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Suppose that $\mathbb{P}$ is a Classical Euclidean Plane (satisfies all five of Euclid's postulates). Can you prove that if $L$ is a line in $\mathbb{P}$, then there is at least one point $p$ in $\mathbb{P}$ such that $p$ lies on $L$. Similarly, can you prove that there is at least one point $q$ such that $q$ does not lie on $L$?

Here's what I've tried so far:

My initial reaction is that there isn't enough information in the postulates to prove either of these results. Here's why:

The most promising postulate from which to prove the answer to the first question is the first postulate. The first postulate states that $\textbf{if}$ $p$ and $q$ are points in $\mathbb{P}$ and $p\neq q$, $\textbf{then}$ there exists a unique line $L$ in $\mathbb{P}$ such that $p$ and $q$ both lie on $L$. I don't think this is enough to prove the answer to our first question, as it only states that a line exists, given two distinct points. It doesn't say that points on a given a line exist.

The most promising postulate from which to prove the answer to the second question is the infamous parallel postulate. It states that $\textbf{if}$ $L$ is a line in $\mathbb{P}$ and $p$ is a point in $\mathbb{P}$ such that $p$ does not lie on $L$, $\textbf{then}$ there is a unique line $M$ in $\mathbb{P}$ such that $L$ is parallel to $M$ and $p$ lies on $M$. Again, I don't think this postulate contains enough information to prove that given a line, there is a point not on that line. It merely states that given a line and a point not on that line, we can find only one line parallel to the first which passes through that point.

For those two reasons, I think that the answer to both questions is 'no'. If I am incorrect about this hypothesis, could someone please show me how to prove these results from the postulates? If I am correct, then it is clear that Euclid's postulates were unsatisfactory for the way we do geometry.

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  • $\begingroup$ You need to read the definitions too (they will make you even more confused). You are not the first to criticise Euclid, but he must have got something right or you wouldn't be worrying about his work more than 2,000 years later $\ddot{\smile}$. $\endgroup$ – Rob Arthan Jan 29 '15 at 23:47
  • $\begingroup$ I have read the definitions (I've been reading them out of 'Euclidean and Non Euclidean Geometry' by M.J. Greenberg), and they aren't terribly confusing. However, it seems clear that lines and points are defined as distinct, primitive concepts in classical Euclidean geometry, and that the only relations between them are found in the postulates. $\endgroup$ – Jay Jan 29 '15 at 23:52
  • $\begingroup$ Have you read the definitions in Euclid? Definitions 3 and 4 both relate lines and points. See aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#defs $\endgroup$ – Rob Arthan Jan 30 '15 at 0:00
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Euclid's postulates have for a long time been known to be incomplete. In fact, in his very first proposition he just assumes that two circles with a common radius in fact intersect.

There have been numerous attempts to rectify this, most famously David Hilbert's axioms of geometry. You can read about them and the history of axiomatic geometry in Greenberg's "Euclidian and non-euclidian geometry" or Hartshorne's "Geometry: Euclid and beyond"

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  • $\begingroup$ Thanks for your response. I actually got this question out of Greenberg's book. So, when you say that the postulates are incomplete, you mean that the answer to both questions is 'no'? $\endgroup$ – Jay Jan 29 '15 at 23:47
  • $\begingroup$ I haven't checked very thoroughly, but as I remember the postulates, none of them ever guarantees the existence of points of any kind. $\endgroup$ – Arthur Jan 30 '15 at 7:27
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Euclid defines a line as a set of points, so that covers your first statement. Euclid's postulate on the existence of a circle which has the segment as a radius should give you the existence of points outside the line.

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  • $\begingroup$ Euclid does not define a line as a set of points. In classical euclidean geometry, lines and points are distinct, primitive concepts. $\endgroup$ – Jay Jan 29 '15 at 23:50
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    $\begingroup$ Euclid defines a line as a "breathless length". See aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#defs. What he meant by that, I do not know, but if we are going to engage in the history of mathematical philosophy on MSE, then please can we be faithful to the original sources. $\endgroup$ – Rob Arthan Jan 29 '15 at 23:55
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Euclid's axioms are incomplete(as has been pointed out), especially on small points like this, Hilbert assumes:

"Each line contains at least two distinct points."

So that gives you more than what you need.

As for the circles intersecting that has been mentioned in other answers, this is a more complicated question, it follows from the completeness axiom. Although I would like to know if it can be proved without completeness.

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