The stories of mathematicians like Srinivasa Ramanujan, who claimed to have derived complex partitions and identities in dreams, have always captivated me. It's as if their minds were tapping into some hidden reservoir of knowledge. I'm curious what drives these intuitive leaps. Was Ramanujan's brain quietly processing patterns during sleep, leveraging its default mode network in ways we're still struggling to understand? Or was it something more fundamental – an emergent property of complex neural networks, perhaps, or even a glimpse into Jung's collective unconscious?
I'm curious to hear how others think about this phenomenon. Do recent advances in neuroscience, AI, or cognitive psychology offer any clues about how innovators like Ramanujan access these hidden sources of insight? Or are we still stuck in the realm of "genius is mysterious"?
Starting from the basics, Ramanujan was known to spend huge amounts of time in the library pouring over mathematical texts. He was also personally and spiritually obsessed with mathematics, thinking it was an expression of divinity. So its quite probable a significant chunk of his memories were already mathematical and random accesses to it were the same.
People are impressed by the seemingly spontaneous origin of genius ideas, but forget all the analytical labor that went into preparing the mind for conceiving such thoughts. Genius doesn't come out of nowhere. The 'stroke of the genius' required a lot of hard work and experience.
And, it works the same with us 'normal' people, even though the results are less spectacular. Want to have a good idea? You need to feed your brain everything you can and obsess over it, then let it stew for a bit. Again, and again. Only then do you stand a chance for a really good idea to just pop out.
As the article mentions ,we was familiar with the literature. He communicated with other mathematicians, read papers, and submitted in journals while in India. he was not some hermit in a cave or something. I think this claim that he just dreamed the results part of mythology that has been built around him. From what I read, he he did a lot of the grunt work deriving these formulas but only published the final results, so it only appears that he conjured them out of nothing. It's not like he could have sent Hardy a book-sized letter of all the steps to derive those results.
This myth of him just coming up with results without all the hard studying he did is often repeated. His results are made to be seen as impossibly hard and his impact is overstated. Galois solved a problem that was open for 350 years, he made significant contributions to new branches of math and he died at 20. I don’t like the whole ranking thing, but with all the myths surrounding Ramanujan I feel compelled to say Ramanujan is overrated and Galois is by far the more gifted and accomplished to just name one person.
There’s a relevant quote from Swami Vivekananda on this
From Karma Yoga, Chapter: I, Karma in its effect on character—
What we say a man “knows”, should, in strict psychological language, be what he “discovers” or “unveils”; what a man “learns” is really what he “discovers”, by taking the cover off his own soul, which is a mine of infinite knowledge.
We say Newton discovered gravitation. Was it sitting anywhere in a corner waiting for him? It was in his own mind; the time came and he found it out. All knowledge that the world has ever received comes from the mind; the infinite library of the universe is in your own mind. The external world is simply the suggestion, the occasion, which sets you to study your own mind, but the object of your study is always your own mind. The falling of an apple gave the suggestion to Newton, and he studied his own mind. He rearranged all the previous links of thought in his mind and discovered a new link among them, which we call the law of gravitation. It was not in the apple nor in anything in the centre of the earth.
Whenever I read about Ramanujan having divine revealing formulas in his dreams, I remember Swami Vivekananda’s quoute on consciousness and mind.
edit: found another relevant quote from Upanishads on tapping the infinite knowledge:
Translation: “The Self hidden in all beings does not shine forth, but it is seen by subtle seers through their one-pointed and subtle intellect.”
Explanation:
The ultimate knowledge or truth is hidden within all beings and is revealed through subtle inner perception. The idea is knowledge is latent within the mind, and it is discovered, not externally found.
Not sure I follow the explanation. And apologies cause I’m not that familiar with these texts, but afaik it talks about “soul” here, not information? Words like Brahman, Atman come to mind.
In case I mistook it, ignore the above.
All knowledge that the world has ever received comes from the mind
Doesn’t it come from the mind’s interaction with the world? I don’t think that any important knowledge comes from doing nothing (apart from psychology due to its nature). It all sounds like a typical religious salad, tbh. “You” discover from your “mind”, but then some seers understand that “you” and “all” and “mind” are the same, etc. Well, you don’t have to be a seer to see it. “You discover from your mind” is a useless semantic loop that creates all this unnecessary complexity, imo.
