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Ramanujan Surprises Again (2015)

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Re: Ramanujan Surprises Again (2015)

#21

Post by nneonneo »

An interesting coincidence: it was recently (2019) discovered that the fastest way to multiply two n-bit integers, in time O(n log n), involves 1729-dimensional Fourier transforms: https://hal.archives-ouvertes.fr/hal-02070778. It is quite surprising that the asymptotically best way to perform such an elementary operation should be tied to Ramanujan’s famous taxicab number.

(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.)

Re: Ramanujan Surprises Again (2015)

#22

Post by hnews_account_1 »

dorchadas wrote: post #11
hnews_account_1 wrote: post #8

Earlier quoted context omitted.

A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one…

What was the quote about Feynman? That he loved to cultivate anecdotes about himself or something similar? Makes a lot of his stories make a lot more sense, too.

I think the part that makes it genuine is that he was comically self aware of himself and his craziness. Even when he was pushing the boundaries just for the sake of it and to make a caricature / character out of himself, he did it in a way that made me think that he didn't really pretend to not be doing it for his ego.

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.

Re: Ramanujan Surprises Again (2015)

#23

Post by pmoriarty »

dannykwells wrote: post #3

The taxi cab story is easily a top-5 math story, and is quintessential Ramanujan. 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.

What are the other 4 top math stories?

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.

[1] - https://en.wikipedia.org/wiki/%C3%89variste_Galois

Re: Ramanujan Surprises Again (2015)

#24

Post by karmakaze »

hnews_account_1 wrote: post #8
v64 wrote: post #7

Great read! When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability. It's interesting to find out that the real coincidence here is that Hardy rode in a taxicab whose number had happened to show up in Ramanujan's investigations of Fermat's last theorem.

A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one…

This reminds me of the von Neumann fly puzzle story:

https://en.wikipedia.org/wiki/John_von_Neumann#Cognitive_abi...

Re: Ramanujan Surprises Again (2015)

#25

Post by gameswithgo »

dorchadas wrote: post #11
hnews_account_1 wrote: post #8

Earlier quoted context omitted.

A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one…

What was the quote about Feynman? That he loved to cultivate anecdotes about himself or something similar? Makes a lot of his stories make a lot more sense, too.

i recall him explaining several shortcuts one can use to solve problems in seemingly impossible speeds by drawing on a breadth of experience from similar problems that you have memorized or are easy to compute and interpolating.

its still genius but not in the sense of actually being able to do huge calculations in ones head the way a computer would.

Re: Ramanujan Surprises Again (2015)

#26

Post by user2994cb »

nneonneo wrote: post #21

An interesting coincidence: it was recently (2019) discovered that the fastest way to multiply two n-bit integers, in time O(n log n), involves 1729-dimensional Fourier transforms: https://hal.archives-ouvertes.fr/hal-02070778 . It is quite surprising that the asymptotically best way to perform such an elementary operation should be tied to Ramanujan’s famous taxicab number. (Technically, it works for any number of d…

In fact, there seems to be a lot of interesting things about 1729: https://en.wikipedia.org/wiki/1729_(number)

Re: Ramanujan Surprises Again (2015)

#27

Post by eesmith »

karmakaze wrote: post #24
hnews_account_1 wrote: post #8

Earlier quoted context omitted.

A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one…

This reminds me of the von Neumann fly puzzle story: https://en.wikipedia.org/wiki/John_von_Neumann#Cognitive_abi...

Or swallow, if you think Wigner's account is more accurate than Halmos'.

Re: Ramanujan Surprises Again (2015)

#28

Post by unlinked_dll »

dr_dshiv wrote: post #9
Vinceo wrote: post #4

He credited his work to his family goddess. From wikipedia: "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.""

Mathematics are the best expression of the transcendental divine. Pythagoras and Plato had the same perspective.

Funny to use the word transcendental there, since the Pythagoreans held ratios to be divine but couldn't figure out irrational numbers, like pi. They had trouble squaring that circle.

Re: Ramanujan Surprises Again (2015)

#29

Post by nexuist »

user2994cb wrote: post #26
nneonneo wrote: post #21

An interesting coincidence: it was recently (2019) discovered that the fastest way to multiply two n-bit integers, in time O(n log n), involves 1729-dimensional Fourier transforms: https://hal.archives-ouvertes.fr/hal-02070778 . It is quite surprising that the asymptotically best way to perform such an elementary operation should be tied to Ramanujan’s famous taxicab number. (Technically, it works for any number of d…

In fact, there seems to be a lot of interesting things about 1729: https://en.wikipedia.org/wiki/1729_(number)

Well, now I know why my Scheme class in uni was called CSE 1729.

Re: Ramanujan Surprises Again (2015)

#30

Post by wenc »

hnews_account_1 wrote: post #8
v64 wrote: post #7

Great read! When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability. It's interesting to find out that the real coincidence here is that Hardy rode in a taxicab whose number had happened to show up in Ramanujan's investigations of Fermat's last theorem.

A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one…

I think most of us are impressed by computational parlor tricks (and indeed raw computational intelligence in general -- being able to process information and compute quickly and accurately), but for me, genius goes beyond that.

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.

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