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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

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?

jerf 9y ago
The answer to your question kinda gets into question-begging about what you mean about "invention". But, nominally, all numbers you know are inventions of humans. Exactly how well they correspond to the real universe is still an open problem. Obviously in many cases the answer is "well enough that you need not worry about the deviations very often", but things like "Is the universe continuous or discrete at the core, or some hybrid?" are open questions, which has direct implications on the question of how "real" any of the numbers are. Even the naturals may be quite artificial.
option 9y ago
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-improbability-transitions-to-functional-impossibility/answer/Oleksii-Kuchaiev https://www.quora.com/Is-there-a-point-where-statistical-imp...
gertef 9y ago
"God created the integers, all else is the work of man." - Leopold Kronecker
hn_throwaway_99 9y ago
Someone else responded to me in a recent Hacker News discussion that really clarified this in my head: A real number essentially has an infinite number of digits after the decimal place - the difference between a rational and an irrational number is that the digits end up in a repeating pattern in a rational number (you can think of rationals that terminate as really having an infinite number of zeros, e.g. 1.5 as 1.5000000...). Thus, to randomly pick a real number, you randomly select digits an infinite number of times after the decimal. Clearly there is no way a random process will produce an infinite number of repeating digits, thus the probability of picking a rational would be 0.
- prmph 9y ago
  Wouldn't it be simpler to explain it this way: there are infinitely many reals between any two different reals, and thus the theoretical probability of picking any number between them at random is 1/infinity, which we think if as 0. But in that case, why is is more probable to pick an irrational number?
  - hn_throwaway_99 9y ago
    I don't think it's simpler that way at all, mainly because the ways in which some "infinities" are larger than others, which explains why it's more probable to pick an irrational number. The rational numbers are countable, while the real numbers are not.

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?

jerf 9y ago

The answer to your question kinda gets into question-begging about what you mean about "invention". But, nominally, all numbers you know are inventions of humans. Exactly how well they correspond to the real universe is still an open problem. Obviously in many cases the answer is "well enough that you need not worry about the deviations very often", but things like "Is the universe continuous or discrete at the core, or some hybrid?" are open questions, which has direct implications on the question of how "real" any of the numbers are. Even the naturals may be quite artificial.

option 9y ago

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-improbability-transitions-to-functional-impossibility/answer/Oleksii-Kuchaiev https://www.quora.com/Is-there-a-point-where-statistical-imp...

gertef 9y ago

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

hn_throwaway_99 9y ago

Someone else responded to me in a recent Hacker News discussion that really clarified this in my head: A real number essentially has an infinite number of digits after the decimal place - the difference between a rational and an irrational number is that the digits end up in a repeating pattern in a rational number (you can think of rationals that terminate as really having an infinite number of zeros, e.g. 1.5 as 1.5000000...). Thus, to randomly pick a real number, you randomly select digits an infinite number of times after the decimal. Clearly there is no way a random process will produce an infinite number of repeating digits, thus the probability of picking a rational would be 0.

prmph 9y ago
Wouldn't it be simpler to explain it this way: there are infinitely many reals between any two different reals, and thus the theoretical probability of picking any number between them at random is 1/infinity, which we think if as 0. But in that case, why is is more probable to pick an irrational number?
- hn_throwaway_99 9y ago
  I don't think it's simpler that way at all, mainly because the ways in which some "infinities" are larger than others, which explains why it's more probable to pick an irrational number. The rational numbers are countable, while the real numbers are not.