Ramanujan Surprises Again (2015)(plus.maths.org) |
Ramanujan Surprises Again (2015)(plus.maths.org) |
(Technically, it works for any number of dimensions >= 1729, but the proof fails for dimensions less than that. Future work might bring the bound down, or better explain why that bound is necessary.)
It is possible to improve the factor K = 1728 appearing in Corollary 5.5, at the expense of introducing various technical complications into the algorithm. In this section we outline a number of such modifications that together reduce the constant to K = 8 + ϵ, so that the modified algorithm achieves M(n) = O(n log n) for any d ≥ 9 (rather than d ≥ 1729).
> 1729 is the natural number following 1728 and preceding 1730.
"A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said the mathematical knowledge he displayed was revealed to him by his family goddess. "An equation for me has no meaning," he once said, "unless it expresses a thought of God.""
People forget how religious newton was and he believed his physics was the discovery of god's physical laws.
Chemistry comes from mystic alchemy. Astronomy derives from astrology.
Just like there is a thin line between genius and madness, the same seems to apply to science and mysticism. Turn the dial a few degrees, you get mysticism. Turn it a few more degrees, you get science.
History of science is just as fascinating as history of politics. It's a shame we focus on the latter so much.
However there are clear possibility of conflicts between scientific approach and institutional religions. Note that people don't need to be anything close to an atheist to get into trouble when their thoughts are going out of the road promoted by the institutional dogma of the day. Giordano Bruno is one famous case of such a human drama.
Of course religion here is more the mean of political control than anything else, and it can be substituted by other means just as detrimental to independent critical free thought.
Recently visited the "Ramanujan Museum":
* Where we could see his family goddess - Sri Namakkal Namagiri Thayaar who he credited for his works:
http://casualwalker.com/wp-content/uploads/2019/08/Ramanujan...
* His desk where he made his early mathematical findings:
http://casualwalker.com/wp-content/uploads/2019/08/Ramanujan...
* Really sadden to see how a Ramanujan's wife lived her life after Ramanujan's demise:
http://casualwalker.com/wp-content/uploads/2019/08/Ramanujan...
* Check the full photo coverage about Ramanujan and his fascinating works at:
http://casualwalker.com/museum-for-the-man-who-knew-infinity...
Thomas Paine wrote on his point of view in "The Age of Reason" [0]:
> Each of those churches show certain books, which they call revelation, or the word of God. The Jews say, that their word of God was given by God to Moses, face to face; the Christians say, that their word of God came by divine inspiration: and the Turks say, that their word of God (the Koran) was brought by an angel from Heaven. Each of those churches accuse the other of unbelief; and for my own part, I disbelieve them all.
> As it is necessary to affix right ideas to words, I will, before I proceed further into the subject, offer some other observations on the word revelation. Revelation, when applied to religion, means something communicated immediately from God to man.
> No one will deny or dispute the power of the Almighty to make such a communication, if he pleases. But admitting, for the sake of a case, that something has been revealed to a certain person, and not revealed to any other person, it is revelation to that person only. When he tells it to a second person, a second to a third, a third to a fourth, and so on, it ceases to be a revelation to all those persons. It is revelation to the first person only, and hearsay to every other, and consequently they are not obliged to believe it.
> It is a contradiction in terms and ideas, to call anything a revelation that comes to us at second-hand, either verbally or in writing. Revelation is necessarily limited to the first communication- after this, it is only an account of something which that person says was a revelation made to him; and though he may find himself obliged to believe it, it cannot be incumbent on me to believe it in the same manner; for it was not a revelation made to me, and I have only his word for it that it was made to him.
> ...
> But some, perhaps, will say: Are we to have no word of God- no revelation? I answer, Yes; there is a word of God; there is a revelation.
> THE WORD OF GOD IS THE CREATION WE BEHOLD and it is in this word, which no human invention can counterfeit or alter, that God speaketh universally to man.
