4 ms·

The most interesting thing about it is that at one point people thought it was a paradox.

by zeroer 10y ago

Someone 10y ago
It took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers.". I even fear that is true for the easier to believe "0.9999999... = 1"
- deleted 10y ago
  [deleted]
- llamaz 10y ago
  There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers. For example, most mathematicians would agree that there are as many reals in [0, 1] as there are in the real line. But according to a different definition of size, namely the most famous measure from measure theory, the Lebesgue measure --- the length of [0, 1] is 1 while the real line is undefined (I think). The person on the street would probably prefer the Lebesgue measure as more intuitive. I guess my ultimate point is that imprecise statements are harder to prove wrong. There's a quote, I don't remember who said it, that you should mistrust precise statements rather than imprecise ones, because it's precisely precise statements that can be proven wrong. edit: found the quote: http://www.brainyquote.com/quotes/quotes/r/raymondsmu190329.html http://www.brainyquote.com/quotes/quotes/r/raymondsmu190329....
  - tanderson92 10y ago
    But the Lebesgue measure is not a counting measure so it cannot be in any way related to the "number of numbers"...?
  - Someone 10y ago
    "There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers." You could call Z[n] ?for any composite n? that (in Z[4], the multiplication table only contains 0, 1, and 2, so 3 is prime there; Z[p] for prime p gives you p different numbers and zero primes) but I think those are the only ones. If you accept that infinity exists you get Hilbert's hotel, which gets you all those paradoxes, which after lots of sleepless nights leads to the only logical conclusion that giving up intuition about infinities is the best way out. If you don't accept that infinities exist, there must be a largest integer M, and you get to decide what M+1 or 2M are. That leads either to Z[n], to K&R's undefined behavior, which is so ugly no mathematician would dare publish it :-), or to some formalized variant of it that isn't Z[n]. I'm not sure I would call the values of Z[n] natural numbers, though, as that feels like it requires having negative numbers, too. Hm, maybe a shifted Z[n] would work. If you replace {0,1,2,3} by {0,1,2,-2,-1} in Z[5], you have two negative and three natural numbers in your universe, none of which is prime. I doubt that any of this kind of mathematical hair-splitting would bring aboard those who have trouble with grasping 0.999999... = 1, though :-)
    - SomeStupidPoint 10y ago
      Can you not make a "tropical Zn" with elements 0 - (M-1), and a "big" element, representing anything M or larger, including infinite values? I dont know that it would be exactly analogous to normal Z[M+1], and might have useful properties for some kind of geometric or combinatorical modeling. (My hunch is any time you want to be capable of carrying a "crossed threshold" flag as well as a value, and have that cascade through calculation.) It would also model systems where you can get to infinity in finite steps, but can't traverse back. Not sure if those are useful in the abstract, though.
      - Someone 10y ago
        IEEE 754 (https://en.wikipedia.org/wiki/IEEE_floating_point https://en.wikipedia.org/wiki/IEEE_floating_point) does that (for floating point, but the integer variant would be easily defined), with a few extensions. You need "small", too, for example, and "don't have the faintest idea" for when somebody subtracts "big" from "big". That's pragmatic, highly useful, but a nightmare for mathematicians. You lose invariants such as x+y-y=x (associativity and communicativity, in general), so you're no longer talking of a group (I don't know of research on 'almost groups') Compiler writers may happily make matters worse by assuming those laws still hold, with the effect that the same computation may overflow or not on different CPUs, under different compilers, compiler settings, or even the same compiler in the same compilation run.
  - msds 10y ago
    The real line is defined, as Lebesgue measure spits out the extended reals.
    - llamaz 10y ago
      You're right. In that case 1 < infinity, so [0, 1] is not the same "size" as the real line.
  - deleted 10y ago
    [deleted]
- amelius 10y ago
  It is because using infinity in place of a number is really a hack. And it is funny to see that mathematicians also get confused by their own hacks :)
- Dylan16807 10y ago
  "as many" is too vague. Density and cardinality are both valid interpretations.

