Foundational status of arithmetic
|
|
In the mathematics of the nineteenth century, the interest in the foundations of mathematics led to what could be called a professional view of the arithmetization of analysis; and a further question about generating arithmetic from more primitive concepts. That is, the status of arithmetic, based on the natural numbers, became a kind of middle term in the foundational debate. The statement of the Peano axioms can be seen in retrospect as closing a chapter on the status of arithmetic.
| Contents |
Historical perspective
The generative methodology, in the form of recursion, is as ancient as the "begats" in the Fifth Chapter of Genesis in The Bible. The familiar successor operation, S(n) = n + 1, n = 0, 1, 2, ..., n = 0, 1, 2, ..., which generates Natural numbers, for example, S(S(S(S(S(S(S(0))))))) = 7, is implicitly the "begat operation", B(n) = n + 1, wherein the nth generation begets the n + 1 generation, beginning with Adam as n = 0.
In the 19th century William Rowan Hamilton created the concept and name of vector and generated complex numbers as vectors (ordered pairs) from the real numbers. This avoided reference to "the square-root of negative one".
Perhaps as a correction of the comment of Plato that "God ever geometrizes", C. G. J. Jacobi said, "God ever arithmetizes". By the end of the period under consideration, a settled point of view was thought to have emerged on foundations. At one of the early international meetings of mathematicians, Henri Poincaré said, "Mathematics has been arithmetized." (Carl Boyer devotes Ch. 25 of his history to "The Arithmetization of Analysis".) And David Hilbert claimed that all of mathematics can be mapped into arithmetic.
There remained questions on axiomatic methodology in mathematics. Was it better than the 'postulational method', attacked by Bertrand Russell in a famous remark?
Generative approaches
All the number extensions of natural numbers -- integers, rational numbers, real numbers -- can be "generated from a vector (ordered pair) basis derived from a simpler system as basis". This means, for example, that "a real number is a Cauchy sequence of ordered pairs of ordered pairs of natural numbers" and "a complex number is an ordered pair of Cauchy sequences of ordered pairs of ordered pairs of natural numbers".
This generative methodology has (erroneously) been described in the literature as the Leopold Kronecker program, since Kronecker said, "God created the integers; all the rest is the work of man." Derivation from natural numbers might have satisfied Kronecker.
The natural numbers form a monoid under addition (or multiplication), but not a group because of partiality (not totalness) of the inverse operation (respectively, subtraction or division). Achieving totalness for these inverses results in the integer (rational number) systems.
Given a difference of natural numbers, namely, minuend - subtrahend, the use of equivalence relations allows the use of the vector form [minuend, subtrahend], subject to showing satisfaction of equivalence properties in the definition process.
Natural number subtraction, namely, a − b = c if, and only if a = b + c. This transforms subtraction into addition, an operation with the property of totalness ("always works. always meaningful") in natural numbers. (The principle involved makes sense in the category theory terms of adjoint functors, too.)
Pedagogy
The advantage of generating arithmetic is that -- instead resorting to the "theft" which Bertrand Russell attributes to the axiomatic method -- able students (by what Russell calls "honest toil") discover, for themselves, the rules of one arithmetic by applying (the "most sacred rule" of) equivalence to an already founded arithmetic. Thus, able students (by "honest toil") discover the rule of signs for integers by satisfying the ("most sacred") requirements of equivalence for products of natural number differences (wherein subtrahend is never greater than minuend).
The axiomatic procedure does not explain the properties of arithmetic as generated from basic properties with informal aspects in daily life, but rather as postulated rules. This has invoked a "School of Social Constructivism" (with many online websites) which argues that any mathematical system is merely a social construction -- on par with table manners.
References
- A History of Mathematics, Carl B. Boyer, Princeton Uiversity Press, Princeton, 1985.
- Mathematical Thought from Ancient to Modern Times, v. 3 (p. 992), Morris Kline, Oxford University Press, New York, Oxford, 1972.
External links
- http://andrew.cmu.edu/~cebrown/notes/leivant.html.
- http://members.fortunecity.com/jonhays/redux.htm.
- http://members.fortunecity.com/jonhays/integralvect.htm.
- http://members.fortunecity.com/jonhays/rationalvect.htm.
- http://members.fortunecity.com/jonhays/compvect.htm.
- http://www.ex.ac.uk/~PErnest/soccon.htm