• albie
    10
    Specifically the notion that you can divide a quantity up into infinite parts.

    Problem: How big are those individual parts?

    They have to be zero in size, hence you are no longer dealing with the quantity in question. Ergo you cannot divide a quantity into infinite parts.
  • Echarmion
    493
    Where do you take that argument from?

    The argument I know is that you can divide a quantity up infinitely. That is, an arbitrarily high amount of times, but never infinite times.
  • hachit
    198
    This is like Zeno paradoxes. I would look at calculus to solve these questions
  • TheMadFool
    3.4k
    Let's take the quantity 1 for simplicity. Define a function y = 1÷x.

    y will grow towards infinity as x approaches zero. But when x is zero, y is undefined. I'm not a mathematician but a graph of the function y will never touch the x axis (will not yield an answer to 1÷0).

    So, it's not zero that makes infinity. It's an arbitrarily infinitesimally small value of x.
  • Metaphysician Undercover
    5.9k
    Specifically the notion that you can divide a quantity up into infinite parts.

    Problem: How big are those individual parts?
    albie

    The parts are infinitely small.
  • Mww
    864


    So your opinion is, because a quantity divided infinitely would have parts with zero size, no quantity can be infinitely divided?

    Not being a math guy, I have to ask.....is there a rule for obtaining a zero size part from any division at all?
  • fdrake
    2.5k
    A part can have different sizes depending on its whole, assuming you allow this discussion to involve sets and elements of sets. EG, the number 1 has size 1 as a cardinal number, but it has size 0 as part of the real line. So long as we understand the sense of size which is currently operative there is no contradiction here.
  • Mww
    864


    I’m trying to picture a guy, standing there chopping off sections of a number line of x units, each part having zero size. I understand doing so is the only possible way to divide infinitely, but you gotta admit....he isn’t really doing anything. So there does appear to be some kind of contradiction.

    Do you agree with the opinion contained in the OP?
  • fdrake
    2.5k


    The idea of chopping something into units requires a countable number of chops. You can half, quarter etc. The real line instead is an uncountable union of real numbers, so the analogy doesn't apply.
  • Mww
    864


    The OP stipulates a infinitely divisible quantity. Number lines do not exist in Nature, but one can be imagined a priori, consisting of an arbitrary, progressively conceivable set of real numbers (the numerical totality of the set cannot be imagined). Because it’s an abstraction, the guy chopping off numbers one at a time is itself an abstraction, but sustains the conclusion he is not chopping off parts of zero size, because the number line must be conceived as getting shorter.

    I’m gonna stop now; I don’t want to be responsible for the math guys hurting themselves laughing at me. (Grin)
  • fdrake
    2.5k
    The OP stipulates a infinitely divisible quantity. Number lines do not exist in Nature, but one can be imagined a priori, consisting of an arbitrary, progressively conceivable set of real numbers (the numerical totality of the set cannot be imagined). Because it’s an abstraction, the guy chopping off numbers one at a time is itself an abstraction, but sustains the conclusion he is not chopping off parts of zero size, because the number line must be conceived as getting shorterMww

    Being unable to shave off parts of zero size is precisely the limitation I spoke about. You can shave off sets of zero size easy, say {x in [0,1] except for 0.5}. I'd say that since it can be done mathematically, and in a consistent manner, it's certainly conceivable, and we shouldn't therefore privilege intuitions of discreteness in nature over intuitions of continuity - what holds where and to what degree is a matter for investigation; conceptual work and experiment.
  • Mww
    864


    Ahhh....that’s what you meant before by involving sets or elements of sets. OK, fine. I can dig chopping off sets of zero size; that’s just an empty set. And by association, the totality of the divisible quantity is undiminished, which seems to sustain the OP.

    Now that you mention it, I am favoring intuitions of discreteness, aren’t I. It never crossed my mind there was any other way to look at the a priori conceptions of “quantity”. Or the infinite for that matter. Apparently, though, I shouldn’t be, with respect to the problem at hand. So....thanks for that.
  • fdrake
    2.5k
    Ahhh....that’s what you meant before by involving sets or elements of sets. OK, fine. I can dig chopping off sets of zero size; that’s just an empty set. And by association, the totality of the divisible quantity is undiminished, which seems to sustain the OP.Mww

    The empty set has size 0, but so does any finite or countable set as a member of the real line. Even the rationals.
  • Mww
    864


    Wha....wait. A finite set is has size zero? So an unpopulated empty set is the same size as a set of countable numbers? In other words, the set is what makes the size, not the members of it. But what is it about a set that determines it’s size?
  • fdrake
    2.5k


    {1} has cardinality 1, but measure 0 in the real line. The size depends on the measure. See this vs this.
  • Mww
    864


    Holy crap on a cracker.....I never even knew there was any of that stuff. Now I see where you’re coming from. I looked up some of the things you brought up, but...obviously....I didn’t get that far.

