Paul Dancstep: The Universal Library; A novel approach to teaching category theory
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- čas přidán 21. 07. 2021
- Title: The Universal Library; A novel approach to teaching category theory
Speaker: Paul Dancstep
Abstract: I will offer a STEM educator’s take on the unique challenges of learning category theory and present an extended thought experiment - roughly derived from the Jorge Luis Borges story The Library of Babel - as a strategy for sharing this subject with a general audience.
Topos Institute Berkeley Seminar
July 20, 2021
topos.site/berkeley-seminar/ - Věda a technologie
I wonder if this could be turned into an Ace Attorney game where the player must convince a LEAN-like proof assistant that the Librarian could in fact have some hidden knowledge without the capacity to read the books??
Incredibly articulate and insightful. I can only hope you'll create more content on the topic.
Re history: the library could exist in a dissociative post-reality setting. The library really does float in some anxiety inducing ontological horror space, and the librarian isn't everything that they seem. You come to find out about the human history of category theory through informative interludes, or actually meeting characters from books.
...At the same time you begin to unravel the mystery of the library, where in spacetime it exists, how you got here, and what the librarian has done to the other humans in your current position
I recognize your name from the maths Facebook group haha
I am now reading Conceptual Mathematics by William Lawvere, its a very good introduction to category theory for those interested in the subject
sweet talk. curious what tools were used to make the visuals.
Did Paul make anymore videos on this subject? I can't find anything else about his library idea.
Simply great, i'm in awe!
awesome presentation, love the slides
Excellent approach, thank you.Questions can not be heard, may be at least make captions for them.
WOOW THE BEST PRESENTATION I EVER SAW!!! AMAZING ANIMATIONS! 🤯
35:00 is the passage from concrete to abstract categories just like the passage of number to variable in our mathematical youth? perhaps this a moment to appreciate the beginner's mind.
what presentation software is that?
This looks to just be permutation theory to me. Is this just tying in the symmetric group in to categorical terms? Composition on permutations and the structure's within? Here it is the super structure like permutation group S_{n,k} = S_n U P(S_{n-1,k}) or whatever(not sure the exact translation but looks to just be just a disjoint sum on S_n).
What I find potentially interesting is that could we not encode the library in to a generating function?
Thanks. That was a very compelling illustration.
I have a little doubt though. Could the librarian be compared to the Laplace's demon in a "category math universe"?
There does seem to be an analogue here, right?
I guess I just wonder why that’s a problem in the first place
@@aaAa-vq1bd Because we’ll never have access to knowledge like the kind Laplace’s demon does. Thinking we might get there one day has the potential to seriously mislead us: czcams.com/video/oDG_7Ame0m0/video.html
@@aaAa-vq1bd "... why that’s a problem in the first place".
I am not sure what you are trying to say exactly.
I mean, why would you phrase a simple random doubt/suggestion as being a problem?
Here are my thoughts.
Category theory belongs to maths and regardless of the many "easy analogies", used to teach it, it doesn't share anything directly with meta-physics which is where the demon's story comes from.
By mixing the two, one will end up doing metaphysics and certainly not doing maths.
And i think that anybody knows that one can say almost anything with meta-physics. Although there may be some people that are confused by the fact that there is the word "physics" in "meta-physics" 😊
So in my view there is no problem here as long as we remember what is the "space" (and its "rules") of our argumentation.
If we mix and match two disciplines, i am not sure that we aren't deceiving oneself. And more often than not, we will be falling in the common pseudo-whatever that is seen everywhere in the net.
That is. When one is introduced to maths as kid, we all add apples and oranges but we should quickly leave those crutches behind if one wants to go further in maths.
Various analogies (and real world applications) can be reused from time to time as an aid to begin to visualise/learn some mathematical ideas, but they should never be the primary base of it if one wants to be anything similar to a "pure" mathematician.
There are some great applications of maths to "practical reality" (science, engineering, ...). And in those disciplines, good analogies, when learning maths, are much more useful and even very important.
And not to forget that in those other disciplines, we have experiment results and peer review that help in choosing "adequate" analogies and avoid endlessly discussing what is the gender of angels.
