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Daniel Geisler (Apr 10 2020 at 01:23):Both mathematics and complexity have well defined examples of hierarchies of existence:
Scott Aaronson - https://complexityzoo.uwaterloo.ca/Complexity_Zoo
Stephen Simpson - Subsystems of Second Order Arithmetic
I'd like to know if there is any formal hierarchy of existence in the physical sciences. Likely there would be a connection with scale beginning with quantum gravity. QFT, chemistry, biology, psychology, economics are a rough example of a hierarchy, but there is a lacking of boundaries insuring a new disciple begins precisely as an old disciple ends.
Johannes Drever (Apr 10 2020 at 18:45):There is a hierarchical model in psychology, or rather neuroscience. The MENS model (Memory Evolutive Neural System) models hierarchical neurons. I just started reading Conciliating neuroscience and phenomenology via category theory, it's seems super interesting.
Sam Tenka (Apr 11 2020 at 19:49):@Daniel Geisler It seems to me that such hierarchies of analytical coarseness reflect something about the abstractions we find pyschologically convenient rather than something fundamental about nature. For example, the abstraction of "resistivity" is convenient for the electrical engineer who prefers not to imagine individual electrons bumping into nuclei. So the hierarchy or directed acyclic graph you seek might be some quotient of the graph of back-references in the definitions in a physics textbook series such as Landau's. Does that sound right to you?
Daniel Geisler (Apr 11 2020 at 20:48):@Sam Tenka (naive student) I'm interested in renormalization where systems greatly simplify. Consider Cooper pairs, two electrons in a superconducting material. Renormalization causes the two electrons to act as a single particle. Superconductivity is not a psychological mechanism, it is reality.
Sam Tenka (May 11 2020 at 20:28):Johannes Drever said:
There is a hierarchical model in psychology, or rather neuroscience. The MENS model (Memory Evolutive Neural System) models hierarchical neurons. I just started reading Conciliating neuroscience and phenomenology via category theory, it's seems super interesting.
I like this! I haven't yet looked at it carefully. It seems to approach something like Minsky's Society of Mind, with the added benefit of being able to invoke arrow-composition and universal properties to make the picture feel less ad hoc. Thanks for sharing!
Daniel Geisler (May 11 2020 at 21:32):I love Neurodynamics of Personality by Grigsby and Stevens. Not based on CT, but I think it could be a good starting place.
Matteo Capucci (he/him) (May 13 2020 at 09:01):Hierarchies are a phenomenon of our descriptions: I can describe resistivity phenomena macroscopically without resorting to a microscopic theory. This means they are 'as real as they can be', since descriptions of phenomena are the only things science can manipulate.
Matteo Capucci (he/him) (May 13 2020 at 09:01):I believe this aspect is quite important in science and amenable to mathematical formalization, too.
Matteo Capucci (he/him) (May 13 2020 at 09:05):OTOH, I think one cannot just deem 'quantum mechanics' at a different hierarchy from 'general relativity'. The distinction has to be context-sensitive and system-dependent. I guess to make sense of the facts one has to a adopt a pedestrian point of view on the matter, formalizing exactly what it's meant by 'physical theory' and 'hierarchy', and this is a crucial part of the formalization effort.
Daniel Geisler (May 13 2020 at 20:53):I have multiple descriptions of the Universe I want to stuff into a single Universe. Plus, I don't make much distinction between mathematical universes and physical universes. Each view has it's champions - John Baez, Terry Tao, Stephen Simpson, Edward Witten, Ken Ono, Stephen Wolfram, Gregory Chaitin, Andrei Linde for example. I'm fasinated in the work of "world builders". These are people whose work is good to follow, although all of them would correctly say my understanding of their work is trivial at best.
My own work is in the area of the Ackermann function and the hyperoperators. Let be the dynamical system of the entire universe that has a Taylor series. Consider the fantastic phenomena across the vast array of distances, from Planck's distance to the radius of the measurable universe. I argue that the Taylor series has significant coefficients across an great range of terms. If physics is comprised of multiple layered systems, then the underlying mathematics should also come in multiple layers.
Stephen Wolfram's work is useful here How can one extend recursive function definitions to continuous numbers? pg 33. While he investigates a number of mathematical models of the universe, only one model is "horizontally" layered - the Ackermann function.
Principle of Alignment - Let's say that we have a second "horizontally" layered system. Due to combinatorics there is a tendency for the first layer to be addition. It's not easy to construct a simple function without using addition or multiplication. Let , then . Another way to say this is that there is a fixed point where time goes to infinity.
a13ph (May 16 2020 at 19:10):Daniel Geisler said:
a connection with scale beginning with quantum gravity. QFT, chemistry, biology, psychology, economics are a rough example of a hierarchy
This reminds me of the term "metasystem transition"
Daniel Geisler (May 16 2020 at 19:13):@a13ph Great reference!