The Unparalleled Genius of John von Neumann (2019) • Hacker News Board

Re: The Unparalleled Genius of John von Neumann (2019)

#41

Post by Koshkin » Wed, Apr 07, 2021, 1:47 AM UTC

jhgb wrote: post #28

pointedset wrote: post #10

Earlier quoted context omitted.

Yes, those two facts about zero/empty cases (and so many more) are definitely related, and this class of facts is one of my favourites! Usually, if you're dealing with something algebraic in flavour (which is a very vague concept, sorry), there will be a sensible way to define the zero/empty case. This is often a good test of whether you have a uniform concept that works for all n without corner cases. It almost irri…

> there will be a sensible way to define the zero/empty case. This is often a good test of whether you have a uniform concept that works for all n without corner cases. And so, begun the array indexing war has.

A polynomial always includes a member (monomial) with the power zero. It seems natural, therefore, to index the coefficients correspondingly. In other situations, 1 may be the more natural starting index.

Re: The Unparalleled Genius of John von Neumann (2019)

#42

Post by adverbly » Wed, Apr 07, 2021, 1:48 AM UTC

pointedset wrote: post #10

hintymad wrote: post #3

I often use von Neumann's quote, "Young man, in mathematics you don't understand things. You just get used to them.", to console myself when I have to read a math book multiple times to really understand something. The quote also has truth in it. I had no problem accepting that 0! = 1 only because I learnt that fact early in school. However, I struggled quite a bit to accept that span({}) = {0}, even though it is not…

Yes, those two facts about zero/empty cases (and so many more) are definitely related, and this class of facts is one of my favourites! Usually, if you're dealing with something algebraic in flavour (which is a very vague concept, sorry), there will be a sensible way to define the zero/empty case. This is often a good test of whether you have a uniform concept that works for all n without corner cases. It almost irri…

Couple more fun examples:

all([]) == True

any([]) == False

Re: The Unparalleled Genius of John von Neumann (2019)

#43

Post by rowanG077 » Wed, Apr 07, 2021, 1:49 AM UTC

Guthur wrote: post #26

I think John von Neumann was a genius but I also think that at the time there was also confluence of great minds across mathematics and science in general which likely feed into and accelerated the advancements. Maybe it's still happening but we no longer hear the names, though potentially it was also just one of those alignments of possibilities.

It's also possible there no longer is an environment that supports these kind of people.

Re: The Unparalleled Genius of John von Neumann (2019)

#44

Post by esjeon » Wed, Apr 07, 2021, 1:50 AM UTC

Guthur wrote: post #26

I think John von Neumann was a genius but I also think that at the time there was also confluence of great minds across mathematics and science in general which likely feed into and accelerated the advancements. Maybe it's still happening but we no longer hear the names, though potentially it was also just one of those alignments of possibilities.

I fully agree with this. While he was definitely a smart person, he was also lucky to be a polymath during the time of big transition for math and science. Knowing multiple fields, polymaths have a big advantage in systemizing a new school of thoughts. Being a grand master of one school will be less advantageous for this specific goal.

Also, I think there are much more polymaths during our time than 1900s, thanks to the internet. A single person can consume much more information than before. This will make it difficult to make a single person get all the spotlight. I believe we have a lot of geniuses in the back stage that the public simply don’t know exist.

Re: The Unparalleled Genius of John von Neumann (2019)

#45

Post by koolba » Wed, Apr 07, 2021, 1:50 AM UTC

Frost1x wrote: post #39

Guthur wrote: post #26

I think John von Neumann was a genius but I also think that at the time there was also confluence of great minds across mathematics and science in general which likely feed into and accelerated the advancements. Maybe it's still happening but we no longer hear the names, though potentially it was also just one of those alignments of possibilities.

I often wonder if there are points in human understanding where there is an opportunistic amount of existing knowledge, current exploration of knowledge, and sets of ideas floating around that lead to these sorts of surges of shared ideas and principles in disciplines. While it's no doubt from accounts Nuemann was brilliant, I've observed other brilliant people who I feel could be ground breaking minds given the righ…

Him and Einstein also did not have to contend with getting jobs optimizing ad placement.

Necessity is the mother of invention. But abstract thought requires time to focus on things outside the mundane.

