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| jonhays |
 Posted: Sun May 11, 2008 10:47 am Post subject: extension vs. intension |
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Forum Freshman
Joined: 11 May 2008
Posts: 6
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Two neglected math concepts: EXTENSION, INTENSION. Easily explicated in set language. Say, in natural number system, N, numbers less than seven. An extension is {0,1,2,3,4,5,6}. An intension is
{x: x in N, x < 7}, "set of any x such that x is in N and less than
7". Seems ok. But Problem: extension is univalent: one referent;
intension may be multivalent: descriptive statement may fit many
referents. Axioms are intensional. The familiar Peano Axioms invoke
the standard integer system. But online is a nonstandard version
fitting Peano Axioms such that its every integer is greater than
any standard integer. (A "stolen identity" case?) Banach-Tarsky
Paradox says Euclidean Geometry Axioms allow moon to be cut into
five parts, fitted together, put in your pocket. Standard Arithmetic
is founded axiomatically: intensionally. But my extensional alternative
exists online: generatics. William Rowan Hamilton rejected standard
definition of a complex number as sum of real number plus real in
product with square root of negative one. Hamilton invented the vector
and its label -- ignored? -- to define complex number as bivector or
reals with appropriate rules. Generatics uses this procedure to start
with naturals, N, recursively (extensionally) defined; define integers,
J, as bivectors of naturals with appropriate rules; define rationals,
Q, as bivectors of integers with appropriate rules; define reals, R,
limits of infinite vectors of rationals (decimal numbers) with
appropriate rules; complex numbers in Hamilton's way. (As a teacher,
I've given my best students clues whereby each independently discovered this.) But the bivectors are hidden by notation. Equivalence reduces naturals bivector to three types, explicated as positive/negative
integers or zero, hidden by signs. Equivalence reduces integer bivectors
to three types, hidden by solidus as fractions. Equivalence reduces
infinite vectors of reals to three types hidden as of decimal numbers.
But something happens with complex numbers. Using concept of "modul" (not module!) in "Introduction to Number Theory" by Oystein Ore, p. 159. A modul is a structure closed under subtraction. Ore notes
naturals numbers aren't closed under subtraction, but integers,
rationals, reals, and complex numbers are closed under subtraction,
so each number forms a modul. Imaginaries form their own modul. Ore
doesn't note that complex numbers form a BIMODUL: one for real part,
one for imaginary part. Significance? VECTORS CAN NO LONGER BE HIDDEN BY NOTATION. Leads to MULTIVECTORS (a.k.a. arithmetic of Clifford Numbers), with Gibbs-Heaviside vectors as special case. Three products (inner, outer, multiproduct) distiguish these. Online in "Arithmetic Redux", I derive these in high school level. But all is extensional: univalent: single referent! I invite comment/argument. |
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| serpicojr |
| Posted: Sun May 11, 2008 11:09 am Post subject: |
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Forum Ph.D.
Joined: 17 Jul 2007
Posts: 871
Location: JRZ
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| You have neat ways of representing things. However, you seem to believe that this is a radical new approach, while I don't know how this is much different from the way people already do things. |
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