Generating arithmetic
Inmathematics ofnineteenth century,interest infoundationsmathematics ledwhat could be calledprofessional view ofarithmetizationanalysis; andfurther question about generating arithmetic from more primitive concepts. That is,statusarithmetic, based onnatural numbers, becamekindmiddle term infoundational debate. The statement ofPeano axioms can be seenretrospect as closingchapter onstatusarithmetic.
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2 Generative approaches 3 Pedagogy 4 References |
Historical perspective
The generative methodology, informrecursion,as ancient as"begats" inFifth ChapterGenesisThe Bible. The familiar successor operation, S(n) = n + 1, n = 0, 1, 2, ..., n = 0, 1, 2, ..., which generates Natural numbers,example, S(S(S(S(S(S(S(0))))))) = 7,implicitly"begat operation", B(n) = n + 1, whereinnth generation begatsn + 1 generation, beginningAdam as n = 0.
In19th century whengreat Irish mathematician William Rowan Hamilton createdconceptnamevectorgenerated complex numbers as vectors (ordered pairs) fromreal numbers'. This avoided reference"the square-rootnegative one".
Perhaps ascorrection ofcommentPlato that "God ever geometrizes", C. G. J. Jacobi said, "God ever arithmetizes". Byend ofperiod under consideration,settled pointview was thoughthave emerged on foundations. At one of early international meetingsmathematicians, Henri Poincaré said, "Mathematics has been arithmetized." (Carl Boyer devotes Ch. 25his history"The ArithmetizationAnalysis".) And David Hilbert claimed that allmathematics can be mapped into arithmetic.
There remained questions on axiomatic methodologymathematics. Wasbetter than‘postulational method’, attacked by Bertrand Russell infamous remark?
Generative approaches
Allnumber extensionsnatural numbers -- integers, rational numbers, real numbers -- can be "generated fromvector (ordered pair) basis derived fromsimpler system as basis". This means,example, that "a real number isCauchy sequenceordered pairsordered pairsnatural numbers""a complex number is an ordered pairCauchy sequencesordered pairsordered pairsnatural numbers".
This generative methodology has (erroneously) been described inliterature asLeopold Kronecker program, since Kronecker said, "God createdintegers; allrest isworkman." Derivation from natural numbers might have satisfied Kronecker.
The natural numbers formmonoid under addition (or multiplication), but notgroup becausepartiality (not totalness) ofinverse operation (respectively, subtraction or division). Achieving totalnessthese inverses results ininteger (rational number) systems.
Givendifferencenatural numbers, namely, minuend - subtrahend,useequivalence relations allowsusevector form [minuend, subtrahend], subjectshowing satisfactionequivalence properties indefinition process.
Natural number subtraction, namely,− b = c if,only if= b + c. This transforms subtraction into addition, an operation withpropertytotalness ("always works. always meaningful")natural numbers. (The principle involved makes sense incategory theory termsadjoint functors, too.)
Pedagogy
The advantagegenerating arithmeticthat -- instead resorting to"theft" which Bertrand Russell attributes toaxiomatic method -- able students (by what Russell calls "honest toil") discover,themselves,rulesone arithmetic by applying (the "most sacred rule" of) equivalencean already founded arithmetic. Thus, able students (by "honest toil") discoverrulesignsintegers by satisfying("most sacred") requirementsequivalenceproductsnatural number differences (wherein subtrahendnever greater than minuend).
The axiomatic procedure does not explainpropertiesarithmetic as generated from basic propertiesinformal aspectsdaily life, but rather as postulated rules. This has invoked"SchoolSocial Constructivism" (with many online websites) which argues that any mathematical systemmerelysocial construction -- on partable manners.
References
A HistoryMathematics, Carl B. Boyer, Princeton Uiversity Press, Princeton, 1985.
Mathematical Thought from AncientModern Times, v. 3 (p. 992), Morris Kline, Oxford University Press, New York, Oxford, 1972.
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
