Apparently there is an actual experiment on six-year-olds taking place [1,2]:
"Experiment. Consider ten children of ages between six and ten and consider ten high-school
teachers of physics and mathematics. The high-school teachers of physics and mathematics will
have all the time they require to refresh their quantum mechanics background, and also to update
it with regard to recent developments in quantum information. The children on the other hand
will have quantum theory explained in terms of the graphical formalism. Both teams will be
given a certain set of questions, for the children formulated in diagrammatic language, and for
the teachers in the usual quantum mechanical formalism. Whoever solves the most problems
and solves them in the fastest time wins. If the diagrammatic language is much more intuitive,
it should in principle be possible for the children to win."
Kindergarten? They just took bra/kets, Feynman diagrams and substituted block-looking shapes. The underlying logic is hardly changed to kindergarten level. The kid better have a masters in physics to follow this.
"...i.e. we need to conjugate the first Hilbert space (although this gives an isomorphic copy). In the above argument
establishing the bijection we used the matrix representation of Hilbert spaces and hence essentially the whole vector
space structure, but in fact we need none of this, and we will show that in any picture calculus we always have..."
Kindergarten Quantum Mechanics: if you ask mom for something, she will say "yes" or "no" (even if she wasn't sure before you asked). Asking her again won't change her answer. At this point, to try to get a different answer to the same question, you have to ask dad.
As meta, I suggest most everyone is vastly underestimating the magnitude of potentially transformative improvement being left on the table by current science education. But that's a discussion for another time.
Imagine putting all of science on a common (diagrammatic) footing, implemented down to the level of types so you can immediately perform calculations...mouthwatering.
"Of course we do not expect you to know about category theory, nor do we want to encourage you here to do so." Ugh. Well, I can encourage you to do so, with some Baez [0][1].
Man everyone seems to be talking about category theory lately and I still have no idea what it is except some general sense in which different types of mathematical objects can operate on each other (which is probably wrong). Looks like I should probably actually put the effort in and learn about it.
As grand-nibling comment suggests, it's primarily about vocabulary and definitions, but also how objects can be put together to form new objects. In combination, we end up with "universal properties", behaviors of objects which are the same in many different contexts.
If you want to get some surface-level mind-blown experiences, then there are some cool pages out there, in addition to the papers I already linked, which have great diagrams. [0] shows a periodic table which relates categories to truth values and sets, which is helpful for understanding what folks mean by "category theory is just as good as set theory for foundational work".
More mind-blowingly, category theory is "formally formal", which means that we can use category theory to formalize its own concepts. This leads to n-category theory. Because of this, we have tables like [1] which relate category theory's own building blocks to type theory and logic.
Category theory was originally the study of natural transformations. Any time that you find enough data to define a natural transformation, then that's interesting; the component categories are then also interesting objects of study. That's still what motivates its presence in physics. There are at least two other perspectives though, a logical perspective (a category is a deductive system for a formal logic) and a spatial perspective (a category is a topological space), and they're both rich as well. But the richest perspective is going to come after you've explored all three.
Normally I'd go with the second, but I'm trying to learn physics as a hobby and, you know, technologically save the world with my messiah complex apparently, and I want to keep tabs on things that may be insightful to know about
Just in case you are serious about having a messiah complex, I am prone to having a messiah complex, thanks to schizoaffective disorder. It took me a while to figure out which medications actually help me, without too many unpleasant side effects, but I am very grateful to have discovered the cocktail that works for me now.
i'm glad to hear you've found something that helps you. I was joking about the messiah complex, though it is something my therapist has made the occasional joke about too when i get over-ambitious about things i wanna do (he did specify i don't seem to actually have it, though - just good old-fashioned depression).
Just a cautionary note - category theory has seemingly been at "There's tremendous potential here! We just need to find a big win to make that clear to everyone else..." for quite a few years now.
I'm obliged, I suppose, to list off a few big wins. From the 50s and 60s, we have classic theorems which use abstract nonsense to generalize big statements about entire classes of objects, like Freyd's adjoint functor theorem [0], Yoneda's lemma [1], and Lawvere's fixed-point theorem [2]. Yoneda's lemma is the slogan that "an object is equivalent to the arrows coming/leaving it", but formal. Lawvere's theorem is a deep and permanent generalization of Gödel, Tarski, Turing, et al. on incompleteness and undecideability.
Starting in the 60s and continuing to the present, there's been a theme of exploring category theory as a prime foundation for maths. There's been complete foundations like the Elementary Theory of the Category of Categories (ETCC) [3], and also much smaller but pointed presentations that focus on just simplifying set theory. I like [4] in particular.
On a deeper level, in terms of structure and philosophy, the entire existence of homotopy type theory (HoTT) relies on categorical presentations and memes. HoTT is, more than any other type theory, an investigation into what equivalence means, and it dovetails wonderfully with 2-category theory and the understanding that algebraic laws can be transformed into transformations.
Those aren't really "big wins" though. To most mathematicians, generalizations and abstraction are only interesting insofar as they are useful, and I don't see any results there someone working in mainstream mathematics would or should care about. (Obviously the Yoneda lemma is used in algebraic topology and geometry, but that's the odd one out on your list, and you don't really need categorical language to state the interesting examples.)
Also, EETC/HoTT are basically junk, as explained in detail by Harvey Friedman in various postings on the foundations of math mailing list. We already have a perfectly good foundation for mathematics and EETC/HoTT don't improve on it in any way (and in fact are worse in many respects).
Friedman and Simpson are, unfortunately, dinosaurs who don't grok categorical concepts. This became clear on the FOM list when Pratt and other abstract algebraists who know category theory but do not depend on it were able to pry apart the problem. Friedman and Simpson deny that certain mathematical objects exist and are equivalent to each other, and while I won't begrudge them their nearsight due to spending so much time with weak/reverse maths, I won't excuse the mistakes.
