Though I've been reading for years, this is my first question here. Believe it or not, I've tried the search feature- apologies if this is a duplicate. The main point of this post can be summarized as:
Does there exist theory that illuminates the intuition that certain paradoxes in logic and geometry are equivalent (or isomorphic or the strongest related term that you find appropriate)? And when I say illuminate I mean, provides a formal structure to explore this (non)relationship in the usual theorem-proof way.
For example, consider some form of the Liar paradox and the Möbius strip. A quick Googling shows that associating the two is not an original thought. If not obviously motivated, thinking of the logical statement as a "surface" and its truth value as its normal bundle also doesn't seem completely foolish.
Similarly, the ideas of self-reference in logic and self-intersection of the Möbius strip immersed in $\mathbf{R}^2$ or $\mathbf{R}P^2 \# \mathbf{R}P^2$ in $\mathbf{R}^3$ might be thought of as "dimensional" issues. Russell's paradox makes clear how self-referential statements in one level of abstraction really should be statements in another, higher level of abstraction (class of all sets, large category of small categories), and of course the Möbius strip and Klein bottle can be embedded in three and four real dimensions respectively.
This is all fun to think about, but again is completely associative, rather than deductive. What I hope for is an appropriate vocabulary to express even the most basic questions such as:
- Is there a way to give a topology to generic logical statements?
- Every non-orientable surface contains a Möbius strip, does every logical paradox of this form contain the Liar Paradox?
A good example of sources I've found is:
http://homepages.math.uic.edu/~kauffman/SelfRefRecurForm.pdf
Very interesting, but not satisfying mathematically.