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How Isaac Newton discovered the binomial power series

quantamagazine.org

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Re: How Isaac Newton discovered the binomial power series

#41

Post by ColinWright »

BeetleB wrote: post #40
ColinWright wrote: post #7

Earlier quoted context omitted.

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 we…

Common in industry as well. I work in a company at the top of its field. A senior engineer once told me that the advice he got while rehearsing for a presentation: "Your slides explain things so well it makes it easy to understand your work. No one will be impressed. The more people have to think to understand your slides, the more impressed they will be. It's OK if most people don't understand much of your presentat…

Generally true, but there are exceptions. For example ...

There was an occasion when we were making a presentation in the early stages of bidding for a contract (complicated industry, don't ask unless you want to know). The business guy had completed his presentation about contracts we'd delivered and systems we could provide, and then it was my turn.

I turned off the projector, retracted the screen, drew a few of blobs on the whiteboard, labelled them "Remote, remote, Command and Control", then turned to the assembled personnel and said: "What would you like to know?"

My intent was to explain clearly what we did in a way they could understand, because I knew that if they understood our capability, they'd buy our kit.

They did, and they did.

So for the right audience, explaining clearly is the right thing to do.

Re: How Isaac Newton discovered the binomial power series

#42

Post by [deleted] »

ColinWright wrote: post #7
ironSkillet wrote: post #4

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.

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 we…

[deleted]

Re: How Isaac Newton discovered the binomial power series

#43

Post by abecedarius »

dotancohen wrote: post #9
ColinWright wrote: post #7

Earlier quoted context omitted.

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 we…

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 a…

Reminds me of a favorite compliment about some of my code: a friend told me he didn't realize the job it did was tricky, after reading it, until later he implemented the same thing.

Re: How Isaac Newton discovered the binomial power series

#45

Post by fakethenews2022 »

stillbourne wrote: post #44

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

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.

Re: How Isaac Newton discovered the binomial power series

#46

Post by kmill »

ColinWright wrote: post #39
kmill wrote: post #38

Earlier quoted context omitted.

Is the first part similar to the "W function" from Tutte's A ring in graph theory (1947)? If I remember correctly, it generalizes the chromatic/flow/Tutte polynomial to yield a linear combination of disjoint unions of bouquets (i.e., a linear combination of ordered lists of natural numbers, describing the dimensions of the cycle spaces of each connected component in the state sum expansion of the chromatic/flow/Tutte…

More abstract than that. It generated an element from an infinitely generated ring. If there was a colouring with colour sets of size x, y, and z, then it included the element P_{x-1}.P_{y-1}.P_{z-1} from the ring. Then it summed over all colourings. So if there is a colouring of $G$ using $n$ colours with induced sets of size 4, 4, 2, and 1, then the function evaluated at $n$ included P_3^2 P_1 P_0. (The "-1" is a t…

This reminds me somewhat of Stanley's symmetric chromatic function, which I guess came about ten years later (1995). Actually, what you're describing seems to be something I rediscovered a couple years ago, where P_k is interpreted as the kth power sum symmetric polynomial in n variables.

I'd be interested in learning more about your invariant. Do you have a copy of your thesis anywhere?

Re: How Isaac Newton discovered the binomial power series

#47

Post by wrycoder »

paulpauper wrote: post #34

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 formu…

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.

Re: How Isaac Newton discovered the binomial power series

#48

Post by ColinWright »

kmill wrote: post #46
ColinWright wrote: post #39

Earlier quoted context omitted.

More abstract than that. It generated an element from an infinitely generated ring. If there was a colouring with colour sets of size x, y, and z, then it included the element P_{x-1}.P_{y-1}.P_{z-1} from the ring. Then it summed over all colourings. So if there is a colouring of $G$ using $n$ colours with induced sets of size 4, 4, 2, and 1, then the function evaluated at $n$ included P_3^2 P_1 P_0. (The "-1" is a t…

This reminds me somewhat of Stanley's symmetric chromatic function, which I guess came about ten years later (1995). Actually, what you're describing seems to be something I rediscovered a couple years ago, where P_k is interpreted as the kth power sum symmetric polynomial in n variables. I'd be interested in learning more about your invariant. Do you have a copy of your thesis anywhere?

I might have a soft copy of the thesis, but I can only guarantee hard copy. If you're interested then the best thing to do is email me so it goes into my work queue, then I can have a quick hunt and see what I can produce easily.

Email(s) in my profile.

Re: How Isaac Newton discovered the binomial power series

#49

Post by tpoacher »

dataflow wrote: post #24
tpoacher wrote: post #20

Earlier quoted context omitted.

Alas, I have reached exactly the same conclusion over the years. You need to strike the right balance between not obfuscating too much such that the reader ragequits in frustration, versus obfuscating things just enough so that the reader has to put a bit of "work" before the insight hits home, lest they fail to appreciate how much effort getting to that insight took in the first place, and dismiss it for a trivial o…

Do you feel like papers should have separate derivations & proofs of correctness? The second one requiring correctness (intuition being secondary), and the first one being intuitive (rigor being secondary).

I never implied that the obfuscation comes from having to present the maths. Clarity and rigour are orthogonal concepts. In fact, the reverse is often the case: make the derivations comprehensive and easy enough to follow, and the reader will feel it was an obvious result.

The argument here is that occasionally you should jump from A to C, skipping B (in the paper, not in your own work!), such that the reader will have to work that extra step to reach C from A, to make them appreciate more that the insight "A leads to C" was not a trivial one (and thus "not novel").

Re: How Isaac Newton discovered the binomial power series

#50

Post by sitkack »

ColinWright wrote: post #7
ironSkillet wrote: post #4

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.

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 we…

I am sorry, that sounds damn scary. The external examiner sounds like a sadist.
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