On standard naturalist views, neither the objective facts of mathematics and morality nor their grounds (e.g., Platonic entities, etc.) have any influence on how matter behaves and hence on how we think. This seems to imply that if our mathematical or moral beliefs happen to be true, that's just a coincidence. But merely coincidentally true belief isn't knowledge (maybe it's Gettiered knowledge). Now consider a response on which:
- Of biological necessity, we have evolved through unguided natural selection to have mathematical or moral beliefs of type N.
- Of metaphysical necessity, most mathematical or moral beliefs of type N are true.
- Therefore, there is no coincidence here and nothing that calls out for further explanation.
But there can. There are two real numbers, x and y, between 0 and 1 with the following property. If you write them out in binary, divide up the bits into groups of eight, and then put the bits into ASCII code, then you actually find a lot of comprehensible text in each. In particular:
- In x, there are infinitely many occurrences of "Consider the following proposition:", and each of them is followed by a well-formed arithmetical sentence (say, written in TeX) and a period. In fact, all possible arithmetical sentence thus occur in x.
- In y, at exactly the same point as each "Consider the following proposition:" string occurs, there instead occurs "That's true" or "That's false."
- Moreover, "That's true" occurs in y precisely when the proposition given in that place in x is true, and "That's false" occurs when the proposition is false.
Now, if we're given two such numbers x and y, the above is an apparent coincidence that calls out for explanation. And maybe an explanation can be given, say in terms of a selection effect: Perhaps the reason we're considering these two numbers is because a logically omniscient being exhibited them to us, and the being chose the two numbers for these remarkable properties. No surprise then!
But what if turned out that x=π and y=e satisfy (4)-(6)? Then we would consider the above coincidence truly remarkable. We would search for some deep mathematical reason for it. But suppose this search fizzled out and we came to conclude that although, of course, it is necessary that π and e have the properties (4)-(6), e.g., it being necessary that Fermat's Last Theorem occur at location 12848994949494888 in π (I assume it doesn't) and "That's true" in e at the same location and so on, mathematically this is just an incredibly unlikely coincidence. That would be a highly intellectually unsatisfying position. So unsatisfying that we would reach for a metaphysical explanation like Descartes' story about God having designed mathematics or a science fictional one like Carl Sagan's novel about aliens having embedded a message in π. We would have good reason to accept such an explanation if it were offered, and if we rejected there being such an explanation, we would have to say we have just a coincidence.
Thus, we can imagine cases of agreement between necessary mathematical facts which genuinely call out for explanation. And we can imagine concluding that although they call out for explanation, there is none, and hence we have a coincidence. Thus we can imagine a coincidence in the realm of necessary truth.