In Euclidean geometry texts, it is always mentioned that point is undefined, it can only be described. Consider the following definition: "A point is a mathematical object with no shape and size." I do not understand what is the problem with this definition. Please give the detailed reasons. Thanks in advance!
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42$\begingroup$ What is shape? What is size? $\endgroup$– egregCommented Aug 9, 2016 at 10:28
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29$\begingroup$ @egreg: for completeness, I guess one could also ask "what is a mathematical object?" $\endgroup$– NefertitiCommented Aug 9, 2016 at 10:56
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3$\begingroup$ If the underlying set is interpreted as a "space" then one calls its elements points. $\endgroup$– Christian BlatterCommented Aug 9, 2016 at 10:57
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13$\begingroup$ There is no problem with this *statement". But the issue is to take it as a (axiomatic) definition. I we are together out in a nice day and I way "the sky is blue and we can see the moon", there is no problem, you will say "yes". But this does not constitute a definition of the sky. $\endgroup$– Jean MarieCommented Aug 9, 2016 at 11:21
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29$\begingroup$ There isn't difficultly in defining a point. Rather, there is flexibility in not-defining a point. This is a feature of geometry, not a bug. $\endgroup$– BlueCommented Aug 9, 2016 at 14:51
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8 Answers
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The exact difficulty is that in mathematics, we define things in terms of other things. We also avoid circular definitions, in other words we do not want to define $A$ in terms of $B$ where $B$ is defined in terms of $C$ and $C$ is defined in terms of $A$.
Because we avoid circularity, we can lay out all our definitions in order so that anything that is defined is defined when it is first mentioned. But if we write out a mathematical theory in this fashion, and look at the very first definition we wrote, the thing it defined was defined in terms of some other things, and those "other things" cannot have been defined previously (since this is our first definition) and will not be defined later (since every definition comes before the first use of the thing it defines).
In short, in order to build a mathematical theory we have to start with some "primitive notions" that we will never define. Everything else can be defined in terms of those notions.
This does not prove that a point must be one of the primitive notions of Euclidean geometry, but it happens that it has been chosen to be one of those primitive notions, and this choice has worked out well.
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$\begingroup$ @Wildcard In mathematics, we assume that primitive notions are understood by others in the same way we understand it. I mean there is some uniformity in thinking process. So, should there not be an "Axiom of uniform understandability" ? $\endgroup$– RajkumarCommented Aug 10, 2016 at 6:47
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2$\begingroup$ @Rajkumar, FYI, you can't tag me on an answer I haven't commented on and didn't write—I don't get a notification; I just happened to see this. $\endgroup$– WildcardCommented Aug 10, 2016 at 7:19
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$\begingroup$ @Rajkumar, there is no mathematics without understanding. Assuming that your thoughts are understood by others is not the point. Communicating your thoughts so that they are understood is the point. It helps to have a grasp of the components of understanding, of course. As I say, your question has to do with epistemology (the study of knowledge) rather than strictly dealing with mathematics. (Mathematics is only one type of knowledge, of course.) $\endgroup$– WildcardCommented Aug 10, 2016 at 7:24
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$\begingroup$ @Wildcard, Sorry for tagging at wrong place. I do not see any option to remove tag now. $\endgroup$– RajkumarCommented Aug 10, 2016 at 7:27
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1$\begingroup$ @Rajkumar, no problem; the only point of tagging is to send the person a notification that they've been mentioned in a comment. I happened to see the comment, but I didn't get a notification so I just thought you should know—you can read more about how it works on the Meta site. $\endgroup$– WildcardCommented Aug 10, 2016 at 7:41
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Under your definition, the following things are points:
- the variable $x$
- the sentence $(\forall x)(\exists y)(x^2 = y)$
- $\int_{\infty}^{\infty} f(x) dx$ where $f$ is unspecified
- the category of all small categories
I'm sure you agree that none of these things are points.