The most interesting thing about it is that at one point people thought it was a paradox.

by zeroer 10y ago

The most interesting thing about it is that at one point people thought it was a paradox.

It took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers.". I even fear that is true for the easier to believe "0.9999999... = 1"

deleted 10y ago
[deleted]
llamaz 10y ago
There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers. For example, most mathematicians would agree that there are as many reals in [0, 1] as there are in the real line. But according to a different definition of size, namely the most famous measure from measure theory, the Lebesgue measure --- the length of [0, 1] is 1 while the real line is undefined (I think). The person on the street would probably prefer the Lebesgue measure as more intuitive. I guess my ultimate point is that imprecise statements are harder to prove wrong. There's a quote, I don't remember who said it, that you should mistrust precise statements rather than imprecise ones, because it's precisely precise statements that can be proven wrong. edit: found the quote: http://www.brainyquote.com/quotes/quotes/r/raymondsmu190329.html http://www.brainyquote.com/quotes/quotes/r/raymondsmu190329....
- tanderson92 10y ago
  But the Lebesgue measure is not a counting measure so it cannot be in any way related to the "number of numbers"...?
- Someone 10y ago
  "There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers." You could call Z[n] ?for any composite n? that (in Z[4], the multiplication table only contains 0, 1, and 2, so 3 is prime there; Z[p] for prime p gives you p different numbers and zero primes) but I think those are the only ones. If you accept that infinity exists you get Hilbert's hotel, which gets you all those paradoxes, which after lots of sleepless nights leads to the only logical conclusion that giving up intuition about infinities is the best way out. If you don't accept that infinities exist, there must be a largest integer M, and you get to decide what M+1 or 2M are. That leads either to Z[n], to K&R's undefined behavior, which is so ugly no mathematician would dare publish it :-), or to some formalized variant of it that isn't Z[n]. I'm not sure I would call the values of Z[n] natural numbers, though, as that feels like it requires having negative numbers, too. Hm, maybe a shifted Z[n] would work. If you replace {0,1,2,3} by {0,1,2,-2,-1} in Z[5], you have two negative and three natural numbers in your universe, none of which is prime. I doubt that any of this kind of mathematical hair-splitting would bring aboard those who have trouble with grasping 0.999999... = 1, though :-)
  - SomeStupidPoint 10y ago
    Can you not make a "tropical Zn" with elements 0 - (M-1), and a "big" element, representing anything M or larger, including infinite values? I dont know that it would be exactly analogous to normal Z[M+1], and might have useful properties for some kind of geometric or combinatorical modeling. (My hunch is any time you want to be capable of carrying a "crossed threshold" flag as well as a value, and have that cascade through calculation.) It would also model systems where you can get to infinity in finite steps, but can't traverse back. Not sure if those are useful in the abstract, though.
    - Someone 10y ago
      IEEE 754 (https://en.wikipedia.org/wiki/IEEE_floating_point https://en.wikipedia.org/wiki/IEEE_floating_point) does that (for floating point, but the integer variant would be easily defined), with a few extensions. You need "small", too, for example, and "don't have the faintest idea" for when somebody subtracts "big" from "big". That's pragmatic, highly useful, but a nightmare for mathematicians. You lose invariants such as x+y-y=x (associativity and communicativity, in general), so you're no longer talking of a group (I don't know of research on 'almost groups') Compiler writers may happily make matters worse by assuming those laws still hold, with the effect that the same computation may overflow or not on different CPUs, under different compilers, compiler settings, or even the same compiler in the same compilation run.
- msds 10y ago
  The real line is defined, as Lebesgue measure spits out the extended reals.
  - llamaz 10y ago
    You're right. In that case 1 < infinity, so [0, 1] is not the same "size" as the real line.
- deleted 10y ago
  [deleted]
amelius 10y ago
It is because using infinity in place of a number is really a hack. And it is funny to see that mathematicians also get confused by their own hacks :)
Dylan16807 10y ago
"as many" is too vague. Density and cardinality are both valid interpretations.