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in Chat

I just signed up to Azimuth Forum for the Applied Category Theory course.

About me: When people used to ask me what I was doing in school, I'd respond, "math or physics, I dunno." They'd ask "why," and I'd shrug and say, "I dunno, to quantize gravity?" But then I learned that Prof. Baez had already uncovered ample dead-ends in that effort, so I figured I should learn from his experience and just start studying more category theory.

Category theory background: I picked up a copy of Lawvere and Schanuel's book a year or two ago; the first time, I read it up until it made my brain hurt, then I read something else for a while. I took a second go at it at some point (a few months later) and got further. Late last year, I picked up a hard copy of Leinster's book and started to do a proper study of it; things grew shaky over time, and somewhere around the material on presheaves I felt like I was just trying to memorize things I didn't understand. So...I'll take another pass at Leinster, sooner or later.

Other things: I also want to reach the point where I understand Prof. Urs Schreiber's work. Between where I am and where I'd need to be to do so, I think there's a bunch of algebraic topology, some algebraic geometry, the whole 'stacks' thing (that I know nothing about yet, generally speaking...but I've bookmarked the stacks project web site) and I'm not sure what all else. (Toposes, but that sort of falls under category theory.) I did a breezy read-through of Rotman's Intro to Algebraic Topology when I got a copy of it several months back, and hope to give it a more thorough read at some point this year. Then, maybe...Bott & Tu, Peter May's two books on algebraic topology, and that one green book on algebraic geometry. I also have a book on homotopy theory and one on k-theory that might be more approachable after Rotman. If all of that prepares one to start studying stacks, then I should be set.

And maybe one of these years I'll actually finish a BS in math and move on to graduate school. (Various 'situations,' namely a health problem and some family complications, are a bit of a wrench in those plans.) If not, then I'll just try to reach the point where I'm writing papers, putting them on arxiv, then nobody reads them because I'm a nobody. \(^^ )/ And then, the sweet embrace of death.

## Comments

Hi! I'm glad to hear my sufferings with quantum gravity have helped someone else avoid similar pain. I immensely enjoyed learning enough physics to understand the problem of quantum gravity... but solving it is a lot harder.

Category theory, on the other hand, is easy. It's all about "the tao of mathematics" - going with the flow, not cutting against the grain.

It's true that learning category theory makes ones brain hurt. I seriously believe that's the feeling of growing new neurons... just like serious exercise makes the muscles hurt as they grow new fibers.

The trick is to keep going back to it. What seems incomprehensible the first few times eventually becomes obvious.

So, welcome to the course!

`Hi! I'm glad to hear my sufferings with quantum gravity have helped someone else avoid similar pain. <img src = "http://math.ucr.edu/home/baez/emoticons/tongue2.gif"> I immensely enjoyed learning enough physics to understand the problem of quantum gravity... but solving it is a lot harder. Category theory, on the other hand, is easy. It's all about "the tao of mathematics" - going with the flow, not cutting against the grain. It's true that learning category theory makes ones brain hurt. I seriously believe that's the feeling of growing new neurons... just like serious exercise makes the muscles hurt as they grow new fibers. The trick is to keep going back to it. What seems incomprehensible the first few times eventually becomes obvious. So, welcome to the course!`

I've been pushing that very suggestion (about math -- and studying -- in general) for years! (And I've also lifted weights as my primary form of exercise since I was about 13, so I've had ample opportunity to ponder this.)

As a kid I was intrigued by the "various kinds of intelligence" talked about by certain neuroscientists, I believe it was. (I can't think of names, but hopefully they were all legitimate ones.) For a while during my teenage years, I took nearly all aspects of life as some soft of "cross-training." (After my teenage years I stopped taking this all quite so literally. But still...)

I had an interesting pair of run-ins with the idea. Let me first mention that I've played various instruments since I was a kid, and have been off-and-on serious about playing the piano. I also started dabbling in studying various languages towards the end of high school, although Japanese is the only (non-native) language that I have any kind of fluency with. But as for the run-ins...

