[Seven years later, I made an edit to this question, see below.]
By delving into topos theory and sheaves one will eventually discover a "deep connection" between logic and geometry, two fields, which are superficially rather unrelated.
But what if I have not the abilities or capacities of delving deeper into topos theory and sheaves? Does the deep connection between logic and geometry have to remain a mistery for me forever?
At which level of abstraction and sophistication can this connection be recognized for the first time?
And which seemingly superficial analogies have really to do with this "deep connection"?
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What's rather easy to grasp is that there is (i) an algebra of logic and (ii) an algebra of geometry. But is this at the heart of the "deep connection"?
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What comes to my mind is, that both logic (the realm of linguistic representations) and geometry (the realm of graphical representations) have to do with – representations. Is this of any relevance?
Edit:
I also wonder if and how graph theory can be related to – or serve as a connection between – logic and geometry, the vertices of graphs representing objects (in the sense of logic), resp. points (in the sense of geometry), the edges representing sentences, resp. line segments.
If there was such a "deep connection" of logic, geometry, and graph theory, the existence and importance of planar graphs might appear in a new light.
Furthermore, I have found this:
"Roughly speaking, category theory is graph theory with additional
structure to represent composition" is a good summary of the
connection between [graph theory and category
theory].Source
So the two possible ways to relate logic and geometry (via categories/toposes/sheaves, resp. graph theory) are related themselves.
Best Answer
Both logic and geometry deal with information. Logic deals with information about the truth of statements, and geometry deals with information about location. Grothendieck toposes connect logic and geometry along this line.
The simplest case it that of the topos of sheaves over a topological space: here the truth value of any proposition is an open subset of the topological space. Thus information on why a proposition is valid is connected to information on where you are in a topological space.