    Any countable set of real numbers has Lebesgue measure 0.
    .....put a measure on any set: the "size" of a subset is taken to be.....

    Back to the OP. Is the opinion correct?
  • fdrake
    2.5k


    Well, because there are sensible ways to think of subsets of sets as having 0 size, that does go against parts (subsets) of wholes (sets) necessarily not having 0 size. Really though the formulation is wrong, because there's not just one size concept which can be neatly applied to everything.
  • Mww
    864


    OK. Agreed. I’m in no position to hold with the things I learned here today, even while appreciating the exposure to them. I think I’m going to stick with what I’ve convinced myself I know, and if somebody comes along and upsets my intellectual applecart as respectfully as you did......so much the better for me.
  • albie
    10
    If we take a block of cheese a foot squared. And we say each infinite point is 1 gram in weight then that means the block of cheese must weigh infinite grams. AS long as each point has a value more than zero you will always get this.
  • albie
    10


    I have a friend who is into physics and he claims because you can divide a quantity up for ever that means that any quantity is made up of infinite points.
  • albie
    10



    And what does infinitely small mean? It is not there.
  • Terrapin Station
    11.6k
    They have to be zero in size, hence you are no longer dealing with the quantity in question.albie

    Say what?
  • Rank Amateur
    1.6k
    physical things have limits, numbers are not physical things

    I have a friend who is into physics and he claims because you can divide a quantity up for ever that means that any quantity is made up of infinite points.albie

    He is correct, between any 2 real numbers, there are an infinite number of real numbers
  • Echarmion
    493
    I have a friend who is into physics and he claims because you can divide a quantity up for ever that means that any quantity is made up of infinite points.albie

    This is possible mathematically. Physically, there can never be an infinity of anything, because observing an infinity is impossible (as it takes an infinite amount of time).
  • Terrapin Station
    11.6k
    Physically, there can never be an infinity of anything, because observing an infinity is impossibleEcharmion

    Not that I'm arguing for extant infinities, but why would whether there's an infinity of anything hinge on observation?
  • Echarmion
    493
    Not that I'm arguing for extant infinities, but why would whether there's an infinity of anything hinge on observation?Terrapin Station

    Well physics describes observable reality. I use it in a narrow sense here, a metaphysical infinity is theoretically possible.
  • SophistiCat
    796
    The things that we have actually observed, in the loosest sense of the word, are a tiny (if not infinitesimal!) fraction of the things that we believe to exist. That goes equally for physical sciences and for everyday observations and beliefs. So are we all wrong in your opinion? Are you some kind of arch-empiricist who will not acknowledge anything that he has not observed?
  • Echarmion
    493
    The things that we have actually observed, in the loosest sense of the word, are a tiny (if not infinitesimal!) fraction of the things that we believe to exist. That goes equally for physical sciences and for everyday observations and beliefs. So are we all wrong in your opinion? Are you some kind of arch-empiricist who will not acknowledge anything that he has not observed?SophistiCat

    No, but there is a difference between things that have not (yet) been observed and things that are unobservable in principle.
  • SophistiCat
    796
    Define "in principle." If you were living on an island with no seafaring vessel, anything beyond the horizon would be unobservable in principle for you. Would you then be obliged to believe that the world ends just at the horizon? If we expand the possibilities implied by "in principle" to anything that is not strictly forbidden by relativistic physics, our horizon would expand to the size of the Hubble sphere centered around Earth. Does the world therefore end there?

    Any way you look at it, it seems that your epistemology puts a priori constraints on the world, in that it can only be such as to be "in principle" observable. It seems strange to make such egocentric demands of the world, which doesn't seem to care about you one wit.
  • Echarmion
    493
    Define "in principle." If you were living on an island with no seafaring vessel, anything beyond the horizon would be unobservable in principle for you. Would you then be obliged to believe that the world ends just at the horizon?SophistiCat

    I am not sure what is unclear about my position, but anyways "in principle" means based on the attributes of the theoretical object. A ship beyond the horizon is still a ship, which means it should for example reflect light. It is observable, even if you cannot practically observe it currently.

    If we expand the possibilities implied by "in principle" to anything that is not strictly forbidden by relativistic physics, our horizon would expand to the size of the Hubble sphere centered around Earth. Does the world therefore end there?SophistiCat

    The world in a practical sense certainly ends there, as far as current knowledge can tell us. You can still make the technical distinction between things that cannot be observed because we cannot get close enough and things that cannot be observed because of their attributes irrespective of their spatial relation to us.

    Any way you look at it, it seems that your epistemology puts a priori constraints on the world, in that it can only be such as to be "in principle" observable. It seems strange to make such egocentric demands of the world, which doesn't seem to care about you one wit.SophistiCat

    I do not put these constraints "on the world". Observable reality can only consist of that which is observable. I am not talking about the nature of objective reality here.
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