I had to go find a blow-by-blow of the drama because it's good. Simpson cannot imagine topoi which don't implement standard set theory [3]. Friedman demonstrates a total misunderstanding of sets vs. categories [4]. McLarty and Feferman claim that categorical FOM make sense once one is used to categories [1]. Finally, the truth is laid bare: Friedman and Simpson simply don't agree with us on whether categorical logic is philosophically valid [2].
For more on this perspective, check out Pratt's take on Yoneda's lemma [0].
(An interesting aside: Simpson is an Objectivist who hates postmodernism! I wonder if this is part of what causes them to reject topos theory, where there are many different logics and collections, with a single barren plain Boolean set theory?)
I read the links you provided (except [0], which I do not have time for at the moment). I do not think Awodey and McLarty come across as well as you think they do, and I do not think Friedman misunderstands anything.
Moreover, you omit some of Friedman's best postings on this topic. The challenge is to show what advantages this proposed theory has over the standard foundations, or to demonstrate some interesting problem or conceptual issue that it resolves. I did not see any serious replies to this.
(Incidentally, the same challenge is my reply to the other comment to my previous post, about the ZX-calculus.)
Here's an analogy. Why do mathematicians consider groups interesting, but general Moufang loops not interesting? I propose the answer is that the additional generality provided by Moufang loops doesn't aid in addressing any interesting questions external to their theory. On the other hand, the utility of the group concept is obvious to anyone who studies a subject with connections to algebra.
This reasoning is roughly why the vast majority of mathematicians don't give two hoots about category theory, beyond picking up the few tricks like Yoneda's lemma that are actually useful. You can – as certain members of the category theory community have shown – take any mathematical concept, consider some generalization, and play various definition pushing games. But that's pointless without a good motivation (e.g. a concrete problem).
The previous two paragraphs concern "actual" mathematics, but similar remarks apply to foundations of mathematics. Friedman is pointing out that we already have a perfectly good foundational theory of mathematics and asking what gain we get by introducing categorical concepts.
In ten words or less: Sets are just 0-categories [7][8]. With more words: Set theory is just 0-category theory. This is obvious today, but two decades ago, Friedman couldn't just go to nCat [0] and educate himself.
Friedman repeatedly demonstrates [1][2] that he isn't interested in grokking why category theory is even a thing; he doesn't see e.g. Pratt's careful explanation that category theory is motivated by studying (natural) transformations. Simpson comes across as a spoiled brat [3] and Friedman comes across as a narcissist who needs to fuel himself by being the bastion of FOM [4]. Seriously, in [5], he has the audacity to simultaneously claim that Lawvere's foundations are the same as Friedman's set-theoretic foundations, and also to ask what a Lawvere theory/sketch is! Unbelievably rude.
I read the entire three-month slapfight again, just to double-check that I hadn't mis-remembered the general outline. Simpson wastes message after message being wrong about Boolean algebras vs. Boolean rings (they're the same picture) and doesn't understand how categorical dualities like Stone duality lead to equivalences. At no point do either of them manage to fully grok a 2-category or how ETCC could be a practical foundations. Tragesser says it well in [5] when he analogizes the entire affair to the sheep and their shop [6], going around and around and always changing the framing but never actually getting to the philosophical meat of the inquiry.
Again, what I see is Friedman and Simpson asking for fairly specific and concrete things, and those things not being provided.
More to the point, what I asked for in my previous comment has also not been provided! What does the knowledge that Sets are "just 0-categories" buy me in concrete terms? Does it facilitate the proof of any theorems in set theory?
Category theory is an essential part of the vocabulary of 20th-century mathematics. Large swaths of algebra, topology, geometry, and logic are fairly inextricably formulated in this language.
Similarly, it seems irreplaceable for certain parts of programming language theory (arguably due to its connection to mathematical logic).
There's certainly a community trying to bring a category-theoretic approach into other fields such as statistics, economics, or other areas of computer science, but it's too early to say it's been "quite a few years".
From [0] 2018: "In [1] we present an entirely diagrammatic presentation of quantum the-ory with applications in quantum foundations and quantum information.This was the result of many years of work by many, and started of as acategory-theoretic axiomatisation motivated by computer science as well asaxiomatic physics. However, I have always felt that the diagrammatic pre-sentation is of great use in its own right, be it to bridge disciplines, makequantum theory more easy to grasp, or, for educational purposes, in [2] wemade the bolt claim that using diagrams high-school kids could even out-perform their teachers, or university students. Now, we will put this claimto the test. To do so, we have written two tutorials [3,4], covering exactlythe same material, but one only using diagrams, while the other containsthe standard Hilbert space presentation. There are corresponding sets ofexamples too."
Yes, they are pretty much the same thing. There are some particular quantum-y elements though, such as the cup & cap. In general relativity these would (i think?) correspond to index raising and lowering operators.
"Experiment. Consider ten children of ages between six and ten and consider ten high-school teachers of physics and mathematics. The high-school teachers of physics and mathematics will have all the time they require to refresh their quantum mechanics background, and also to update it with regard to recent developments in quantum information. The children on the other hand will have quantum theory explained in terms of the graphical formalism. Both teams will be given a certain set of questions, for the children formulated in diagrammatic language, and for the teachers in the usual quantum mechanical formalism. Whoever solves the most problems and solves them in the fastest time wins. If the diagrammatic language is much more intuitive, it should in principle be possible for the children to win."
[1] https://twitter.com/coecke/status/1285531026270367744 [2] https://arxiv.org/abs/0908.1787