1st run-in: At one point several years back, I was retaking calculus 2 after my progress in school had been interrupted for a few years. One day, I was doing some math homework (involving integrals, as I recall), and my brain was getting tired so I took a break; I sat down at the piano and opened some sheet music to a piece I was just barely starting to learn. Just moments into playing, my brain cramped up and essentially said, "What did I just say? I need a break from this." At that moment, I conjectured that the symbology-to-muscle-movement of sight-reading music was using the same 'muscles' in my brain as the symbology-to-muscle-movement that I had long felt was key to doing math problems. (Side note: The tasks of reading math, pondering math -- which for me usually means chatting away as if I were talking to someone -- or doing visualizations all seem to be different sets of brain muscles, for me.) So I closed the sheet music and worked on a piece that was already memorized and that was in the polishing phase, and my brain didn't object.

2nd run-in: Years back, I picked up the Rosetta Stone software for Japanese. One day, I had just been playing the piano before I started doing listening lessons in Rosetta Stone, and as I did a lesson I felt as though I was able to hear the language (distinguishing phonemes, that sort of thing) more clearly than usual. I had long suspected a connection (along the lines of the math homework/sight-reading connection) but that was the first time I felt that I had experienced it so distinctly. A little Google'ing suggested that this was a known phenomenon, so ultimately I made it part of my language study routine. Ultimately I concluded that it was "focused listening" of music that seem to grease the wheels of language; that kind of listening comes from either playing an instrument and listening to one's playing -- that is, not just mindlessly doing exercises -- or reading along with sheet music while listening to music.

A friend I met after high school and I were both a bit enamored with the idea of "brain cross-training," and at some point we dubbed math, music and language to be the "holy triumvirate of brain activation," as I believe we put it. (Meant to be tongue-in-cheek.) And math has always been the "heavy lifting" component.

-Steve

`>It's true that learning category theory makes ones brain hurt. I seriously believe that's the feeling of growing new neurons... just like serious exercise makes the muscles hurt as they grow new fibers. I've been pushing that very suggestion (about math -- and studying -- in general) for years! (And I've also lifted weights as my primary form of exercise since I was about 13, so I've had ample opportunity to ponder this.) As a kid I was intrigued by the "various kinds of intelligence" talked about by certain neuroscientists, I believe it was. (I can't think of names, but hopefully they were all legitimate ones.) For a while during my teenage years, I took nearly all aspects of life as some soft of "cross-training." (After my teenage years I stopped taking this all quite so literally. But still...) I had an interesting pair of run-ins with the idea. Let me first mention that I've played various instruments since I was a kid, and have been off-and-on serious about playing the piano. I also started dabbling in studying various languages towards the end of high school, although Japanese is the only (non-native) language that I have any kind of fluency with. But as for the run-ins... 1st run-in: At one point several years back, I was retaking calculus 2 after my progress in school had been interrupted for a few years. One day, I was doing some math homework (involving integrals, as I recall), and my brain was getting tired so I took a break; I sat down at the piano and opened some sheet music to a piece I was just barely starting to learn. Just moments into playing, my brain cramped up and essentially said, "What did I just say? I need a break from this." At that moment, I conjectured that the symbology-to-muscle-movement of sight-reading music was using the same 'muscles' in my brain as the symbology-to-muscle-movement that I had long felt was key to doing math problems. (Side note: The tasks of reading math, pondering math -- which for me usually means chatting away as if I were talking to someone -- or doing visualizations all seem to be different sets of brain muscles, for me.) So I closed the sheet music and worked on a piece that was already memorized and that was in the polishing phase, and my brain didn't object. 2nd run-in: Years back, I picked up the Rosetta Stone software for Japanese. One day, I had just been playing the piano before I started doing listening lessons in Rosetta Stone, and as I did a lesson I felt as though I was able to hear the language (distinguishing phonemes, that sort of thing) more clearly than usual. I had long suspected a connection (along the lines of the math homework/sight-reading connection) but that was the first time I felt that I had experienced it so distinctly. A little Google'ing suggested that this was a known phenomenon, so ultimately I made it part of my language study routine. Ultimately I concluded that it was "focused listening" of music that seem to grease the wheels of language; that kind of listening comes from either playing an instrument and listening to one's playing -- that is, not just mindlessly doing exercises -- or reading along with sheet music while listening to music. A friend I met after high school and I were both a bit enamored with the idea of "brain cross-training," and at some point we dubbed math, music and language to be the "holy triumvirate of brain activation," as I believe we put it. (Meant to be tongue-in-cheek.) And math has always been the "heavy lifting" component. -Steve`

You know, I've been noticing that the questions I've been asking myself have been growing more and more fundamental, ever since I started thinking more about category theory.

