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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

#31

Post by mjthrowaway1 »

EGreg wrote: post #26
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.

I want to write a book on the history of science, where we actually go througy what people thought (luminiferous ether, phlogiston, the humours) until some experiments convinced them otherwise (michelson morley etc). Who knew what (Arabs and algebra), and how did it spread (Jusuits)? I think it is really needed but I haven’t found any approach like this ! It is all about the shrinkwrapped final results and modern bel…

Emperor of All Maladies is this but for 20th century cancer research.

I too want to read your planned book.

Re: How Isaac Newton discovered the binomial power series

#32

Post by bawana »

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…

This human tendency towards obfuscation is encouraged by the need to survive in a capitalist market. If others dont see value in what you do, why should they share the fruits of their own labor? It has resulted in many Rube Goldberg constructs in our world- the health care system, the legal system, the military-industrial complex, office bureaucracy, etc. Silicon valley has its own version and even labeled it - technical debt.

I cant wait for AGI to show us how meaningless we all are. Perhaps when a machine can outvalue all human endeavor, we will be forced to recognize a greater truth and invent a new mode for valuing our existence. Something that values a sense of of goodwill and willingness to help our neighbor. Because his plight might be our own.

Re: How Isaac Newton discovered the binomial power series

#34

Post by paulpauper »

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?

Re: How Isaac Newton discovered the binomial power series

#35

Post by copperx »

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…

What was your PhD in? Math?

Re: How Isaac Newton discovered the binomial power series

#36

Post by ColinWright »

copperx wrote: post #35
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…

What was your PhD in? Math?

Pure Maths, specifically Combinatorics and Graph Theory.

The first part was a generalisation of the Chromatic Function on graphs to provide information of the "Shapes" of colourings, and not just the number.

The second part showed that a plausible (hand-wave hand-wave) location for a counter-example to an open conjecture about partially-ordered sets did not, in fact, hold a counter-example. In other words, I proved a special case of an open conjecture on posets.

More details on request.

Re: How Isaac Newton discovered the binomial power series

#37

Post by ISL »

psychphysic wrote: post #21
plopilop wrote: post #12

Earlier quoted context omitted.

I clearly despise academics (and the papers they write) who think that everything should look novel by hiding your thought process. Sadly, especially in fields where conferences are favored over journals, this seems to be common: you only have a single chance of impressing the reviewer, so you lose little by hiding your thought process, and if possible make your result look more complicated that it actually is. Many…

I think there is a balance to be struck and certainly don't want to say oh well there should be another journal for that. But before the internet elaborate pedagogical papers simply took up to much paper real estate and so brief punchy result dense work was produced. Im won over by the article in preprint then elaboration blog posts theme we see more in maths.

Physics has the same strategy: Important and timely results appear in the four-page (now four-page-ish) PRL and the long-term paper wrapping it up appears in PRD (or PRA, PRB, PRC, or PRE).

Only drawback is that people primarily read PRL, so people try to squash long-form results into PRL alone.

Re: How Isaac Newton discovered the binomial power series

#38

Post by kmill »

ColinWright wrote: post #36
copperx wrote: post #35

Earlier quoted context omitted.

What was your PhD in? Math?

Pure Maths, specifically Combinatorics and Graph Theory. The first part was a generalisation of the Chromatic Function on graphs to provide information of the "Shapes" of colourings, and not just the number. The second part showed that a plausible (hand-wave hand-wave) location for a counter-example to an open conjecture about partially-ordered sets did not, in fact, hold a counter-example. In other words, I proved a…

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 polynomial).

Re: How Isaac Newton discovered the binomial power series

#39

Post by ColinWright »

kmill wrote: post #38
ColinWright wrote: post #36

Earlier quoted context omitted.

Pure Maths, specifically Combinatorics and Graph Theory. The first part was a generalisation of the Chromatic Function on graphs to provide information of the "Shapes" of colourings, and not just the number. The second part showed that a plausible (hand-wave hand-wave) location for a counter-example to an open conjecture about partially-ordered sets did not, in fact, hold a counter-example. In other words, I proved a…

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 technical thing that only feels obvious once you've played with it enough).

You end up with what the values should be, but the clever part was using "umbral evaluation" on the polynomial, and not just simple substitution. It might be possible to find the details on the web, but this was 35 years ago.

Re: How Isaac Newton discovered the binomial power series

#40

Post by BeetleB »

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…

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 presentation. "

Specific advice included: Pack your slides with content (charts, etc). The audience should feel overwhelmed.

From what I've experienced, the advice is sadly spot on.

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