Universal Stack

For the discussion of the sciences. Physics problems, chemistry equations, biology weirdness, it all goes here.

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Pfhorrest
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Universal Stack

Postby Pfhorrest » Wed May 15, 2019 7:01 am UTC

I was unsure whether to post this in Forum Games or here, but I decided that since this is more about applications of real science knowledge it would be better put here. It also overlaps a lot with something more appropriate for the mathematics forum, but I guess here is good enough.

The initial idea was to "play a game", of sorts, by starting with me saying what something is made out of (e.g. "humans are made out of cells"), and then the next poster saying what that in turn is made out of (e.g. "cells are made out of molecules"), and so on building down to the fundamental objects of the universe (quantum fields?), and then the mathematical objects those are equated to (special unitary groups?), and then the mathematical objects that those are made out of, and so on and so on until we get down to empty sets.

But then I thought that maybe some people along the way might get some steps wrong, and a better idea would be for me to just start with stating my understanding of this "Universal Stack", and then subsequent posters to post more and more corrected versions of that (inserting missing steps, or correcting missteps, etc), with notes explaining their corrections as needed.

So, I guess here is my first go at it, and I know that this gets really weak in the middle with the math stuff between numbers and special unitary groups:

Humans are made of cells
Cells are made of molecules
Molecules are made of atoms
Atoms are made of electrons and nuclei, bound together by photons (mediating the electromagnetic interaction)
Nuclei are made of nucleons (protons and neutrons) bound together by gluons (mediating the strong nuclear interaction)
Protons and neutrons are made of up and down quarks bound together by gluons (mediating the strong nuclear interaction)
Up and down quarks, gluons, photons, and electrons, like all fundamental particles, are all excitations of quantum fields.
Quantum fields are special unitary groups, different sizes (of their constituent matrices) for different fields.
Special unitary groups are unitary groups with determinant 1. (I don't fully understand what that means).
Unitary groups are Lie groups of unitary matrices.
Unitary matrices are complex square matrices whose conjugate transpose is also its inverse. (I don't fully understand what that means).
Complex square matrices are matrices of complex numbers with equal width and height.
Matrices are sets of numbers that can be ordered in two dimensions.
meanwhile...
Lie groups are groups that are also a differentiable manifolds.
Differentiable manifolds are manifolds that are smooth and continuous in such a way that calculus can be done on them.
Manifolds are spaces that are locally Euclidean, i.e. approaching flat on the smallest scales.
Spaces are sets of connected points.
Points are the equivalence classes of all of the numbers that could be mapped to them by different coordinate systems.
Equivalence classes are set of things that are equivalent to each other in some way.
meanwhile...
Groups are sets that are closed and assocciatve under some operation, with an identity element and an inverse element.
meanwhile...
Complex numbers are ordered pairs of real numbers.
Ordered pairs are sets of two things with an order assigned to them.
Real number are Dedekind cuts of the rational numbers. (I'm not sure I fully understand what that means).
Dedekind cuts are special kinds of ordered pairs of sets (subsets of a larger set)?
Rational numbers are equivalence classes of pairs of integers equal to one integer divided by the other.
Integers are equivalence classes are pairs of natural numbers equal to one natural number subtracted from the other.
Natural numbers are sets of sets, with each next number/set equal to the previous number/set unioned with a set containing only itself.
Zero, the first natural number, is just the empty set.
Forrest Cameranesi, Geek of All Trades
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Eebster the Great
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Re: Universal Stack

Postby Eebster the Great » Wed May 15, 2019 7:50 am UTC

Humans are made of cells, but they are also made of atoms. It's not clear how we decide how many intermediate steps to insert. For instance, human cells are made of organelles, cytosol, membranes, and other components. Organelles themselves consist of several structures the way a house has walls, a floor, a roof, and other parts. Each of those is comprised of many different types of molecules including protein complexes with several levels of organization. Molecules themselves can be grouped into functional parts. And so on.

