It occurred to me this morning that my criticism of the journalism may rest on a platonist assumption. What if the journalist were some sort of intuitionist (Brouwer, et al.) who held that the existence of a number is identifiable with its being constructed or generated?
That's an interesting question. I'm not completely sure, but I suspect even intuitionists would agree that there is no largest prime number. The result does not rely on proof by contradiction or other dubious platonic methods. We have a constructive algorithm that, given any prime number n, will generate a larger yet prime. Consider m:= n! + 1. Either m is prime, or can be factorised into prime factors larger than n.
I assume that constructivists only require an algorithm we know to be in principle computational. (I hope the above is indeed computational in principle. Expanding a factorial surely is - that's just repeated multiplication. But how about checking for prime factorizations? I suppose it should be, as a computer could just do an exhaustive series of divisions by x [for all x: n < x < m] to check that m/x never yields a whole number?) The mere fact that we cannot yet actually compute it, e.g. due to hardware limitations, might not be a problem. Does anyone know for sure: do intuitionists (constructivists) require only that we demonstrate something to be constructable, or must it actually be constructed?