Ramanujan's Easiest Formula
johncarlosbaez.wordpress.com
johncarlosbaez.wordpress.com
he often wouldn't (or wasn't able to) prove them. they were just assertions that, much more often than not, turned out to be accurate.
Surely there would be a method for coming up with such assertions (maybe only known to him though)
Describing one of his many insightful math dreams, Ramanujan said:
"While asleep I had an unusual experience. There was a red screen formed by flowing blood as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of results in elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing...""
I think maybe because it flies in the face of everyone that claims 10x developers are not possible. Or that some people are just genetically gifted and there's nothing you could possibly do that would get you within a hundred yards of them no matter how much hard work you put in, or how good your teachers were, etc. (thinking about John von Neumann here).
It's sort of depressing to think that there's almost a different species of man amongst us. It's undeniable though.
While genetics play a huge role, nurture plays a large role as well [1].
[1] https://www.psychologytoday.com/us/articles/200507/the-grand...
Maybe one day (in a distant future most likely) the general population will have access to such mental prowess. This would then be another kind of singularity!
Oh, there is an easy recipe : just emulate the universe as it was back then and observe Ramanujan. As we know, having a recipe doesn’t mean we have the resources to implement it.
There are several movies on Ramanujan as well as biographic books. Apart from the tragic path of is life, these material also depict him not only as gifted but as a compulsively working his maths. So that’s no wonder he would even dream about it. And of course, you can model that as the result of unconscious thoughts throwing the result of problems that were fed to the mind during awaken time.
So, the magic recipe is practice, practice, and practice even more.
> I think maybe because it flies in the face of everyone that claims 10x developers are not possible.
On what metrics? The ability to be able to throw impressive "magical" code and being able to work efficiently with the rest of a team will not necessarily come together, for example.
> Or that some people are just genetically gifted and there's nothing you could possibly do that would get you within a hundred yards of them no matter how much hard work you put in, or how good your teachers were, etc.
“Be yourself; everyone else is already taken.” is generally attributed to Oscar Wilde.
Every life is genetically gifted, but not all lives encounter the environment that enables to thrives this gift – unfortunately.
Also no one is perfect. Many aspect of human life are not only dependent on how performant you are individually, so being really "too ahead" can be actually a severe handicap – at least if it doesn’t come with equal excellence into convincing other to trust you. And so depression is not something you can expect to be out of the realm of gifted evoked here.
In addition to hard work Ramanujan was exceptionally gifted, and likely stumbled on some mental processes at an early age that gave him an edge, and spent the rest of his life perfecting them.
Real shame that he passed so early, due to a disease that would have easily been avoidable or curable today. Imagine what else he would have been able to achieve if given another 10-30 years.
According to quoteinvestigator there is no evidence that Oscar Wilde said that
I'm reminded of a documentary on Tiger Woods on one of the Discovery channels where the voice over is going, "How did Tiger Woods become such a genius? Was it his family's Buddhist traditions?", and it went on to list a bunch of possible factors, all the while showing a clip of Tiger Woods playing golf with absolute perfection AT THE AGE OF FOUR! I had to laugh at the mental effort being expended to avoid the horrific conclusion that he was born with it.
I also find it fun to consider that the population is 7x larger today than it was in 1800, and there's generally much better access to education across the board. Around 1900 you've got a bunch of absolute giants in mathematics: Hilbert and Ramanujan, for starters, but also Riemann and Lobachevsky (whose noneuclidean geometry work was basically Einstein minus the physics). So, suppose O(5) world-changing geniuses. On raw population alone, one would expect about 35 people of similar caliber to be active today. But also mathematics as a profession, access to phd programs, etc, is many times larger per capita than it would have been in 1900. So I would personally guess that there's likely to be a couple hundred people currently active in math of a similar caliber.