Apart from that, molecules are not the only relevant chemical constituent of cells. Cells are mostly aqueous solutions, and important species in aqueous solutions include ions. The compounds of which our bone is made are not entirely molecular. Most organic matter is molecular, granted, but not absolutely all of it.

You then sort of shift away from what things are made of to what they are defined as. I'm not sure what the motivation is for the change of direction. Special unitary groups, like all groups, are made of a set X of elements and a binary function • on that set called the group operation. The elements are usually themselves sets of some sort, though exactly which sets is entirely arbitrary, but in a set theory with urelements they can be that. The group operation is a binary operation, which means it is a function •: X×X → X, which means it is a triple (X, X, Z), where Z is a collection of ordered triples (x,y,z)∈X³ where no two triples have the same x and y. Ordered triples are made of elements like the underlying set X, except they have only three each.

What matters for tuples and groups and the like is not what they are "made of" though, because you can make them out of whatever you like, but the formal structure assigned to those constituents.

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SuicideJunkie
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Re: Universal Stack

Postby SuicideJunkie » Wed May 15, 2019 4:53 pm UTC

Quantum fields are special unitary groups, different sizes (of their constituent matrices) for different fields.
Special unitary groups are unitary groups with determinant 1.
Every step has to drop something important during the simplification, but I'm pretty sure you switched from listing simpler things to listing simpler descriptions of things somewhere around that step.

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Pfhorrest
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Re: Universal Stack

Postby Pfhorrest » Wed May 15, 2019 5:38 pm UTC

Eebster the Great wrote:Humans are made of cells, but they are also made of atoms. It's not clear how we decide how many intermediate steps to insert.

I guess I'm going for "however many are needed for clear understanding". It seems like "humans are made of quantum fields" skips over so much detail that a reader would be left with no understanding of how they're made of them, so more intermediate steps are needed.

You raise a bunch of good points for possible missing or incorrect steps, so feel free to post a corrected version of the stack with those changes made to it. That's the 'game' here.

SuicideJunkie wrote:
Quantum fields are special unitary groups, different sizes (of their constituent matrices) for different fields.
Special unitary groups are unitary groups with determinant 1.
Every step has to drop something important during the simplification, but I'm pretty sure you switched from listing simpler things to listing simpler descriptions of things somewhere around that step.


The map is not the territory, unless the map is perfectly accurate 1:1 scale map, in which case it's just a replica of the territory, and a territory in its own right.

Whatever mathematical object perfectly describes the fundamental constituents of reality is thereby indistinguishable from them and so identical to them.

It is of course always an open question whether that map is perfectly accurate or not, but I'm just looking for what the current best guess at that is.

I discuss this at more length in an essay I recently wrote, which is also what prompted me to think about the topic of this thread.
Forrest Cameranesi, Geek of All Trades
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SuicideJunkie
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Re: Universal Stack

Postby SuicideJunkie » Wed May 15, 2019 8:45 pm UTC

Another way to consider it is in the reverse:
It is pretty easy to see how assembling a bunch of cells in the right order gets you a human. (and arranging the right types of humanoids together could make a family or society)
A variety of molecules can build up to cells.
It might be worth injecting organs and molecule chains in there, but I think it is close enough to common experience that people can make the leap without much issue.

Maybe it is me but, I feel there is a bigger discontinuity when going from groups and matrices to actual quantum fields that can spawn fundamental particles. Can you really arrange them up in the correct order and get a live universe?
Perhaps it is Time that is the missing ingredient here?

Complex numbers are ordered pairs of real numbers.
Ordered pairs are sets of two things with an order assigned to them.
Real number are Dedekind cuts of the rational numbers. (I'm not sure I fully understand what that means).
I feel the middle is redundant. All? the other steps are implicitly assuming you get multiple, varied instances of the lower items to assemble into a higher item.

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Pfhorrest
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Re: Universal Stack

Postby Pfhorrest » Wed May 15, 2019 9:10 pm UTC

SuicideJunkie wrote:Maybe it is me but, I feel there is a bigger discontinuity when going from groups and matrices to actual quantum fields that can spawn fundamental particles. Can you really arrange them up in the correct order and get a live universe?

