**from the**

**how-to-learn-mathematical-thinking**dept.Technology Review is running an unusual book review -- books about learning math, pure math, not applied math, https://www.technologyreview.com/2023/04/24/1071371/book-reviews-math-education/

The author admits to being adrift,

As a graduate student in physics, I have seen the work that goes into conducting delicate experiments, but the daily grind of mathematical discovery is a ritual altogether foreign to me. And this feeling is only reinforced by popular books on math, which often take the tone of a pastor dispensing sermons to the faithful.

An initial attempt led to a MasterClass by a "living legend of contemporary math", but the master is seated in a white armchair with no blackboards, pens or paper and does not enlighten.

A side story covers a writer for the New Yorker who plans a year to go back and learn the high school algebra/geometry/calculus that escaped him, but mostly fails. For backup he has a niece who is a math professor...but after months without getting it, he complains. Her answer?

"For a moment, think of it as a monastic discipline. You have to take on faith what I tell you." Where his niece and others see patterns and order, he perceives only "incoherence, obfuscation, and chaos"; he feels like a monk who sees lesser angels than everybody around him.

I won't spoil the end, but the author does make some progress with books by mathematician and concert pianist Eugenia Cheng, starting with "Cakes, Custard and Category Theory", where each chapter starts with an analogy to baking.

Unfortunately, for the SN audience, the article does not include any car analogies...

**The Fine Print:**The following comments are owned by whoever posted them. We are not responsible for them in any way.

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## (Score: 5, Interesting) by JoeMerchant on Sunday June 11 2023, @02:35PM (25 children)

>think of it as a monastic discipline. You have to take on faith what I tell you

My high school geometry teacher put it this way:

"You can't use that for a proof because I haven't taught that yet. It doesn't matter if it's true or not, we do this the same exact way every time."

Then at some point she proceeded to skip ahead and start using theorems that she hadn't taught, because that's what she wanted to do.

One lesson of many in: grades indicate how well you follow instructions, repeat what your are told and toe the line. Thinking is not only optional, but actually discouraged.

Then, if you have reached the pinnacle, regurgitated all the litanies correctly, you are permitted to attempt to show something new that builds on top of all that exists. And, like the einstein tile guy, be prepared for the world to piss in your cornflakes when you do show them something new. I wonder if he had the non-flipping einstein in his back pocket and just released the flipper first to see what would happen?

🌻🌻 [google.com]

## (Score: 2) by krishnoid on Sunday June 11 2023, @03:24PM (1 child)

I hope someone called her out on that and meted out a proportionate punishment [youtu.be], possibly with follow-ups if she continues in her intransigence. But seriously, having to answer for that inconsistency (at least asking after class is over) would probably reveal something about educational methods for a classroom of multiple skill levels, or maybe just about rigor in delivering the material and method -- considering "we do this the exact same way every time".

Parent## (Score: 2) by JoeMerchant on Sunday June 11 2023, @05:02PM

She was teaching high school 10th graders, she looked to be in her late 50s / early 60s, shaped like a tomato with a tight belt on. In 1982 her eyeglasses were high fashion from 1958. She was clearly there for the paycheck, and a little power tripping, nothing more.

🌻🌻 [google.com]

Parent## (Score: 5, Insightful) by VLM on Sunday June 11 2023, @03:26PM (2 children)

School is seen as vocational training for the workplace.

Parent## (Score: 2) by JoeMerchant on Sunday June 11 2023, @07:14PM

My workplace actively encourages us to innovate, both products and our internal processes.

The real test is continuing to do what is required to collect the paychecks as the decades roll by and you watch 90% of the innovations sit on the shelf while the most active 5% of process innovations swing back and forth in one reorganization after another.

It's nice to be non-redundant. I keep trying to cross train my skills to others but the more I do that the more essential I seem to become.

🌻🌻 [google.com]

Parent## (Score: 0) by Anonymous Coward on Monday June 12 2023, @12:33AM

Sounds correct.

Shut up. Sit at a desk. Accept you are being graded. When the angry man at the front yells at you, bow your head. "Because I say so" is all the motivation you need.

