Ramanujan's lost notebook | Hacker News Reader

Ramanujan's lost notebook

en.wikipedia.org

306 pointsbenbreen94 comments

In the spirit of openness which is the hallmark of FOSS such as GNU/Linux, would this book be scanned and published for all to see? That would be great. Like FOSS, math too belongs to the world. I heard someone say that the university has not published many of his works. That would be sad. If published someone like Terrence Tao could write formal proofs for them like he and his team did some improvements for the work by Yitang Zhang.

> If published someone like Terrence Tao could write formal proofs for them like he and his team did some improvements for the work by Yitang Zhang.

From the second paragraph:

> George Andrews and Bruce C. Berndt (2005, 2009, 2012, 2013, 2018) have published several books in which they give proofs for Ramanujan's formulas included in the notebook

The notebook (like Ramanujan's three previously known notebooks) has already been published, in 5 volumes by Bruce Berndt and George Andrews, with extensive annotation (e.g. just flipping at random, in the third volume Chapter 6 is called "Theorems about the Partition Function on Pages 189 and 182", and occupies 24 pages, and indeed contains formal proofs etc). A raw scan of the notebook(s) is even available online: http://ramanujan.sirinudi.org

Unfortunately the actual notebook links on that site appear broken

The page number selector appears to link to working PDFs.

Just enter a page number; it works fine. Or if you wish, here are the direct links:

http://ramanujan.sirinudi.org/Volumes/01/01.pdf

http://ramanujan.sirinudi.org/Volumes/02/02.pdf

http://ramanujan.sirinudi.org/Volumes/03/03.pdf

http://ramanujan.sirinudi.org/Volumes/04/04.pdf

Ramanujan's total known/preserved output today is 37 published papers and these four notebooks; all of it is available on the site. The notebooks have been transcribed and published in several volumes with explanation (as mentioned above); those are the only thing not included on this site (and arguably, not "by" Ramanujan).

This sounds exactly like the thing I’ve kind of wanted—a compilation/overview of Ramanujan’s work, ideally with commentary. I’ll have to take a closer look at this. What I’d really like is something akin to Yves Hellegouarch’s Invitation to the Mathematics of Fermat-Wiles which seems to be not only out of print, but rather rare now (it’s listed on Amazon at over 400 bucks although that could be algorithmic madness among the sellers).

I think Bruce Berndt's Number Theory in the Spirit of Ramanujan (2006) may be along the lines of what you're looking for, and also possibly G. H. Hardy's Ramanujan: Twelve Lectures on subjects Suggested by His Life and Work (originally 1940, but republished 1991/1999 by AMS with further commentary by Berndt). I'm not qualified to say for sure, as they get too hard for me pretty fast. :) [Some of Ramanujan's easier papers I was able to read (very slowly) for quite a few pages though, such as the one on highly composite numbers.]

FWIW, https://archive.org/details/invitationtomath0000hell .

That is quite possibly the most disappointing URL in the history of URLs. Invitation to Math Hell, indeed.

His penmanship is quite aesthetic

Yeah, when I read "more than one hundred pages written [...] in Ramanujan's distinctive handwriting" in the Wikipedia article, I was thinking "almost indecipherable scrawl", but this borders on calligraphy...

All those links 404 for me

They are http ... some browsers may default to not loading them.

Thanks, I'd set my browser to always switch to https

FOSS != piracy

And, frankly, I find it pretty upsetting that you seem to negate that equation.

Sorry for nitpicking, but just trying to clarify what seems like a double negative - do you mean "to affirm that equation", i.e. that the author claims they are equal?

I guess you're right, negating an inequality is likely the same as affirming an equality? I am not a native speaker.

Is that not the same logically in every language?

No, in Spanish double negations are not positive statements.

I think the parent commenter was using the word 'equation' in its colloquial sense to refer to any mathematical expression or statement, despite their statement containing a 'not equals to' symbol, rather than actually equating them (which would have an equals sign).

He's been dead for 104 years, where's the piracy?

I expect future AI models to be like Ramanujan. Has tremendous intuition but can't accurately explain it's reasoning for arriving at a solution.

Perhaps this will be true, for the types of insights that also require tremendous intuition for people to see. One difference is that it will be possible to examine the full state of the AI which lead to the insight, and perhaps learn from it.

(edit: rewrote to flow a bit better)

It seems like a misconception that Ramanujan could not explain his reasoning.

