Mathematical primitive

https://proofwiki.org/wiki/Definition:Formal_Language/Alphabet/Primitive_Symbol

Mathematical primitive:

• space

• number

• set

• empty set

Primitives are undefined notions or objects. A mathematical primitive is not defined in terms of previously defined concepts, but described informally by appealing to one's intuition or experience, and expected to be taken for granted. Many fundamental mathematical objects (number, set, space) are primitives.

Models are used to approximate notions or objects by identifying and encoding patterns in the data.

Mathematics uses many types of spaces, but doesn't define the notion of "space".

Mathematics uses several types of numbers (natural, integral, rational, real, complex), but "number" is not used as a mathematical notion and has no definition.

Mathematical primitive is a concept not defined in terms of previously defined concepts. Primitives, lacking a formal definition, are to be taken for granted, intuitively understood.

Regarding the lack of proof, mathematical primitives are similar to axioms in a formal system, since axioms do not require proof.

Primitive notions cannot be avoided because every process/concept needs to start somewhere, with some, possibly undefined, terms.

Otherwise, we'd be stuck, forever defining concepts in terms of previously defined ones, and so on, until singularity.

The two tools to allow us to cut this regression chain are primitives and axioms.

And sometimes a concept just doesn't have a formal definition (e.g. "set", "space").

In mathematics, logic and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience.

In an axiomatic theory or other formal system, the role of a primitive notion is analogous to that of axiom. In axiomatic theories, the primitive notions are sometimes said to be "defined" by one or more axioms, but this can be misleading.

Formal theories cannot dispense with primitive notions, under pain of infinite regress.

Alfred Tarski explained the role of primitive notions as follows:

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. At the same time we adopt the principle: not to employ any of the other expressions of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions of the discipline whose meanings have been explained previously. The sentence which determines the meaning of a term in this way is called a definition.

An inevitable regress to primitive notions in the theory of knowledge was explained by Gilbert de B. Robinson:

To a non-mathematician it often comes as a surprise that it is impossible to define explicitly all the terms which are used. This is not a superficial problem but lies at the root of all knowledge; it is necessary to begin somewhere, and to make progress one must clearly state those elements and relations which are undefined and those properties which are taken for granted.

The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics:

• Set theory: The concept of the set is an example of a primitive notion. As Mary Tiles writes: The "definition" of set is less a definition than an attempt at explication of something which is being given the status of a primitive, undefined, term. As evidence, she quotes Felix Hausdorff: "A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit."

• Naive set theory: The empty set is a primitive notion. To assert that it exists would be an implicit axiom.

• Peano arithmetic: The successor function and the number zero are primitive notions. Since Peano arithmetic is useful in regards to properties of the numbers, the objects that the primitive notions represent may not strictly matter.[citation needed]

• Axiomatic systems: The primitive notions will depend upon the set of axioms chosen for the system. Alessandro Padoa discussed this selection at the International Congress of Philosophy in Paris in 1900.[4] The notions themselves may not necessarily need to be stated; Susan Haack (1978) writes, "A set of axioms is sometimes said to give an implicit definition of its primitive terms."

• Euclidean geometry: Under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweeness, and incidence.

• Euclidean geometry: Under Peano's axiom system the primitive notions are point, segment, and motion.

• Philosophy of mathematics: Bertrand Russell considered the "indefinables of mathematics" to build the case for logicism in his book The Principles of Mathematics (1903).