Parent## (Score: 2) by krishnoid on Sunday June 11 2023, @03:52PM

They won't if you show them something just new enough [youtu.be].

LOL, passive-aggression in modern mathematics. Nice.

Parent## (Score: 5, Interesting) by Anonymous Coward on Sunday June 11 2023, @04:03PM (6 children)

We certainly had different high school math experiences. My geometry teacher was friendly and let several of us better students sit in the back. She posted the next day's assignment at the beginning of class--and the four of us in the back would take the first bit of class to get that done (so we rarely had homework to do at home). Then we quietly passed notes back and forth while the others in the class took in her lecture. She saw exactly what was going on and let us goof off (quietly) because she also knew that we understood the material.

Many years later I read her obit. Turns out she was a PhD math major at a top university and had done some original math research--but was locked out of an academic career by the glass ceiling for women back then. She never discussed her misfortune, although she had every reason to be bitter.

Now, AP Calculus (senior year high school) was another matter all around. It was taught by the football coach who was (by his own admission) barely qualified. He more or less said, "Here's the book, go to it." At that point I really needed some help and didn't get it. The result is that, while I can intellectually make the jump from the slope of a small interval to an infinitesimal (differential, for one example), I never really believed it. Without that jump, the motivation to understand the rest of it wasn't there.

Parent## (Score: 5, Interesting) by HiThere on Sunday June 11 2023, @04:51PM (4 children)

It's not really clear that you SHOULD believe in continuity, infinitesimals, infinities, etc. They follow from a certain set of assumptions, but I, personally, don't believe them. I believe that the universe is discontinuous at around the Planck length. This has certain problems in it's own right, but they are different problems than the ones that continuity entails. (And I think Cantor's red and blue number lines is just goofy. You can't assign colors to numbers, only to a more complex structure. What he really did was create two instances of the same [infinite] set, but then his reasoning isn't about the set, but rather about instances of the set. Either that, or he was creating a struct {int value; color bool;} and reasoning about that rather than about integers.)

Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.

Parent## (Score: 1) by shrewdsheep on Monday June 12 2023, @05:02PM (3 children)

Mathematics makes no claims about the universe. The deep point about calculus is about limits. Limits mean that you assign a single number to an infinite sequence of numbers. As a matter of fact most numbers are defined as sets of infinite sequences (equivalence classes). Mathematics is a game that allows you to manipulate sequences of symbols - no more, no less.

Parent## (Score: 1, Interesting) by Anonymous Coward on Monday June 12 2023, @08:02PM

> Mathematics is a game that allows you to manipulate sequences of symbols - no more, no less.

Interesting comment, thanks. If true, then my difficulty with math is common with my lack of interest in most games. I just don't get why most people like games.

Parent## (Score: 3, Interesting) by HiThere on Monday June 12 2023, @09:11PM (1 child)

Yes. And that's why I said "they follow from a certain set of assumptions". I'm not denying that they *do* follow from those assumptions, just that the assumptions properly describe the universe. (They do come pretty close, though.)

Javascript is what you use to allow unknown third parties to run software you have no idea about on your computer.

Parent## (Score: 4, Insightful) by hendrikboom on Tuesday June 13 2023, @01:48AM

And that's the difference between math and physics.

And that's why physicists still use math.

Parent## (Score: 3, Insightful) by JoeMerchant on Sunday June 11 2023, @05:06PM

The teacher makes all the difference. If you can teach yourself, with enthusiasm, that's a teacher you never lose (at least not until senility sets in...)

🌻🌻 [google.com]

Parent## (Score: 3, Interesting) by mhajicek on Sunday June 11 2023, @04:49PM (4 children)

I had a very similar experience in high school geometry. Also, for me, I need to know an application in order to be motivated and really understand something. When I asked the teacher how this might be used in the professional world, she said "Trust me, you'll use it."

Then I went to tech school for machining, and I had applications for everything I learned. Vector math was a breeze, because I knew it was needed for polar coordinate programming on a live-tool lathe, for example.

