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Comment by mbbbackus | Hacker News Reader

I've been reading the author's book, Mathematica, and it's awesome. The title of this post doesn't do it justice.

He shows that math skill is almost more like a sports talent than it is knowledge talent. He claims this based on the way people have to learn how to manipulate different math objects in their heads, whether treating them as rotated shapes, slot machines, or origami. It's like an imagination sport.

Also, he inspired me to relearn a lot of fundamental math on MathAcademy.com which has been super fun and stressful. I feel like I have the tetris effect but with polynomials now.

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Sounds really cool.

It reminds me of programming, that moment when new code starts to really sync up and code goes from being a bunch of text to more intuitive structures. When really in the zone it feels like each function has its own shape and vibe. Like a clean little box function or a big ugly angry urchin function or a useless little circle that doesn't do anything and you make a note to get rid of. I can kinda see the whole graph connected by the data that flows through them.

There's a lot of interesting discrete math that can supercharge programming at different levels of scale. What's pretty cool is that it reveals a layer of understanding when I watch my toddlers learn math from counting.

One of the interesting things is being able to exactly describe how something is an anti-pattern, because you have a precise language for describing constraints.

I would love to learn about some of these anti-pattern proofs if you have any examples or references you can share!

I don't have proofs. I haven't gone into any level of rigor with any of this.

Here is a recent example: https://files.spritely.institute/papers/petnames.html

The idea of a naming system can be (1) decentralized, (2) globally unique, (3) human meaningful. It talks about the onion DID names which achieves decentralized and globally-unique, and proposes a petname system that maps local names to achieve all three when combined with the onion names.

It sounds similar to me to the mathematical concept of an atlas. Atlases originally came about trying to map a non-Euclidian topology to a local, Euclidian topology. No Euclidian topology can fully describe the non-Euclidian topology, but a set of those can, and together would form an atlas.

Someone with more math chops than I can prove (or disprove) that the petname system forms an atlas over the set of globally-unique names (identifiers). The biggest anti-pattern I can see coming out of it is when people using this attempt to make the local petnames globally unique instead of working with it as a local mapping that can never fully describe the global space of unique names.

> rotated shapes, slot machines, or origami

Or gears (like Seymour Papert), or abacus beads, or nomograms, or slide rules, etc etc. Anyone have any more, throw them out!

Is mathacademy good? I have been thinking of giving it a month of a try. You say "stressful", which I'm not sure is a mis-type or not.

I ordered Mathematica at my local library by the way, and can now forget about it until I get an SMS one day informing me of its arrival. Thank you for confirming that it's worth it!

I've had a MathAcademy subscription for some time and it's quite good. I'd say it's best at generating problems and using spaced repetition to reinforce learning, but I think it falls short in explaining why something is useful or applicable. I don't know, most math education seems to be "here's an equation and this is how you solve it" and MathAcademy is undoubtedly the best at that, but I wish there were resources that were more like "here's how we discovered this, what we used to do before, why it's useful, and here's some scenarios where you'd use it."

I have so wanted such resources for years. I have found some and should make a list.

The first time the difference between understanding some math, and understanding what the math meant, was after high school Trig. The moment I started manually programming graphics from scratch, the circle as a series of dots, trigonometry transformed in my mind. I can't even say what the difference was - the math was exactly the same - but some larger area of my brain suddenly connected with all the concepts I had already learned.

While ordering the "Mathematica: A Secret World of Intuition and Curiosity" I came across these books, which looked very promising in the "learning formal math by expanding intuition" theme, so I bought them too:

Field Theory For The Non-Physicist, by Ville Hirvonen [0]

Lagrangian Mechanics For The Non-Physicist, by Ville Hirvonen [1]

The Gravity of Math: How Geometry Rules the Universe, by Steve Nadis, Shing-Tung Yau [2]

Vector: A Surprising Story of Space, Time, and Mathematical Transformation, by Robyn Arianrhod [3]

[0] https://www.amazon.com/dp/B0CN7HMTJN

[1] https://www.amazon.com/dp/B0CN7HMK38

[2] https://www.amazon.com/dp/1541604296

[3] https://www.amazon.com/dp/0226821102

Excited to read each (based on their synopses & ratings), and if I will get compounding fluency across both math and physics between all five books.

