6 ms·

How Isaac Newton discovered the binomial power series

140 points by anthelios 4y ago

mellosouls 4y ago
"How Isaac Newton Discovered the Binomial Power Series" Current title ("Isaac Newton Discovered the Binomial Power Series") implies something rather different :)
- contravariant 4y ago
  If I recall correctly HN automatically removes words like 'How' from the title (I think it also removes 'Why', not sure if it removes all interrogative words) Technically the current title is not false.
  - mellosouls 4y ago
    Yeah I thought it might be an automatic thing.
    - ColinWright 4y ago
      Interestingly, if,after submission and the initial word stripping, you then edit the title, you can restore it yourself.
- dang 4y ago
  Rehowed. Thanks!
ironSkillet 4y ago
I love articles like this. I wish more research papers and textbooks actually explained how people came up with their ideas/proofs in the first place.
- ColinWright 4y ago
  There's a reason they don't. I failed my PhD first time around, with the external examiner recommending I not even go to the viva stage, saying that the results were obvious and trivial, not worthy of a PhD. He recommended I be given a Masters and not even permitted to re-write or re-submit. My internal examiner insisted we go to viva, wherein I managed to convince them (unaware of all the above) that the results were surprising, interesting, novel, and worth a PhD. The problem was that in the first version I explained far too clearly all my thinking, and how I came to the results. As a result, it all seemed obvious and inevitable. To publish, results need to be novel, surprising, or otherwise "worthy", and if you explain too clearly how you got there, your results can easily be dismissed as "trivial". Beware of explaining things clearly.
  - mjburgess 4y ago
    That sounds like a profoundly immature external examiner. Any seasoned person here is aware of this effect. In my view the "logical sciences" all aim for triviality: the hardest thing is to specify the problem in such a way that the solution is trivial. All the work is done, precisely, so that it is trivial.
  - dotancohen 4y ago
    I have a similar experience in industry (software development) that I relive weekly, but I never learn. When I help someone solve a problem, I like to show them how I got to the solution. In the end, my hour invested was "we did it together" and not appreciated by anyone, least of all the manager. But on the few occasions that I just bang out a function quickly for somebody because I don't have the time, I'm hailed as a lifesaver critical to the business and my name is mentioned in the Daily Standup the next day. Explaining how we work trivializes our work. It's better to remain mysterious.
    - vmilner 4y ago
      Mr. Jabez Wilson laughed heavily. “Well, I never!” said he. “I thought at first that you had done something clever, but I see that there was nothing in it after all.” “I begin to think, Watson,” said Holmes, “that I make a mistake in explaining. ‘Omne ignotum pro magnifico,’ you know, and my poor little reputation, such as it is, will suffer shipwreck if I am so candid. - "The Red Headed League", (The Adventures of Sherlock Holmes)
laichzeit0 4y ago
Well this flies in the face of everyone who thinks the biggest stumbling block to learning mathematics is the “obscure notation”. There’s nothing obscure in this discovery, but there is genuine invention involved. Something more than can be taught by rote, or using different notation.
fud101 4y ago
if this appeals check out the writings of morris kline.
paulpauper 4y ago
finally a quanta article in which the math is easy to understand The binomial series is just algebra. You take something like (a+b)^3 and then you generalize the rule for exponents to any value and , presto done. My question is, when did math become so much harder? I think the quntic formula was when math finally became what It is today. This required a whole new conception of math . How do you go from a simple formula a high school student can grasp, which made Newton a genius because he co-discovered it, to 60+ pages of dense notion at phd level?
- wrycoder 4y ago
  This was posted at least five times, without much action, until the full title was used. It turns out that the Quanta editor was good at picking titles.
stillbourne 4y ago
Veritasium did a video on this: https://www.youtube.com/watch?v=gMlf1ELvRzc https://www.youtube.com/watch?v=gMlf1ELvRzc
- fakethenews2022 4y ago
  There are also several somewhat more worked out versions of the article and one of them shows how he arrived at an infinite series for sine. The cleverness here was just a small part of a lot more.