The spacelike surfaces of time foliations can have a cusp at the surface of discontinuity. - P. Hajicek

Parent## (Score: 3, Interesting) by JoeMerchant on Sunday June 11 2023, @07:21PM (3 children)

I don't know that I have ever used high school geometry type of proofs. I know that I have never used path integrals around a singularity despite studying them for two semesters. I sort of forced diffeqs into service a couple of times but could easily have ignored them.

Integrals: yes, definitely, particularly discrete summations. Trigonometry: a lot. Discrete mathematics proofs by induction and similar... Not really.

🌻🌻 [google.com]

Parent## (Score: 1, Funny) by Anonymous Coward on Sunday June 11 2023, @10:18PM

Q: Why did the mathematician name his dog Cauchy?

A: Because it would leave a residue at every pole.

Parent## (Score: 2) by aafcac on Sunday June 11 2023, @11:47PM

They are a complete and utter waste of effort at that point. It was mostly a matter of being able to refer to various proofs in the book and those mostly didn't have standardized names, so if you weren't working out of the same book, you probably wouldn't be able to understand the proofs. It's wonderful that there's been more focus on word problems, as those at least attempt to deal with a real problem people have. It's rare to be handed an equation to be solved, usually, you have to write and solve your own problem. The word problems in math tend to be rather artificial, but there's not much way around that if you're expecting to not require methods that haven't been taught.

Parent## (Score: 0) by Anonymous Coward on Monday June 12 2023, @12:39AM

Well it's all fucking irrelevant now we have rockstar CEO scientists that sell their visions to funding agencies and oversee a sweatshop of grad students on visas under threat of failing to meet the standards of a committee full of CEO mentors.

Parent## (Score: 4, Interesting) by RamiK on Sunday June 11 2023, @05:18PM (4 children)

A professor in uni forced me to retake a test over using (yet to be taught) Laplace transforms to solve a test in ordinary (surely it wasn't complex ones as that would have been unavoidable? can't recall...) differential equations. So, in the redo, I developed the transforms I needed from the axioms that were on the material and they were forced to accept the answer.

I don't know how your teacher would have responded to that but I'm guessing they wouldn't have objected too fiercely...

compiling...

Parent## (Score: 3, Interesting) by JoeMerchant on Sunday June 11 2023, @05:42PM (2 children)

I took every (Calculus II and up) math class offered in my uni math department, then I took one more from the physics department that taught three subjects: Green's functions (a way to solve some differential equations) something about closed path integrals around singularities, and something else about differential equations. The final exam counted as 60% of the final grade, and was pre-announced to be: select 5 of 11 questions. I never did learn how to use Green's functions, but probability told me that the likelihood of 6/11 questions being on Green's functions was slim. The prof also handed us sample previous exams to study, which I did.

Exam day arrives and 5/11 questions explicitly stated "using Green's functions..." and one of the remaining questions was a "solve this differential equation" which could have used Green's functions. However, I recognized the equation as being a simple transform of a question from a previous exam, so I proved my "solution by inspection/recognition" and flew through the other four questions. 5/5 perfect answers, and yet I still received a B...

Seems that the prof saw through my shortcut and, having failed to demonstrate knowledge of Green's functions 100% earns a B.

🌻🌻 [google.com]

Parent## (Score: 2) by RamiK on Sunday June 11 2023, @05:52PM (1 child)

I think you sorta vindicated your high-school teacher there... :D

compiling...

Parent## (Score: 3, Funny) by JoeMerchant on Sunday June 11 2023, @06:34PM

By the time I was taking Mathematics for Physics 517, I had learned how little the difference between an A grade and a B matters - to me, to my future, etc. I certainly enjoyed waterskiing on Tuesdays and Thursdays much more than mastering Green's functions could ever have done for me.

🌻🌻 [google.com]

Parent## (Score: 2) by bzipitidoo on Monday June 12 2023, @03:14PM

I found the approach of deriving things took too long for tests. The timed format forces the student to memorize a lot more and consequently understand less, in order to be fast enough to finish the tests.