He did not write it down on his notebooks (and before going to England he mostly lacked the language of formal proofs to do it), but for the most part the equations are arranged in a logical order, and when meeting with Hardy he had no trouble explaining where each equation came from (though a lot of his reasoning was too unrigorous for Hardy's liking).

I’m an atheist, but when Ramanujan claims to derive all of their formulas from god, I ask how we can make it easier for them to listen to god, rather than feel the urge to argue against them.

If you meet him, and he thought discussing god was interesting, would you then do it? If he wanted to discuss your faith in god, would you do it then?

Discussing is always interesting but I would probably interpret it as him not being able to explain his own abilities like some people just "see" good chess moves or just "know" that there are actually 48 matches on that floor

I would wonder what the truth of the matter is, but each to their own, the unknown is not popular/pleasant.

If you look the origin of the word genius I think there is a good parallel as thinking that your own genius is an external spiritual entity whispering you solutions

I feel like his statements on god are very similar to this one:

> Simon P. Norton, an expert on the properties of the monster group, is quoted as saying, "I can explain what Monstrous Moonshine is in one sentence, it is the voice of God."

https://en.wikipedia.org/wiki/Monster_group#Moonshine

Solid.

Pun intended.

Goddess also, actually.

https://en.m.wikipedia.org/wiki/Srinivasa_Ramanujan

Check section "Personality and spiritual life".

I feel like this comment and comments by western atheists make me think that the west has this idea of judeo-christian god that they just bring into any conversation without thinking deeply. For ramanujan god is more like an emotion.

I'll never figured out whether Einstein thought that the god as described by Spinoza is a real thing? Since there's so many misattributed quotes of him which frames him as talking about the judeo-Christian god as most people know by modern standards.

Also, there have been other early religions like Gnosticism in the west, which also falls into this category. Point is, even western religion has way more nuances.

I would suggest that the answer to that is "yes" in that the Spinozan God is... for lack of a better word the Universe.

(Kaizō in 1923 https://books.google.com/books?id=vLm4oojTPnkC&pg=PA262#v=on... )

> Scientific research can reduce superstition by encouraging people to think and view things in terms of cause and effect. Certain it is that a conviction, akin to religious feeling, of the rationality and intelligibility of the world lies behind all scientific work of a higher order.

> This firm belief, a belief bound up with a deep feeling, in a superior mind that reveals itself in the world of experience, represents my conception of God. In common parlance this may be described as "pantheistic" (Spinoza).

https://www.gutenberg.org/files/3800/3800-h/3800-h.htm

(Ethics was one of the hardest reads I had back in modern philosophy class)

And while it is a gross simplification of Ethics, from news://rec.humor.funny ( https://everything2.com/title/Existence+of+the+System+Admini... )

    1. The System Administrator is defined as the most perfect user possible.
    2. The property of necessary existence means that anything which possesses it must necessarily exist.
    3. If existence is better than non-existence (see the ontological proof), then necessary existence is better still.
    4. Any perfect user must possess the property of necessary existence.
    5. Therefore the System Administrator must necessarily exist.

    However:

    6. Being perfect, the System Administrator cannot make mistakes, delete the wrong account, trash the root directory, mess up a tape load, etc.
    7. Being perfect, the System Administrator can not be capable of goal-directed action, because such action would imply that the network is somehow less than perfect in its current state.
    8. Therefore, the System Administrator is really more of a force of nature within the system.
    9. Arguably, then the System Administrator *is* the system itself.

    Counter-argument:

    1. None, since the System Administrator has been defined to the point where it is a totally useless concept, there's no point in arguing.

    At least this resolves one of the major issues: the Spinozist argument proves that *if* the System Administrator does exist, it cannot be intelligent.

---

The God of Spinoza and Einstein is the magnificence of the universe as it reveals itself to us. The universe is real as is its majesty.

Re 6. Listing so many mistakes the System Administrator surely never does is Blasphemy!

I was with you right up until the word "emotion." Although "Hinduism" is vast and covers quite a few disparate beliefs and philosophies, none of them (to my knowledge) even approximately equate God with an emotion. Even bliss (ananda) is only one aspect.

I don't claim to know the particular religion of his village and/or lineage, but I understand his God to be something closer to the Consciousness that (supposedly) gives rise to all of reality.

>Although "Hinduism" is vast and covers quite a few disparate beliefs and philosophies, none of them (to my knowledge) even approximately equate God with an emotion.

Bhakti does.

https://en.m.wikipedia.org/wiki/Bhakti

>covers quite a few

Tons.

