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Alfred Tarski explained the role of primitive notions as follows: [4]. When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable; the expressions in this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings.
Primitives (undefined terms) are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while a fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are undefined.
They understand the role of undefined terms, definitions, axioms and theorems in Euclidean geometry. However, students at this level believe that axioms and definitions are fixed, rather than arbitrary, so they cannot yet conceive of non-Euclidean geometry. Geometric ideas are still understood as objects in the Euclidean plane. Level 4.
Although these terms are not further defined, Euclid uses them to construct more complex geometric concepts. [5] Whether a particular function or value is undefined, depends on the rules of the formal system in which it is used. For example, the imaginary number is undefined within the set of real numbers.
A good example is the relative consistency of absolute geometry with respect to the theory of the real number system. Lines and points are undefined terms (also called primitive notions) in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems. [citation needed]
In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. [69] Hilbert, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms.
To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.
Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry. [2] He selected a small core of undefined terms (called common notions) and postulates (or axioms) which he then used to prove various geometrical statements.
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