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Comment by igravious | Hacker News Reader

a real number is "a point on the number line"

These posts are always stimulating.

My understanding of a line is that it is delimited by two points, but does not contain any points. To elaborate, no point could be "on" a line because a point has no extension, whereas a line does. This is the crux of the matter. Therefore a line is not "made up of" points. (By analogy a plane could not be made up of lines.) This begs the question, what are lines made up of? Are they made up of anything? Is a point really where two (or more) lines would intersect if they could intersect. Is this what is meant by a Dedekind cut?

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The "point on the number line" definition has always been non-rigourous. It is meant to imply the intuition that real numbers are what we typically think of as "numbers", notably that they extend to infinity, are ordered, and are dense (for any two distinct real numbers, there exists a real number between them). Of course from a rigorous perspective, this does not even suggest a difference between the reals and the rationals.

The line you are talking about in the rest of your post seems to be an 'unrelated' object that is used in geometry. I am not familiar with the formal definition of line that is used in geometry, but one way of defining a line is as the set of all points which satisfy "y=mx+b", for a given (m,b). A line segment would be the above definition with restrictions on the domain: x_0<x<x_f.

"My understanding of a line is that it is delimited by two points"

That is not how Euclid defined it and how it is still seen in geometry today. What you describe is called a (line) segment (http://en.wikipedia.org/wiki/Line_segment)

"but does not contain any points"

Lines extend indefinitely in two directions (if you go past Euclidean geometry, that 'indefinitely' changes meaning a bit)

One talks of a point being _on_ a line in geometry. 'contains' is something from set theory: "the set of all points on line l contains point P" is a perfectly valid expression (but "P is on l" is way shorter)

Excuse me. Of course. I was using line and line segment interchangeably there. Which I should have not been doing if I am aiming for clarity but I think my point (ahem) applies to line segments and lines that extend indefinitely in one or two directions. Presumably people will contend that even a line segment "contains" an infinite number of points. But if points have zero extension then even an infinity of them cannot sum to anything greater than zero. So I ask you again, does it make sense to think of lines (or line segments) as composed of points, I reckon it does not.

Would it make a difference if you substituted "infinitesimal extension" for "zero extension"?

That's the thing though. As I understand it, or as it's said to be: points have zero extension. So, no amount of points, not even an infinity of them could ever have extension. But, it should make sense for a line to be composed of entities with infinitesimal extension as you say. I have seen the term linelet used before for these entities.

Charles Sanders Peirce said in 1903, ”Now if we are to accept the common idea of continuity […] we must either say that a continuous line contains no points or […] that the principle of excluded middle does not hold of these points. The principle of excluded middle applies only to an individual […] but places being mere possibilities without actual existence are not individuals.”

> That's the thing though. As I understand it, or as it's said to be: points have zero extension. So, no amount of points, not even an infinity of them could ever have extension.

This isn't true for an uncountably infinite set of points, assuming by 'extension' you mean what is usually called 'meausre' in modern mathematics. Modern theory is perfectly fine with saying that a line of nonzero length contains an infinite number of points of zero length, and trying to draw on Euclidean notions definitions of 'point' and 'line' to find conclusions about real analysis is going to be unhelpful.

I'm not sure what that Peirce quote is trying to say.

> But if points have zero extension then even an infinity of them cannot sum to anything greater than zero.

Can you make this rigorous? Because using the standard definitions, this statement is not true. It's true that a countable number of points must have total length zero (and you can even give a rigorous proof of this) but not necessarily true for a non-countable number of points. The study of "lengths of sets of points" is called measure theory.

I think it is unnecessary, however, to bring in the whole concept of length when defining lines. For example, we could simply define a line as a set of points obeying some special properties.

Perhaps your intuition changes if you think about the line (in a plane) coinciding with the x-axis. Don't you think it is reasonable to define that line as the set of points (in the plane) having y=0? Also, should that line not be identical (isomorphic) to the set of real numbers - which is just a set of numbers? And should not all other lines in the plane be identical (isomorphic) to the first line?

"But if points have zero extension then even an infinity of them cannot sum to anything greater than zero."

Infinities are weird; anybody who wants to learn math has to accept that. 0.99999… does equal 1, there are as many even numbers as integers, etc. these things are 'true' not because they make sense initially, but because they make the most sense of all the other things we have thought of so far. Similarly, a set of Aleph-0 points can completely cover a line.

"a set of Aleph-0 points can completely cover a line"

Aleph_0 is the cardinality of the integers. I don't think that'll cover a line. For that, you need the cardinality of the reals, C, which may or may not be Aleph_1.

OOPS. Thanks

Infinities are weird.

> My understanding of a line is that it is delimited by two points, but does not contain any points.

A line is (or can be viewed as) an infinite set of points.

> To elaborate, no point could be "on" a line because a point has no extension, whereas a line does.

That seems to be a consequence of an unusual definition of "on".

I suspect you are talking about formal formulations of abstract geometry. That's not what we're talking about here. Here we are talking about lines as being sets of points in the plane that satisfy an equation of the form ax+by=c. Solutions (x,y) of that are said to be a line, although they themselves are points.

You can deal instead with Euclid's axiomatization of geometry, and there "line" is an abstract thing defined by two points. Different animal, although seldom explained clearly by teachers, who often themselves don't really understand what's going on. (Although some do, and don't get the chance to explore these things because of the pressure of the curriculum, and students who don't care, but need to pass.)

All too often people get confused about this and are told to shut up by their teacher, whereas in fact the student has had an insight, and demonstrated deeper understanding.

Your argument is more philosophical than mathematical. Lines are traditionally defined as the set of all points which satisfy some critera. In this case, a line is precisely made up of points.

In linear algebra, lines are sets of points, but in modern geometry, points and lines remain undefined terms, implicitly defined by their incidence relations (which points are "on" which lines).

You may find the Fano plane (a three-dimensional finite projective space) interesting:

  * http://en.wikipedia.org/wiki/Fano_plane (brief description)
  * http://math.ucr.edu/home/baez/octonions/node4.html (connections with higher math)

In geometry it is quite common to define a line as an infinite set of points (namely, those which satisfy a linear equation). Likewise with other figures, like a circle.

this is kind of a nonsense, circular definition: what is a line? A set of points that can be mapped onto the reals. The more meaty answer is down below, with dedekind cuts (although "a set that fulfills the arithmetic axioms and least upper bounds" is also sufficient).

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