Why is symmetry so important in particle physics?
mfaizan.github.io
Why is symmetry so important in particle physics?
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Re: Why is symmetry so important in particle physics?
#2This idea shows up in essentially all scientific fields. It’s the idea of adjointness. Together with norm, they give you the idea of fixed points, (invariants, spectra, embeddings, braids etc).I'm
Lawvere's fixed point theorem is I think the best formulation of the idea
https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore...
I've been putting together a brain dump on the topic
https://github.com/adamnemecek/adjoint/
Join the discord https://discord.gg/mr9TAhpyBW
Re: Why is symmetry so important in particle physics?
#3I wonder how much of this can be ELI5'ed with diagrams and intuitions instead of algebra.
Re: Why is symmetry so important in particle physics?
#4This idea shows up in essentially all scientific fields. It’s the idea of adjointness. Together with norm, they give you the idea of fixed points, (invariants, spectra, embeddings, braids etc).I'm Lawvere's fixed point theorem is I think the best formulation of the idea https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore... I've been putting together a brain dump on the topic https://github.com/adamnemecek/a…
Wouldn’t adjointness be distinctly not “symmetry”, given it relies on a forgetful functor?
Re: Why is symmetry so important in particle physics?
#5I wonder how much of this can be ELI5'ed with diagrams and intuitions instead of algebra.
Most of it.
Or conversely, a lot of it can be explained without having to resort to category theory or similarly dense terminology.
I've found that Mathematicians like to come up with the most general, tersest definitions possible, where every symbol is overloaded with layer upon layer of meaning.
You end up with language that looks like:
a = b c
This is never useful as a method of pedagogy, yet this is about the only type of content you will ever find online in places like Wikipedia or nLab.
Any attempt to clarify with an example is resisted, because it's not "general", or "not the definition" of the concept. In the end, all practicality is erased, leaving definitions so pure that they could be referring to almost anything.
Re: Why is symmetry so important in particle physics?
#6I wonder how much of this can be ELI5'ed with diagrams and intuitions instead of algebra.
PBS Spacetime has a very good video[0] that helps visualize what's going on in the equation, recent too!
Re: Why is symmetry so important in particle physics?
#7I wonder how much of this can be ELI5'ed with diagrams and intuitions instead of algebra.
Most of it. Or conversely, a lot of it can be explained without having to resort to category theory or similarly dense terminology. I've found that Mathematicians like to come up with the most general, tersest definitions possible, where every symbol is overloaded with layer upon layer of meaning. You end up with language that looks like: a = b c For some absurd percentage of the statements. Sometimes the equal sign…
Because to expand the whole expression, such as the Einstein field equations, would fills many lines or pages.
Re: Why is symmetry so important in particle physics?
#8Earlier quoted context omitted.
Most of it. Or conversely, a lot of it can be explained without having to resort to category theory or similarly dense terminology. I've found that Mathematicians like to come up with the most general, tersest definitions possible, where every symbol is overloaded with layer upon layer of meaning. You end up with language that looks like: a = b c For some absurd percentage of the statements. Sometimes the equal sign…
Because to expand the whole expression, such as the Einstein field equations, would fills many lines or pages.
As someone that also doesn't like the status quo - I would love for there to be a better mathematical notation. If we could get some kind of functional programming-like syntax it would be nice and consistent.
Also what's up with the acceptance of single letter variables in math?
Re: Why is symmetry so important in particle physics?
#9I wonder how much of this can be ELI5'ed with diagrams and intuitions instead of algebra.
Most of it. Or conversely, a lot of it can be explained without having to resort to category theory or similarly dense terminology. I've found that Mathematicians like to come up with the most general, tersest definitions possible, where every symbol is overloaded with layer upon layer of meaning. You end up with language that looks like: a = b c For some absurd percentage of the statements. Sometimes the equal sign…
At the extreme end of this you get category theory, with Mathematicians affectionately refer to as "abstract nonsense".
I checked the homepage of nLab and found this description [0]:
> This is a wiki for collaborative work on Mathematics, Physics, and Philosophy — especially, but far from exclusively, from the n-point of view: with a sympathy towards the tools and perspective of higher algebra, homotopy theory, type theory, category theory and higher category theory.
And following to the description of n-point of view:
> In particular, the nLab has a particular point of view, which we may call the nPOV, the higher algebraic, homotopical, or n- categorical point of view.
So, nLab seems to have made a very deliberate choice to cater to the style you are talking about.
I think your complaint is more valid for Wikipedia, which seems to want to be more general purpose. However it is not specific to math. Most articles on Wikipedia have a tendency to assume a high level of subject knowledge on the part of the reader.
The math community is well aware of the pedagogical concerns. You don't see category theory until grad school, where it is taught with the assumption that you are well versed in many concrete instances of categories.
You don't even see abstract algebra until the later mart of a undergrad math major. Again, it is taught with the assumption that you are already well versed in several concrete instances of the algebraic structures.
If you get a math textbook, you will find that they almost never take the most generic approach; and instead tend to take the most concrete approach that would cover the subject matter.
Re: Why is symmetry so important in particle physics?
#10I wonder how much of this can be ELI5'ed with diagrams and intuitions instead of algebra.
Most of it. Or conversely, a lot of it can be explained without having to resort to category theory or similarly dense terminology. I've found that Mathematicians like to come up with the most general, tersest definitions possible, where every symbol is overloaded with layer upon layer of meaning. You end up with language that looks like: a = b c For some absurd percentage of the statements. Sometimes the equal sign…
When math “started” paper and writing utensils were harder to acquire. One had to transmit idea’s succinctly.
Still it would take many more books to capture it all in English than a simpler glyph set. One will still go further in math learning the meaning of the glyphs than reading it all explained in “human language”.
It’s not so hard when one accepts math is the four common operations (add, sub, div, mult) on a variety of data structures though. “Square root of” is a shorter coded language for “use numeric result of this sequence of operations against this given number” (the data structure).
Math isn’t hard. Government mandate we teach students in speaking traditions instead of educate them to measure for themselves has resulted in piss poor education in math.