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

quantamagazine.org

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

#371

Post by llm_trw »

Tainnor wrote: post #370
llm_trw wrote: post #368

Earlier quoted context omitted.

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.

Godel starts by the assumption that the system is free from contradictions which makes his papers largely irrelevant here.

Again, feel free to explain how you can have a second order logical system without axiomatic limitations that doesn't contain contradictions.

Hell, go by the more popular version of first order logic + set theory if you're more comfortable with that.

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

#372

Post by cutemonster »

davidbessis wrote: post #311
cutemonster wrote: post #297

Earlier quoted context omitted.

> document the secret "oral tradition" of top mathematician > A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao. Sounds like people mostly living in different bubbles? What do they know about the world? They aren't hanging out with the kids who fail in school because maths and reading and writing is to hard,…

> Sounds like people mostly living in different bubbles? What do they know about the world? Well, they do know something about math — in particular that it requires a certain "attitude", something that no-one told them about in school and they felt they only discovered by chance. Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were spec…

> they do know something about math ... that it requires a certain "attitude"

Of course. That does not mean that intelligence doesn't play a (big) role.

> Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were special because of this "attitude" and not because of what people call "intelligence"

That doesn't make sense. Back when they were active, intelligence, IQ tests and the heritability of intelligence hadn't been well studied. They didn't have enough information, like we do today: https://en.wikipedia.org/wiki/Heritability_of_IQ#Estimates "Various studies have estimated the heritability of IQ to be between 0.7 and 0.8 in adults and 0.45 in childhood in the United States."

And, evolution and genetics weren't these peolpe's domains. Does it make sense to assume they were authorities in genetics and inheritance, because were good at maths and physics?

Sometimes they were wrong about their own domains. Einstein did say "Genius is 1% talent and 99% hard work" (I can understand how it makes sense from his own perspective, although he didn't know enough about this animal species, to say that).

But he also said "God does not play dice" and was wrong about his own domain.

> Why do you bring "kids who fail in school" and "start selling drugs" into this conversation?

It was an example showing that the researchers live in bubbles.

That they're forming their believes about humans, based on small skewed samples of people. There's billions of people out there vastly different from themselves, whom they would have left out, if thinking about about others' abilities to learn.

In fact, now it seems to me that you too live in a bubble, I hope you don't mind.

> Usually, competency in one domain is presumed to make you a bit more qualified than the random person on the internet when it comes to explaining how this domain operates.

1) Maths and 2) evolution, DNA, genetics, intelligence, learning and inheritability are not the same domains.

Anyway, best wishes with the book and I hope it'll be helpful to people who want to study mathematics.

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

#373

Post by cutemonster »

margorczynski wrote: post #312
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-…

> In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. Sure they don't. Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work. It runs completely contrary to the findings of modern biology and psychology, most of our intellectual potential at adulthood is genetic. The floor and ceiling you…

> Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work

I suppose that makes sense from their own personal perspectives (but that doesn't make them right), in that they had to put in lots of time and work, but didn't do anything to become bright people.

> The floor and ceiling you will operate on in your life

Interesting that what you wrote got downvoted. Lots of flat-earthers here? (figuratively speaking)

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

#374

Post by Tainnor »

llm_trw wrote: post #371
Tainnor wrote: post #370

Earlier quoted context omitted.

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.

Godel starts by the assumption that the system is free from contradictions which makes his papers largely irrelevant here. Again, feel free to explain how you can have a second order logical system without axiomatic limitations that doesn't contain contradictions. Hell, go by the more popular version of first order logic + set theory if you're more comfortable with that.

Nobody has found a contradiction in (first or second order) ZFC in over a hundred years. You're out of your depth.

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

#375

Post by llm_trw »

Tainnor wrote: post #374
llm_trw wrote: post #371

Earlier quoted context omitted.

Godel starts by the assumption that the system is free from contradictions which makes his papers largely irrelevant here. Again, feel free to explain how you can have a second order logical system without axiomatic limitations that doesn't contain contradictions. Hell, go by the more popular version of first order logic + set theory if you're more comfortable with that.

Nobody has found a contradiction in (first or second order) ZFC in over a hundred years. You're out of your depth.

>Nobody has found a contradiction in (first or second order) ZFC in over a hundred years.

