# Logic and geometry

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?

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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... – amWhy Oct 31 '12 at 22:58

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@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. – Hans Stricker Oct 31 '12 at 23:27

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That's why I didn't post my comment as an answer; I am simply sympathetic with your question and interested in viewing answers... – amWhy Oct 31 '12 at 23:45

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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". – Zhen Lin Nov 1 '12 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.) – Shawn Henry Nov 1 '12 at 15:02
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## 1 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.

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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. – Zhen Lin Nov 12 '12 at 19:28
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