answered Aug 9, 2016 at 10:36
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20$\begingroup$ The last one has a size! Its size is "large". $\endgroup$ Commented Aug 9, 2016 at 14:58
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1$\begingroup$ @BenMillwood Well, for that matter, a point has a size: it's "infinitesimal" (admittedly in an informal sense rather than category theory's "large"). But as I understand it, "large" just means "there isn't a set of it", rather than explicitly being a size restriction, although of course the easiest way to be "large" is to be too big for your elements to fit in a set. $\endgroup$ Commented Aug 9, 2016 at 15:09
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$\begingroup$ Wait a minute, a point is an indivisible, not an infinitesimal. $\endgroup$ Commented Aug 9, 2016 at 15:20
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3$\begingroup$ I think @Yakk is touching upon Christian Siever's answer, which is also something I heard as early as high school: you could apply Euclidean geometry to things that don't "look" geometric at all as long as you can match the undefined terms "point", "line", etc. to your new objects in a way that satisfies the postulates. Of course the teacher who said this then proceeded to do nothing of the sort. $\endgroup$– David KCommented Aug 9, 2016 at 21:54
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1$\begingroup$ @Remy Christian Siever's answer is correct, but he wasn't talking about real-life applications. The idea of applying the same set of axioms to problems that "look" very different is a common theme in higher mathematics, although one doesn't normally see much of that in high school math. $\endgroup$– David KCommented Mar 1, 2017 at 16:26
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In a famous comment in the context of a discussion of axiomatisations, David Hilbert noted that these undefined objects might as well be beermugs. The basic conceptual problem here is that mathematicians are tempted to look for ultimate foundations for mathematics. This naturally leads one to a stalemate because whatever foundation you declare to be ultimate, will contain undefined terms which you can in turn ask for even more rigorous foundations for.
The solution is to abandon foundationalism altogether together with a quixotic search for ultimate rigor, and view axiomatisations for what they are, namely convenient tools for clarifying the relations among mathematical entities one is interested in. And of course I will refuse to define a mathematical entity just I would refuse foundationalism.
answered Aug 9, 2016 at 15:15
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1$\begingroup$ If any two beermugs are on exactly one line, you don't need any further definition. A point is the beermug at which two given lines intersect. $\endgroup$– SpencerCommented Aug 10, 2016 at 2:20
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1$\begingroup$ @Spencer, precisely. What interests us is the relation between points and lines and how it plays itself out with regard to procedures and theorems of affine or projective geometry, etc. What interests us less is to get to the exact ontological bottom of how it feels to be a point or a line. $\endgroup$ Commented Aug 10, 2016 at 7:46
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$\begingroup$ Except that with the axioms, we are at the ontological bottom -- points and lines are already defined by the axioms. $\endgroup$– SpencerCommented Aug 10, 2016 at 11:33
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$\begingroup$ @Spencer, you can't convince a mathematician to stop worrying about foundations any more than you can convince a child to stop believing in santa klaus, but what you could do is point out that there are other interesting questions, namely that procedures are more interesting than ontology; see this article. $\endgroup$ Commented Aug 10, 2016 at 11:38
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It may attract controversy, but I will go ahead and present an alternative view on this question.
The other answerers have done an exemplary job showing why a mathematical primitive cannot be mathematically defined.
The trick that is missed is that in order to communicate with someone at all, you must have some shared reality with them. You must have some form of agreement on how you are communicating or they will simply never be able to understand you.
Mathematics takes the view that anything which is not precisely defined, can't be considered valid mathematically—except for primitives. I've observed that some authors and mathematicians are uncomfortable with this exception, but they see no way around it so they learn to live with it.
In point of fact we do share a common reality, or you would not be able to read this answer at all. We have a shared reality of the physical universe. All language is rooted ultimately in this shared reality, and you can observe that babies learning the language (any language, not just English) do so by observing real objects and real actions, and gradually work up to high levels of abstractions.
So the answer to this question falls in the realm of epistemology rather than mathematics.
Through mathematics it is possible to symbolize and reason about universes with entirely different rules from the universe we live in. However, in order to communicate about these other universes or abstractions, the fundamentals must be explained in a way that can be agreed with and understood by people in this universe—otherwise there will be no way to communicate about all these beautiful abstractions. Higher-level abstractions in these other universes (or "rules sets") can be explained in terms of primitives—but the primitives must still be defined.
Now you've mentioned Euclidean Geometry, so let's take that up specifically. Euclidean Geometry has a high degree of applicability to this universe—but it is not this universe; it is an idealized abstraction which can be applied to this universe.
Therefore the above statement about definitions of primitives applies to Euclidean Geometry: the primitives must be defined in terms which can be understood by the realities of this universe. The rest of the terms can be defined in terms of the primitives and do not, strictly speaking, exist in this (or any) universe except as an abstraction.
As a final note, I will add that from a pedagogical standpoint, I believe that the ubiquitous introduction to mathematics textbooks by stating, "Here are some terms that cannot be defined by anyone," is a gigantic mistake—and may even account for a large portion of the population who regard mathematics as complex and incomprehensible.