And the thing that drives me crazy is that they remind me of the sort of questions I used to ask when I was much younger...but, at the time, I had no idea that they were worthwhile questions. As a kid in San Jose (in California), I learned that asking the questions that came to mind were rarely going to welcomed by math teachers; they were usually either "off-topic" or "probably not something important." (And, on occasion, provoked hostility.) (Which, sadly, is why I basically tuned school out from the ages of about 12 to 17.)

At one point during my sr. year of high school, I was taking AP calculus AB, and asked a question in class, along the lines of, "When we take a derivative...is something changing? Like, something about the context in which the math was happening?" The teacher (who was a student teacher, although the regular teacher wasn't any more helpful) simply said "I have no idea" as a response. It wasn't until years later, when I was reading Tu's Intro to Manifolds, that he gave a short example at one point, about "...derivations as a functor from the category of pointed manifolds to the category of tangent spaces" (hopefully I got that right). It felt like an answer that had been a long time coming, and it was probably the thing that made me see category theory as the place to look for answers I had been wondering about.

I had picked up Tu's book as part of where my curiosity was taking me, after I first took linear algebra. I had long heard suggestions that linear algebra had "tremendous depth" and that it was "fundamental to all sort of things," but at that point in my life I had no idea what that meant. I had picked up a Dover book on linear algebra just after high school, but it was mostly matrix math, a statement of the vector space axioms, then an at-the-time-inexplicable chapter on eigenvectors and eigenvalues. Looking back, the book did give "polynomials as a vector space" as an example at one point, but at the time I saw that as little more than an interesting coincidence, rather than the "jumping-off point to the abstract side of math" that I would probably characterize it as, nowadays.

I was working full-time when I took linear algebra (as an evening class, at the community college near where I worked -- although that community college just happened to have on staff a fellow who had done his PhD under Bott, but sadly he wasn't the one that taught the linear algebra class I took...although I did have him for introductory DiffEq, and his insights are probably the thing the set me on the path away from "clueless about the abstraction" to towards studying math for real). Even at the end of the semester, I still had no sense of the abstract side of linear algebra (as that didn't come until the aforementioned DiffEq class). One thing that stuck with me, however, was how the work "linear" seemed to vanish (to some extent) from the math literature after linear algebra, which didn't make sense to me at the time, since it was purportedly so fundamental. (Mind you, this was in the mid/late 90s; I'm sure the internet would have given me a clearer picture, had it been then what it is today). I'm not sure what pointed me toward modern algebra as the next step, but I picked up a Dover book (Pinter's A Book of Abstract Algebra) and, before long, was seeing a generalization of linear algebra that was almost astonishing to me, at the time. (And, for a while, if asked where the word 'linearity' went, I would have responded "it starts being referred to as 'homomorphism,'" though I wouldn't phrase it that way nowadays.)

And then it was an off-hand remark from Prof. Andrejs Treibergs at Univ. of Utah (where I started going after I left my last job), when I asked him where to look to find out what tensors were, and he responded "oh, you want to look at differential geometry for that." (I think I had said something about wanting to understand them in the context of general relativity, but that would have only been because I had no idea what role they play in algebra, at the time.) And, for whatever reason, I wound up getting a copy of Tu's Intro to Manifolds as my "starting point towards differential geometry" (I think because Tu described it as being a book accessible to those with only some linear algebra experience, which sounded perfect for me at the time). And, as I mentioned, that eventually prompted me to start looking into category theory.

...which I could probably go on about, for a while. I'll stop, though. I didn't really mean for this to get so long-winded.