But i would say that the two (those disciplines vs their maths expression) share a relation more akin to a language/toolbox describing/predicting/calculating some "things" than them being the same altogether.
Sometimes I like to remember a formerly famous saying which helps me in maintaining a little bit of sanity in the midst of all the chaos of knowledge and information:
"the map is not the territory".
@@estranhokonstain general, Laplace’s demon is based on particular assumptions about *thermodynamics*, particularly reversibility. There’s no reason why we need to make our ontological maps irreversible though. By the way, by even engaging in category theory you are doing metaphysics. In fact, higher mathematics is one of the very best examples of metaphysics
pronouncing Banach from Banach-Tarski with a [tʃ] sound is like pronouncing Loch Ness or Johann Sebastian Bach with a [tʃ] sound
"Being and Time" by Martin Heidegger would perfectly blend into the library of Babil.
If Jorge Borges watched this, he must have become a category theorist.
For category popular divulgation, there should be many ways and "levels" that can be used. In particular with the universal library, i think that recapturing Jorge Luis Borges's story driven way would be an excellent solution. Any library's question would be related to the needs and motivations of the story's protagonists. The abstractness would be extremely diluted in this way.
Here some general rules to write a "captivating" book.
First the reader establish a contract with the writer which is usually called 'suspension of disbelief'. As long as the writer do not abuse it to much , like presenting things without sufficient preparation (gun in a mantel, logical argument, character emotional motivation, etc), the reader will give his good will to the story.
The story neds a protagonist that enable the reader to create a "empathic" relation. The reader will then be moved as the protagonist is moved by the story. see the famous Joseph Campbell's hero's journey for a reference.
Etc, etc, etc
Gamification is also a very common way of engaging a public. but it really needs more interaction from it. It can be done successfully in a book but the audience will not be as wide.
I think Eugenia Cheng heard you with her new book, the Joy of Abstraction
Are Paul Dancstep and Paul Stepahin the same person?
"Paul Dancstep grew up in San Diego but has never been on a surfboard. He studied physics in college but is still baffled by things like zippers." -- Exploratorium website.
"Exhibit developer Paul Stepahin grew up in San Diego but has never been on a surfboard. He studied physics in college but still can't understand how zippers work." -- Exploratorium CZcams channel.
Paul and Paul appear to be closely related.
Make a library!
the reason I will never get category theory is: how is this not just the same as the usual theory of sets and functions? what's the difference? I already know about sets and functions.
Basically you might have set theory, group theiry, ring theory, etc. Which are all separate branches of math with different theorems and such. Category theory lets you take one result in a theory and generalize it to many (sometimes unexpected) places all at once.
5:45 Fibonacci-Tarski? Surely Banach-Tarski paradox!
I think the Librarian should be a robot to make the character make more sense lol
The talk had a captivating introduction and held my attention until about 20 minutes into it, when it started moving too fast for me to follow, and it became hard to listen and think about what was just said. Once people began asking questions at the end, I couldn't hear the questions very well, and the video became unwatchable. The Q & A doesn't help the video and should be edited out. Then, it needs a solid conclusion to summarize all the new definitions you told us we would need to learn in order to understand the subject. Your presentation of the material made it very interesting, but I didn't achieve my goal of understanding what this subject is about.
"We can make up any definitions of science fiction, fantasy and horror that we want. We can draw our boundaries and make our labels, but in the end it's still the same old story, the one about the human heart in conflict with itself.
The rest my friends is funiture.
The House of Fantasy is built of stone and wood and furnished in High Medieval. Its people travel by horse and galley, fight with sword and spell and battle-axe, communicate by palantir or raven, and break bread with elves and dragons.
The House of Science Fiction is built of duralloy and plastic and furnished in Faux Future. Its people travel by starship and aircar, fight with nukes and tailored germs, communicate by ansible and laser, and break protein bars with aliens.
The House of Horror is built of bone and cobwebs and furnished in Ghastly Gothick. Its people travel only by night, fight with anything that will kill messily, communicate in screams and shrieks and gibbers, and sip blood with vampires and werewolves." - G. R. R. Martin
You're gonna have a hard time getting students to focus on the category theory aspect and not all the other baggage this example brings with it...