Burn Math Class follows that tradition, although it starts pretty basic, so it requires some patience.

https://a.co/d/fZnWUU8

If you're interested in how vector calculus developed, and who was instrumental, all the way from Newton/Leibnitz to Dirac or so, by way of Hamilton, Maxwell, Einstein and others, then Robyn Arianrhod's 'Vector' is brilliant.

But be warned, it gets progressively harder, along with the concepts, so unless you're conversant with tensors, at some point you will have to put on your thinking cap.

The reviews on Goodreads – including my own – are worth reading to get a flavour: https://www.goodreads.com/book/show/202104095-vector

I really want to try MathAcademy.com. How quickly do you think someone doing light study could move from a Calc 1 -> advanced stuff using that site? In my case I could put in at least 30 minutes to an hour a day.

I can't speak to the advanced stuff but here's my stats on Fundamentals I:

Total time on site (gathered from a web extension): 40h 30m Total days since start: 32

Total XP earned: 1881

Since "1 XP is roughly equivalent to 1 minute of focused work", I "should have" only spent 31 hours. I did the placement test and started at ~30%, and now I'm at 76%. I'd say 75% is stuff I learned in HS but never had a great handle on, 25% I never knew before.

Overall, I'm quite happy with the course. I'm learning a lot every day and feel like I have stronger fundamentals than I did when I was in school. The spaced review is good but I do worry I'll lose it again, so I'm thinking of ways I can integrate this sort of math into my development projects. It's no Duolingo, you really do have to put in effort and aim for a certain number of Xp per day (I try for 60 XP rather than time).

Hard to say but this should give you an idea [0].

At that rate, less than a year is reasonable.

[0] https://www.justinmath.com/what-is-the-highest-sustainable-d...

Would you say the book ventures more into the practical side of learning this stuff or is it closer to the tone of this article? I found this article hard to gain anything from. A lot of just motivational cliche statements and nothing really groundbreaking or mind altering. If the book is better at that, I'd love to read it. If it's stories and a lot of fluff, I'd rather skip. So I'm curious what you are getting from it and how practical and applicable it feels to you?

Agree. The article turned me off as well. No specific example, felt like an ad.

Yeah, I quit reading it because it didn't talk about the book, it felt like a meta article.

>>It's like an imagination sport.

Honestly speaking I think this is a wrong way to teach people to think about Math. Math is just one of those things which feels hard because people struggle to hold long trials of manipulations in their head. Especially if they are manipulations to something very large, evolved slowly over hundreds of steps. People are not coming short, its just how the human mind works.

IMO, the right way to teach Math is to teach people that its just base axioms, manipulation rules. And after that its how you evolve the base axiom using rules. People need to be taught how to make one valid change at a time. Of course this means tons of paper work and patience. But that is what Math actually is. Its taking truth and rules, to make new ones.

Im teaching this to my kid, and she often goes like this is it?? its really just laborious paper work??

Im using this method and LLM help at times these days to learn Algorithms and Data Structures. When you start working things from base conditions and build from there. A lot of Algos that otherwise seem like the domain of novel inventions just seem to follow from the manual steps you just worked, and then translated into a program.

When you remove all the fluff, Patience and Paper work is all there is to Math.

The author (and Grothendieck, liberally quoted in the book) disagree with you.

I think the reason you disagree is that it sounds like you’re teaching your child to be good at math class (a perfectly valid and good thing to do). Being good at math class requires being good at rational/logical thinking and computation. It also has only glancing similarities to anything that the author would recognise as mathematics.

>>It also has only glancing similarities to anything that the author would recognise as mathematics.