>Hinduism

There is no such thing.

The term is a Western invention, by complacent sneering so-called scholars about India.

You are confusing using emotion to reach God with emotion being God.

And of course there is "such a thing." You even described what it refers to, a definition which is sufficient for the purposes of this thread.

There you go with your weasel words again.

Hey academic philosopher, God is everything, and there is nothing else, so God can be emotion and emotion can be God :) :) :)

Don't you know about Brahman?

There is nothing else ;)

https://en.m.wikipedia.org/wiki/Brahman

QED.

> For ramanujan god is more like an emotion.

for most jewish, christian, and muslim mystics and some theologians, as well.

the 'new atheist' conception of what "all christians" think about when they think about god is, well, not very representative.

True. I was just commenting on the lack of depth in some modern discussions in the west

The lack of depth in modern discourse is not limited by geography, I'd think.

> I feel like this comment and comments by western atheists make me think that the west has this idea of judeo-christian god that they just bring into any conversation without thinking deeply. For ramanujan god is more like an emotion.

From https://en.wikipedia.org/wiki/Srinivasa_Ramanujan#Personalit...

> 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 elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.

That doesn't seem like an emotion at all. It's a very visual, concrete image. It's closer to the "western atheists" view of god that just an abstract feeling of some sort you were describing, wouldn't you agree?

Its a bit of a psychedelic experience. If someone was on LSD and seeing stuff i would compare it an emotional experience. Of course it goes deeper but just explaining the reasoning behind the word emotion

I had a similar experience while in college, although not as dramatic! While studying CS I was working on problems every night late into the night. I came a across a issue I could not figure out, and went to sleep frustrated. That night I had a dream where a very specific solution was told to me(a voice with text) and it was in a way I had never thought of. I tried it out the next day when I woke up and it worked. It always freaked me out as I don't know how that solution was given to me, my assumption is maybe the subconscious is more powerful than we thought in terms of reasoning.

In the early stages of learning something totally new (a new language, etc) I will be completely confounded by something, only to wake up and find that I fully understand it. It's a spooky feeling because it really doesn't seem like you had the prerequisites for that understanding before you slept (although of course you must have).

Atheism is an accepted school of thought in Hinduism [1]

Indian philosophy is diverse and vast. See the comparison of these philosophies [2]

[1] https://en.wikipedia.org/wiki/Charvaka

[2] https://en.wikipedia.org/wiki/Indian_philosophy#Comparison_o...

Sure is. Doesnt invalidate my argument that western atheism lacks depth

“Western atheism” doesn’t lack depth, it’s more that the people actively arguing with others that call themselves atheists tend to lack knowledge of their own position. Dawkins, Hitchens, etc. are notoriously uninformed on the topic of religion.

[reference needed]

https://old.reddit.com/r/askphilosophy/search?q=dawkins&rest...

https://en.wikipedia.org/wiki/New_Atheism#Criticisms

No one in the philosophy world takes the works of the New Atheists seriously. They are pop-culture writers, not serious philosophers (other than Daniel Dennett.)

If you want a serious thinker's thoughts on atheism, you are better off picking up a book by Nietzsche, Charles Taylor, David Bentley Hart, the Oxford Handbook of Atheism, or any other number of thinkers, or browsing the articles here: https://plato.stanford.edu/search/searcher.py?query=atheism

Yeah, Hinduism has enough room for non-believers too. I’m not religious at all, so when someone asks me if I’m keeping a fast for such and such occasion, or celebrating some festival, I simply tell them that I don’t believe in these things and I’m usually left alone.

Although this aspect of Hinduism seems to be changing and becoming a bit more hardline

Why would you have to believe in these things as "real" to still keep a fast? Why would you not answer a simple "no" instead of the answer you give?

Then you need to understand it from a theoretical perspective first. Firstly, when people use the word "god" they honestly ought to define what they refer to by using that word, and what the abilities of that god are supposed to be.

If we're in a physical universe with energy, then it's been explained that a creator-like god can be argued to have 'created' the principle of existence, itself, on which all other things then evolved with energy. So then, what is that principle, stated concretely? Modern physics has already found out almost everything about it.

What is left of an entity if it becomes capable (at least beyond some meaningful threshold) of functionally discarding / disobeying the influence of untruth?

Is someone (in properly controlled circumstances) going to be more or less perceptive of 'what is' when they are more truthful?

I have a personal theory about human intuition.