Re: The Unparalleled Genius of John von Neumann (2019)

#46

Post by smnplk » Wed, Apr 07, 2021, 1:52 AM UTC

hintymad wrote: post #3

I often use von Neumann's quote, "Young man, in mathematics you don't understand things. You just get used to them.", to console myself when I have to read a math book multiple times to really understand something. The quote also has truth in it. I had no problem accepting that 0! = 1 only because I learnt that fact early in school. However, I struggled quite a bit to accept that span({}) = {0}, even though it is not…

0! = 1 is not hard to accept, since it follows a rule. You just need to look at it backwards. To get the previous factorial (Ni-1)! you need to divide the N! by N

4! = 24 ;; 24 / 4 = 6

3! = 6 ;; 6 / 3 = 2

2! = 2 ;; 2 / 2 = 1

1!= 1 ;; 1 / 1 = 1

0! = 1

Re: The Unparalleled Genius of John von Neumann (2019)

#47

Post by Koshkin » Wed, Apr 07, 2021, 1:52 AM UTC

sellyme wrote: post #36

hintymad wrote: post #3

I often use von Neumann's quote, "Young man, in mathematics you don't understand things. You just get used to them.", to console myself when I have to read a math book multiple times to really understand something. The quote also has truth in it. I had no problem accepting that 0! = 1 only because I learnt that fact early in school. However, I struggled quite a bit to accept that span({}) = {0}, even though it is not…

> Young man, in mathematics you don't understand things. You just get used to them. Not quite in the same weight class as von Neumann, but Matt Parker's "There's a trick for dealing with that in mathematics, called 'not really worrying about it'" when discussing results that don't mesh with our intuitive understanding is another nice one.

I’d say, rather, try and build your intuition in accordance to what mathematics, in fact, tells you (which is achieved by doing exercises).

Re: The Unparalleled Genius of John von Neumann (2019)

#48

Post by smaddox » Wed, Apr 07, 2021, 1:52 AM UTC

Rochus wrote: post #13

> von Neumann proposed a description for a computer architecture now known as the von Neumann architecture Well, he actually described an already existing design by Eckert and Mauchly under his name and this paper was illegally disclosed. For the ones interested in this and other fascinating stories about ENIAC, here is a good book: https://www.amazon.com/Eniac-Triumphs-Tragedies-Worlds-Compu... ; worth reading is al…

Wow... How is this not more well known? It's not like von Neumann doesn't have plenty of other important contributions to his name.

Re: The Unparalleled Genius of John von Neumann (2019)

#49

Post by practicalpants » Wed, Apr 07, 2021, 1:54 AM UTC

Always in awe to read about assemblies of great minds like at Princeton's IAS.

Does another such assembly exist today? I don't think it does but want to know if others can point to one.

Did it happen then because the discoveries were "low hanging" enough that they would be found eventually, such that today we've basically exhausted the major discoverable parts of nature given human limitations, like a depleting orebody where what's left are marginally economic residuals? Or was there something in particular about Western societies a century or two ago that produced such brilliance (or even specifically the Habsburgs's HRE where von Neumann was from), which perhaps we can learn from to steer things today?

Re: The Unparalleled Genius of John von Neumann (2019)

#50

Post by pointedset » Wed, Apr 07, 2021, 1:56 AM UTC

hintymad wrote: post #27

pointedset wrote: post #10

Earlier quoted context omitted.

Yes, those two facts about zero/empty cases (and so many more) are definitely related, and this class of facts is one of my favourites! Usually, if you're dealing with something algebraic in flavour (which is a very vague concept, sorry), there will be a sensible way to define the zero/empty case. This is often a good test of whether you have a uniform concept that works for all n without corner cases. It almost irri…

I was also puzzled by det(0x0) being 1, because I had built an intuition that determinant of a matrix was the volume of the parallelepiped represented by the matrix. I made my peace by accepting that my intuition on volume implies that volume is defined in a space that has positive dimensions, and by treating zero space as an algebraic construct.

Now you're reminding me of a wacky math conversation I had at Mathcamp [1] with a much smarter guy, who was talking about more esoteric definitions of volume in euclidean space. Something like:

- n-dimensional volume is a function from (some) subsets of space to real numbers

- it should be additive under union

- it should scale by t^n when you scale the space by a factor of t

I think the upshot of the conversation was that 0-dimensional volume of a shape should be its Euler characteristic. In the simple case of a finite set of points, the "volume" would be the number of points.

And by your earlier comment, span({}) consists of a single point, so its volume should be 1. It all works!

[1] https://www.mathcamp.org/