What Is Point?
Point, together with line, is a basic concept of elementary geometry. The idea of point is an abstraction that distills our understanding of the concept of location. You can't get too wrong if you say that point is a mere location, location without width, breadth or length. But you won't be able to get very far either because, given leisure time and sufficient curiosity, you will next ponder about the concepts I just mentioned: location, width, breadth, length. What are they?
It makes no point (a pun intended) to give a definition without a purpose. So, before becoming impatient with my reluctance to give a definition for the term point, try to think of what point should be good for. I believe that the most plausible property a point may be expected to have is its ability to be joined to other points. If so, it is practically impossible to define point without mentioning the joints that may go between the points. And this is how we come around to talking about points and - simultaneously - lines. To paraphrase Alfred E. Neuman, It takes line to know point -- and vice versa.
The common practice is to leave the terms point and line undefined but just start using them in the first principles.
There is a single line through two points. There is a single point at the intersection of two lines.
An article in S. Schwartzman's The Words of Mathematics describes etymology and supplies some additional information:
point: a French word meaning "dot, point, period (the punctuation mark)," from Latin punctus, past participle of pungere "to prick, to punc- ture." The Indo-European root is peug- "to prick." Related borrowings from Latin include pugilist and pugnacious. When you puncture something, you use a sharp object to make a tiny hole in it. That tiny hole, especially from a distance, looks like a dot, so a point is a dot. Metaphorically speaking, if you're punctual, you arrive right on the dot. In mathematics a point is assumed to be dimensionless, but of course any physical representation of a point must be of some size. A point is often represented in textbooks by the smallest of all printing symbols, the period. In fact printers' type sizes are measured in units called points: the fact that 72 points make an inch tells us the size that printed periods commonly used to be. The verb to point developed from the use of the noun point to refer to the tapered end of an object like a stick or a pencil. Such objects were and still are used for pointing to things. In mathematics, two vectors of equal length may be distinguished by the direction in which each one points.
A Look Ahead
Imagining a point as a location may be congenial to our way of thinking but in mathematics beyond the elementary this may become a hindrance. Very often a point is just an element of a set regardless of its nature. What is a function over a line segment is a point in the space of all functions defined on that segment. In projective geometry, the terms point and line are completely interchangeable. This does not mean that there is no difference between the two but that there are objects of two classes and it does not matter which you designate points and which lines, meaning that the theory remains the same regardless of your choice. In fact making a choice is absolutely, unconditionally an arbitrary affair.
- S. Schwartzman, The Words of Mathematics, MAA, 1994