4 ms·

Those infinities are conceptual only. They don't exist. If infinity was part of the integer domain then (infinity + 1) > infinity which it isn't. I know about

Those infinities are conceptual only. They don't exist. If infinity was part of the integer domain then (infinity + 1) > infinity which it isn't. I know about IEEE infs and nans, and neither are numbers, just useful conventions.

enriquto 7y ago
I do not understand what your words mean. All numbers are conceptual. None of them "exists", they are all just useful conventions.
- throwaway_pdp09 7y ago
  > All numbers are conceptual Agreed, neet to be precise here, so let's de-abstract away from numbers. Which is greater, 3 apples or 8 apples? Now give me infinity apples. Finite apples exist, infinite apples don't except as an idea. Personally, infinities make me uncomfortable.
  - layoutIfNeeded 7y ago
    How do you know that 3 apples is less than 8 apples? How do you know when I give you a basket of apples that it contains 3 apples or 8?
  - tsimionescu 7y ago
    The apples example is bad, because while I can agree that 3 apes and 8 apples exist in a very real sense, I would not agree that Graham's number of apples exists any more than an infinity of apples. Also, I can't have pi apples or i apples. As for mathematical notions of numbers, there are of course mathematics with all kinds of infinite numbers. You have the transfinite ordinal and cardinal numbers, with the famous 2^Aleph0 = Aleph1 value (2^the number of natural numbers = the number of real numbers). You have infinitesimals as well. These are all just as well defined as the naturals and rationals, and exist just as much.
    - lmm 7y ago
      Maybe you should be dubious about Graham's number. It seems like you should always be able to add an apple to a pile of apples and get a pile with more apples in, but that's not actually the case; at some point, well before Graham's number, your pile would collapse into a black hole.
      - tsimionescu 7y ago
        If we start defining numbers by physical quantities ('what really exists') we'll have a pretty hard to work with numerical system. Because MAX_APPLES is of course smaller than MAX_DUST_MITES, so they define two different number systems. And I can probably pretty easily subdivide until maybe a hundredth of an apple, but a hundredth of a dust mite is much less physically plausible (at human scales). Not to mention that you could say: well sure, I can imagine a trillion trillion apples, but they don't really exist, they're just a fantasy. In reality we only have MAX_REAL_APPLE apples, and damn it, Johnny just ate one so the largest 'really existing' number just went down. Overall, we can play this game a lot, but the reality is that pi is as real as Aleph0 which is as real as 1 and 2 and 0.0000...1 and +Inf etc.
        lmm 7y ago
        > If we start defining numbers by physical quantities ('what really exists') we'll have a pretty hard to work with numerical system. Because MAX_APPLES is of course smaller than MAX_DUST_MITES, so they define two different number systems. And I can probably pretty easily subdivide until maybe a hundredth of an apple, but a hundredth of a dust mite is much less physically plausible (at human scales). Sure. So a simplified model is probably the right tradeoff to make. But we should not forget that it is a tradeoff; by working in a simplified world of notation we run the risk of forming constructions that can't actually be carried back to the real world of counting apples or dust mites. The map is not the territory. > Overall, we can play this game a lot, but the reality is that pi is as real as Aleph0 which is as real as 1 and 2 and 0.0000...1 and +Inf etc. "All models are wrong, but some are useful" - but not all models are equally wrong, and not all are equally useful. Including pi gives us a model that's easier to work with, but introduces a certain risk of forming a construction that doesn't translate back to reality. Including infinities brings a substantially bigger version of that risk. It may still be the right choice, but it's a choice that should be made carefully and deliberately; blindly assuming that all mathematical models are appropriate to a given situation is quite unwise.
  - Joker_vD 7y ago
    Well, 3 big apples are greater than 8 small apples. If you think otherwise, don't get into business of farm produce retail.
  - uryga 7y ago
    "3 apples" is just a rough (though often useful!) and very context-dependent abstraction anyway. real-world apples differ in size, so you might want to count a small apple as half an apple; and does a big "twin"/"conjoined" apple count as two apples? at best you're using some arbitrary size treshhold. it gets hairy/edge-casey pretty fast... i know this is nitpicking, but i think when you really look at it, "infinity apples" isn't that much more of an "idea" than "3 apples". (even if the latter is more useful in day-to-day life)

enriquto 7y ago

I do not understand what your words mean. All numbers are conceptual. None of them "exists", they are all just useful conventions.

