Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

How real are real numbers? (2004)

arxiv.org

Post reply on HN
Print view View original

71–80 of 275 posts

Re: How real are real numbers? (2004)

#71

Post by option »

textfile wrote: post #65
option wrote: post #55

Note that the probability of randomly picking a rational number from [0,1] is exactly 0. That is, with P=1 you will end up with an irrational number, not representable in any modern computer.

This is a mind-blowing. So rational numbers are an invention of humans. That is to say, natural numbers simply don't exist unless and until they are explicitly defined. Am I understanding correctly?

The most mind-blowing part is that P(E)=0 does not imply E is impossible. My statement above is just a consequence of that. See my Quora's answer to related question https://www.quora.com/Is-there-a-point-where-statistical-imp...

Re: How real are real numbers? (2004)

#72

Post by option »

jerf wrote: post #68
7373737373 wrote: post #59

Earlier quoted context omitted.

Really? Interesting. I would have expected it to be something like epsilon.

I'm not sure if this is entirely mathematically valid, but it's at least intuitively valid. Imagine we're in decimal (for familiarity). Imagine we're generating our random number by drawing digits out of a bag containing the ten digits, then replacing and drawing again. This is of course a magical perfectly uniformly distributed bag. In order to draw a rational number, you must randomly select a number that has a rep…

it is entirely mathematically valid. See here for some math https://www.quora.com/Is-there-a-point-where-statistical-imp...

The catch, here is "selecting a number randomly" - that isn't constructively defined. And to me this looks kind of similar to the notion of "measurement" in quantum theory - not precisely defined either.

Re: How real are real numbers? (2004)

#73

Post by gertef »

mbid wrote: post #45
btilly wrote: post #30

Earlier quoted context omitted.

The proof of the reals being uncountable depends on the idea that one can build a number that depends on being able to make an infinite number of choices, each of which depends on the absolute truth or falseness of a statement. But what happens if we open it up to have statements be true, false, or currently unknown? That is we develop a system of mathematics that could be in principle done inside of a Turing machine…

Mathematician here. If you mean to say that you cannot constructively prove "the real numbers are not countable", then you're wrong. As a rule of thumb, you can usually prove negative statements constructively as you would prove them classically. A constructivist would probably state the result more positive (and stronger, constructively): To every countable set M of real numbers, there is a real number not contained…

"the real numbers are not countable" is a different statement from "the real numbers are constructible"

Re: How real are real numbers? (2004)

#74

Post by protonfish »

whatshisface wrote: post #3

"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never…

Numbers are not physical things, they are symbols that are part of a system that has changed significantly over years. We can make a metaphorical link between a number and a measurement of our universe, but that does not mean that numbers are part of the physical world.

Re: How real are real numbers? (2004)

#75

Post by SeanLuke »

Richard's Paradox for some reason reminded me of the proof that there is an infinite number of interesting whole numbers. The proof goes like this. Assume instead that the number is in fact finite. Consider the first number higher than any of the interesting numbers, and thus bounding them. Now that's an interesting number! QED.

Re: How real are real numbers? (2004)

#76

Post by gertef »

stymaar wrote: post #62
mcherm wrote: post #51

Earlier quoted context omitted.

Consider the set of all real numbers which you can actually specify IN ANY FASHION AT ALL, so that the person writing a paper about it and the person reading the paper are talking about the same number. This includes easy ones like "3" or "Square root of Pi". It includes oddballs like Chaitin's constant -- the probability that a randomly constructed program will not get stuck in an infinite loop (for some particular…

> Consider that great big set. It's still countable. You didn't prove that statement, and actually Cantor's diagonal arguments [1] proves you wrong: Consider the set of «all real numbers which you can actually specify IN ANY FASHION AT ALL». If you can count it, you can order them in a certain fashion: n0, n1, n2, n3 … It's easy to specify a number X which is not part of this set (which is in contradiction with the d…

The number you are paraphrasing cannot be precisely identified in finite time.

Re: How real are real numbers? (2004)

#77

Post by unit91 »

DanBlake wrote: post #9

Unrelated, but I read a article a while back which said something similar, but it was based on the fact that our entire mathematical system is designed around "base 10" and as such is only relevant in many constructs to our specific human 'ten fingered', interpretation of math.

Ironically the publisher of that article used a variety of radices to make the claim that 10 is the only radix!

2 - to enter and transmit data on the machine

10 - radix in question

16 - color description on the page

etc.

Re: How real are real numbers? (2004)

#78

Post by gertef »

CogitoCogito wrote: post #40

The continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions. One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers.…

Rationals are conintuous, but not a continuum. you don't need real numbers to have continuous functions.

http://math.stackexchange.com/a/672151

Re: How real are real numbers? (2004)

#79

Post by gertef »

textfile wrote: post #65
option wrote: post #55

Note that the probability of randomly picking a rational number from [0,1] is exactly 0. That is, with P=1 you will end up with an irrational number, not representable in any modern computer.

This is a mind-blowing. So rational numbers are an invention of humans. That is to say, natural numbers simply don't exist unless and until they are explicitly defined. Am I understanding correctly?

"God created the integers, all else is the work of man." - Leopold Kronecker

Re: How real are real numbers? (2004)

#80

Post by gertef »

7373737373 wrote: post #59
option wrote: post #55

Note that the probability of randomly picking a rational number from [0,1] is exactly 0. That is, with P=1 you will end up with an irrational number, not representable in any modern computer.

Really? Interesting. I would have expected it to be something like epsilon.

The measure of epsilon is 0.

Proof: what else could it be? If it's not 0, there's a smaller number, contradicting your (intuitive) definition of epsilon.

Post reply on HN