Usually, a point is said to divide a line.

So e.g. a point 3cm along a 5cm line will divide it into 2 lines, one 3cm and one 2cm.

Can a point divide two lines?

I'm thinking no, because the point cannot belong to both the lines it bisects.

However, I feel this may raise some fantastic and entirely unforeseen consequences, which would be absurd. I think that then two different dimensions could not reduce to the same thing. How then could time and space both depend upon the body?

  • Shhhh. A point has position only. It has no length, width or thickness. (Don't ask, that's what my teacher told me, don't ask).
    – Gordon
    Sep 2 '17 at 3:44
  • Don't ask about lines either, they just have length. Or surfaces, width, length, no thickness
    – Gordon
    Sep 2 '17 at 3:51
  • 6
    I'm voting to close this question as off-topic because it appears to be a question about basic math rather than philosophy. If there's some really deep angle that I'm not seeing, someone explain why this is a philosophical problem.
    – virmaior
    Sep 2 '17 at 4:48
  • @virmaior i'll edit the question to seem really deep and philosophical, then, ha
    – user28474
    Sep 2 '17 at 4:54
  • I am NOT a mathematics expert. I think there is a "problem" here in philosophy of math maybe, with so-called undefined terms, which the geometers proceed to (rather humorously) define, imo. But as far as I can tell, this was not intended to to present just that problem.
    – Gordon
    Sep 2 '17 at 4:58

Here is Tobias Dantzig, one of Einstein’s favourite mathematicians.

"Herein I see the genesis of the conflict between geometrical intuition, from which our physical concepts derive, and the logic of arithmetic. The harmony of the universe knows only one musical form – the legato; while the symphony of numbers knows only its opposite, – the staccato. All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato may appear to our senses as legato. Yet our intellect will always brand such attempts as deceptions and reject such theories as an insult, as a metaphysics that purports to explain away a concept by resolving it into its opposite."

He goes on...

"The axiom of Dedekind – “if all points of a straight line fall into two classes, such that every point of the first class lies to the left of any point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions” – this axiom is just a skilful paraphrase of the fundamental property we attribute to time. Our intuition permits us, by an act of the mind, to sever all time into the two classes, the past and the future, which are mutually exclusive and yet together comprise all of time, eternity: The now is the partition which separates all the past from all the future; any instant of the past was once a now, any instant of the future will be a now anon, and so any instant may itself act as such a partition. To be sure, of the past we know only disparate instants, yet, by an act of the mind we fill out the gaps; we conceive that between any two instants – no matter how closely these may be associated in our memory – there were other instants, and we postulate the same compactness for the future. This is what we mean by the flow of time.

Furthermore, paradoxical though this may seem, the present is truly irrational in the Dedekind sense of the word, for while it acts as partition it is neither a part of the past nor a part of the future. Indeed, in an arithmetic based on pure time, if such an arithmetic was at all possible, it is the irrational which would be taken as a matter of course, while all the painstaking efforts of our logic would be directed toward establishing the existence of rational numbers."

These words come from the second reference here, all three being worth reading on this topic.

Bell, John L, ‘Hermann Weyl on intuition and the continuum’. http://publish.uwo.ca/~jbell/Hermann%20Weyl.pdf

Dantzig, Tobias, Number – The Language of Science, (Pearson Education 2005 (1930)

Weyl, Hermann, The Continuum: A Critical Examination of the Foundations of Analysis, Dover (1987)

  • Enjoyable read. I've not heard of Dantzig's book. I have big problems with much of what he says here, but that's beside the point and some of it may have something to do with the time it was written. However, it is hard to excuse his use of Dedekind Cuts to model time and identify the present, which is clearly problematic even if it is clever.
    – nwr
    Sep 2 '17 at 18:31
  • I found the article by John Bell a useful survey of Weyl's view, and it expands on Dantzig's view as expressed here. The main point would be that whatever the OP meant by his question the continuum is a major philosophical problem.
    – user20253
    Sep 3 '17 at 10:47

Since an infinite number of lines passes through any given point, it follows that a point not only can divide two lines, but that it effectively divides any given number of lines.



The point of intersection of two lines that cross divides both of the lines that intersect there.

So deducing anything, however abstruse, from this 'impossibility' is not possible. I cannot make any sense whatsoever out of the rest of the question, but it must not be valid logic because it proceeds from a false premise.

  • can i have a reference pls. thank you for the reply, tho
    – user28474
    Sep 2 '17 at 5:24
  • 4
    I am not going to go find a reference for 5th grade math.
    – user9166
    Sep 2 '17 at 5:25
  • do you think you can't make sense of it because you disagree with its premise? i'm not sure
    – user28474
    Sep 2 '17 at 5:25
  • No, but valid logic cannot proceed from a false premise. Period. There is no issue with agreeing, math doesn't work that way.
    – user9166
    Sep 2 '17 at 5:26
  • 1
    @luke. Oh, heavens. You edited it. It's getting late. I need to go to bed.
    – Gordon
    Sep 2 '17 at 6:03

Suppose line AB runs from north to south, and line CD from east to west. Suppose AB intersects CD at point E. If so, E divides AB into AE and EB, and E divides CD into CE and ED. Nothing wrong with that!

Still not sure if it belongs to two lines, though.

  • 1
    It does not "belong" to anything. Points, lines, and surfaces merely get the ball rolling in geometry. It's part of the rules of the game of geometry. Like Monopoly(TM), you have these rules.
    – Gordon
    Sep 2 '17 at 6:00
  • i was speaking at length over several years to an expert in the infinitely small. they say that points can belong to the end or the beginning, or perhaps neither. ps please do vote my edit thru ... :)
    – user28474
    Sep 2 '17 at 6:02
  • Any point along a line can be said to belong to the line. So since E is part of AB as well as CD, it 'belongs' to both. And if AE, EB, CE and ED are considered to all include E, E 'belongs' to all of them as well. If those rays are open at E, then they don't include E by definition (in which case E doesn't belong to the rays). In any case, it would help for you to rigorously define the term belongs. E.g. P is said to 'belong' to line segment MN if and only if P lies between M and N, and MP and PN are co-linear.
    – Lawrence
    Sep 2 '17 at 12:50

If a point cannot belong to two ends simply by definition, then what is the property of a point that is preventing it from doing so? If that property is that a point is infinitesimally small in all dimensions, then it does not actually exist in space and is only a concept, as a concept it can have whatever property we define it as having, including that of being able to belong to two objects at the same time.


Can a point divide two lines? Yes.

Now to the question of belonging posed by Luke in his fourth sentence.

If we were to snap the lines into four at this point of intersection, must we dicide which end to stick the point onto? No, and I am unanimous in this. Whence "belonging"?

The point is gone, there is no longer any position to mark. All the point ever was was a position at the intersection of two lines, each line is just length. There is never any sense of a point belonging at all. A point has position only.

The geometers say we can only describe the point, line and surface; let no one say we define them. That's what they say. All "blessings" flow from this trinity, which begin the process of definition and underlie the definitions of all other geometric terms.

  • and anyway, what of zeno?
    – user28474
    Sep 2 '17 at 9:48

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