4 ms·

No human or computer has ever used an actual real, transcendental number in any computation. It would take infinitely long.

ttctciyf 4y ago
I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?
- Someone 4y ago
  I don’t think so. Programs that use numbers approximate e and π. It’s only programs that manipulate symbols that can use their exact values.
  - gpm 4y ago
    All programs manipulate symbols. All bits are is symbols. It's just sometimes we use those symbols to represent a subset of the integers (e..g. with the popular binary notation)... or a subset of the rationals (e.g. with the popular floating point notation)... or a subset of the reals that happens to include a transcendental number because we decided that some symbol (or combination of symbols) represents some particular transedental number.
- moloch-hai 4y ago
  That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.
  - defrost 4y ago
    I absolutely agree with you from the outset . . . . . . and then I start thinking about that damn Bailey–Borwein–Plouffe formula. Although ( handwave ) all the transcendentalness is magically manipulated away leaving a simple computation for the immediate reveal of any arbitrary n-th (hexadecimal) digit of π But surely that's over simplification? Is this magic by wizards?
    - moloch-hai 4y ago
      You have an abstract definition of a sequence of digits, and an algorithm that reveals an actual digit from among them. It is surprising that you don't need to approximate an infinite series to get it, but everybody agrees that digit would show up there if you did one. You would still need to do an infinite amount of work to get the rest of them. It is magic of a kind by wizards of a kind, or anyway indistinguishable from it.
      - defrost 4y ago
        Just to be clear, for any third party spectators, the surprise isn't that more work is required to get more digits .. the WTF moment for some is that no matter how large N is there's no need to do an increasing amount of work as N grows (there's no cost to "skipping to (not quite) the end"). It's another of those intuition challenging moments in math.
        moloch-hai 4y ago
        It's the wackiest result I know of, and seems overwhelmingly more profound than BT.
  - ttctciyf 4y ago
    > Wherever it comes down to actual numbers Wait, are you saying e and pi aren't "actual numbers"?
    - moloch-hai 4y ago
      I am not a Platonist, or any sort of theist. e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world. I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no better than 113/355 meters long.
      - drdec 4y ago
        > e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. Thank you! I was beginning to think I was the only one who believes this. So many people seem to mistake the map for the terrain.
        moloch-hai 4y ago
        I probably confused matters with my undefined expression "actual numbers". e and pi are as actual as i or 0. I think Turing introduced "computable numbers" which are a lot like the reals, but countable. You can write a program that produces each. So it includes integers, rationals, polynomic irrationals, and lots of transcendentals. But the set any particular person (or computer) operates on over the course of their existence is not just countable, but finite and not very large. You have to have expressed a representation of each in it at least once. We can only express a small number of numbers over the course of a life. Computers can do more of them. You might claim all those that programs you wrote have expressed. And, of course, the number has to have finite expression. Representing pi as a summation is finite. But presenting a numerical computation involving it has to be an approximation if it ever to finish.
        4y ago
  - camjw 4y ago
    It feels to me like you're redefining what a number is to be very different to what anyone with a maths background would say a number is. Essentially you're saying that neither e nor pi are numbers?
    - moloch-hai 4y ago
      They are not numbers you have ever used in an actual calculation. You always approximate them anyplace they don't cancel out. There is a countable subset of the reals called "computable numbers". We only ever use a subset of computable numbers in real calculations.
    - fellowmartian 4y ago
      in my view they never were "numbers" (as in constants), they are functions
    - andrewaylett 4y ago
      The only numbers that may exist in nature are the non-negative rationals. Negative numbers are widely useful, but they don't exist in any meaningful way: they're an abstraction over reality. Irrational numbers definitely don't exist, and don't get me started on so-called "imaginary" numbers -- they're no less present in nature than the negative numbers are. But the concepts are useful, and the symbols have meaning. The article mentions normal words having different meanings in mathematical language, and that impacts even every-day use of maths.

ttctciyf 4y ago

I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?

Someone 4y ago
I don’t think so. Programs that use numbers approximate e and π. It’s only programs that manipulate symbols that can use their exact values.
- gpm 4y ago
  All programs manipulate symbols. All bits are is symbols. It's just sometimes we use those symbols to represent a subset of the integers (e..g. with the popular binary notation)... or a subset of the rationals (e.g. with the popular floating point notation)... or a subset of the reals that happens to include a transcendental number because we decided that some symbol (or combination of symbols) represents some particular transedental number.
moloch-hai 4y ago
That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.
- defrost 4y ago
  I absolutely agree with you from the outset . . . . . . and then I start thinking about that damn Bailey–Borwein–Plouffe formula. Although ( handwave ) all the transcendentalness is magically manipulated away leaving a simple computation for the immediate reveal of any arbitrary n-th (hexadecimal) digit of π But surely that's over simplification? Is this magic by wizards?
  - moloch-hai 4y ago
    You have an abstract definition of a sequence of digits, and an algorithm that reveals an actual digit from among them. It is surprising that you don't need to approximate an infinite series to get it, but everybody agrees that digit would show up there if you did one. You would still need to do an infinite amount of work to get the rest of them. It is magic of a kind by wizards of a kind, or anyway indistinguishable from it.
    - defrost 4y ago
      Just to be clear, for any third party spectators, the surprise isn't that more work is required to get more digits .. the WTF moment for some is that no matter how large N is there's no need to do an increasing amount of work as N grows (there's no cost to "skipping to (not quite) the end"). It's another of those intuition challenging moments in math.
      - moloch-hai 4y ago
        It's the wackiest result I know of, and seems overwhelmingly more profound than BT.
- ttctciyf 4y ago
  > Wherever it comes down to actual numbers Wait, are you saying e and pi aren't "actual numbers"?
  - moloch-hai 4y ago
    I am not a Platonist, or any sort of theist. e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world. I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no better than 113/355 meters long.
    - drdec 4y ago
      > e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. Thank you! I was beginning to think I was the only one who believes this. So many people seem to mistake the map for the terrain.
      - moloch-hai 4y ago
        I probably confused matters with my undefined expression "actual numbers". e and pi are as actual as i or 0. I think Turing introduced "computable numbers" which are a lot like the reals, but countable. You can write a program that produces each. So it includes integers, rationals, polynomic irrationals, and lots of transcendentals. But the set any particular person (or computer) operates on over the course of their existence is not just countable, but finite and not very large. You have to have expressed a representation of each in it at least once. We can only express a small number of numbers over the course of a life. Computers can do more of them. You might claim all those that programs you wrote have expressed. And, of course, the number has to have finite expression. Representing pi as a summation is finite. But presenting a numerical computation involving it has to be an approximation if it ever to finish.
        4y ago
- camjw 4y ago
  It feels to me like you're redefining what a number is to be very different to what anyone with a maths background would say a number is. Essentially you're saying that neither e nor pi are numbers?
  - moloch-hai 4y ago
    They are not numbers you have ever used in an actual calculation. You always approximate them anyplace they don't cancel out. There is a countable subset of the reals called "computable numbers". We only ever use a subset of computable numbers in real calculations.
  - fellowmartian 4y ago
    in my view they never were "numbers" (as in constants), they are functions
  - andrewaylett 4y ago
    The only numbers that may exist in nature are the non-negative rationals. Negative numbers are widely useful, but they don't exist in any meaningful way: they're an abstraction over reality. Irrational numbers definitely don't exist, and don't get me started on so-called "imaginary" numbers -- they're no less present in nature than the negative numbers are. But the concepts are useful, and the symbols have meaning. The article mentions normal words having different meanings in mathematical language, and that impacts even every-day use of maths.