@jmorrice I never processed this properly at the time so I am boosting you manually:
"Turing is a nightmare for Platonists. The point of an ideal is that it's separate from the shadows on the wall - that it's nature is contained within itself.
Turing shows that in a universal machine, any form of reasoning can emerge through determinate rules..."
We know now that there are infinitely many such machines. Thus, for any ideal you present to me, I can find an equivalent representation of it using different terms.
This means that the uniqueness of an ideal is not given by its singular form, but by the constraints imposed on it by other ideals - i.e. invariants/theorems which hold regardless of the representation chosen.
In this light, the platonist has three unattractive options:
One: formulate a new Cave metaphor which captures the suspicion about the phenomenal world's unreliability while granting the underdetermination of reason: that there is no one true shape, that it's shadows all the way down.
Two: accept that logical distinctness is illusory or at least secondary, since everything is interlinked (by computability). This is really just Pre-Socratic mysticism again.
Three: realism. The shadows are all there is. Ideals are only tools.
Gödel: [Tarski and I both stress] the great importance of the concept of... Turing's computability... this importance is largely due to the fact that, with this concept, one has for the first time succeeded in giving an absolute notion to an interesting epistemological notion, i.e., one not depending on the formalism chosen
..."the wider phenomenon that Gödel's Theorem was pointing to: *number theory is already a universal computer*. Or more precisely: when we ask whether a given equation has an integer solution, that's already equivalent to asking whether an arbitrary computer program halts."