How to fit an elephant (2011) • Hacker News Board

Re: How to fit an elephant (2011)

#21

Post by dahart » Thu, Oct 12, 2017, 3:58 PM UTC

Y_Y wrote: post #19

dahart wrote: post #18

Earlier quoted context omitted.

Pairing functions only work on countable sets. This is funny because Cantor is the same person who proved real numbers are uncountable, and that there is no pairing function between 1 real number and naturals, let alone 2. https://en.m.wikipedia.org/wiki/Countable_set

I'm claiming a "pairing function" between single reals and pairs of reals. They have respective cardinalities 2^N0 and 2*2^N0=2^N0 where N0<2^N0 is the cardinality of the naturals.

You're right. I was wrong. I found an explanation of how to make interleaving method work. https://math.stackexchange.com/a/183383

Re: How to fit an elephant (2011)

#22

Post by wolfgke » Thu, Oct 12, 2017, 4:10 PM UTC

Y_Y wrote: post #20

wolfgke wrote: post #12

Earlier quoted context omitted.

> Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example. But this (set) isomorphism between R and C is not continuous. Indeed one can show that there exists no continuous epimorphism f: R^n -> R^m, where m > n, since for every such continuous map f the image f(R^n) has a measure of 0 with respec…

Sure, but did we need continuity? Also, if you want to be awkward, you can get around this by using the discrete topology, I don't think we needed the metric structure of R^n.

> Sure, but did we need continuity? Also, if you want to be awkward, you can get around this by using the discrete topology

This is indeed possible - but this is clearly not the topology that "ordinary people" and physicists mean when talking about continuity of functions from R^n to R^m.

Re: How to fit an elephant (2011)

#23

Post by [deleted] » Thu, Oct 12, 2017, 4:14 PM UTC

Y_Y wrote: post #20

wolfgke wrote: post #12

Earlier quoted context omitted.

> Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example. But this (set) isomorphism between R and C is not continuous. Indeed one can show that there exists no continuous epimorphism f: R^n -> R^m, where m > n, since for every such continuous map f the image f(R^n) has a measure of 0 with respec…

Sure, but did we need continuity? Also, if you want to be awkward, you can get around this by using the discrete topology, I don't think we needed the metric structure of R^n.

[deleted]

Re: How to fit an elephant (2011)

#24

Post by mamon » Thu, Oct 12, 2017, 7:56 PM UTC

pvg wrote: post #14

scott00 wrote: post #11

Earlier quoted context omitted.

Source please. Must learn more.

Von Neumann was renowned for his great prowess at mental maths. A famous (if also not entirely serious) story: "When posed with a variant of this question involving a fly and two bicycles, John von Neumann is reputed to have immediately answered with the correct result. When subsequently asked if he had heard the short-cut solution, he answered no, that his immediate answer had been a result of explicitly summing the…

The article you linked to provides a trivial solution:

"the trains take one hour to collide (their relative speed is 100 km/h and they are 100 km apart initially). Since the fly is traveling at 75 km/h and flies continuously until it is squashed (which it is to be supposed occurs a split second before the two oncoming trains squash one another), it must therefore travel 75 km in the hour's time."

So if von Neumann was solving it by explicitly summing the series, as the anecdote claims, then he was doing it wrong :)

Re: How to fit an elephant (2011)

#25

Post by pvg » Thu, Oct 12, 2017, 8:47 PM UTC

mamon wrote: post #24

pvg wrote: post #14

Earlier quoted context omitted.

Von Neumann was renowned for his great prowess at mental maths. A famous (if also not entirely serious) story: "When posed with a variant of this question involving a fly and two bicycles, John von Neumann is reputed to have immediately answered with the correct result. When subsequently asked if he had heard the short-cut solution, he answered no, that his immediate answer had been a result of explicitly summing the…

The article you linked to provides a trivial solution: "the trains take one hour to collide (their relative speed is 100 km/h and they are 100 km apart initially). Since the fly is traveling at 75 km/h and flies continuously until it is squashed (which it is to be supposed occurs a split second before the two oncoming trains squash one another), it must therefore travel 75 km in the hour's time." So if von Neumann wa…

I assume you're trying to beat the original responder in brazen literalism - now we have a citation request for a throwaway joke, an explanation (with citation) and a literal interpretation of the joke in the citation.

Re: How to fit an elephant (2011)

#26

Post by xelxebar » Fri, Oct 13, 2017, 1:48 AM UTC

mamon wrote: post #24

pvg wrote: post #14

Earlier quoted context omitted.

Von Neumann was renowned for his great prowess at mental maths. A famous (if also not entirely serious) story: "When posed with a variant of this question involving a fly and two bicycles, John von Neumann is reputed to have immediately answered with the correct result. When subsequently asked if he had heard the short-cut solution, he answered no, that his immediate answer had been a result of explicitly summing the…

The article you linked to provides a trivial solution: "the trains take one hour to collide (their relative speed is 100 km/h and they are 100 km apart initially). Since the fly is traveling at 75 km/h and flies continuously until it is squashed (which it is to be supposed occurs a split second before the two oncoming trains squash one another), it must therefore travel 75 km in the hour's time." So if von Neumann wa…

Well, if you're familiar with sums, then the "shortcut" isn't all that much faster; it's mostly a function of how quickly you can grok the salient points:

The fly travels 3/2 times the speed of a train, so every bounce the fly travels 3/5 of the remaining track and leaves 1/2 * 2/5 = 1/5 track to travel, so we just compute the geometric sum

    3/5 * \sum 1/5^r = 3/5 * 1/(1 - 1/5)
                     = 3/5 * 1/(4/5)
                     = 3/5 * 5/4
                     = 3/4.

One nice thing about this sum is that it encodes a bit more insight about the fly's flight path than the shortcut method.

Re: How to fit an elephant (2011)

#27

Post by xelxebar » Fri, Oct 13, 2017, 2:16 AM UTC

oddeyed wrote: post #3

I can't help but feel like a complex number is two parameters (real&imag / mod&arg) - so really this is 8 parameters.

This is a good point. If we're posing Occam's Razor-like arguments against models for having "too many free parameters", then we should probably really be comparing something more precise, like the models' Kolmogorov complexities.

Otherwise, it's just too easy to hide a lot of complex machinery inside a "single parameter". In fact, from this perspective it's arguable that an arbitrary real number is actually a (countably) infinite set of parameters, since it takes that many bits to uniquely specify any real number.