The mind quickly falls apart when interaction-deprived (at least human one). This whole “comes from the mind” is just a metaphor, because the mind is also the arriving point, so it’s like “knowledge is defined by and comes from where it arrives to, on itself”. This semantic no-op nonsense originates in taking this metaphor literally, which one should not do.
What physically happens is that a mind accumulates and (sometimes) systematizes information from outside, and knowledge is the result of this concentration process. Akin to a puddle collecting water from higher grounds. But it doesn’t synthesize water by simply being concave.
Blowing ideas out of poor metaphors is a regular thing in religious texts, so be careful with that if you want to build some practical world view.
I think you're too hung up on the mind part. I don't understand this as saying the mind synthesizes knowledge from nothing. That's why I said, there is no fundamental difference for me - whether the mind conceives of something new, or synthesizes knowledge from outside information is not very interesting a difference, since a mind is required to grasp this knowledge.
Maybe a better term would be information vs. knowledge.
That’s beautiful, thanks for sharing. I’ve seen that firsthand growing up. It’s like when you hear a song as a kid, and it just sounds cool. But then, later in life, you hear the same song, and it hits you on a whole different level—like the meaning was always there, but you had to live through stuff to really get it.
> Translation: “The Self hidden in all beings does not shine forth, but it is seen by subtle seers through their one-pointed and subtle intellect.”
That doesn't look sound to me. If the "seers" are seeing "The Self", are they beyond and separate from "The Self"? If they do so with their "subtle intellect", is that intellect outside of "The Self"?
If affirmative, then "The Self" is something external to the seer, making the term a misnomer. And furthermore, there is something outside of The Self (that which is seeing The Self), which remains to be explicated.
I'll try to slightly dive into Advaita philosophy here.
Both the Upanishads and the Yoga Vasistha affirm that the distinction between the seer and the Self is an illusion created by the mind. When this illusion is dispelled, the oneness of the Self is realized. The intellect, which seems to function as a separate tool, is ultimately part of the same illusion. True knowledge is realizing that there is no separation—everything is the Self.
Quoting one from Katha Upanishad 1.3.10:
"Indriyebhyaḥ parā hy artha, arthebhyas ca param manah, manasas tu parā buddhir buddher ātmā mahān parah"
Translation: "Beyond the senses are the objects, beyond the objects is the mind, beyond the mind is the intellect, beyond the intellect is the Self."
This quote emphasizes that the intellect is still a part of the illusion. Beyond even the intellect lies the Self, which is one and undivided.
Likely because it's far more complicated than Newtonian gravity and he wouldn't have been aware of the shortcomings of his theory e.g. Mercury's orbit. Also, the study of light was only just beginning in Newton's era - Ole Rømer's measurement of the speed of light was in 1676, but Maxwell's theories didn't come out until 1865.
If you spend an enormous amount of time (like, too much) working on a single piece of software, you'll come up with solutions in your dreams and wake up and write them down. It's not that rare. Now... the solutions might be for loops so I'm not comparing such situations to Ramanujan, but it's not some extremely rare phenomenon.
I am also intrigued by this question: What was different for guys like Ramanujan, and how were they able to tap in to this hidden reservoir of knowledge. And how can we replicate it
One guy able to tap into this knowledge in dreams is an indication that it is possible. Now, how do we make this the default for everyone is the question I wonder about
The way we found one variant of wheat in Mexico that was resistant to bacteria, and replicate that to the whole world -- can we do something like that for humans
( even I don't like the sound of it, but I hope you get the feeling )
> how were they able to tap in to this hidden reservoir of knowledge.
My guess would be by filling the reservoir first. After filling it, then you can stir it and extract new knowledge. In other words: you get better at math by studying, reading, and thinking a lot about math. I don't think there has been a brilliant mathematician who has not immersed himself in math at first.
We all already do solve problems in our dreams. Even if you haven't awoken with a Eureka, surely you've occasionally found that something that was difficult to solve yesterday is obvious with a pair of fresh eyes.
how do we make this the default for everyone is the question I wonder about
You delete stupid prejudices from society, then it allows itself to do that. One group of people forcibly practicing it for idiotic reasons now plagued the whole field with its label, amplified by religious ideas. The irony is, it’s that same limitation that disallows to address itself and continues to propagate.