Math is one of my favorite subjects, because it is objective. The right answer is the right answer. Teachers who hate smart students have much less room for subjective bull excuses to justify giving a good student a bad grade. They can still do some crap ("you didn't show your work!") but as your anecdote shows, that's harder.

Parent## (Score: 1) by khallow on Monday June 12 2023, @01:48AM

All the einstein tile guy had to do was deliver what they promised. It wasn't a high bar. And note that they actually did some weeks later.

Parent## (Score: 1, Touché) by khallow on Monday June 12 2023, @04:23AM

Parent## (Score: 2) by VLM on Sunday June 11 2023, @03:24PM (4 children)

I don't know if copyrighted expensive books are the right way to learn math in 2023. Too many out of copyright or cheap books. Too many youtube videos of lectures some by pretty famous names. Too many "big name" online courses. Too many wikipedia and mathworld and similar competitors. Too many AI type services to ask questions and get a questionable answer in response.

## (Score: 3, Interesting) by looorg on Sunday June 11 2023, @05:10PM (3 children)

A good book is hard to replace. If nothing else you could get some rework or translation of Elements by Euclid. That should have expired copyright wise a while back since it's a bit over 2000 years old. The reworks might not tho. I wish they would have used that in elementary school instead, I didn't get to learn about it until university. It opened a lot of eyes on how things worked and why instead of just "use formula X, put in the numbers and compute the answer ... you don't need to know why!".

Parent## (Score: 1) by pTamok on Sunday June 11 2023, @08:21PM (1 child)

Translation?

One of the more interesting websites is one that aims to teach you how to read Euclid in the original Greek. There are times I wish I had more than one life so I could do all the things I think are interesting.

Reading Euclid: This course combines Greek and Geometry to show how to read Euclid's Elements in the original language. [du.edu]

Parent## (Score: 2) by looorg on Sunday June 11 2023, @08:31PM

Yes. It had to be a translation. It was a rework of his work. I can't tell if it explained things better then Euclid or not but it was a real eyeopener as far as geometry and axiomatic theory was concerned.

Right. If only I had the time to spare. That does appear to be quite interesting. Bookmarked for future reference and reading. If only the day had more hours ... There is always retirement, decades away if ever ...

Parent## (Score: 2) by VLM on Monday June 12 2023, @12:34PM

Yeah there's good books but there's no good new books, no textbook written by boomers or younger is worth buying, generally.

There's an entire business model around "I'm going to teach linear algebra as a new prof, so I'll write a new textbook, charge at least $200 for it, and require my students to buy it to make a stack of cash in a corrupt manner". On the other hand the best linear algebra book you could buy for undergrads, at least recently, was probably Strangs book. So just buy Strang's book for $75 and call it good. No need to make some lesser known professor writing a lower quality book an extra $125. His students have to suffer and pay an extra $125 but no one else has to suffer.

The main determinant (oh the bad puns) of success at learning linear algebra or anything else is the willingness to put in the time and sweat, not the name of the author on the book, anyway. So just get a cheap old book, ideally the best book of the last half of the previous century.

Parent## (Score: 3, Funny) by Mojibake Tengu on Sunday June 11 2023, @03:25PM (2 children)

Recently, I was told "everything is a hypergraph".

So, "everything is a category" is out, hypergraphs are just in.

I didn't object against it. Incidence matrices are fun, coding them on recent architectures is fun too.

Though I can provide a car analogy for soylentils: Suppose you have some cars. Some of them are red, some are Ford, some are electric, some broken. That's a hypergraph.

Rust programming language offends both my Intelligence and my Spirit.

## (Score: 0) by Anonymous Coward on Sunday June 11 2023, @04:07PM (1 child)

The hypergraph sounds like tags in Gmail -- some posts are archived as is, others have one or more tags added to them before archiving. Tags can be any short string. Search includes, "All posts with this tag."

For comparison, how about a car analogy to category theory?

Parent## (Score: 2) by Mojibake Tengu on Sunday June 11 2023, @04:23PM

In category theory, you just have a category of cars. That's all...

