6 ms·

Why does the existence or absence of a finite name affect the reality of a number? You can put every real number in a one-to-one correspondence with itself. Or

by kortex 4y ago

Why does the existence or absence of a finite name affect the reality of a number? You can put every real number in a one-to-one correspondence with itself. Or a one-to-one correspondence of f(x). This is a really interesting thought experiment, and I think in some ways reframes the problem of "exists" (which is nebulous) to "has a description which is finite (or countably infinite, but not uncountable)". But I think if you could prove a way to countably label an arbitrary real number, you could probably inductively prove that you could do it for any. But you run face-first into the interesting number paradox and/or incompleteness. https://en.wikipedia.org/wiki/Interesting_number_paradox https://en.wikipedia.org/wiki/Interesting_number_paradox

thethirdone 4y ago
If unicorns exist, but humans can never observe them do you really care whether they exist or not? I generally am reducing questions that are only relevant to philosophy such as "Do real numbers exist?" to questions which you can actually answer. To make something clear, rationals have a finite description. If you allow infinite descriptions, all real numbers can be described by Cauchy sequences. When I talk about countably many names, I am talking about the set of finite names humans use for numbers. You cannot meaningfully have infinitely many sets of finite names that humans agree on so you cannot cover the real numbers with names. The interesting number paradox only works for countable sets unless I am mistaken. You can have "interesting" real numbers.
- dragontamer 4y ago
  When the sets of "finite names" you discuss requires more space than the universe has atoms and/or plank-lengths, this entire discussion feels incredibly arbitrary. Consider Graham's number for example. Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely. Despite this random number being a finite value describable by a finite number of digits, there's simply not enough space in the known universe to actually write down the hypothetical random number chosen. I've chosen numbers so large that its impossible for you to literally describe these finite numbers.
  - chrismonsanto 4y ago
    Your insistence here & in the other sub-thread that these numbers are represented as a string of digits suggests that you don't understand the argument being made. If you don't insist on this, then let me give a quick counterexample to "All the numbers larger than say... a googleplex": googleplex + 1
    - dragontamer 4y ago
      Sure. But my own understanding of your stance has evolved throughout today. My current counter argument is: > Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely I know you cannot do this, and I presume you also know you cannot do this. Just because there exists a finite representation doesn't mean you can tell me that representation. ------- I insist upon this because your requirement of "finite representation" is seemingly arbitrary to me. We can enumerate all important numbers as say... all numbers represented by algebra (addition, subtraction, multiplication, division, roots, exponents, variables, logs, sine, cosine, integral, derivatives, Knuth's notation, and other functions... etc. etc.) that can be described in fewer than 1-million symbols. And now we have a significant number of "irrational" and "i" numbers, specifically the set that we'll figure out with modern mathematics. Its a finite set (by bounding it by 1-million symbols, we have a finite number of numbers), and arguably the more important set of numbers that represents how modern math functions. -------- I dare say that all "important" numbers follows my (relatively arbitrary) definition above (all numbers describable in 1-million modern mathematical symbols or fewer), and is certainly more important than say... most of the numbers between googleplex and Graham's number.
      - chrismonsanto 4y ago
        >your stance Just to be clear, I'm not the person you were originally replying to. You can describe what you're talking about by writing a program--whether or not it terminates in our lifetime does not change that it is in fact a representation of the number.
        4y ago
- kortex 4y ago
  > If unicorns exist, but humans can never observe them do you really care whether they exist or not? I do, actually. It depends what you are using the unicorns for. If you use them for their blood for extending your lifespan, no they don't exist. If you are using the metaphor of unicorns as "the go-to thing which is used as a counter-example of existence", then they do exist. "Existing" is nebulous. > The interesting number paradox only works for countable sets unless I am mistaken. You can have "interesting" real numbers. Ah yeah, right, that relies on on sequencing.

by kortex 4y ago

thethirdone 4y ago

If unicorns exist, but humans can never observe them do you really care whether they exist or not? I generally am reducing questions that are only relevant to philosophy such as "Do real numbers exist?" to questions which you can actually answer. To make something clear, rationals have a finite description. If you allow infinite descriptions, all real numbers can be described by Cauchy sequences. When I talk about countably many names, I am talking about the set of finite names humans use for numbers. You cannot meaningfully have infinitely many sets of finite names that humans agree on so you cannot cover the real numbers with names. The interesting number paradox only works for countable sets unless I am mistaken. You can have "interesting" real numbers.

dragontamer 4y ago
When the sets of "finite names" you discuss requires more space than the universe has atoms and/or plank-lengths, this entire discussion feels incredibly arbitrary. Consider Graham's number for example. Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely. Despite this random number being a finite value describable by a finite number of digits, there's simply not enough space in the known universe to actually write down the hypothetical random number chosen. I've chosen numbers so large that its impossible for you to literally describe these finite numbers.
- chrismonsanto 4y ago
  Your insistence here & in the other sub-thread that these numbers are represented as a string of digits suggests that you don't understand the argument being made. If you don't insist on this, then let me give a quick counterexample to "All the numbers larger than say... a googleplex": googleplex + 1
  - dragontamer 4y ago
    Sure. But my own understanding of your stance has evolved throughout today. My current counter argument is: > Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely I know you cannot do this, and I presume you also know you cannot do this. Just because there exists a finite representation doesn't mean you can tell me that representation. ------- I insist upon this because your requirement of "finite representation" is seemingly arbitrary to me. We can enumerate all important numbers as say... all numbers represented by algebra (addition, subtraction, multiplication, division, roots, exponents, variables, logs, sine, cosine, integral, derivatives, Knuth's notation, and other functions... etc. etc.) that can be described in fewer than 1-million symbols. And now we have a significant number of "irrational" and "i" numbers, specifically the set that we'll figure out with modern mathematics. Its a finite set (by bounding it by 1-million symbols, we have a finite number of numbers), and arguably the more important set of numbers that represents how modern math functions. -------- I dare say that all "important" numbers follows my (relatively arbitrary) definition above (all numbers describable in 1-million modern mathematical symbols or fewer), and is certainly more important than say... most of the numbers between googleplex and Graham's number.
    - chrismonsanto 4y ago
      >your stance Just to be clear, I'm not the person you were originally replying to. You can describe what you're talking about by writing a program--whether or not it terminates in our lifetime does not change that it is in fact a representation of the number.
      - 4y ago
kortex 4y ago
> If unicorns exist, but humans can never observe them do you really care whether they exist or not? I do, actually. It depends what you are using the unicorns for. If you use them for their blood for extending your lifespan, no they don't exist. If you are using the metaphor of unicorns as "the go-to thing which is used as a counter-example of existence", then they do exist. "Existing" is nebulous. > The interesting number paradox only works for countable sets unless I am mistaken. You can have "interesting" real numbers. Ah yeah, right, that relies on on sequencing.