Yes, which is exactly why we have guardrails like ZFC in place. We got all sorts of exciting results without them. We of course have to keep adding axioms to ZFC - or replacing them all together - because there is a lot of math out there which is outside it's purview.

>You're out of your depth.

You're being incredibly abrasive for no reason.

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

#376

Post by jiggawatts »

vundercind wrote: post #166
penguin_booze wrote: post #139

To my mind, the premature formalization of the math is the principal contributor to gas lighting and alienation of people from maths. The reduction of concepts to symbols and manipulation thereof, is an afterthought. It's misguided for them to be introduced to people right at the outset. People need to speak in plain English [0]. To some mathematicians' assertion that English is not precise enough, I say, take a hike…

I'm an adult who's been programming computers professionally for 20 years, and went to school for it, and I've lost most mathematical skills past what I'd learned by 6th grade or so, from lack of use. People who aren't even working in a field that's STEM-adjacent have even less use for stuff past simple algebra and geometry (the latter mainly useful just for crafting hobbies and home-improvement projects) and a handf…

For nostalgia I keep a copy of Mathematica on my laptop, so I can pretend to be an an actual computer scientist and not just an overpaid button-presser.

Ten years ago I used it to fit a nonlinear model to some performance metrics to predict the behaviour of a disk array past the maximum load level I was allowed to use for testing.

That’s the last time I did something that Excel couldn’t handle.

In university I learned how matrix exponentiation can be used to calculate the maximum throughput of a mesh network. In real life everyone just buys 10x the bandwidth they need.

It’s depressing.

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

#377

Post by tptacek »

cutemonster wrote: post #372
davidbessis wrote: post #311

Earlier quoted context omitted.

> Sounds like people mostly living in different bubbles? What do they know about the world? Well, they do know something about math — in particular that it requires a certain "attitude", something that no-one told them about in school and they felt they only discovered by chance. Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were spec…

> they do know something about math ... that it requires a certain "attitude" Of course. That does not mean that intelligence doesn't play a (big) role. > Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were special because of this "attitude" and not because of what people call "intelligence" That doesn't make sense. Back when they were…

Current estimates of the "heritability" of intelligence are far, far lower than "0.7 or 0.8"; they're probably below 0.1, and that's before digging into what "heritability" means, which is not generally what people think it does.

I'd guess the person you're responding to has thought more carefully about this issue than the median HN commenter has.

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

#378

Post by beezlewax »

gsabo wrote: post #15

I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things. I've been working a lot on my math skills lately (as an adult). A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's e…

I'd like to improve my mathematical knowledge (which has degraded to basically zero).

Can you talk about the path or resources you are using to improve here?

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

#379

Post by antegamisou »

tgv wrote: post #364
antegamisou wrote: post #231

Earlier quoted context omitted.

> 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 eff…

> I doubt that.

Not even the most vicious polemicists of communist states deny that i.e. the superiority of the Soviet-era education system.

https://www.amazon.com/What-Ivan-Knows-Johnny-Doesnt/dp/4871...

which is also why the following statement

> Most people suck at math, and will always suck at it.

is strictly US-centric.

> 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.

No, you came up with that cause you have a very poor understanding of what constitutes a good musician which, like any other typical HNer believes, is another LeetCode type of thing where the more problems like a good little monkey you can solve, the smarter you become. And I already stated that there are people striving in the arts, even more than the ones with the supposedly 'better' skills according to absurd and clueless standards you set, i.e. no, those who can't access specific repertoire easily are not people that can't do a certain task or a bunch that has mediocre skills.

> And maths is no different from sports or music in that sense.

Let me guess, you also think that if painters can't draw photorealistically they're not deserving the artist title and lack talent? Or that the opera is all about who can sing the highest note?

Anyone who lumps every discipline within one other like that without realizing they require completely different things to be considered successful at and believe everything boils down to some supposedly 'objective' absurd video-game like character strategy always serves to remind that the US today has nothing to offer other than hi-tech bombing technology and horrific subculture.

Nowhere did I deny the existence of some people having the innate ability to absorb skills faster and better than the others and of course this is an interdisciplinary fact. But it definitely doesn't hold the same weight for every single discipline for one to strive.

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