Rust programming language offends both my Intelligence and my Spirit.

Parent## (Score: 3, Insightful) by looorg on Sunday June 11 2023, @05:16PM (1 child)

https://xkcd.com/435/ [xkcd.com]

The old classic XKCD Purity cartoon. If you split the mathematician character in two it would make the strip much wider as the pure math people would be far far away from the rest of us. Those people are out there. While I do have a fairly extensive university education in math, it's main applied math. The pure math people still make my head hurt from time to time when I just think about it.

## (Score: 2) by ChrisMaple on Tuesday June 13 2023, @04:23AM

The problem with pure math is that people keep finding ways to apply it.

Parent## (Score: 4, Informative) by istartedi on Sunday June 11 2023, @05:45PM (3 children)

There are actually some really good YouTubers who make "pure math" interesting. I watched a lot of them during the pandemic. What is pure math anyway? You never know what applications might emerge. Boolean algebra, for example, was invented well before any kind of practical computer and probably had very limited if any applications. Today there's a bazillion Boolean algebra calculations going on in your pocket.

For starters, go ahead and type "math is incomplete" in there. I forget which one I watched that introduced me to this whole idea of assigning a number to equations (and this type of encoding leads to the assignment of astronomical numbers), and the bizarre conclusion that not only must there be things we don't know, there must also be things of which we are uncertain!

I'm probably not explaining that properly. It's been a couple years since I watched it now...

Appended to the end of comments you post. Max: 120 chars.

## (Score: 2, Insightful) by pTamok on Sunday June 11 2023, @08:28PM (1 child)

That's Gödel's Incompleteness Theorems [wikipedia.org].

A fun read that gives a feel for Gödel is

Gödel, Escher, Bach: an Eternal Golden Braidby Douglas Hofstadter [wikipedia.org]. I've lent out two copies, and never got them back. Which is kind of OK, but means I need to buy another for my own enjoyment (I like re-reading books).Parent## (Score: 2) by istartedi on Sunday June 11 2023, @09:18PM

IIRC, it came out in the 80s. I seem to remember it being heavily advertised in the science magazines I read back then. I had a feeling it would go over my head, and as a tween/teen it probably would have. I should probably give it a read some day, now that a bunch of stuff like calculus is in the rear-view mirror.

Appended to the end of comments you post. Max: 120 chars.

Parent## (Score: 2) by VLM on Monday June 12 2023, @12:48PM

This will probably be considered handwavingly bad, but "everyone knows" there's theorems WRT computer science that the fastest way to prove if a program will halt is to run it and watch. There's no "algebra" or "compiler optimization" that is guaranteed to turn all terminating programs into a single NOP instruction (although there's nothing stopping it from working 99.9999% of the time, just to be optimistic)

So if you wrote a perfect computer algebra system that searched for and output proofs all day, it turns out that is doubly cursed in that there's no way to prove the program will ever terminate and return a result AND there exist finite statements you can attempt to prove and the proof search will never terminate it'll search forever and never make a decision its provable or not. The universe of what we CAN prove seems to be very large, infinite, probably, but there are unprovable math statements, which is kind of depressing. Before 1900 or so people really thought a powerful enough computer or powerful enough list of integration equations would literally be able to prove every possible math statement one way or the other. Whoops guess not.

The second Godel theorem is an extension of the above, if there are unprovable statements in a system, then you can't prove a set of rules for that system are adequate.

There's a lot of handwaving about "sufficiently complicated system" where its a "no brainer" to prove a pile of NAND or NOR gates can emulate any other gate or combo of gates, but a complicated enough system like an imaginary theoretical "prove the sound of a tree falling in a forest with no one to hear it" or "prove the existence of God using lots of 7400 series logic chips" can't be proven...

The output of a true random number generator executed as code can be mathematically proven to choke a general purpose computer or a system of math axioms, which is interesting. Not just "the odds are bad" but there are abstract mathematical proofs a RNG will break those calculation systems.

