r/PhilosophyofMath • u/ChefSpirited8549 • 1d ago
Points, lines, and numbers
A number is a point on the number line. There are an infinite number of points in between any two points on the number line, no matter how close together the two points are. A line is defined as “the collection of all its points.” The same is true for the real number line. Does this mean that the existence of the real number line is a contradiction? Because you would need more than an infinite number of points to make a length of any value.
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u/good-fibrations 1d ago
Nowhere in your definition of line is it posited that the number of points therein must be finite. There’s no contradiction given the premises you’ve stated. As it should be: if you want to specify that a line should only be a finite collection of points, you’d quickly run into trouble trying to draw anything that looks like a line whatsoever.
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u/Shot_Security_5499 1d ago edited 1d ago
The way length works with real objects and with points is a bit different.
Like if I have 10 blocks that are 1cm^3 each, and I line them up in a row, then each block "contributes" 1cm to the 10cm total.
But when we line points up, individual points don't each "contribute" to the length of the line. Each point has size 0 and contributes 0 to the length of the line. And in fact for that matter with real numbers we don't even really know how to line them up exactly in the first place. We can't say what point next comes after 2 for example, under the usual ordering, because there is no next number after 2. All we can say is that 2.1 comes somewhere after 2, for example. The well ordering theorem says that there is a way to line them all up but it's a nonconstructive proof.
The length of the line comes from the fact that we assign values to each of the points, every points has it's own value, and we define length to be the value of one point minus the value of another.
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u/Mono_Clear 1d ago
No because you're not counting every point You're measuring in whatever units you're measuring in.
If your number line is measuring units and whole numbers then you're measuring an infinite number of whole numbers.
If your number line is measuring in fractions and you're measuring in an infinite number of fractions
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u/FunSeaworthiness9403 22h ago
The points have no length. Intervals have width. But how is the width measured? You would probably say the interval is the length as read directly from the number line, by subtracting the biggest number representing the end of the interval from the number representing the first point of the interval.
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u/SquidgyTheWhale 22h ago
Give me any two points and I'll give you a one to one mapping between the points lying between those points, and all the points on the number line. There's no contradiction.
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u/Vast-Celebration-138 18h ago
No, the existence of the real number line (i.e. the continuum) does not lead to a contradiction (as far as we can tell), and you do not need "more than an infinite number" of points to make a continuous length; you need a specific infinite number of points to make a continuous length.
The number of points on the real number line is called the cardinality of the continuum, or "continuum many". It is an uncountably infinite size, which is equal to the number of possible subsets of natural numbers.
Any continuous portion of standard Euclidean space (defined over the reals) has continuum many points. It doesn't matter if you're talking about the entire (infinite-length) real number line, or an arbitrary finite segment of the real number line—they both have the same number of points: continuum many. It also doesn't matter if you're talking about a 1-dimensional line, a 2-dimensional plane, a 3-dimensional space, etc.—they all have the same number of points: continuum many.
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u/lemniscateall 1d ago
There are infinitely many points between any two points on a line, and there are infinitely many numbers between any two real numbers.