If you cloned Euler, and magically gave the clone a new upbringing that resulted in the same math skills, Euler!clone wouldn't be becoming famous for proving e^iπ + 1 = 0, because he'd have learned that in high school like the rest of us. Instead, he might go off and do something like Terence Tao and hack away at the Twin Prime conjecture in a series of papers that require a PhD in mathematics just to understand the abstract. It's a lot harder to become famous that way, even if the work being done is much harder in some sense.
I don't want to diminish the genius of pa-h mathematicians, because Euler is still a legitimate genius by any measure, but part of the reason why he could get around the way he did is that he was metaphorically working in a field where he could pluck ripe fruit off the ground. Similar geniuses exist today, but even as geniuses they still need ladders to get to the fruit now, and that just takes more time.
I'm not lamenting this, celebrating it, or judging it... it just is the way it is.
And I wouldn't want to grow up be a giraf either,if climbing a tree works as well, or just shake it up.
Oh boy, these metaphors are useless.
Let's put it another way. Picking the low hanging fruit has just become either, and it is still necessary to get there. As you mention Terry Tao, to imply at the same time that we had never heard of him is rather disingenious. What is the point?
It has been noted that science, for lack of a better word, is becoming increasingly specialized on the individual level and that we have no polymaths today as it were. Maybe you are still correct insofar as we from in the midst of it cannot yet really tell what combination of skill will take the crown.
But then let's drop a few names, Peter Shor, Noam Chomsky, Frederik Kortlandt. Oh that's right, you never heard of Kortlandt, probably my favorite Indo-Europeanist.
But the thought is ultimately naive. The pedastol we put these people on doesn't grow with the number of people alive. I don't know what model of a social network you would need to make that calcjlation. The measure of genious that should obviously be part of that theory, but I am sure such measure does not exist objectively.
Besides, I'd argue that with an evergrowing body of knowledge and tooling necessary to weild it, the requirements and constraints for a genious today are different from the times that you want to compare. For a measure and linear growth you would need a linear space to begin with.
Really though, I just wanted to say that there is a ton of smart people out there. Shoulders of giants and all. Which should be a humbling thought.
As an amateur chess player, I don't find it depressing -- although more likely I use it as an excuse. I know that there are kids out there who started out with a significantly-higher ranking within a year of playing than I will ever get to as an adult. (That said, many those same kids will eventually go on to put in tens of thousands of more hours than me.)
I guess the part that's slightly depressing is that, even if I slog and slog, and raise my ranking up by a bit, it will always feel like winning by perspiration, and not by divine inspiration. But I think it's fine for me to say "this isn't my game, I will play it so long as it's fun, but I won't ever be a genius at it."
* 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...
* Check the full photo coverage about Ramanujan and his fascinating works at:
http://casualwalker.com/museum-for-the-man-who-knew-infinity...
"Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper."
So he did his derivations and proofs on the blackboard, and just wrote down the result in his notebook and then erased the blackboard.
He knew he didn't have room for anything else in his notebooks.
Source: https://www.quantamagazine.org/three-puzzles-inspired-by-ram...
There is a method behind it. The formulas are derived from the study of relatively simple ODEs, in this example f'(x) = x.f(x) + 1 and g'(x) = x.g(x) − 1. The latter was solved 100 years prior by Jacobi. While the solutions are non-trivial, they are fairly compact, accessible to a high schooler. The presentation linked from the blog is pretty good at unveiling the magic.
This is to say that, with proper guidance, the kid down the street can too become as cool as Ramanujan.
https://math.ucr.edu/home/baez/ramanujan/ramanujan_whittier_...
Here's the direct link to the presentation the post author links to in the blog
https://math.ucr.edu/home/baez/ramanujan/ramanujan_whittier_...
https://golem.ph.utexas.edu/category/2020/08/chasing_the_tai...
https://golem.ph.utexas.edu/category/2020/09/chasing_the_tai...
In other words, this presentation doesn't seem to teach me how to think like a mathematician, it seems more like showing me how mathematicians can find solutions that nobody else could ever find in a million years.