Parent## (Score: 5, Insightful) by Rosco P. Coltrane on Sunday June 11 2023, @06:19PM (4 children)

The problem with math, physics, geography, history, CS or anything else taught in school is this: you only see the beauty of it after years or decades of drilling down into the particular subject matter.

Take history for example: what you're taught in school is essentially a litany of dates and events with no apparent connections between them. Most history cursi try to explain the how and why events are connected to some extent, and some history teachers are actually fairly good at it. But essentially, history is a boring-ass subject. But as you age, as you read more about the nitty gritty details of history, suddenly all those disconnected events make sense as a complete whole.

And then that knowledge of history spills over into the arts: it gives meaning to the boring-ass works of literature you were forced to read in high-school: you suddenly realize why an author wrote what he wrote at the time they wrote it in the style they wrote it. It gives background and meaning to the works. Music too often makes sense with regard to the era and the circumstances that gave rise to certain styles.

Or take CS: my teachers in U tried to teach me that newfangled object-oriented programming thing that was all the rage back then. I hated it. It literally took me decades to understand how elegant and simple OO really is, and how utterly fucked up almost all implementations of it are and that's the real reason why OO sucks.

That's the curse of schools: what they teach is godawful, boring, nasty and stressful. And yet, if you ever want to see the beauty of anything they teach, they have to forcibly and aggressively pump your head head full of obtuse and boring shit for years. And I'm not certain there's any other way to go about giving someone an education in a finite number of years.

So yeah, I bet the author of that book is already at the point where she totally sees the beauty of pure math. And I can totally relate to her need to share the joy. But I'd be susprised if more than a handful of readers get the revelation and suddenly see the beauty she sees. Still, cool effort.

## (Score: 2) by JoeMerchant on Sunday June 11 2023, @10:36PM (3 children)

>That's the curse of schools: what they teach is godawful, boring, nasty and stressful.

Doesn't have to be. I've had both kinds of teachers / professors. Mostly like you describe, but also the occasional gem who shows the beauty and makes you want to learn the fundamentals underpinning it. Some have it easy with the subject matter, like Marine Biology, but I've seen it done for math (differential equations, in particular) too.

🌻🌻 [google.com]

Parent## (Score: 2) by aafcac on Sunday June 11 2023, @11:51PM

It doesn't have to be, but we're too cheap to pay for teachers that have the necessary knowledge to even attempt that. So often they're teaching stuff that is just below their highest level of education and not thinking too deeply about it because there's a ton of things that they're expected to to.

Parent## (Score: 0) by Anonymous Coward on Monday June 12 2023, @12:49AM (1 child)

I think by the time you get to grad school, the lessons are more about whether the prof even knows his subject or is bullshitting and/or getting a psychological kick out of crushing your soul. All the faculty seem to do is namedrop about social connections and how it's all about "soft skills", while being 95% useless at hard skills. I think at some point in the last 20 years the part where a researcher actually had to become expert got skipped in favor of wearing a suit and switching their language to contracts and deliverables.

Parent## (Score: 2) by JoeMerchant on Monday June 12 2023, @03:10AM

Don't forget to marry money, the real rock PhDs I know married money then attempted to prove their worthiness through some entrepreneurial venture. The ventures were a crap shoot, but the inlaw inheritance worked every time.

🌻🌻 [google.com]

Parent## (Score: 3, Interesting) by AlwaysNever on Sunday June 11 2023, @10:05PM (1 child)

I was good at math in school, as in I aced the exams where you applied learned theorems or techniques to solving math problems.

I, however, never understood math as in what it means and how it relates to things. I was just good at doing math exams after dedicating hours to drill example math problems.

Math remains an impenetrable mystery to me.

## (Score: 2) by JoeMerchant on Sunday June 11 2023, @10:44PM

In grad school I took an analog filters design class ~12 students, 10 who basically copied the verbal lecture presentation into notes and hoped to interpret them later, and two of us who actually kept up with the prof and could really follow along with his presentation.