`>It's all about "the tao of mathematics" You know, I've been noticing that the questions I've been asking myself have been growing more and more fundamental, ever since I started thinking more about category theory. And the thing that drives me crazy is that they remind me of the sort of questions I used to ask when I was much younger...but, at the time, I had no idea that they were worthwhile questions. As a kid in San Jose (in California), I learned that asking the questions that came to mind were rarely going to welcomed by math teachers; they were usually either "off-topic" or "probably not something important." (And, on occasion, provoked hostility.) (Which, sadly, is why I basically tuned school out from the ages of about 12 to 17.) At one point during my sr. year of high school, I was taking AP calculus AB, and asked a question in class, along the lines of, "When we take a derivative...is something changing? Like, something about the context in which the math was happening?" The teacher (who was a student teacher, although the regular teacher wasn't any more helpful) simply said "I have no idea" as a response. It wasn't until years later, when I was reading Tu's Intro to Manifolds, that he gave a short example at one point, about "...derivations as a functor from the category of pointed manifolds to the category of tangent spaces" (hopefully I got that right). It felt like an answer that had been a long time coming, and it was probably the thing that made me see category theory as the place to look for answers I had been wondering about. I had picked up Tu's book as part of where my curiosity was taking me, after I first took linear algebra. I had long heard suggestions that linear algebra had "tremendous depth" and that it was "fundamental to all sort of things," but at that point in my life I had no idea what that meant. I had picked up a Dover book on linear algebra just after high school, but it was mostly matrix math, a statement of the vector space axioms, then an at-the-time-inexplicable chapter on eigenvectors and eigenvalues. Looking back, the book did give "polynomials as a vector space" as an example at one point, but at the time I saw that as little more than an interesting coincidence, rather than the "jumping-off point to the abstract side of math" that I would probably characterize it as, nowadays. I was working full-time when I took linear algebra (as an evening class, at the community college near where I worked -- although that community college just happened to have on staff a fellow who had done his PhD under Bott, but sadly he wasn't the one that taught the linear algebra class I took...although I did have him for introductory DiffEq, and his insights are probably the thing the set me on the path away from "clueless about the abstraction" to towards studying math for real). Even at the end of the semester, I still had no sense of the abstract side of linear algebra (as that didn't come until the aforementioned DiffEq class). One thing that stuck with me, however, was how the work "linear" seemed to vanish (to some extent) from the math literature after linear algebra, which didn't make sense to me at the time, since it was purportedly so fundamental. (Mind you, this was in the mid/late 90s; I'm sure the internet would have given me a clearer picture, had it been then what it is today). I'm not sure what pointed me toward modern algebra as the next step, but I picked up a Dover book (Pinter's A Book of Abstract Algebra) and, before long, was seeing a generalization of linear algebra that was almost astonishing to me, at the time. (And, for a while, if asked where the word 'linearity' went, I would have responded "it starts being referred to as 'homomorphism,'" though I wouldn't phrase it that way nowadays.) And then it was an off-hand remark from Prof. Andrejs Treibergs at Univ. of Utah (where I started going after I left my last job), when I asked him where to look to find out what tensors were, and he responded "oh, you want to look at differential geometry for that." (I think I had said something about wanting to understand them in the context of general relativity, but that would have only been because I had no idea what role they play in algebra, at the time.) And, for whatever reason, I wound up getting a copy of Tu's Intro to Manifolds as my "starting point towards differential geometry" (I think because Tu described it as being a book accessible to those with only some linear algebra experience, which sounded perfect for me at the time). And, as I mentioned, that eventually prompted me to start looking into category theory. ...which I could probably go on about, for a while. I'll stop, though. I didn't really mean for this to get so long-winded.`

It's fun to read someone thoughtfully reflecting on a life of learning math and playing music - two of my favorite activities as well, though lately I've slacked off on playing music and mainly just

imagine it in my head.I haven't spent much time seriously trying to learn languages since high school. Basically math and physics eat up all my brain power, because having put so much work into them they are much more rewarding than any other form of thought, for me: it's pretty easy to come up with really cool new discoveries, and that's very addictive.

Yes, I think "linear maps" in linear algebra become "homomorphisms" when you go to modern algebra and then simply "morphisms" when you go to category theory. I bet you'll have a lot of fun with category theory.

`It's fun to read someone thoughtfully reflecting on a life of learning math and playing music - two of my favorite activities as well, though lately I've slacked off on playing music and mainly just _imagine it in my head_. I haven't spent much time seriously trying to learn languages since high school. Basically math and physics eat up all my brain power, because having put so much work into them they are much more rewarding than any other form of thought, for me: it's pretty easy to come up with really cool new discoveries, and that's very addictive. Yes, I think "linear maps" in linear algebra become "homomorphisms" when you go to modern algebra and then simply "morphisms" when you go to category theory. I bet you'll have a lot of fun with category theory.`