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30 Apr

### Is a Point a Part of a Line?

Is a point a part of a line? The question touches on the fundations of geometry. To be able to answer it, one should probably first clarify the notions that are involved in the question. According to the present day understanding, the notions of point and line are left undefined in geometry. We just get used to them by making derivations from the axioms, and then derivations from the derivations, and so on. Lines, if intersect, define a point, while two points define a line. It often helps to have a visual representation of either, but, such a visualization may prove treacherous. For example, in projective geometry, due to the Principle of Duality, the notions of point and line are entirely intercheangiable. Desargues' Theorem supplies a clearest example of that duality.

So, when two distinct lines intersect, they define a point. It is a commonly accepted norm to declare that the point of intersection of the two lines lies on both lines. More accurately, we express this idea by saying that the point is incident to the lines. Does it mean that the point of intersection is a part of both lines? Sure, if "being a part" is used as a substitute for "being incident to" - neither more nor less.

If, on the other hand, treating "being a part" in a material sense, as if the points - like building blocks - were glued together to form a straight line, leads to apparent paradoxes. An example of such treatment can be found in Augustus De Morgan's work (1831):

The beginner may perhaps imagine that a line is made up of points, that is, that every line is the sum of a number of points ... But taking the mathematical definitions of a point, which denies it all magnitude, either in length, breadth, or thickness, and of a line, which is asserted to possess length only without breadth or thickness, it is easy to show that a point is no part of a line, by making appear that the shortest line can be cut in as many points as the longest, which may be done in the following manner.

De Morgan illsutrates his argument with the following diagram

The projection from D of the points on a finite segment AB covers an infinite ray from A rightwards. De Morgan alerts the reader that imagining a line as "the sum of a number of points" leads to a contradiction. The alternative - as he suggested - is to deny that points lie on a line.

So, what do you think? Almost two centuries have passed from the time A. De Morgan gave his answer to the question at hand. Now, as then, common words find their way into mathematical vocabulary. Group, rational, perfect, neighborhood, ..., part...

Since the time of De Morgan, mathematicians learned the importance of assigning the words a definite meaning to escape the ambiguities of the everyday usage. They also learned that not everything needs to (or may) be formalized. "How to divide a segment into 3 equal parts?" is a legitimate question although the word "part" could not be found in any mathematical dictionary I am aware of. Here they choose to rely on common sense: every one knows when something is a part of a whole and when two parts are equal. The question whether a point is a part of a line sounds strange nowadays and may raise some eyebrows. In any event, De Morgan's answer is definitely not acceptable but mostly because we now know - in an around-about way - that in this case we deal with underlying infinities which requires caution and a stricter defined terminology. As De Morgan's argument shows, a line consists (in a certain sense) of the same number of points, regardless whether it is finite or infinite. Whether a point is a part of a line is an ill-defined question in so far as the notion of a part has not been settled in the math community.

It would have been exciting to see De Morgan's bewilderment, nay, consternation, at learning of the Banach-Tarski decompositions.

### References

1. A. De Morgan, On the Study and Difficulties of Mathematics, Dover, 2005, pp 194-196

#### 8 Responses to “Is a Point a Part of a Line?”

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Santo D'Agostino Says:

I believe it is standard in North American education nowadays to consider the point of intersection of two lines to be part of both lines.

One can infer this from the language that is typically used. For instance, one often reads in textbooks that a line is "a collection of points," which is a slight but significant modification of the phrase "a line is the sum of a number of points," which De Morgan did not like.

The standard definition of a relation in R^2 is a collection of ordered pairs. Each ordered pair corresponds to a point, so it is natural for a North American student to think of a curve in the plane as a collection of points. In terms of set theory, if one interprets "part of" to mean "element of," then it is natural to accept that a point is a part of a line, and that a line is a set of points.

It's insightful to see how our predecessors struggled with concepts that we take for granted!

2. 2
Maya Incaand Says:

@Santo D'Agostino

I would take the article to mean that we should not take things for granted.

Our poredecessors struggled with the concepts where we are apparently content to sweep them under the rug (eg Banach Tarski is just an interesting paradox, duh!).

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In a tweeter.com discussion, Gary Davis observed that a set of axioms may have consequences not intended at the outset. As an example, he gave finite geometries., where a point forms an apparently substantial portion of a line so that it could warrant calling it "a part". Curiously, finite geometries of certain kinds, e.g., projective, exist only for certain number of points, such that removing a point, destroys not only a line or two but the whole geometry. I'd think that in such circumstances, a point is a more essential attribute of a line than just "a part". Gary also gave another example of a three-point "incidence geometry".

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I actually concur with Maya Incaand. We are still, especially in elementary mathematics, use ill-defined concepts - being a part, is just one example. I could also think of introduction of infinite decimals in middle school.

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Santo D'Agostino Says:

I agree with you, Maya, that careful definitions are important, and I was certainly not trying to sweep anything from this excellent and thought-provoking article under the rug.

I guess the point I was trying to make (perhaps not very well) is that nowadays it is usual to adopt the perspective that points are locations, and that if one defines a line to be a set of points that satisfies a certain property (i.e., relation between x and y is linear), then this is precise enough to remove the difficulty. (For if a line is defined to be a set of points, then one of the points from the set is certainly a part of the line, in the sense that the point is an element of the set that represents the line. And the intersection of two lines can be though of in the set theory sense, so that indeed the point of intersection of the two lines is an element of each of the two sets.)

However, I absolutely agree with the author that the idea that a point is material is still a common misconception for students, even for university students, and teaching in a way that clears up this particular misconception (or ideally does not introduce it in the first place) is a significant difficulty. The idea that the real numbers can be modelled by an infinite number of material points in a line, one next to the other, is unfortunately a very current error. The rationals are erroneously thought of as just the same, except that a few (!) points are removed here and there.

I was happy to learn De Morgan's perspective on these matters, and maybe this is another of the author's points: Perhaps the best way to help students correct their misconceptions is to present them with articles such as this one, so that they can see for themselves that thinking of a point as being material leads to inconsistencies.

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Another point that could be made from reading history is that the concepts that the students are supposed to except from their experiences are not at all trivial. Famous mathematicians that have created significant mathematics themselves had difficulties with some mathematical concepts. It took decades for certain ideas to percolate into mathematical mainstream and crystallize to the extent that now they are treated as obvious. Many are not naturally so.

This is close to Santo's interpretation.

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erdal abayli Says:

i strongly think that a point is not a part of a line. a line is 0\100 of a line (by length). a point is not a portion of a line (since it is 0\100 by length. 0\100 doesnt mean a portion). at first glance we see length of a line but we never realize 0 length portion of a line (a point). no one can concentrate on zero length parts of a line. maybe they even doesnt exists on a line. as in the line paradox , infinite number of point can only form of a line which has zero length. but on the contrary , a line seems to be total of points which is a paradox called line paradox. maybe we can solve this paradox by saying not zero length but saying "without length". if a point doesnt have length than there is no way for it to be a part of a line. it is not proportional to a line. it seems like an imaginary thing.whether it exists or not doesnt mean anything for a line. if we take 1000000 points from a line and than measure the length of a line, we will see that the length of the line is not changed. as a result , number of points on the line is again same, not reduced. did we take any part of the line? if so why it remains same. a point is not a part of a line. it is clear.

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