I think “deep fundamental theory” is deeper than just “powerful abstraction that is useful in a lot of domains”.
Part of what makes a Deep Fundamental Theory deeper is that it is inevitably relevant for anything existing in a certain way. For example, Ramón y Cajal (discoverer of the neuronal structure of brains) wrote:
Before the correction of the law of polarization, we have thought in vain about the usefulness of the referred facts. Thus, the early emergence of the axon, or the displacement of the soma, appeared to us as unfavorable arrangements acting against the conduction velocity, or the convenient separation of cellulipetal and cellulifugal impulses in each neuron. But as soon as we ruled out the requirement of the passage of the nerve impulse through the soma, everything became clear; because we realized that the referred displacements were morphologic adaptations ruled by the laws of economy of time, space and matter. These laws of economy must be considered as the teleological causes that preceded the variations in the position of the soma and the emergence of the axon. They are so general and evident that, if carefully considered, they impose themselves with great force on the intellect, and once becoming accepted, they are firm bases for the theory of axipetal polarization.
At first, I was surprised to see that the structure of physical space gave the fundamental principles in neuroscience too! But then I realized I shouldn’t have been: neurons exist in physical spacetime. It’s not a coincidence that neurons look like lightning: they’re satisfying similar constraints in the same spatial universe. And once observed, it’s easy to guess that what Ramón y Cajal might call “economy of metabolic energy” is also a fundamental principle of neuroscience, which of course is attested by modern neuroscientists. That’s when I understood that spatial structure is a Deep Fundamental Theory.
And it doesn’t stop there. The same thing explains the structure of our roadways, blood vessels, telecomm networks, and even why the first order differential equations for electric currents, masses on springs, and water in pipes are the same.
(The exact deep structure of physical space which explains all of these is differential topology, which I think is what Vaniver was gesturing towards with “geometry except for the parallel postulate”.)
That’s when I understood that spatial structure is a Deep Fundamental Theory.
And it doesn’t stop there. The same thing explains the structure of our roadways, blood vessels, telecomm networks, and even why the first order differential equations for electric currents, masses on springs, and water in pipes are the same.
(The exact deep structure of physical space which explains all of these is differential topology, which I think is what Vaniver was gesturing towards with “geometry except for the parallel postulate”.)
Can you go into more detail here? I have done a decent amount of maths but always had trouble in physics due to my lack of physical intuition, so it might be completely obvious but I’m not clear about what is “that same thing” or how it explains all your examples? Is it about shortest path? What aspect of differential topology (a really large field) captures it?
(Maybe you literally can’t explain it to me without me seeing the deep theory, which would be frustrating, but I’d want to know if that was the case. )
There’s more than just differential topology going on, but it’s the thing that unifies it all. You can think of differential topology as being about spaces you can divide into cells, and the boundaries of those cells. Conservation laws are naturally expressed here as constraints that the net flow across the boundary must be zero. This makes conserved quantities into resources, for which the use of is convergently minimized. Minimal structures with certain constraints are thus led to forming the same network-like shapes, obeying the same sorts of laws. (See chapter 3 of Grady’s Discrete Calculus for details of how this works in the electric circuit case.)
I think “deep fundamental theory” is deeper than just “powerful abstraction that is useful in a lot of domains”.
Part of what makes a Deep Fundamental Theory deeper is that it is inevitably relevant for anything existing in a certain way. For example, Ramón y Cajal (discoverer of the neuronal structure of brains) wrote:
At first, I was surprised to see that the structure of physical space gave the fundamental principles in neuroscience too! But then I realized I shouldn’t have been: neurons exist in physical spacetime. It’s not a coincidence that neurons look like lightning: they’re satisfying similar constraints in the same spatial universe. And once observed, it’s easy to guess that what Ramón y Cajal might call “economy of metabolic energy” is also a fundamental principle of neuroscience, which of course is attested by modern neuroscientists. That’s when I understood that spatial structure is a Deep Fundamental Theory.
And it doesn’t stop there. The same thing explains the structure of our roadways, blood vessels, telecomm networks, and even why the first order differential equations for electric currents, masses on springs, and water in pipes are the same.
(The exact deep structure of physical space which explains all of these is differential topology, which I think is what Vaniver was gesturing towards with “geometry except for the parallel postulate”.)
Can you go into more detail here? I have done a decent amount of maths but always had trouble in physics due to my lack of physical intuition, so it might be completely obvious but I’m not clear about what is “that same thing” or how it explains all your examples? Is it about shortest path? What aspect of differential topology (a really large field) captures it?
(Maybe you literally can’t explain it to me without me seeing the deep theory, which would be frustrating, but I’d want to know if that was the case. )
There’s more than just differential topology going on, but it’s the thing that unifies it all. You can think of differential topology as being about spaces you can divide into cells, and the boundaries of those cells. Conservation laws are naturally expressed here as constraints that the net flow across the boundary must be zero. This makes conserved quantities into resources, for which the use of is convergently minimized. Minimal structures with certain constraints are thus led to forming the same network-like shapes, obeying the same sorts of laws. (See chapter 3 of Grady’s Discrete Calculus for details of how this works in the electric circuit case.)