It feels like answers come "from outer space" or "from God" but it's really just a highly optimized and efficient part of the brain for a very specific function. For that narrow function your brain has an effective IQ of something like 2000. As our brains have a much lower average IQ, we don't have the context to understand how we found the answer so we experience it and describe it as intuition.

If our entire brain operated at an IQ of 2000, we wouldn't experience it as intuition or a flash of genius, we would just say "I figured it out".

What I have read about Ramanujan sounds like he had some form of high functioning savant syndrome. https://en.wikipedia.org/wiki/Savant_syndrome

https://en.wikipedia.org/wiki/The_Origin_of_Consciousness_in...

Its been a while since I read about it but there is some theorizing on the split halves of the brain and the corpus callosum that connects the two halves.

One theory is that the connection between the two halves is a recent evolution and that people used to and some still do perceive the other half of their brain as a separate entity that communicates with them. Possibly being due to the corpus callosum not being the same in everyone. The greater theory being that this is basically the root of all religion.

> However, Wilson died in 1935 and Watson seems to have lost interest in the project in the late 1930s.[3] After Watson's death in 1965, J. M. Whittaker examined Watson's papers (which were in disarray, due to be incinerated in a few days) and found Ramanujan's notebook

Not expecting too many comments here. These formulas are complicated and it took a single person in the later 20th century to re-derive a good chunk of them.

If you are talking about Bruce Berndt, Bruce did not rederive Ramanujan's results alone; he has a big network of collaborators and students who have worked with him, and of course many independent researchers have derived some of his formulas. His books contain all the relevant attributions and history (I talked with him extensively and got some of his classes when I was a phd student in UIUC).

yes that is him. Same for Peter Borwein

can we please stop spending more ink on this hack?

What's hacky about Ramanujan?

So we dont know anything about his math/theorem generation process. By his own admission he saw visions of math/theorem god, which is a BS claim since god doesn't exist. So "god whispered theorems in my sleep" is just a proxy for "I coped it from someone". Many of his so called theorems are false (by Hardy's own admission). Lot of progress in math depends on the "process", not just the end result. But he shares nothing about the process.

> By his own admission he saw visions of math/theorem god, which is a BS claim since god doesn't exist

Shit reasoning like this from atheists like you is why I cringe to admit I'm an atheist when people ask me about my religious beliefs. It is entirely conceivable that he saw visions of math, produced by his own subconscious, and attributed these visions to a god. It would not be terribly dissimilar from any artist who attributes their creative visions to a muse. Do you accuse any such artist of plagiarism too?

He didnt have formal training in math. So where did his subconscious "learn" the advanced stuff in his visions?

Lacking formal training does not equate cannot understand something. As far as copying something goes, there should have been an equivalent mathematician nearby from whom he can copy, right? Who is it?

As an atheist myself, I do not believe in god and I do not believe Ramanujam got it from god. However, it does not change the fact Ramanujan thought he got it from god because he could not necessarily explain how he made those intuitive leaps.

The results of his collaboration with Hardy (the approximations for the partition function and the circle method) make it clear he knew well what he was doing. Nobody was doing anything similar at the time.

A feline can make a precise jump between two points and be utterly unable to explain how it computed the force signals to send to its legs to make the jump. It probably doesn't even know force or muscles. Formal training isn't everything.

"we don't know anything about his math generation process" My man just look at the notebooks. He did not use formal proofs but he was very organized and most of his equations had quite a logical buildup from equations that he wrote before. And even the ones that came out of nowhere are obviously worthy discoveries.

You seem so sure of yourself. How did you end up like that?

> ... is just proxy for "I coped it from someone".

How can you be so confident in this claim?

Lets just say that I feel "he copied" is more plausible than "theorem god whispered to me in my sleep"

I see that. But you're making it sound like you know that for sure. Do you see that there's a middle ground? I.e. "I had a dream where it seemed that God was describing to me these new theorems"?

explain yourself

I almost feel like there is a random ramanujan posting generator. Every day it runs on a cron job, and with some probably something about him is posted. Much like the dunning krueger effect. Some topics arise with such consistent regularity. Almost makes you wonder what the reasoning is

Because he’s a fascinating and mysterious character on top of being the most brilliant mathematician since Euler.

One quarter thought out software project, mild overthinking conspiracy theory, and a non sequitur misspelled reference to Dunning-Kruger. This is like a parody of a Hacker News comment

From https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu... it looks likethere are like 5 or 6 post per year.

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