Primitive notion
In
regress problem
).
For example, in contemporary geometry, 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
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:
- Set theory: The concept of the set is an example of a primitive notion. As Mary Tiles writes:[6] [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.
- 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.[7]
- Arithmetic of real numbers: Typically, primitive notions are: real number, two binary operations: addition and multiplication, numbers 0 and 1, ordering <.
- Hilbert's axiom systemthe 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.
Russell's primitives
In his book on
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
- Axiomatic set theory
- Foundations of geometry
- Foundations of mathematics
- Logical atomism
- Logical constant
- Mathematical logic
- Notion (philosophy)
- Natural semantic metalanguage
References
- ^ More generally, in a formal system, rules restrict the use of primitive notions. See e.g. MU puzzle for a non-logical formal system.
- ^ Euclid (300 B.C.) still gave definitions in his Elements, like "A line is breadthless length".
- ∃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.
- ^ Alfred Tarski (1946) Introduction to Logic and the Methodology of the Deductive Sciences, p. 118, Oxford University Press.
- Gilbert de B. Robinson (1959) Foundations of Geometry, 4th ed., p. 8, University of Toronto Press
- ^ Mary Tiles (2004) The Philosophy of Set Theory, p. 99
- CiteSeerX 10.1.1.218.9262.
- ^ 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
- ISBN 9780521293297