140 points by anthelios 4y ago

"How Isaac Newton Discovered the Binomial Power Series" Current title ("Isaac Newton Discovered the Binomial Power Series") implies something rather different :)

contravariant 4y ago
If I recall correctly HN automatically removes words like 'How' from the title (I think it also removes 'Why', not sure if it removes all interrogative words) Technically the current title is not false.
- mellosouls 4y ago
  Yeah I thought it might be an automatic thing.
  - ColinWright 4y ago
    Interestingly, if,after submission and the initial word stripping, you then edit the title, you can restore it yourself.
dang 4y ago
Rehowed. Thanks!

ironSkillet 4y ago

I love articles like this. I wish more research papers and textbooks actually explained how people came up with their ideas/proofs in the first place.

ColinWright 4y ago
There's a reason they don't. I failed my PhD first time around, with the external examiner recommending I not even go to the viva stage, saying that the results were obvious and trivial, not worthy of a PhD. He recommended I be given a Masters and not even permitted to re-write or re-submit. My internal examiner insisted we go to viva, wherein I managed to convince them (unaware of all the above) that the results were surprising, interesting, novel, and worth a PhD. The problem was that in the first version I explained far too clearly all my thinking, and how I came to the results. As a result, it all seemed obvious and inevitable. To publish, results need to be novel, surprising, or otherwise "worthy", and if you explain too clearly how you got there, your results can easily be dismissed as "trivial". Beware of explaining things clearly.
- mjburgess 4y ago
  That sounds like a profoundly immature external examiner. Any seasoned person here is aware of this effect. In my view the "logical sciences" all aim for triviality: the hardest thing is to specify the problem in such a way that the solution is trivial. All the work is done, precisely, so that it is trivial.
- dotancohen 4y ago
  I have a similar experience in industry (software development) that I relive weekly, but I never learn. When I help someone solve a problem, I like to show them how I got to the solution. In the end, my hour invested was "we did it together" and not appreciated by anyone, least of all the manager. But on the few occasions that I just bang out a function quickly for somebody because I don't have the time, I'm hailed as a lifesaver critical to the business and my name is mentioned in the Daily Standup the next day. Explaining how we work trivializes our work. It's better to remain mysterious.
  - vmilner 4y ago
    Mr. Jabez Wilson laughed heavily. “Well, I never!” said he. “I thought at first that you had done something clever, but I see that there was nothing in it after all.” “I begin to think, Watson,” said Holmes, “that I make a mistake in explaining. ‘Omne ignotum pro magnifico,’ you know, and my poor little reputation, such as it is, will suffer shipwreck if I am so candid. - "The Red Headed League", (The Adventures of Sherlock Holmes)

laichzeit0 4y ago

Well this flies in the face of everyone who thinks the biggest stumbling block to learning mathematics is the “obscure notation”. There’s nothing obscure in this discovery, but there is genuine invention involved. Something more than can be taught by rote, or using different notation.

fud101 4y ago

if this appeals check out the writings of morris kline.

paulpauper 4y ago

finally a quanta article in which the math is easy to understand The binomial series is just algebra. You take something like (a+b)^3 and then you generalize the rule for exponents to any value and , presto done. My question is, when did math become so much harder? I think the quntic formula was when math finally became what It is today. This required a whole new conception of math . How do you go from a simple formula a high school student can grasp, which made Newton a genius because he co-discovered it, to 60+ pages of dense notion at phd level?

wrycoder 4y ago
This was posted at least five times, without much action, until the full title was used. It turns out that the Quanta editor was good at picking titles.

stillbourne 4y ago

Veritasium did a video on this: https://www.youtube.com/watch?v=gMlf1ELvRzc https://www.youtube.com/watch?v=gMlf1ELvRzc

fakethenews2022 4y ago
There are also several somewhat more worked out versions of the article and one of them shows how he arrived at an infinite series for sine. The cleverness here was just a small part of a lot more.