To say nothing of the youtube audience who are not math students. The mega library is a powerful visual image, that's most of what I got out of this, and I don't even know if it's an accurate metaphor.
Gee, a book introducing a mathematical topic expects it's readers to have a background in mathematics -- who would have thought.
Scott sounds like a really obnoxious guy
I dunno man. This is the "intermediate" level you were aiming at. But the acid test of a great expositor would be to allow a ninth grader to understand n-Cats. It's possible. Category theory is mathematics seen through arrows (processes), while Set theory is mathematics seen through membership (counting). The nerds make it seem harder than it needs to be. There's a unifying bridge too, which is the notion of "sameness" (which elite nerds obscure in terms like homotopy type theory and univalent foundations). It's all stuff a ninth grader can understand, and that's how a graduate student should be required to exposit on their topics, so that lunch time mathematics seminars aren't a boring piece of sh111te. (John Baez I think tries to champion dual level seminars: one for the οἱ πολλοί (hoi polloi) and one for the experts who would not fall asleep with the jargon.)
Ninth graders can easily comprehend different sorts of "equals". Doug Hofstadter has a wealth of amusing examples in his creative analogies publications and whatnot. So it can even be made fun so that a ninth grader might even enjoy learning about category theory.
It is not that the math does not already exist, it is its abstraction and codification in to a precise language. That is what you seem to fail to grasp. Even cave men understand "calculus" as they have to understand rates of change when hunting or they would always miss their targets with their spears and arrows.
What makes it mathematics is the abstract language that codifies "universal" things in to a template that then can be applied to anything(not just the cave man's arrow).
Category theory is way more than just objects and arrows. Category theory is a recursive language and hence a fractal that breaks down all structure in to objects and arrows and then analyzes the abstract structure of all things.
Yes, 9th graders can understand the basics of category theory but they will never completely understand it, no one can. It is something that the more you learn the more structure you'll see using it. It's about having a language that can be used for understanding anything rather than having to invent new concepts and terms to constantly describe new things we come across in existence.
Yes, on some level "nerds" make it harder than it is but that is not their fault. First, most mathematicians when discovering/inventing category theory saw it as a very abstract mathematical thing not so much realizing how powerful it really was and how applicable it would be(e.g., same with the internet, computers, medicine, etc) so in their minds it is harder than it is now(they learned the hard way but they are also the ones who present it easier for others to so they don't have to waste their life trying to figure it all out).
Also, by "easily comprehend" it depends on what you mean. Even a baby can comprehend the difference between "equalities" after a few bites... but that is very different from comprehending the abstract mathematical meaning of equality that all things have. Every kid knows what dogs but they don't know what really makes the category of dogs. Recognition != understanding. The first step is recognition and it can be simple or hard.... understanding, though, is an order of magnitude harder and that is just for a first level understanding. Understanding is actually a life time process(and including recognition).
Mathematics has traditionally been a boring subject for a very important reason... and that is to avoid emotions obscuring reality. This allows for very deep analysis that, with your 20/20 hindsight might seem rather trivial but so was building the space shuttle when you could ignore the actual entire process such as the 10's of thousands of engineers that had to get up every morning, drink coffee, take a shower, kiss the wife, drive the car to the buildings(including dealing with the flat tires, breakdowns, snow, etc), etc, etc, etc, etc).
My suggest is that you calm down on the nerds a bit and if you think you have a better way to present things or whatever then no one is stopping you from trying to show others a better way. If you don't then it's just all talk. Things might not be as simple as you think or maybe they are simple because everything is simple after one understands it.
Now, sure there is a better way so have at it. Clearly, at least when I was taught mathematics(most of it self learned but) it was very formal and at the time I was clueless because I didn't understand why it was important to do what I was told to do. It was all so abstract because it was presented only as definitions, theorems, and proofs without any real point. For me, it was a terrible way to learn and very time consuming and I always felt like I was missing something. That is mainly an issue of education though and not mathematics. Good mathematicians generally are very poor teachers. In fact, most teachers are poor teachers. It is changing though as humans learn how to do a better job at things through trial and error and discussion. It's really not that big of a deal, one day you will be dead and so it's not really going to matter so you might as well enjoy it the best you can while you can.