> …
> It is only in the CREATION that all our ideas and conceptions of a word of God can unite. The Creation speaketh an universal language, independently of human speech or human language, multiplied and various as they may be. It is an ever-existing original, which every man can read. It cannot be forged; it cannot be counterfeited; it cannot be lost; it cannot be altered; it cannot be suppressed. It does not depend upon the will of man whether it shall be published or not; it publishes itself from one end of the earth to the other. It preaches to all nations and to all worlds; and this word of God reveals to man all that is necessary for man to know of God.
> Do we want to contemplate his power? We see it in the immensity of the Creation. Do we want to contemplate his wisdom? We see it in the unchangeable order by which the incomprehensible whole is governed! Do we want to contemplate his munificence? We see it in the abundance with which he fills the earth. Do we want to contemplate his mercy? We see it in his not withholding that abundance even from the unthankful. In fine, do we want to know what God is? Search not the book called the Scripture, which any human hand might make, but the Scripture called the Creation.
> The only idea man can affix to the name of God is that of a first cause, the cause of all things. And incomprehensible and difficult as it is for a man to conceive what a first cause is, he arrives at the belief of it from the tenfold greater difficulty of disbelieving it. It is difficult beyond description to conceive that space can have no end; but it is more difficult to conceive an end. It is difficult beyond the power of man to conceive an eternal duration of what we call time; but it is more impossible to conceive a time when there shall be no time.
> In like manner of reasoning, everything we behold carries in itself the internal evidence that it did not make itself Every man is an evidence to himself that he did not make himself; neither could his father make himself, nor his grandfather, nor any of his race; neither could any tree, plant, or animal make itself; and it is the conviction arising from this evidence that carries us on, as it were, by necessity to the belief of a first cause eternally existing, of a nature totally different to any material existence we know of, and by the power of which all things exist; and this first cause man calls God.
> ...
> That which is now called natural philosophy, embracing the whole circle of science, of which astronomy occupies the chief place, is the study of the works of God, and of the power and wisdom of God in his works, and is the true theology.
---
Has there been a genius of his kind since? Maybe Terry Tao, but his work also lacks the ease and lack of machinery that Ramanujan had. Truly amazing.
For me one of them has to be of Évariste Galois[1], who, legend has it, hastily wrote fragments of his last mathematical discoveries on his shirt sleeves before fighting the duel that would end his life.
It's hard to compare mathematicians, but I suppose Erdős[1] would be in the conversation.
But honestly it’s kind of a silly game to rank mathematicians this way.
How does that work? Who can explain this to me?
[0] https://www.youtube.com/watch?v=YuIIjLr6vUA [1] https://www.youtube.com/watch?v=jcKRGpMiVTw
That threw me for a loop and I started believing shit like no one's smarter than I was etc. Then I just ... grew up, I guess. And I remembered this story by Feynman and I realised that despite his absolutely undoubtable genius, he'd have appeared godlike to me if I was his classmate back in the day.
Ramanujan's brain worked even faster by most accounts. He dreamed in math, I think. So there are multiple stories where people ask him a puzzle and he'll answer with an equation that solves it for the entire family of problems that the puzzle could come from.
Genius is about having rare and useful insights that the rest of us are incapable of, and that a computer is unable to easily replicate.
For instance, there was this thing on Twitter recently about all percentages being reversible (7% of 50 is equal to 50% of 7, but the latter is easier to mentally calculate). Most of us are aware that multiplication is commutative, but it takes genius to recognize and frame that insight in a useful way.
https://en.wikipedia.org/wiki/John_von_Neumann#Cognitive_abi...
This is the way the story is always presented, and I think that's usually how it's intended, but I think it's quite misleading for another reason too. If you've ever made or looked at a table of cubes, the famous fact really jumps out (in base 10). I'm serious, look:
n n³
--------
1 1
2 8
3 27
4 64
5 125
6 216
7 343
8 512
9 729
10 1000
11 1331
12 1728
(Yes, I've commented on this before: https://news.ycombinator.com/item?id=21165031)
Hard work and obsessive work effort on a specific area makes it appear innate.
