# Logic, geometry, and graph theory

[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"?

• 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"?

• 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.

• I think the Pythagoreans would very much agree with your intuition of a deep connection (as I share)...And Clearly, Euclid was an early exemplar of systematic axiomatic methods/constructs. No doubt there's more, but just wanted to note that your sentiments, I suspect, were shared by many of our ancient predecessors... Oct 31, 2012 at 22:58
• @amWhy (= WhoAmI?): My question was not for or about intuitions and sentiments but for specific arguments - even if I should not have been able to make this clear. Oct 31, 2012 at 23:27
• That's why I didn't post my comment as an answer; I am simply sympathetic with your question and interested in viewing answers... Oct 31, 2012 at 23:45
• The connection is nothing so superficial. On the one hand, to every logical theory of a certain type is associated a "classifying topos", which is a geometric object whose "points" correspond to models of the theory, and many properties of the classifying topos correspond to properties of the theory. This is at the heart of the "bridges" technique expounded by Olivia Caramello. On the other hand every topos is a mathematical universe unto itself and gives rise to new interpretations of intuitionistic logic, the so-called "Joyal–Kripke semantics". Nov 1, 2012 at 7:49
• There are other connections between logic and geometry beyond topos theory. For instance, you might take a look at Hrushovski's proof of the Mordell-Lang conjecture, which uses cutting-edge techniques from model theory. (Though I personally find this stuff even more mind boggling.)
– user45071
Nov 1, 2012 at 15:02

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.

• The point of view in your second paragraph is somewhat misleading: the internal logic of the topos of sheaves on a space does not perceive itself as having propositions which are "true" in some places and "false" in others. Rather, if we take the internal point of view, sheaves are sets "smeared out" over the base space; it is very difficult to formulate things like "if such-and-such is true at point $P$, then so-and-so is true at point $Q$" within the internal logic. Nov 12, 2012 at 19:28
• I "discovered" your answer only today, sorry for that. But it hits the nail on the head... Sep 10, 2013 at 14:36
• I think, this is what I had in mind with representation (= information?) Sep 16, 2020 at 6:09

I wonder if this analogy goes into the right direction:

• Logic lies at the heart of set theory. In set theory you may define numbers as equivalence classes of sets wrt "having a bijection", together with a prototypical representative (e.g. $\{\{\}\}$ for all sets with 1 element).

• In geometry you define numbers as equivalence classes of lines wrt "having same length", together with a prototypical representative (e.g. the line $\overline{01}$ for all lines of length 1).

Other "natural" representatives (but being other types of objects) are:

• the circle with center 0 that contains 1

• the point 1 itself.