One day the prof stopped and asked: "So what's the gain at the center frequency in this case?" and it was some godawful complex design with multiple stages and I just kind of estimated the stages and came to "5" which I said out loud. My counterpart was more methodical, cranking out the exact answers doing the integrals on the fly with pencil on paper, prof waited for him about 30 seconds as he scratched away and finally said: 4.98... yeah, five.

I always figured there would be computers and calculators for that stuff, better to be able to judge when the computer is wrong (meaning: somebody screwed up something in the inputs.) My physics prof in 1985 had been teaching this for years: when you solve a physics problem, don't solve the problem using numbers, derive the equation and put the numbers in at the end. That way, you can play with the inputs: if I double the mass does the result react like I expect? How about cutting the distance in half? etc. It's way too easy to drop a negative sign somewhere, regardless of which method you use, but combining the numbers along the way to the answer erases your ability to check the result near the end.

🌻🌻 [google.com]

Parent## (Score: 3, Insightful) by khallow on Monday June 12 2023, @04:15AM

Math is unique in that everything is certified - usually highly rigorously. Sure, to the ignorant it may seem like a church, but it's not. The only reason this writer had to take it on faith, was that he couldn't understand the certifications. And his perceptions are merely the typical perceptions of the innumerate. It's not a reflection of what math is.

## (Score: 4, Insightful) by aliks on Monday June 12 2023, @07:22AM

and it was god awful boring for years.

Just pumping iron. Again and again.

I learnt nothing about why these wierd moves could be useful. It was only years later when my body was properly balanced up that I realised how it all fitted together.

And I went down to the Language Lab to learn French. and it was god awful boring for years.

Just memorising grammar patterns and repeating sentences. Again and Again.

Nothing I was learning was remotely useful in everyday life and holding a conversation was impossible.

It was only years later than it started to click and I could go on holiday in France and enjoy myself.

So I got interested in Photography. Hours clicking rubbish photos. Again and again.

It was only years later . . . well I think you get my point.

To err is human, to comment divine

## (Score: 3, Informative) by hendrikboom on Tuesday June 13 2023, @02:44AM (2 children)

The Joy of Abstraction is a lovely book, written by a lovely author.

She says that she spent years working through an advanced degree in mathematics, and she learned a lot about a lot of things.

But it wasn't until near the end of her degree work that she encountered category theory and then all the stuff she had already learned fell into place and made sense as a coherent whole.

Then she wished that she had have learned about category theory at the beginning of her study of mathematics instead of at the end.

So she decided to teach category theory

withoutrequiring a background in abstract algebra, topology, functional analysis, differential equations, etc.And the book "Joy of Abstraction" resulted from that effort, including a stint of teaching category theory to a college of art.

The book starts with her explanation of how and why she ended up writing the book, and then expounds on how mathematics works, always exemplified by everyday examples.

Gradually she gets to the idea of what a category is.

And she shows a large variety of different kinds of categories.

A category is made up of objects and arrows that satisfy certain properties, but there is a huge variety of different kinds of things that can serve as these objects and arrows, making the theory applicable to a wide variety of situations. (it's often advertised as a way to convert problems in one field of mathematics to other fields of mathematics where they may be easier to solve. But I think it's more than that.)

And then, about halfway through the book she starts the technical development of category theory itself, with precise definitions, but always relating these definitions to the intuitive concepts she provided before.

What's more, she has set up an online book club [topos.site], which is still running as I write this. Every week or so she covers another chapter of the book. Members of the club read the chapter and send her questions about the content. Then she makes a youtube video in which she answers questions.

Look, I'm a mathematician who has spent most of his life messing with computers. I've looked at the theory behind anything from logic to quantum mechanics. Despite my seven decades of mathematical life, her answers in the youtube videos still occasionally give me new insights.

When you get through the book, you will know the basics of what category theory is and why it is useful.

-- hendrik

## (Score: 2) by hubie on Tuesday June 13 2023, @04:25AM (1 child)

Thank you. I have an open Amazon browser tab where I've been trying to figure out what to spend gift card money on and you've sold me on this. :)

Parent## (Score: 2) by hendrikboom on Tuesday June 13 2023, @03:02PM

You are quite welcome.

Parent