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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)?

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  • $\begingroup$ I have used the soft question tag just in case; feel free to delete it if you think it doesn't apply. $\endgroup$ – Anon Jan 31 '17 at 6:30
  • $\begingroup$ Also, I thought this was a reasonable question for MSE rather than specifically for philosophy.SE since it seems to apply very specifically to mathematics. Just let me know if you disagree. $\endgroup$ – Anon Jan 31 '17 at 6:31
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    $\begingroup$ For what it's worth, I think this question should be asked here because it does pertain to mathematics, but also because it is my experience that philosophy.SE does not have the broad areas of knowledge applicable to the subject. But I think it is on topic - this site has hosted questions about Hegel as it pertains to category theory in the past, and certainly this is just as topical as that is. $\endgroup$ – Alfred Yerger Jan 31 '17 at 6:40
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    $\begingroup$ The question is "unsolved" at least from Plato's time. See at least Indispensability Arguments in the Philosophy of Mathematics and Naturalism in the Philosophy of Mathematics. $\endgroup$ – Mauro ALLEGRANZA Jan 31 '17 at 7:15
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    $\begingroup$ @MauroALLEGRANZA Thank you for the links. Note that the problem being unsolved only puts it in a class with most philosophical problems. Nevertheless, usually on such questions there is a division of opinion into schools of thought (e.g. positions on the problem of mathematical ontology being divided into intuitionists, formalists, realists etc.); it was the schools of thought on this problem (since there are clearly more than one) that I was specifically asking for. $\endgroup$ – Anon Feb 5 '17 at 21:42
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I don't know that there's a clear universal definition of knowledge. I could define knowledge in such a way that somebody knows a mathematical statement is true after they mentally figure out that it's true. The problem with that definition is that people sometimes make careless mistakes in a mental computation. I think it's still possible to define knowledge in such a way that those with a different type of brain who never make careless mistakes really know that they figured out the right answer and other people just don't know that that person knows it. 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.

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  • $\begingroup$ This question didn't even have an answer before I wrote this one. How can I make my answer better. Do you think maybe it can't be answered well so it should be closed to get fixed up? $\endgroup$ – Timothy Aug 19 '18 at 4:38
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    $\begingroup$ Thank you for the answer; nevertheless I don't think it really addresses the problem I was presenting. Regardless of whether there is a person $x$ who can perform mental arithmetic and get the same answer as a calculator, this doesn't solve the epistemological problem of how we can be sure that the 2 physical processes (calculator and $x$'s mental arithmetic) correspond to the mathematical object they are attempting to emulate... $\endgroup$ – Anon Aug 21 '18 at 22:56
  • $\begingroup$ ...I was asking for the different schools of thought on this problem; examples include Deutsch's empirical position and the stricter argument of impossibility articulated in my question. Arguing along the lines of the latter approach one might ask how we can trust that the calculator (still less the melee of $x$'s brain) indeed corresponds to our desired mathematical object; in this approach it is little help to bring in my own mental arithmetic or intuition since it arises out of the still more hidden environment of my brain... $\endgroup$ – Anon Aug 21 '18 at 22:56
  • $\begingroup$ ...Deutsch might argue on the other hand that it is adequate to pragmatically trust my own intuitions that the calculator has been designed appropriately and use a 'good enough' empirical justification (this is probably closer to what most mathematicians do in practice)... $\endgroup$ – Anon Aug 21 '18 at 22:57
  • $\begingroup$ ...Another approach might be to argue probabilistically that it is not sufficient to justify my own intuitions, but that in the agreement of several proof-checking mathematicians' brains sufficient certainty is found (which is still open to the possibility that all mathematicians have systematically deviant brains). I am not sanguine enough to ask for a definite answer, rather I am asking for the different schools of thought on this problem. $\endgroup$ – Anon Aug 21 '18 at 22:57

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