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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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    And for a very simple instance where logic touches on geometry, check out the Ax-Kochen theorem.2012-11-01

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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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    I "discovered" your answer only today, sorry for that. But it hits the nail on the head...2013-09-10
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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.