As others say - just study a lot of math. You will not be Ramanujan, but you will be better at math.
That's not to say there isn't something spiritual and mystical in knowledge. I think the fact that he was from a different background gave him a different perspective. But yeah you cannot skip all the math studying.
The recent book Mathematica by David Bessis attempts to describe this exact topic. One core theme of the book is that math is done by building mental models and using our intuitions that come from. Formalism is then used to shape and refine these models. By spending time developing these models, insights eventually become obvious. These mental models are the essence of mathematics, not theorems. The author explicitly uses neural networks as an example to describe how this negative feedback could work to make these changes to our mental models happen in our brains.
The core premise of the book is to describe how mathematicians work and think and to show that this is a process that everyone can do (although some will be better at it than others). It includes interesting accounts of Grothendieck, Bill Thurston, and Descartes as well as from the author's own research career at Yale and École normale supérieure. The book is targeted at the general reader and at times reads a little like a self-help book, especially in the first third or so. However, I found it to be an enjoyable and fascinating read. It provoked a lot of interesting questions about the nature of learning and provided a framework to begin to answer them (e.g. "How can I have proved something and yet feel no understanding of it?", "How can some people solve problems orders of magnitude faster than other smart people, as if they don't even have to think about it?", "Why do I sometimes watch a presentation on a new topic, follow every step, and come away feeling like I've learned nothing?"(* see except below)). I don't think I'm doing it justice here, so I'll stop by saying I highly recommend it based on your comment.
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* I'll use this as an excuse to provide a related excerpt featuring Fields Medalist and Abel Prize winner Jean-Pierre Serre:
One day, I had to give a lecture at the Chevalley Seminar, a group theory seminar in Paris. I didn't have substantial new results to announce, but it was an opportunity to make a presentation even simpler than usual. [...] A couple of minutes before the talk was to start, Serre came in and sat in the second row. I was honored to have him in the audience, but I let him know right off that the presentation might not be very interesting to him. It was intended for a general audience and I was going to be explaining very basic things.
What I didn't tell him, of course, was that his presence was intimidating. Still, I didn't want to raise the level of my talk only to keep him interested. I just kept an eye out to see if he'd taken off his glasses, which would mean he was getting bored and had stopped listening. No worries there—he kept his glasses on till the end.
I gave my presentation as I would have without him there, speaking to the entire audience, especially the students seated in the back, whom I was pleased to see listening and looking like they understood.
It was a normal presentation, fairly successful, not very deep but well prepared, clear, and intelligible. At the end of the seminar, Serre came up to me and said—and here I quote verbatim: "You'll have to explain that to me again, because I didn't understand anything."
That's a true story, and it plunged me into a state of profound perplexity.
Apparently, Serre wasn't using the verb to understand the way most people use it. The concepts and reasonings of my talk couldn't really have caused him any difficulty. I'm sure he wanted to say that he understood what I had explained, but he hadn't understood why what I had explained was true.
There are two levels of understanding. The first level consists of following the reasoning step by step and accepting that it's correct. Accepting is not the same as understanding. The second level is real understanding. It requires seeing where the reasoning comes from and why it's natural.
In thinking again about Serre's comment, I realized that my presentation had too many “miracles,” too many arbitrary choices, too many things that worked without my really knowing why. Serre was right; it was incomprehensible. His feedback helped me become aware of a number of very big holes in my understanding of the objects and situations I was working on at the time. In the years that followed, research into explanations for these various miracles allowed me to fill in some of the holes and achieve some of the most important results of my career. (However, some of the miracles remain unexplained to this day.)
But the most troubling aspect was the abruptness, the frankness with which Serre had overplayed his own incomprehension.
My intuition is that some people are able to develop abstractions on top of abstractions (compression is a key part of intelligence) that allow them to traverse the search space much, much faster than without these, and with enough repetition this becomes routine, even faster in the brain.
I don't think we have a good theory of how this works, or at least I haven't come across it.
Neural network guided search is somewhat similar but I think we are missing several key pieces
I'm curious to hear how others think about this phenomenon. Do recent advances in neuroscience, AI, or cognitive psychology offer any clues about how innovators like Ramanujan access these hidden sources of insight? Or are we still stuck in the realm of "genius is mysterious"?