Everyone is capable of, and can benefit from, mathematical thinking • Hacker News Board

Re: Everyone is capable of, and can benefit from, mathematical thinking

#361

Post by tgv » Fri, Nov 22, 2024, 9:30 PM UTC

hilbert42 wrote: post #232

sourcepluck wrote: post #193

Earlier quoted context omitted.

Imagine our world was extremely similar to how it is now in any way you'd care to imagine, except two things were different. 1. Everyone (young, old, poor, rich) thinks that maths is interesting and fun and beautiful and important. Not important "to get a good job" or "to go to a good college" or "to be an impressive person", but rather important because it's fun and interesting. And maybe it also helps you think cle…

"Kids are told week in week out that maths is stupid, that they are stupid. …." Come on, how often are kids exposed to such stupid talk? I suspect very infrequently. My grandmother, who wasn't stupid by any means but who knew only basic arithmetic, would never have uttered such nonsense. And I'd stress, like many of her generation and background, her knowledge of mathematics was minimal, if she'd been ask what calcul…

> how often are kids exposed to such stupid talk? I suspect very infrequently

Pretty often, by other kids. And kids listen to kids, not well-meaning, mealy-mouthed adults.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#362

Post by llm_trw » Fri, Nov 22, 2024, 9:41 PM UTC

Tainnor wrote: post #352

llm_trw wrote: post #306

Earlier quoted context omitted.

In arithmetic. There is a lot more to math than arithmetic.

Square roots are fundamental to (real and complex) analysis and to algebra (in the study of polynomials), so the two major branches of modern mathematics.

They are fundamental to the arithmetic used in analysis and algebra.

One can get along quite well without them once you replace the arithmetic expressions with more general objects.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#363

Post by llm_trw » Fri, Nov 22, 2024, 9:53 PM UTC

Tainnor wrote: post #357

llm_trw wrote: post #146

Earlier quoted context omitted.

Mathematics is the conversion of a large number of object languages in to a single meta language that lets us talk about all of them. The sin of modern mathematics is that it's meta language is so ill define that you need towers of software to manipulate it without contradiction. Rewriting all of it into s-expressions with a term rewriting system for proofs under a sequent calculus is an excellent first step to makin…

Weirdly enough, mathematicians have been manipulating expressions and writing proofs for centuries without (routinely) stumbling into contradictions all without the need of formal proof calculi or s-expressions. I have nothing but admiration for projects like Lean and Coq and working in them can be a lot of fun (coupled with a lot of frustration when "obvious" things sometimes take an inordinate amount of time to pro…

It wasn't until the 1930s that people realised second order logic with arithmetic will always lead to contradictions without guardrails. Before then all mathematics was done in the object language of whatever the field in question was and only translated to the meta language for succinctness after the fact.

The magic of modern maths is that we can now work only with the meta language and get results free from contradiction. For this we absolutely need a modern notation to do the new type of maths since we are no longer grounded by the reality of the object language.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#364

Post by tgv » Fri, Nov 22, 2024, 9:56 PM UTC

antegamisou wrote: post #231

tgv wrote: post #121

Earlier quoted context omitted.

I cannot agree. It's just "feel-good thinking." "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained. And the biological underpinning of our capabilities doesn't magically stop at the brain-blood barrier. We all do have different brains. I've taught math to psychology students, and they just don't…

> Or let's try other topics, e.g. music. Conservatory students study quite hard, but some are better than others, and a select few really shine. "Everyone is capable of playing Rachmaninov"? I don't think so. Bad example, it's much more likely to create a musical prodigy by providing early and appropriate guidance. Of course this is not easy as it assumes already ideal teaching methods and adequate motivation to the…

> the vast majority of people in the Soviet Union were very math literate

I doubt that.

> although indeed most became engineers

And that is demonstrably false.

Anyway, most of your argument boils down to: there's a bunch of people that can't do a certain task, there's a bunch that has mediocre skills, a few that are good, and a handful that's really good. There's no argument, just observation, not even related to effort, which is what this discussion is about.

And maths is no different from sports or music in that sense. Most people suck at math, and will always suck at it. The things described in the article are personal reflections of elite mathematicians. They have no bearing on development of knowledge and skills of us mortals, if only because those reflections have no truth value. It's all just "feel good" thoughts, no data, nothing provable, etc.

