aahhh just another one of Ramanujan's random identities that he divined from the minds of the gods. reading these has been a reliable way for mathematicians to humble themselves since 1900.
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.
That’s very fascinating. I guess it still is the case for many people in the world, that paper is too expensive. Whereas for many of us it’s readily available in massive, one might even say infinite, quantities.
How would a mere mortal could possibly divine equations like this?
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.
"Ramanujan said that, throughout his life, he repeatedly dreamed of a Hindu goddess known as Namagiri. She presented him with complex mathematical formulas over and over, which he could then test and verify upon waking. Once such example was the infinite series for Pi:
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 this kind of explanation irritates many people for some reason. As if things just can't be that way, there must be a repeatable, reproducible recipe that someone else could follow, and if they did, they would get the same results as Ramanujan.
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.
There have been quite a few such people in history. It's really humbling to know that Galois did what he did at just 20, Gödel was just a year older than me when he broke PM, Neumann could solve problems faster than what it took a person to write a program on the computer to solve the problem and the computer running.
While genetics play a huge role, nurture plays a large role as well [1].
Yes, it goes against the cult of Effort, "10000 hours of deliberate practice" and all that. But I believe you're 100% right: those at the absolute right end of the distribution are often gifted. They work hard, too, but that is akin to sharpening the knife. It is not the knife.
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.
It's also the case that genius alone isn't sufficient. One also needs hard work and, as Ramanujan illustrates, access to institutions that can publish+legitimize the work.
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.
Bear in mind that if someone of Euler's level is operating today, you very likely wouldn't know unless you were in the field, and it might not necessarily be obvious even then.
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.
Cool, I commented to the same effect, but your food-based metaphor brings to mind the notion that we have less lead in the water today, better health care, a handle on air pollution etc. p. p. I am doubting that as I write it, thinking of the hopelessness of CO2 reduction and all the things to come after we have already passed the point of no return.
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.
Interesting thought. We are after all posting under a link to John C. Baez.
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.
What cult of effort are you talking about? I think usually people outside a field underestimate both the talent required and the effort. Children can be even more obsessed than adults also and be completely laser focused.
> I think this kind of explanation irritates many people for some reason. As if things just can't be that way, there must be a repeatable, reproducible recipe that someone else could follow, and if they did, they would get the same results as Ramanujan.
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.
I think it’s just the case that some people are born with intuition or better yet everyone is but some people are born with intuition that is game changing. Ramanujan had a natural intuition for infinite series as Einstein did for relativity and even though their work would have eventually been discovered one way or another I believe their intuition does give the impression they did what would have taken many people to do in a short time in comparison.
yes, I think this is the likely explanation - your brain processes information when you are asleep and many times you may get insights from the unconscious this way in the morning.
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.
I for one find the fact that such people existed very exciting. It means that some possible configuration of our physiology allows for amazing capabilities. Maybe nurturing has to do with it too, but the point is that it is possible.
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!
...or the 4-year-old chess prodigy, or the 5-year-old composer.
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."
I've had times when during a dream, I've heard a melody, and when I woke up, I picked up an instrument and was able to replicate the melody I heard when dreaming. If I can come up with good sounding melodies in my sleep despite being a fairly unpracticed musician, I can only imagine what geniuses in their respective fields are able to figure out unconsciously! Obviously the presentation within the above described dream is fairly dramatic, but that might just have been his mind's way of making sure he remembered it when he woke up.
Reminds me of TempleOS, "created by American programmer Terry A. Davis, who developed it alone over the course of a decade after a series of prophetic episodes that he later described as a revelation from God."
I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly.
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?
When you see some infinite nested pattern, you try to capture its symmetries in some way. Differential equations are one such way. Or looking for symmetries. Induction can also help, or looking for fixed points. They can all give information. It’s a bag of tools. One could go and then look at the differential equation in more detail and relate it geometrically to the series and maybe learn some cool things (but that would be a posteriori, which you don’t want!)
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.
> he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly
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.
There are two ways to solve a math problem and you described one of them: hit the nut with a hammer as hard as you can until it breaks. Grothendieck preferred the second method of slowly raising the ocean and soaking the nut for a few years, until one day it opens all by itself.
Homeopathic approaches can feel just as arbitrary, though. “How did the mathematician know to take a bazillion tiny steps to get here?”, “Which steps in this process are the ones that do anything?”, and “Why can’t they just get to the point?!” are all questions one might ask oneself while in the midst of reading such material.
I'm reminded of this answer, regrettably on Quora, about how mathematicians have an unfortunate habit of tearing down the scaffolding they used to build their results.
> I am not a fan of these types of proofs. Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly.
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.
Unfortunately sometimes mathematics just uses tricks to do certain things. As you move along in your math learning you pick up a bag of tricks to deal with certain situations that you’ve seen before.
>Baez didn't explain how one would go about searching for this proof, he just happened to know that repeatedly using the differential operator on a seemingly random differential equation would create a pattern that resembles the original equation after using substitution repeatedly.
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.
The source of the derivation of the continued fraction part is https://mobile.twitter.com/duetosymmetry/status/130202147174... and it was an attempt to simplify a derivation due to Jacobi. Perhaps he explains how he comes up with it, though you’ll need to be able to read Latin. Possibly this is a relatively standard technique for solving differential equations with continued fractions which people no longer use.
There is a lot of pattern-based theorem proving in the works of Ramanujan, Jacobi and Euler. This is beautiful mathematics, but this is not how usual mathematical research is done. Thinking like a mathematician often involves generalization, finding analogies between seemingly different theorems etc.
"Good mathematicians see analogies between theorems or theories. the very best ones see analogies between analogies." Stefan Banach.
> 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."
It is weird, very weird. A very nice explanation of the proof is in a video by Mathologer "Ramanujan: Making sense of 1+2+3+... = -1/12 and Co." https://www.youtube.com/watch?v=jcKRGpMiVTw
[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.
Honestly, I don't buy the proof. When you take a formula that's valid for a certain range of inputs and just use it outside that range anyways, you're no longer in the land of proof. At that point, you're in the land of "if we extend out mathematical system, we get this". And that's fine, but you don't wind up with "sum(1...inf) === -1/12", you wind up with "in this other mathematical system, sum....".
I'm not going to blame you. As the video explain, this is not a straightforward and obvious generalization of summation.
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.
Niels Abel: "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes."
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.