In mathematics, "undefined notions" refer to concepts or terms that cannot be rigorously defined within the context of a particular system or framework. Instead, they are understood intuitively and serve as the foundational building blocks for more complex concepts. Here are some common examples of undefined notions in mathematics:
-
Point: In geometry, a point represents a location in space, but it has no size, dimension, or definition beyond its existence as a position.
-
Line: A line is often defined as a straight one-dimensional figure that extends infinitely in both directions, but it cannot be defined in purely formal terms without reference to points.
-
Plane: A plane is a flat, two-dimensional surface that extends infinitely in all directions, yet like points and lines, it is intuitive rather than formally defined.
-
Set: While sets can be described and manipulated, the basic idea of a "set" is often taken as an intuitive concept that serves as a foundational idea in set theory.
-
Number: The notion of a number, such as a natural number, integer, or real number, can be defined in various ways (e.g., using set theory) but is often accepted as an intuitive concept.
-
Space: The concept of space, whether in terms of physical dimensions, abstract spaces in mathematics, or other contexts, can be fundamental yet intuitively understood.
These undefined notions are crucial because they allow mathematicians to develop axioms, theorems, and other mathematical structures based on shared intuitions. For example, in Euclidean geometry, points, lines, and planes act as foundational elements upon which other properties and theorems are built.