Descarte has been lauded for putting together geometry and algebra, and his achievement allowed the invention of calculus by Leibniz & Newton and allowed its efficacious and explosive development by subsequent mathematicians & physicists in contrast to the rudimentary and primitive steps taken by Archimedes in integration and the Keralan school in power series.

Now various propositional Logic can be algebraised:

classical propositional logic -> boolean algebras

intuitionistic propositional logic -> heyting algebra

modal logic -> modal algebra

The question: is there a significant geometric form of these logics? Significant, simply in not being just a translation into geometric form, as in Venn Diagrams for boolean algebras (first being represented as some system of sets), but that allows for something deeper to be said about logic itself?

  • Would you consider topos theory geometric? What about the fact that open sets of a space are a Heyting? Sep 21 '13 at 19:59
  • yes, I would. Its what inspired this question. I'm really looking for alternative geometric characterisatons, or within topoi themselves a geometric significant way of talking about logic. Sep 21 '13 at 20:06
  • Interesting question. You might want to ask this in math.se also (if you do, please share the link). I'm looking forward to the answers here as well as there. Sep 21 '13 at 20:43
  • @Rostomyan: I've asked the question on math.se following your suggestion. Sep 26 '13 at 10:13

The question: is there a significant geometric form of these logics?

Here, "these logics" refers to Boolean algebras, Heyting algebras and modal algebra. The various representation theorems for these algebras as set-algebras related to certain topological spaces seem to provide a positive answer to this question. These representation theorems are non-trivial, since they are equivalent to the Boolean prime ideal theorem, which cannot be derived from the axioms of Zermelo–Fraenkel set theory without the axiom of choice.

It is well known that every Boolean algebra can be represented as the set-algebra of clopen sets of its associated Stone space.

There are multiple closely related representation theorems for Heyting algebras. First note that a Heyting algebra is bounded and distributive as a lattice, and every bounded distributive lattice can be represented as the set-algebra of clopen upper sets of its associated Priestley space. In case of a Heyting algebra, this associated Priestley space is an Esakai space, which is a Priestley space for which the downward closure of each clopen set is clopen. There are also representation theorems for bounded distributive lattices and Heyting algebras using pairwise Stone spaces instead of Priestley spaces, and representation theorems using spectral spaces.

The wikipedia article on modal algebras says

Stone's representation theorem can be generalized to the Jónsson–Tarski duality, which ensures that each modal algebra can be represented as the algebra of admissible sets in a modal general frame.

I haven't tried to understand this in detail, but some wikipedia authors apparently believe that this is a geometric representation:

The general frame semantics combines the main virtues of Kripke semantics and algebraic semantics: it shares the transparent geometrical insight of the former, and robust completeness of the latter.

  • I wanted to add to my previous comment that I'm especially looking forward to a response from Thomas Klimpel, but didn't want to put you on the spot. Thanks. Sep 21 '13 at 23:17
  • +1 for a nice answer. I am a little vague on frames though - I know of two kinds of frames: a Kripke frame which is a set with an accessibility relation and which acts as a model for modal logic, and a frame as the opposite of a locale (which is a 'pointless' generalisation of a topological space). I've not associated the two together before, are you implying that they are in fact connected? Sep 22 '13 at 21:18
  • @MoziburUllah You're right, I confused Kripke frames with frames from pointless topology here. The Jónsson–Tarski duality refers to Kripke frames with an additional structure, and has nothing to do with pointless topology at all. So there probably is a nice representation theorem related to topological spaces also for model algebra. Sep 22 '13 at 21:43
  • @Klimpel: I thought that might have been the case, but I hesitated to say since the Jonsson-Tarski duality suggests by analogy to Stone duality that there may be a way of interpreting kripke frames (with additional structure) in some generalised topological way. Sep 22 '13 at 23:00

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