I guess that's the point when the topic is about Ramanujan, maybe?
Also, the differential equation doesn’t come from nowhere/is not random, it’s derived in the presentation from differentiating the function and seeing that it still resembles the original function in some way, allowing you to describe it with a differential equation.
What he describes as a trick with solving the differential equation can be explained - if you have f’(x)=A(x)f(x)+b, that’s a strong hint there’s an exponential there somewhere; if A is x, then the chain rule hints that you have and x^2 in the exponential, etc...
A lot of it (depending on what mathematician) can boil down to pattern matching and having a big-enough bag of tricks.
https://www.quora.com/How-did-Strassen-derive-his-matrix-mul...
"Good mathematicians see analogies between theorems or theories. the very best ones see analogies between analogies." Stefan Banach.
Indeed. But without knowing what really happend here, I suspect the following:
Somebody (I guess Ramanujan), came up with the random differential equation, then found two equal solutions that looked structurally different. So he asked how you can prove that they are equal.
So, I'm not sure here, but in general, this is how many mathematical puzzles are created: You come up with a random proof by starting somewhere in the middle that then transforming your equation in two totally different directions. If you now throw away the middle part, you have a hard to prove fact.
Yep. One of the main differences of advanced Math is that the problem gives no clues about the solution. You look at the problem, you pick one of the standard tricks from your backpack of trick, and try to hit the problem as hard as you can. Sometimes the trick solves the problem. Sometimes the trick simplifies the problem. (Sometimes it is not obvious that it is a simplification). Sometimes the trick does nothing, so you just pick another trick from your backpack of tricks...
If that doesn't work, you call a friend that has another backpack full of tricks ...
The idea is that in a Math BS or PhD you see a lot of tricks, and get some advice about where each one can be useful. Transforming an infinite sum to a function is an standard trick. Transforming that to a differential equation is not so standard, but I've seen it before.
And sometimes no trick solves the problem, so you must invent a new trick. After a few year, if the trick is useful in other problems it will become popular and it will be added to the standard curriculum of a major in Math, or to the advanced classes for PhD, or just be a standard trick in a small niche.
If you're not used to working with continued fractions, the techniques are very counterintuitive. None of the usual calculus tricks for infinite sequences work normally. It's probably not best to read that part as an introduction to the theory of continued fractions. As Baez himself admitted, even he had a hard time understanding Laplace's proof.
The rest of the development, I assume, was not so confounding. It's a little of both: here's something you might understand, there's some black magic.
> I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
Who is the author of the paper?
My favorite Ramanujan’s formula: 1 + 2 + 3 + 4 + ⋯ = −1/12
[spoiler alert]
You can extend finite summations to infinite summations. The first extensions are nice and you get intuitive results, and are the extensions studied in Calculus in the university.
But these extensions are not enough for this sum. You must make more bold extensions, and the results are not as intuitive, and not useful outside some special applications.
I think that the generalization that use averages are fine, and some analytic continuations using power series are fine. Analytic functions are just too good to ignore them. It is not another mathematical system. It it just the extension of our mathematical system.
Anyway, to get a result for this sum, you must drop too many of the obvious and expected properties of summation. For example if you add a zero in front of the sum, the result changes :(. There was a nice blog post I can't find, but there is a hint of the problem in https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B... . So the result is ... polemic.
Anyway, in some courses of a Math degree, one of the important ideas is that you have many possible definitions of convergence. For example, if you have a sequence of functions, you have https://en.wikipedia.org/wiki/Uniform_absolute-convergence and https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_spac... and many many many more.
So you don't write
lim f_n -> f,
lim_{something} f_n -> f
The trick here is to hide the nasty problem of the multiple definitions of convergence in the dots. So the correct statement of the problem is
1+2+3+4+..._{with a weird convergence}=-1/12
but it doesn't look as nice as
1+2+3+4+...=-1/12
[1] https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...
[2] https://medium.com/cantors-paradise/the-ramanujan-summation-...