throwaway_pdp09 7y ago
> All numbers are conceptual Agreed, neet to be precise here, so let's de-abstract away from numbers. Which is greater, 3 apples or 8 apples? Now give me infinity apples. Finite apples exist, infinite apples don't except as an idea. Personally, infinities make me uncomfortable.
- layoutIfNeeded 7y ago
  How do you know that 3 apples is less than 8 apples? How do you know when I give you a basket of apples that it contains 3 apples or 8?
- tsimionescu 7y ago
  The apples example is bad, because while I can agree that 3 apes and 8 apples exist in a very real sense, I would not agree that Graham's number of apples exists any more than an infinity of apples. Also, I can't have pi apples or i apples. As for mathematical notions of numbers, there are of course mathematics with all kinds of infinite numbers. You have the transfinite ordinal and cardinal numbers, with the famous 2^Aleph0 = Aleph1 value (2^the number of natural numbers = the number of real numbers). You have infinitesimals as well. These are all just as well defined as the naturals and rationals, and exist just as much.
  - lmm 7y ago
    Maybe you should be dubious about Graham's number. It seems like you should always be able to add an apple to a pile of apples and get a pile with more apples in, but that's not actually the case; at some point, well before Graham's number, your pile would collapse into a black hole.
    - tsimionescu 7y ago
      If we start defining numbers by physical quantities ('what really exists') we'll have a pretty hard to work with numerical system. Because MAX_APPLES is of course smaller than MAX_DUST_MITES, so they define two different number systems. And I can probably pretty easily subdivide until maybe a hundredth of an apple, but a hundredth of a dust mite is much less physically plausible (at human scales). Not to mention that you could say: well sure, I can imagine a trillion trillion apples, but they don't really exist, they're just a fantasy. In reality we only have MAX_REAL_APPLE apples, and damn it, Johnny just ate one so the largest 'really existing' number just went down. Overall, we can play this game a lot, but the reality is that pi is as real as Aleph0 which is as real as 1 and 2 and 0.0000...1 and +Inf etc.
      - lmm 7y ago
        > If we start defining numbers by physical quantities ('what really exists') we'll have a pretty hard to work with numerical system. Because MAX_APPLES is of course smaller than MAX_DUST_MITES, so they define two different number systems. And I can probably pretty easily subdivide until maybe a hundredth of an apple, but a hundredth of a dust mite is much less physically plausible (at human scales). Sure. So a simplified model is probably the right tradeoff to make. But we should not forget that it is a tradeoff; by working in a simplified world of notation we run the risk of forming constructions that can't actually be carried back to the real world of counting apples or dust mites. The map is not the territory. > Overall, we can play this game a lot, but the reality is that pi is as real as Aleph0 which is as real as 1 and 2 and 0.0000...1 and +Inf etc. "All models are wrong, but some are useful" - but not all models are equally wrong, and not all are equally useful. Including pi gives us a model that's easier to work with, but introduces a certain risk of forming a construction that doesn't translate back to reality. Including infinities brings a substantially bigger version of that risk. It may still be the right choice, but it's a choice that should be made carefully and deliberately; blindly assuming that all mathematical models are appropriate to a given situation is quite unwise.
- Joker_vD 7y ago
  Well, 3 big apples are greater than 8 small apples. If you think otherwise, don't get into business of farm produce retail.
- uryga 7y ago
  "3 apples" is just a rough (though often useful!) and very context-dependent abstraction anyway. real-world apples differ in size, so you might want to count a small apple as half an apple; and does a big "twin"/"conjoined" apple count as two apples? at best you're using some arbitrary size treshhold. it gets hairy/edge-casey pretty fast... i know this is nitpicking, but i think when you really look at it, "infinity apples" isn't that much more of an "idea" than "3 apples". (even if the latter is more useful in day-to-day life)