How Isaac Newton discovered the binomial power series
quantamagazine.org
quantamagazine.org
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.
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 papers I read look needlessly complicated, and I cannot help but wonder why the authors chose this style. I've already been rejected because reviewers found my contributions too simple, however I'd say that the majority prefer to highlight the fact that the paper is "easy to understand". (Of course, sometimes you also need to get told that your contributions are minimal).
Sometimes, I wonder if I have this point of view because some academics like to hide their simple result under a mountain of useless rigor, or because I'm not smart enough to comprehend the necessity of the above rigor and I'm just a dumb engineer who chanced into research. I'd say both are true, but usually not at the same time: some papers are clearly hiding a small contributions, while some are just above me.
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.
Only drawback is that people primarily read PRL, so people try to squash long-form results into PRL alone.
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.
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.
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)This time, maybe I've learned. Thank you.
Carmack recently said that all the key ideas to create AGI will probably fit in a couple pages, with a basic implementation only being only a few thousand lines of code. I'm very much inclined to believe him.
Once you know how the trick is done, of course it'll seem obvious. But if it was so obvious why didn't you come up with it yourself?
Some people feel the need to write incredibly long texts to explain what they're doing, because it makes the topic seem more complicated.
"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.
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.
Much better than articles that make stuff sound fancy to make it more convincing (e.g. I've seen articles that used $r^n \pi^(n/2) / \Gamma(n/2 + 1)$ instead of the more sensible 'volume of an n-sphere').
The "understanding your audience" bit is repeated again and again and again, but seems never to be heard or heeded.
To be fair, most PhDs in Pure Maths don't contain earth-shattering, field altering results. Mostly they are small things that are genuinely new, but not very substantial. The author then has to make it clear which wide they fall in the tension between "trivial observation" and "genuine progress". I didn't do that the first time.
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 one.
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").
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.
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.
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.
I'd be interested in learning more about your invariant. Do you have a copy of your thesis anywhere?
Email(s) in my profile.
https://www.amazon.com/Mathematics-Its-History-Undergraduate...
which goes through the history of maths while actually _doing_ some maths at undergraduate level.
Also the lectures based on it by N J Wildberger
It just happen that I was sharing minutes ago, the start of my very weak personal attempt at building a math "tech tree" out of that idea ( Someone suggested me the name ). Starting with euclidian geometry.
here is the link: http://mathuvue.pythonanywhere.com/ (I apologize in advance , I am not a native english speaker, and it's only the kernel of what I really want to do)
Everything around mathematics is a lonely endeavor. I would love to get in touch with anyone that might be interested in anyway.
My email in my profile !
Where can I get this info? Anyone know of anything close to this?
I too want to read your planned book.
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?
Current title ("Isaac Newton Discovered the Binomial Power Series") implies something rather different :)
Technically the current title is not false.