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.
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.
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.
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]
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.
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).
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?
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.
+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
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.
I'm a fullstack web developer from Stanford who's looking to work at a fast growing company, preferably in AI but any company with interesting problems to solve will excite me.
I live in the part of Arlington that this article talks about the most and it's so fascinating to learn about the history of how the area got developed. It really does feel like the county has nailed the the development of Ballston and Clarendon. The only thing I'd complain about is parking. Street parking has costed me over a thousand dollars lol (my car got shot by a bb gun and insurance didn't pay so that's mostly why, but otherwise it feels super safe here)
Really excited to see if this approach can help reduce hallucinations overall. Process supervision combined with the browsing models could drastically improve how logically sound anything that's generated is. Creating consistently valid logic is the harder part so this is awesome.
Is there some formal/objective definition of what exactly constitutes a 'hallucination' as opposed to other types of errors? At a high-level it only seems relevant with respect to questions for which there is some objectively true answer, or where 'facts' are included in answering a question that are false.
Good point. You can create a formal definition of hallucinations in formal language, like math and logic, but probably not for natural language. There are bound to be edge cases where natural language defies the rules of formal logic without being incoherent. But, while you might not be able to have a simple formal definition, maybe you could create a model that's trained to recognize these edge cases, and the model would be a sort of approximation of a formal definition.
The paper states that hallucinations are logical reasoning errors... that confused me. For me a hallucination is the conjuring up of facts, things, references, characters etc. as is convenient for the given context.
Semantics aside: I once gave it a math problem, and its hallucinations really did present as logical errors. I was a TA for several years, and it was reminiscent of the mistakes students make. Something that sounds fairly plausible, but still incorrect, with logical errors or unjustified steps.
The trouble (probably not for basic math or even low-level analysis) is that what we refer to as "logic" or a logical reasoning error isn't uniform across different axiomatic systems, and the specific interpretations is a fairly human, social activity which cannot be readily "checked" in many cases for correctness against some baseline notion of "logic". The symbol φ, for instance, has a variety of significations in different context (the family of sets of all functions, probably something in physics idk), which our interpreto-bot (GPT) might not be able to both logically integrate and loosely interpret at the same time. Humans have the capacity to apprehend both consistent and complete logical systems: they interpret at the level of the text (at the level of the weave of signification), and any generally intelligent AI would have to mimic that behavior of the constant, on-the-fly dynamic changes to its network at the appearance of every new signifier in the same way as a human does.
It's really hard to formally define hallucination, for me it's a "know it when you see it" situation.
In my view, saying "10 + 4 = 15" is not a hallucination but inventing a citation that doesn't exist is one. There is a big grey area between these, like what if it cites a real paper but gets the year wrong? Is that a hallucination (because the paper as cited is fake), or just an incorrect statement (because the year is just wrong)?
It’s a sliding scale that depends on the use case and different uses will want levels of hallucination that other uses would find unacceptable.
In summarization, it’s literally anything that’s not in the context. If it’s used for writing historical fiction or scifi, very little would count as a hallucination as long as it’s following the prompt.
In the recent case of the Texas lawyer, the citations follow a specific format and can be checked against a database, so hallucinations are easy to define w.r.t. citations.
Best article I've read on here in a while. From now on, whenever there's dead air in a conversation, I'm just gonna drop "I get weirded out when couples treat their dogs like babies."
Also, this is great for relationship advice too. Personally, I always struggle with asking too many yes/no or how many questions rather than asking why/how questions. Givers vs Takers aside, if I had to guess, I'd say that "doorknobs" can be created by saying or asking something the gets you to explore each others' opinions and experiences.
I also thought the part about how sharing boring mutual memories is more fun than exciting individual memories is interesting, but maybe not a hard rule to follow. I've been more interested in scuba diving recently because of some of the awesome stories my friends have told me.
> I've been more interested in scuba diving recently because of some of the awesome stories my friends have told me.
By the Thursday of our honeymoon cruise, I was ready to go scuba diving (we were in Grand Cayman) and my wife was ready for a spa day. We each enjoyed our days and telling each other about them, but she's still envious and wishes she'd come along to see the large groupers hanging out with their jaws open getting their teeth cleaned by shrimp (like the cartoon shrimp in 'Finding Nemo.')
Note: the primary purpose of this post is to get you to spend a lot of money on scuba certification and gear, prompted by the belief that it's worth it!
A friend of mine was afraid of the sea. Swimming in any water deeper than 1m, basicaly. Probably some thing from childhood... No amount ot prodding or laughing would make him do it. But anyway. One day he was told that scuba diving may help overcome the fear. And he got interested. And he tried in some club.
And it worked.
Now the gear collects dust, but we go swimming together..
In my experience, the commitment to read every day is more important than how much. If I’m in a spacey mood, I can still power through a page in a meaningful way. Plus, sometimes I’ll get momentum from that and then the next few pages will pull me in.
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.