Dialethism and platonism
'Logic takes care of itself; all we have to do is to look and see how it does it. --Wittgenstein Notebooks 1914-1916
It's no surprise that Wittgenstein reveals himself as a platonist, at least as far as logic is concerned. He's saying that logic is there to be discovered, it is not something invented.
No surprise because Wittgenstein was a logician - but, as he was a philosopher, unusual. Philosophers tend to be sceptical of platonism, not being mathematicians, or logicians, they do not have the gut-feel or instinct, or experiential knowledge that would have them incline towards believing in platonism, so, being sceptical generally, they are, reasonably, sceptical about platonism and usually believe, unlike mathematicians, that maths is invented. ( There's an article on objections to platonism here: http://www.iep.utm.edu/mathplat/ )
Where does dialetheism fit into this?
Most mathematicians, and, I expect, most logicians, are platonists. It comes with the territory - rather literally, because, as working mathematicians, they are familiar with the experience of 'discovering' something unexpected, and they know that they could not have 'invented' it. Things in maths could not be otherwise than as they are.
Some might dispute this. What about the 5th postulate? For a long time it was seen not to fit in, it didn't seem provable as a theorem, but it wasn't obvious enough to be an axiom. Riemann discovered, by assuming the opposite in order to arrive at a proof by contradiction (one particular case where the search for a proof by contradiction was particularly fruitful!), he had a completely consistent geometry. Indeed it proved that there are three geometries possible, one with the fifth postulate as Euclid had it, one in which parallel lines diverge 'at infinity' and one where they converge 'at infinity'. All these can be described in one geometry where the curvature of the space defines how they behave.
So, in the end, the platonic expectation that there was a consistent mathematical realm was satisfied.
Wittgenstein would not have been a dialetheist - or we'd have known. Dialetheism was not completely unknown, but the prevailing wisdom, until only a handful of years ago, was that it, as Aristotle had claimed, was nonsense. Many still are stuck with the Aristotelian view.
Would Plato be happy with dialethea? This isn't quite the same question as whether dialetheism is platonic, but it's an interesting one nevertheless. Heraclitus with 'everything flows, nothing stands still' would have been comfortable with dialetheism, but the sort of certainty that Plato saw in the ideal realm probably would have meant that he'd have rejected it.
A platonist today, though, I think, would accept that, once genuine dialethias have been discovered, as they have, they are then part of the platonic realm - that's the point of the realm, really, it is how things are, and you discover it, it's not up to you to say whether something ought, or ought not to be there, they simply are, or not.... or. could this be relevant to the platonic realm itself? Could there be ideal objects that both are in the platonic realm, and are not?
So you could have a true statement that 'X exists in the natural numbers', and the true statement that 'X does not exist in the natural numbers'. If you could, then that would, indeed, be the end of platonism. You could not have a platonic realm that was fuzzy at the edges like that, it'd not be a realm any more. There would no longer be a metaphysical gap between human minds and the mathematical realm because it'd become a matter of opinion, taste, or invention whether you decided what was going to be in the realm - not that the realm itself stood alone and separate. This 'X' would not be a 'necessary truth'.
Could this then be a scientific test for platonism? That is, could it, as a theory, now have the advantage of being disprovable? Theories that can't be disproved are not really useful theories. All that's needed is a counter-example, something that is not a necessary truth of mathematics.
It's probably worth pointing out that the common argument used against platonists, the Löwenheim-Skolem Theorem, relies upon the axiom of choice, and the axiom of choice implies the law of the excluded middle, rendering, for the dialetheist, the argument void - until, of course, there's an alternative proof for the Löwenheim-Skolem Theorem that doesn't rely on the axiom of choice. Dialetheism and the axiom of choice