How do mathematics define a point?

Question

I have a serious doubt. How do mathematicians define a 'point' in a space or a plot? If we have a clear explanation for a 'point' , I think my doubt on infinitesimals and infinity will be clarified.

Answer 1

Your serious doubt has been shared by many great thinkers throughout history! This is a heavily philosophical issue, and there are smart people who would disagree with much of what I'm about to write.

In a nutshell, a geometric point is an abstraction of the idea of a location in space. However, once abstracted, the geometric point becomes a non-physical entity. It exists and is described only by its mathematical properties. In other words a mathematical point has no "essential nature" in the sense that we can say this is what a point is, rather we say these are the properties of points. For example, "between any two distinct points there is a line." This is a very deep philosophical issue, and depending on your level of sophistication and experience, you may have trouble grasping it.

That's the traditional, geometric, axiomatic notion of point. Another view, more and more popular these days, is to take the real numbers as your "primitive" notion, and then define a point in terms of its coordinates. In other words, for plane geometry, you might see a definition such as "a point is an ordered pair of real numbers".

These ideas are like, for example the physical locations on a sheet of paper when we draw coordinate axes. But that is best thought of as a physical approximation to an ideal, abstract mathematical structure. Or, inversely, the mathematical structure is the abstracted, idealized version of the physical system.

It's worth noting that "point" has other, more abstract meanings in higher branches of mathematics, but I don't think that's relevant to your question.

Edit: In reply to @Henry's comment, simply quoting Euclid's definition "a point is that which has no part": much can be said about why Euclid wrote this and how it was an improvement on his predecessors. Regardless, that ancient "definition" is sorely lacking, as has been noted time and again in commentaries. The most resounding objection, though not the most damning, is this: ought we really to to define something so fundamental to an entire discipline, solely in terms of what it is not?

Answer 2

The name “point” is just another name of a member of a set which one can call a “space”, that's all. So a better question would be: “What do we call a space?” To stay vague, any set that allows to apply geometrics notions. It's just a way to speak intuitively about otherwise “abstract” objects. It makes talking and reasoning easier.

You have to keep in mind that mathematician don't usually invent new words for the objects they're dealing with: they just borrow them from everyday language and give them a new precise notion.

In your life as a mathematician your notion of what a point is will extend more and more.