical approaches to this problem, either claiming that we could never verify (or hence use) $Omega$'s digits if we thought we had them, or else claiming that an empirical justification would be sufficient; I wonder whether there are other approaches to the problem? Some degree of empiricism seems to be required regardless. Note that this problem is not the same as the general epistemological problem of how we know anything. It is the problem of how, assuming we already know a mathematical object, we can be sure that a physical process corresponds to it. The ubiquity of computational devices (including our brains) in mathematics and the rise of computer-assisted proof seem to make this important.

My question is thus: What are the different mathematical schools of thought with regards to the problem of how we can be sure that physical processes give us the 'right' answer (i.e. correspond to the mathematical objects we think they do)?

I don't know that there's a clear universal definition of knowledge. I think there are some people who have a really good memory that other people trust to have a good memory. Of those, some of them are the type of person who can do mental multiplication of any two numbers of at most a certain length by retaining the results they figured out in the middle of the computation and using them to keep figuring out new statements until they figure out the answer to the question. They could figure out a product mentally first then check whether the calculator gives the same result and then claim that they know the calculator gave the correct answer because they already figured it out, and then other people would agree that it's correct to say that person really knows it.