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!
2026-03-26 22:15:03.1774563303
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What is the exact difficulty in defining a point in Euclidean geometry?
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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.
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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.
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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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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.
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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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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.
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.