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I wonder if these texts really want to stress that it is dificult or impossible to define what a point is. I guess they just want to emphasize that they didn't even try, because it is useful to only describe the relations points have with other undefined entities such as lines. That is because if you have a theorem that just uses these axioms and you have some objects that behave like points and the other objects of this axiomatic description, then you can apply that theorem to them, even if they don't look like anything that you would normally consider points and lines etc.
answered Aug 9, 2016 at 10:52
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1$\begingroup$ I agree. But I am not aware of any axiomatic definition or axiomatic description of point. One just assumes this object as known to everyone. $\endgroup$– RajkumarCommented Aug 9, 2016 at 11:29
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$\begingroup$ Usually you have axioms like "for every two different points there is exactly one line that contains both points", without defining what points and lines actually are. People usually have a concept of lines and points and agree (or not), but they could as well just say: "interesting stuff you're talking about, let's see what follows from these axioms". $\endgroup$ Commented Aug 9, 2016 at 12:20
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7$\begingroup$ @Rajkumar It is not unusual that their is no aximatic definition of a point. Likewise there are no axiomatic definitions of vectors, numbers, etc. Rather, we axiomatize structures such as vector spaces and rings and fields, etc because we are not only interested in their elements in isolation but, rather, how those elements are related to each other other by certain natural operations and relations. $\endgroup$ Commented Aug 9, 2016 at 17:52
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No problem
What is the exact difficulty in defining a point in Euclidean geometry?
There is no difficulty and no problem, it's just the way it is defined.
Meaning
"A point is a mathematical object with no shape and size."
The definition literally means that a point has no "shape and size" (i.e., it occupies no space, it has no body, no interiour, no color, no temperature, no feature, whatever other word you would like). It literally has nothing except the few numbers that define/describe it.
"A point is a mathematical object" means that it exists purely mathematically; there is no physical equivalent to a point. It is a mathematical object, which means it is a purely abstract thing which we argue about in the frame of some mathematical system.
For example, in the simple 2-dimensional plane we are used to when discussing these things in a school setting, a combination of two numbers (which also are mathematical objects, not physical). Or in the setting of Euklid, by just reasoning about the relationships between points, lines etc. in an abstract plane without even giving coordinates to them.
Not a point
As a comparison: what is not a point? If you take your pencil and make a little impression on your piece of paper, this is not a point. That is a pencil mark, or a circle if you so wish. It has non-mathematical properties, for example the minuscule thickness of the layer of paint you addded to the paper, or the radius of the circle, its color, the type of molecules making it up, etc. etc. Even if you go to the lowest possible scales, i.e., placing a (classical, not quantum-mechanical) atom somewhere with a raster tunnel microscope, that atom will still take up a spherical amount of space and will not be a "point".
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4$\begingroup$ When describing a point in Euclidean space by means of coordinates, you are already working with a very specific model of Euclidean space, like $\Bbb{R}^n$. Euclidean geometry does not come with coordinates; coordinates were only introduced almost two millennia after Euclid's Elements. $\endgroup$– ServaesCommented Aug 9, 2016 at 15:08
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$\begingroup$ Thanks, I have split that example away and added your point (sic) about the original Euclidian way of going at these things. $\endgroup$– AnoECommented Aug 9, 2016 at 15:34
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In my mind, a point is a context-specific term which means an element of the set being considered.
All the following would make sense to me:
- $sin(x)$ is a point in the space of functions
- (1,2) is a point in $R^2$
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Thanks are due to Wildcard for the complete lesson in the philosophy of the slippery slope that is (higher) mathematics. Perhaps my notion is only pedagogical.
This begins with the definition of a line as length with no width. Just as one can imagine the length going on forever one can see that the width as a compliment can be vanishingly small. It is only represented by the bulky collection of pencil lead on wrinkled paper. The pencil lead is not the line, it only represents the line.
Rather than a collection of points (slippery again) the line is a boundary between one side or idea and another. Perhaps as with the horizon between the earth and the sky they are not in contact but the boundary is there. The boundary is being represented and is all you need to consider.
With this in mind a natural question is what happens when two lines intersect. A simple example is this "+". Here are just two marks representing two lines of no width in themselves are crossing. The cross sectional area they have in common is itself infinitely small. As such, any number of lines can pass through it with the only limit being the amount of pencil lead you can put up with to represent it on the increasingly wrinkled paper. Such an infinitely small point can be indicated by plusses or periods "+, ." with only what it represents in mind. With points represented as such you can do all of geometry, analytic and otherwise without undue philosophical angst.
One will be vexed in trying to define these ideas out of order. One often begins with the smallest object, the point, and moves to the larger, the line. The mistake is starting with the smallest object, not the simplest. That is when you are stuck for a definition of the first for which you need the second. Not an uncommon problem.
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$\begingroup$ A good pedagogical approach! I only just noticed this answer. I’m glad my answer helped! $\endgroup$– WildcardCommented Feb 5, 2018 at 10:00