Yes, effort improves skill, but everyone has a limit, and the ones with the high limits we call talented.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#365

Post by blharr » Fri, Nov 22, 2024, 10:23 PM UTC

tgv wrote: post #223

llm_trw wrote: post #164

Earlier quoted context omitted.

You'd also be hard pressed to find one who doesn't know how to flush a toilet. Neither trivia has anything to do with being good at mathematics as done by mathematicians.

Are you an LLM? You brought up the point of mathematicians not knowing what a square root is yourself. Anyway, the square root is is so many levels below maths as done by mathematicians, it's laughable.

Pretty sure you followed the thread wrong. They only mentioned the square root in response to you?

Re: Everyone is capable of, and can benefit from, mathematical thinking

#366

Post by llm_trw » Fri, Nov 22, 2024, 10:56 PM UTC

hyperthesis wrote: post #350

davidbessis wrote: post #189

Earlier quoted context omitted.

I'm the author of what you've just described as clickbait. Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion. The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line. Highly-…

This beats TFA. Interesting relation between cumulativeness and distribution ("Yule process"). But how does this explain variation is how quickly children pick up maths - would you argue it's due to prior exposure e.g. parental tutoring? Any comments on the "10x programmer"?

There is math the abstract field and math the concrete example you're working on.

Current education is _extremely_ biased to concrete arithmetic and a bit of algebra. If you have a predisposition to either you will do extremely well. If you don't you won't.

Those have little to do with how math is done by mathematicians.

In short: education needs to catch up to what's happened since the 1920s in maths. Parents are conservative and don't want their kids to learn something they themselves don't understand, so we're stuck with what we have until enough generations pass and 20th century math is absorbed by osmosis into the curriculum.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#367

Post by Tainnor » Fri, Nov 22, 2024, 11:13 PM UTC

llm_trw wrote: post #363

Tainnor wrote: post #357

Earlier quoted context omitted.

Weirdly enough, mathematicians have been manipulating expressions and writing proofs for centuries without (routinely) stumbling into contradictions all without the need of formal proof calculi or s-expressions. I have nothing but admiration for projects like Lean and Coq and working in them can be a lot of fun (coupled with a lot of frustration when "obvious" things sometimes take an inordinate amount of time to pro…

It wasn't until the 1930s that people realised second order logic with arithmetic will always lead to contradictions without guardrails. Before then all mathematics was done in the object language of whatever the field in question was and only translated to the meta language for succinctness after the fact. The magic of modern maths is that we can now work only with the meta language and get results free from contrad…

None of what you wrote is true.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#368

Post by llm_trw » Fri, Nov 22, 2024, 11:15 PM UTC

Tainnor wrote: post #367

llm_trw wrote: post #363

Earlier quoted context omitted.

It wasn't until the 1930s that people realised second order logic with arithmetic will always lead to contradictions without guardrails. Before then all mathematics was done in the object language of whatever the field in question was and only translated to the meta language for succinctness after the fact. The magic of modern maths is that we can now work only with the meta language and get results free from contrad…

None of what you wrote is true.

Feel free to explain why.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#369

Post by Tainnor » Fri, Nov 22, 2024, 11:29 PM UTC

llm_trw wrote: post #362

Tainnor wrote: post #352

Earlier quoted context omitted.

Square roots are fundamental to (real and complex) analysis and to algebra (in the study of polynomials), so the two major branches of modern mathematics.

They are fundamental to the arithmetic used in analysis and algebra. One can get along quite well without them once you replace the arithmetic expressions with more general objects.

Just come on. The square root of 2 is the easiest example of an irrational number, this has been known since Ancient Greece. You can't compute distances in Euclidean spaces without the square root. "Solving equations by roots" is the bread and butter of algebra. Adjoining roots to a field is how you get Galois Theory. Several algorithms related to number theory have complexity O(sqrt(n)). And so on.

You chose an extremely poor example and now you're trying to die on that hill. Please don't die on that hill.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#370

Post by Tainnor » Fri, Nov 22, 2024, 11:36 PM UTC

llm_trw wrote: post #368

Tainnor wrote: post #367

Earlier quoted context omitted.

None of what you wrote is true.

Feel free to explain why.

Your very first sentence is simply wrong, I don't know what more there is to say other than that you clearly don't understand what Gödel's results are about (hint: they're called "incompleteness theorems", not "contradiction theorems"). Maybe read a textbook? I could recommend several good ones.

The other sentences basically fall in the category of "not even wrong", i.e. basically nonsensical.