Primitive notion

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In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. [1] Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem).

Contents

For example, in contemporary geometry, point , line, and contains are some primitive notions. Instead of attempting to define them, [2] their interplay is ruled (in Hilbert's axiom system) by axioms like "For every two points there exists a line that contains them both". [3]

Details

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. 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. [5]

Examples

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

Russell's primitives

In his book on philosophy of mathematics, The Principles of Mathematics Bertrand Russell used the following notions: for class-calculus (set theory), he used relations, taking set membership as a primitive notion. To establish sets, he also establishes propositional functions as primitive, as well as the phrase "such that" as used in set builder notation. (pp 18,9) Regarding relations, Russell takes as primitive notions the converse relation and complementary relation of a given xRy. Furthermore, logical products of relations and relative products of relations are primitive. (p 25) As for denotation of objects by description, Russell acknowledges that a primitive notion is involved. (p 27) The thesis of Russell’s book is "Pure mathematics uses only a few notions, and these are logical constants." (p xxi)

See also

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References

  1. ↑ More generally, in a formal system, rules restrict the use of primitive notions. See e.g. MU puzzle for a non-logical formal system.
  2. ↑ Euclid (300 B.C.) still gave definitions in his Elements , like "A line is breadthless length".
  3. ↑ This axiom can be formalized in predicate logic as "∀ x1,x2 ∈ P. ∃ y∈L. C(y,x1) ∧ C(y,x2)", where P, L, and C denotes the set of points, of lines, and the "contains" relation, respectively.
  4. ↑ Alfred Tarski (1946) Introduction to Logic and the Methodology of the Deductive Sciences, p. 118, Oxford University Press.
  5. ↑ Gilbert de B. Robinson (1959) Foundations of Geometry, 4th ed., p. 8, University of Toronto Press
  6. ↑ Mary Tiles (2004) The Philosophy of Set Theory, p. 99
  7. ↑ Phil Scott (2008). Mechanising Hilbert's Foundations of Geometry in Isabelle (see ref 16, re: Hilbert's take) (Master's thesis). University of Edinburgh. CiteSeerX   10.1.1.218.9262 .
  8. ↑ Alessandro Padoa (1900) "Logical introduction to any deductive theory" in Jean van Heijenoort (1967) A Source Book in Mathematical Logic, 1879–1931, Harvard University Press 118–23
  9. ↑ Haack, Susan (1978), Philosophy of Logics, Cambridge University Press, p. 245, ISBN   9780521293297
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