I’m old-school and still believe there’s some fact of the matter about what the multiverse is.
I enjoyed this sentence.
If we found a better mathematical way to capture this, presumably your limitations would expand and you would then care about more than you do now.
Only to a point! Suppose Coscott has a policy for accepting new math which applies a critical eye and only accepts things based on a consistent criteria. That is: if he would accept X upon inspection, then, he would not have accepted not-X via his inspection.
Then consider the “completion” of Coscott’s views; that is, the (presumably uncomputable) system which is Coscott in the limit of being taught arbitrarily many new math techniques.
Now apply Tarski’s Undefinability. We can construct a more powerful math which Coscott could never accept.
Therefore, if Coscott is a sufficiently careful mathematician, then his math powers seem to have ultimate limits already, rather than mere current limits.
On the other hand, if there is no such limit, because Coscott could accept either X or not-X for some X depending on which is presented to him first, then there is hope! A “stronger” teacher could show Coscott the way to the more powerful math. Yet, Coscott is also at risk of being misled.
Here’s a riddle: according to Coscott, is there any way Coscott could be mislead? Coscott has established that physical existence is meaningless to him; but is there mathematical truth beyond provability? Is there a branch of the multiverse in which the axiom of choice is true, and one where it is false? (It seems clear that there is a subset of the universe where the axiom of choice is true, and one where it is false, but I don’t think that’s what I mean...)
I enjoyed this sentence.
Only to a point! Suppose Coscott has a policy for accepting new math which applies a critical eye and only accepts things based on a consistent criteria. That is: if he would accept X upon inspection, then, he would not have accepted not-X via his inspection.
Then consider the “completion” of Coscott’s views; that is, the (presumably uncomputable) system which is Coscott in the limit of being taught arbitrarily many new math techniques.
Now apply Tarski’s Undefinability. We can construct a more powerful math which Coscott could never accept.
Therefore, if Coscott is a sufficiently careful mathematician, then his math powers seem to have ultimate limits already, rather than mere current limits.
On the other hand, if there is no such limit, because Coscott could accept either X or not-X for some X depending on which is presented to him first, then there is hope! A “stronger” teacher could show Coscott the way to the more powerful math. Yet, Coscott is also at risk of being misled.
Here’s a riddle: according to Coscott, is there any way Coscott could be mislead? Coscott has established that physical existence is meaningless to him; but is there mathematical truth beyond provability? Is there a branch of the multiverse in which the axiom of choice is true, and one where it is false? (It seems clear that there is a subset of the universe where the axiom of choice is true, and one where it is false, but I don’t think that’s what I mean...)