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Everyone knows that a point on any plane has zero magnitude, height, width, or volume. A line is a collection of points, and a plane is a collection of lines, while a cube is a collection of planes and so forth.

How is it then that any line, surface or cube can be made of points that have no magnitude area etc.

Monte Carlo
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  • Note that this problem is inductive - lines, too, have no area, and planes have no volume. – Loki Clock Sep 29 '13 at 16:12
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    Today we have definitions that allow us to make sense of this, but this was a problem that even mathematicians found troubling. At the end of the XIX century, it was one of the objections to some of the proposed formalizations of analysis (the real line could not consist of points, because we would run into the questions you just mentioned). This was also related to the question of whether infinitesimal quantities "exist". See for example Gordon M. Fisher, The Infinite and Infinitesimal Quantities of du Bois-Reymond and their Reception. Arch. Hist. Exact Sci., 24 (2), (1981), 101--163. – Andrés E. Caicedo Sep 29 '13 at 16:28

4 Answers4

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In order to calculate the "sum" of lengths/areas/volumes etc. for a collection of subsets, you need the collection of subsets to be countable. But if you take a curve or surface etc. and look at all the points on the set, it is uncountable. So you can't reason that the total length/area/volume etc. is $0 + 0 + \ldots = 0$. If anything, you have to consider that e.g. for length of $S$ along the line you get $dx + dx+ \ldots$, "summed" over all points in $S$, which is indeed zero for a countable point set $S$ but for all points on a line segment you get an integral (not countable sum) of $dx$, which gives the length of the line segment.

user2566092
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Because length, area, volume, etc. are not ways of counting points, but ways of counting sizes relative to a unit. The structures of a topological space or measure space assign the property of being a blob of space to collections of points, rather than assigning a makeup of points to them. When you have a finite set of points in what you want to be blobs of space, the counting measure is appropriate. But not when you get to infinite sets. Consider the map that sends the $n^{\text{th}}$ odd number to $n$ and the $n^{\text{th}}$ even number to $-n$, starting from the $0^{\text{th}}$ even number. Clearly the number of points is equal between the natural numbers and the integers. But you might find it more appropriate for the natural numbers to be half the size of the integers. Furthermore, any two intervals, by number of points, would be equal in length if you were measuring space as the number of points you can count in a region. The point is, if you're axiomatizing your notion of relative size, that's not the same as acquiring a notion of point content where every point contributes the same amount to the size.

Loki Clock
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  • A good illustration of this is the fact that the Cantor set is a perfect space, yet has (Lebesgue) measure 0 in the reals. – Loki Clock Sep 30 '13 at 21:22
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The issue you raise here is similar/related to things like:

Archimedes axiom: http://en.wikipedia.org/wiki/Archimedean_axiom and Zeno's Paradoxes http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

Joseph Malkevitch
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I think the idea is that lines are sets of points. The length of a line is equal to the cardinality of the set of the points that comprise it.

Said otherwise, let line segment $P_{1}P_{3}$ designate the set $\langle P_1,P_2,P_3 \rangle$. The line segment is 3 units long.

In this way, assigning actual measures, such as inches, to units isn't any different than assigning actual things to numbers. For example, $2$ apples.

Math is abstract. Abstraction involves leaving things out (alternatively, concretion involves adding things in; however you prefer). Leaving out physical things allows us to generalize, which is a reason that math is so broadly applicable.

Hal
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    The length of a line is certainly not equal to the cardinality of the set of points making it up. Why on earth do you think that? – John Gowers Jun 23 '15 at 23:46
  • @Donkey_2009 Take a Cartesian plane, set the references lines at maximal resolution (i.e. one x-line and one y-line intersecting each point). It seems to me that a line-segment along the x-axis of that plane is identical to the points that it crosses. Why on earth would you think otherwise? – Hal Jul 12 '15 at 17:57
  • I repeat - the length of a line is not equal to the cardinality of the set of points that comprise it. A line segment of length $1$, for example, has uncountable cardinality - certainly not equal to $1$! – John Gowers Jul 12 '15 at 17:59
  • "I repeat". What does repeating your statement do to make your argument more convincing? A line segment that is as long as a point (length 1 as defined in my example) is just a point. If you claim otherwise, please tell me how they differ. – Hal Jul 12 '15 at 18:52
  • A point has length $0$, not $1$. An example of a line segment of length $1$ is the closed interval $[0,1]$. – John Gowers Jul 12 '15 at 19:41
  • Again, you've only repeated yourself. As the OP notes, the notion of a point as an object of 0 is incompatible with the conception of a line as a set of infinitely many points, such that each is contiguous with exactly two others. Characterizing a point as something infinitesimally short along every dimension both, satisfies Euclid's characterization of a point as 'that which has no part', and is compatible with the aforestated conception of a line. Your reply must appeal to intuition or logic, you cannot appeal to a definition: a point is undefinable. Continue to restate yourself if you wish. – Hal Jul 13 '15 at 20:49
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    I am not repeating myself. I am repeating hundreds of years' worth of thought and careful work by some of the greatest minds of all time. Your ideas are interesting, but the standard definitions are: any (topological) space has an underlying set, and the elements of this set are called points (we will ignore pointless topology for now). Length (of lines) is more complicated; this subject belongs to measure theory and analysis. These definitions are perfectly well motivated by both intuition and logic. – John Gowers Jul 13 '15 at 21:00
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    By contrast, your statement that 'the length of a line is equal to the cardinality of the set of points that comprise it' makes no intuitive sense. Does the line segment $[0,1]$ not have length $1$, and are there not more than one points contained inside? Are there not lines whose length is not an integer? How could that length possibly be interpreted as the cardinality of any set? – John Gowers Jul 13 '15 at 21:02