If you like, think of it as we're building a simulation of a universe in a programming language that only deals with sets. You can build numbers and groups out of sets, and spaces and and matrices out of numbers, and if you build the right kind of space that is also a group of matrices, you've now got a thing that behaves identically to a quantum field, and so from there you've got a perfect simulation of the universe, with its particles, atoms, molecules, cells, and humans.

Complex numbers are ordered pairs of real numbers.
Ordered pairs are sets of two things with an order assigned to them.
Real number are Dedekind cuts of the rational numbers. (I'm not sure I fully understand what that means).
I feel the middle is redundant. All? the other steps are implicitly assuming you get multiple, varied instances of the lower items to assemble into a higher item.

That step was to make clear that an ordered pair is a kind of set, just like the step saying an equivalence class is a kind of set. Not all of the other steps are "this is made of multiples of these", though it would be nice if they were. I describe what makes a manifold differentiable and what makes a space a manifold before I get to what spaces are made out of, though maybe it would be better to go straight from special unity groups to those being (differentiable, locally-euclidian, connected) sets of matrices (all whose conjugate transposes are also their inverse, and which have determinant 1) that form groups (under matrix multiplication), but it seems like that's too many qualifiers to pack into one step just to get from special unity groups to the things they're made of multiples of (matrices) in one step, so I made it several. I guess, in the case of "humans are made of cells", there are a lot of qualifiers on what kinds of cells and how they need to be arranged that I didn't bother describing, so maybe it's better to just say "special unity groups are made of matrices" and skip the qualifiers entirely? And then matrices are made of numbers that are made of other numbers that are ultimately made of sets of sets of empty sets.
Forrest Cameranesi, Geek of All Trades
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Eebster the Great
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Re: Universal Stack

Postby Eebster the Great » Wed May 15, 2019 10:30 pm UTC

The smashing together of physical and mathematical reductionism here is definitely philosophically suspect... and a little confusing.

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Zamfir
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Re: Universal Stack

Postby Zamfir » Thu May 16, 2019 8:42 am UTC

If you like, think of it as we're building a simulation of a universe in a programming language that only deals with sets.

But that point, the physical reduction would describe what happens inside the hypothetical universe-simulation machine. Unless we assume that the machine is built in Plato-space from ideal Lie groups - but that is quite a stretch, and a break from what the previous steps do.

p1t1o
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Re: Universal Stack

Postby p1t1o » Thu May 16, 2019 10:45 am UTC

I think the jump from "list" to "universe simulation machine" is a little...bonkers, to put it very mildly

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Pfhorrest
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Re: Universal Stack

Postby Pfhorrest » Fri May 17, 2019 7:28 am UTC

Yeah well it's a thing and not my original thing so whatever if you don't like the premise of this 'game' don't 'play'.
Forrest Cameranesi, Geek of All Trades
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The Codex Quaerendae (my philosophy) - The Chronicles of Quelouva (my fiction)

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Eebster the Great
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Re: Universal Stack

Postby Eebster the Great » Fri May 17, 2019 7:43 am UTC

Well there is more than one axiomatization of QFT, so which should we use? More to the point, since we know the standard model is not exact and does not agree with all observations, the formalisms also need to change. Formally speaking, we can construct these theories in infinitely many different ways. Which is "right"?

You are mixing observable facts with mathematical descriptions. Nothing we learn about quantum gravity will change the fact that we are made of cells which are made of molecules, but any arbitrary change in representation will completely change the rest of the chain after elementary particles. So ontology aside, you are still mixing two epistemologically very different things.

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Re: Universal Stack

Postby p1t1o » Fri May 17, 2019 9:09 am UTC

Pfhorrest wrote:Yeah well it's a thing and not my original thing so whatever if you don't like the premise of this 'game' don't 'play'.


That mathematics can be used to describe the universe does not make a list into a machine, but fair point, the deeper you dive into physics and maths, the closer you get to abstract principles and philosophy. But that doesnt give meaning to everything abstract. On the other hand, as a thought exercise I admit it can have merit, so dont take my grouchiness to heart.


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