2000 years later, Kepler had faith in a harmonious cosmos, and charged his model of harmony so it could fit the evidence. He elipsed the circles, instead of squaring them.
Fun fact #1: it is impossible to square a circle [1]
Fun fact #2: the Pythagoreans conducted the first attested scientific experiment in Western history (according to a recent PhD thesis at UMich [2])
They most certainly do! Grothendieck’s work is heavy on definitions, but the essence of his work is that the right definitions obviate (and are seen to be right because they obviate) the need for heavy machinery. See the famous quote, taken from Wikipedia (https://en.wikiquote.org/wiki/Alexander_Grothendieck#Quotes_...) because that’s the first place I found it:
> I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.
Therefore, these are not the most boring facts about 1729.
;)
[0] https://en.wikipedia.org/wiki/Interesting_number_paradox
To be fair, with most shortcuts, it’s possible to construct difficult cases (17% of 23 is difficult in either order) but where it applies (when one of the pairs is a common percentage), exploiting commutativity can be quite useful. Plus the mental overhead of remembering the rule is extremely minimal.
I'd say the real mistake he made was that he lifted the veil off of how he did things, leading people to say "oh even I could have done that".
its still genius but not in the sense of actually being able to do huge calculations in ones head the way a computer would.
There's a (quite possibly apocryphal) story about Niels Henrik Abel in primary school, where his teacher supposedly wanted time to do some grading and assigned the students the busywork of adding up all numbers from 1 to 100. Abel supposedly quickly found the well known formula n(n+1)/2 and gave the teacher the answer within minutes, and the teacher supposedly believed he'd somehow "cheated" because he could not imagine any of them could figure it out.
I have no idea if the story is real (I grew up in Norway, so Abel was a popular subject for stories like this) - it was told to me in high school by a maths teacher after giving us the modified task of seeing if we could find any shortcuts to doing the sums, and seeing what we'd come up with. I found the formula quickly, but at that age that's nothing special, especially not when prompted to find an alternative solution.
But the overall idea the teacher was trying to get us to understand was how to pause and think about how to decompose a problem rather than just picking the most obvious alternative, and learning to be "lazy" in the sense of relentlessly looking for an easier way to do things is a large part of what got me into software development..
And I looked it up- Yes, the same possibly apocryphal story is on his Wikipedia page: https://en.m.wikipedia.org/wiki/Carl_Friedrich_Gauss
It's like 4 levels of thinking somehow merged in his actions: 1) be normal and look at the crazy people, 2) be a crazy person, 3) be a crazy person and be aware of your craziness, 4) be a crazy person, be aware of it and let others know that you're aware of it. It feels like one of those thought spirals I go into if I have weed. It's right on the boundary of crazy but probably also (in his case) inside the realm of genius.
Several clips from Gell-Mann's Web of Stories interview (late 1990s) pertain to his on-again off-again collaboration with Feynman.
https://www.youtube.com/watch?v=o2sEW4ggVlA&list=PLVV0r6CmEs...
Undoubtedly a great thinker and genius, but that doesn't say very much about personality traits.
"Interestingness" is often as good a heuristic as any when looking for paths that lead to useful developments, although the path is often not a straight one or short one.
I also like the idea of secondary and tertiary effects. One simple example: By "playing" with cute yet fun ideas that are highly likely to not lead to anything immediately interesting, we can build up skillsets and capabilities that lead to very useful results for other problems. Perhaps this is somewhat akin to how the young of predator species "play" around in a way that prepares them to actually hunt when they are older.
(something like treating the digits as coefficients of a series of powers of the number base, defining a function f(b,n) that returns the sum of the digits of n in base b, setting that equal to the product of the forward and reverse representation of that result in base b, and seeing if the equation looks interesting)
E.g. you can use a similar technique to show why the finger counting method of multiplying by 9 works, or why multiples of 3 have digits that sum to a multiple of 3 (same for 9)
some (n mod 9) can be found by (is congruent to) (sum of the digits mod 9) instead is the most obvious example
True only in base 10 although similar congruences exist for other bases and also involve adding the digits