Nah, these are the same things. Trying to make Math look like is for people who are 'geniuses' i.e people with massive capabilities of holding large thought trials and changelogs in their head is how you arrive at making people look stupid doing math and eventually make them hate the subject.

Math is paper work. Approach it that way and all of a sudden doing a 100 page proof is within everyones reach. If you ask people to hold a 100 page proof in their head, and more importantly make changes to that in random places and fix the entire changelog trial, probably 2 - 3 people on earth will be able to do it, and you will just convince everyone else its not for them.

I have a hunch that big mathematical breakthroughs in history have happened around and after renaissance era due to paper getting cheap and ubiquitous. There is only that much you can do in your brain alone.

Through all of this, don't get me wrong, the rigorous application of rationality that it takes to step-by-step construct a proof is very important and an incredibly useful skill. Also, I agree that basically no-one can hold more than 3 things in their head at once.

The book also agrees vehemently that math is NOT restricted to "geniuses" and even argues that those don't really exist in the way that culture thinks they do.

However! His assertion is that the (to him) tedious, laborious, error-prone, paperwork is not the fundamental output of "doing math". For him, symbolic written mathematics is akin to sheet music. It would in principle be possible to teach students to read and write sheet music and even do manipulations like transposing it to different keys, without ever letting them listen to music. It would be hard and boring. Some students would find the memorization and application of rules satisfying but most would struggle.

In such a classroom, there might be one student who by chance figures out for herself that you can kind of "hear" these symbols in your brain and suddenly all the arbitrary rules seem obvious and natural and she doesn't even have to go through the tedious steps at all to answer questions. "Of course this is in a minor key." she might say. "No, I didn't rigorously check each chord, it's just... obvious".

Such a student would be labeled a "prodigy" or "genius", and would struggle to explain to others that no, what she's doing isn't harder than the her classmates laboriously doing the rote work, it's actually much easier.

Of course... this is not to denigrate sheet music. It's a wonderful invention that makes it possible to transmit music out of one person's brain to the brains of an orchestra.

Just like written mathematics.

The author's contention is that, like the contrived example above, no-one ever talks about "the music" of mathematics, just the sheet music, and therefore things are much harder than they need to be.

One of the simple mathematical examples he uses is to ask: Can you imagine a circle in your head (unironically an amazing thing to be able to do!). Then to ask a question like: Can a straight line intersect a circle in 3 places?

You likely have an immediate, intuitive response to this highly non-trivial mathematical problem. That's the music. Now, try to write that down in mathematical language for someone who can't see circles. Oof, it's going to be a slog.

>>Through all of this, don't get me wrong, the rigorous application of rationality

Much of this is just talking to oneself and testing it to see if our idea holds under test conditions.

I was once watching a video on how chess grandmasters think and work. Most of it is-

1. Do we know a pattern of moves, even if done, in series that is known to score some win/check. If so, lets do it.

2. Are any pieces under attack, If gone can effect point 1. eventually? If yes, lets protect them.

3. What can all possible moves of our pieces prevent opponent from having successfully execute their own point 1. And can we force opponent into point 2? Lets do it.

Basically every our move and its possible outcomes(Known through prior study of patterns of previous games seen), every move of our opponent.

A strong internal monologue and testing imaginary moves.

Math is just this except over paper.

I actually heard about this book very recently, and it's coming up soon on my (never-ending) reading list.

Happy to hear you're enjoying it, makes me even more confident that I should read it :)

+1 for Math Academy. I’ve been using it daily for over a year now (started October 2023). I summarized my experiences after 100 days here in case it helps anyone: https://gmays.com/math

Thanks for sharing this. I was debating buying the book.

This sounds like a book I needed for one of my early comp sci classes in college. It was called something like Think Like a Programmer: An Introduction to Creative Problem Solving. Maybe it was this, maybe it was something like this.

I mean to say, just applied scientific thinking is important. Even if you never get into pure math or computer programming, applying concepts like "variables", "functions" or "proofs" is universally useful.

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