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Great talk and ideas. But the idea of representing mathematical equations in the physical form is like supporting the same medium of symbols and just representi

Great talk and ideas. But the idea of representing mathematical equations in the physical form is like supporting the same medium of symbols and just representing it in a different way. But fundamentally math was invented using pen and paper, so the centuries of mathematical thoughts were already slaves to the 2D medium. Maybe it would be interesting to think about, what would happen if we take away 2D surfaces and ask people to reinvent math in a medium that Bret is talking about. How would someone go about discovering, solving, proving mathematical systems without pen and paper? How would you represent 4, 5 or 20 multi-dimentional spaces in a 3D environment ? Maybe pen&paper was kind of abstracting out limitations of physical world so that mind can explore even further.

orbifold 12y ago
The progress in n-category theory is largely about that. Instead of just studying objects, operations and equations, like "a group is a set g, together with multiplication + certain axioms" one studies objects, relations between objects, relations between relations etc. To study this up to level two, you visualize objects as regions in 2d space, relations between objects as lines between them and relations between relations as points where lines meet. Equations are then equalities between such pictures. If you look at those equalities as a dynamic process, which starts at time t=0 with one picture and time t=1 with the other, you get a picture in three dimensional space, which you can visualize as a movie (to be mathematically precise after the choice of a morse function you get a movie). One of the easiest examples would be the zig-zag identity. See for example https://golem.ph.utexas.edu/category/2006/10/classical_vs_quantum_computati_4.html https://golem.ph.utexas.edu/category/2006/10/classical_vs_qu... and scroll down. One person who has completely gone overboard with that is Jacob Lurie, he has practically single handedly written down a foundational work of higher algebra and algebraic topology, which essentially precisely corresponds to what you are asking. How does mathematics look like if we weren't constrained to 2d paper to write down equations. Of course to actually write down those things you have to somehow project the equations to several essentially 1d sections or invoke very abstract methods to reason about them. Of course algebraic topology already has tools for that. The early impetus came from Grothendieck I believe, who began to create a theory of "higher groupoids" in several thousand pages, ordinary groupoids being categories in which every morphism is an isomorphism.

orbifold 12y ago

The progress in n-category theory is largely about that. Instead of just studying objects, operations and equations, like "a group is a set g, together with multiplication + certain axioms" one studies objects, relations between objects, relations between relations etc. To study this up to level two, you visualize objects as regions in 2d space, relations between objects as lines between them and relations between relations as points where lines meet. Equations are then equalities between such pictures. If you look at those equalities as a dynamic process, which starts at time t=0 with one picture and time t=1 with the other, you get a picture in three dimensional space, which you can visualize as a movie (to be mathematically precise after the choice of a morse function you get a movie). One of the easiest examples would be the zig-zag identity. See for example https://golem.ph.utexas.edu/category/2006/10/classical_vs_quantum_computati_4.html https://golem.ph.utexas.edu/category/2006/10/classical_vs_qu... and scroll down. One person who has completely gone overboard with that is Jacob Lurie, he has practically single handedly written down a foundational work of higher algebra and algebraic topology, which essentially precisely corresponds to what you are asking. How does mathematics look like if we weren't constrained to 2d paper to write down equations. Of course to actually write down those things you have to somehow project the equations to several essentially 1d sections or invoke very abstract methods to reason about them. Of course algebraic topology already has tools for that. The early impetus came from Grothendieck I believe, who began to create a theory of "higher groupoids" in several thousand pages, ordinary groupoids being categories in which every morphism is an isomorphism.