Kingdom of Infinite Number: A Field Guide - Softcover

Bunch, Bryan H.

Softcover

ISBN 10: 0716744473 ISBN 13: 9780716744474

Verlag: W.H.Freeman & Co Ltd, 2001

Inhaltsangabe

Treats mathematics as if it were a biological kingdom as it introduces the various kinds of numbers, from the natural numbers used in counting to infinite numbers, and explains how mathematicians use them in sets and proofs.

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Über die Autorin bzw. den Autor

Bryan Bunch has extensive experience as a writer and editor in the math and science fields. The editor of The Family Encyclopedia of Diseases, he is also the co-author, with Alexander Hellemans, of The Timetables of Science and the author of Mathematical Fallacies and Paradoxes and Reality's Mirror: Exploring the Mathematics of Symmetry, which was named one of the year's best science books by Library Journal. He lives in Pleasant Valley, New York.

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The Kingdom of Infinite Numbers

Kingdom Number

Genus Natural (Counting Numbers)

In some societies, counting language has not evolved beyond 2 or 3 and a word that means "many" for higher numbers. From a linguistic point of view the counting numbers, at least those greater than 2 or 3, may be considered a human invention.

Yet there is something natural about the genus Natural. If the concept of matching, or one-to-one correspondence, is taken as a fundamental idea, the natural numbers--1, 2, 3, ...--would appear to be its logical outgrowth; a natural number describes the set of all sets that can be matched a member at a time with a given set, without regard to order.

From this point of view, 1 might be considered the characteristic that all the sets which match with me myself, considered as an individual, have in common. The number 2 is characteristic of collections that match my eyes or hands or legs or ears. Then 3 is part of the description of sets that match the spaces between the knuckles on one hand, 4 the sets that match the fingers without the thumb, 5 with the thumb, and so forth. You see why some societies stopped using natural numbers after 2 or 3, since we quickly begin to exhaust the observables shared by all.

Although the idea of one-to-one correspondence may seem natural--it is pretty much what we teach to young children as the meaning of number--it is not very mathematical as most mathematicians for the past 200 years haveenvisioned the field. Mathematical ideas must be pinned down; it is not enough to say that we abstract a concept from experience. The definition of a mathematical entity must be "operational"--that is, there must be some operation that produces the entity. Furthermore, the operation needs to emerge from within mathematics. The ancient Greeks were satisfied, for example, with the idea that geometric figures are congruent if one may be picked up and superimposed on another. This intuitive idea is rejected today because superimposition is not a mathematical operation. Congruence is established instead with a series of definitions based on such ideas as equality of lengths or angles, which are part of mathematics generally. So it is with the numbers--instead of the "natural" interpretation of numbers, mathematicians use the "counting" interpretation. The nineteenth-century German mathematician Richard Dedekind, the first to define the genus Real without recourse to geometry, wrote: "I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting."

Several mathematical definitions based on counting have been proposed for genus Natural. The most common and in some ways the most acceptable was developed by the Italian mathematician Giuseppe Peano [1858-1932] in 1889. It combines the idea of counting with a very general principle called mathematical induction that applies nicely to counting numbers. The Peano definition is a set of five axioms that begins simply and becomes increasingly less intuitive, but still manages to have the axiomatic property of seeming inevitable (although a certain amount of reflection is required to grasp the essence of the last two). Those who are familiar with Euclid's five postulates for geometry will notice a certain parallel.

In ordinary language, Peano's definition is as follows:

The number 1 exists.

You can always count further by adding 1 more.

No amount of counting will bring you back to 1 again.

If two numbers reached by counting are equal, then the numbers just before them are also equal.

If you define a set of numbers so that: (a) if the defining property applies to some counting number, then it also applies to the number that is 1 more; and (b) if 1 has the property, then all the counting numbers also have the property.

The last rule in the definition is hard to follow, with its two ifs and two thens, so I have tried to emphasize the parts. Peano himself would never have stated these rules in anything like this language, however. He believed strongly in symbolic logic and stated all his conclusions using special symbols, so his five-volume book explaining why this definition describes the counting numbers is almost all symbols and unreadable in the ordinary sense. He also insisted on writing out all his lectures using symbolic logic, which caused the students in his classes to rebel. Peano tried to quell the rebellion by promising the students that they would all pass, no matter what, but this attempt at pacification failed. As a result, Peano was forced to resign from teaching.

The terms "counting number" and "natural number" refer to the same set of numbers, but emphasize different aspects of them. The genus name for these numbers is Natural because that is the name mathematicians usually use, but the definition that formal theory prefers is based on counting rather than the "natural" idea of matching sets. Thus, it would be more logical to speak of the "counting numbers" rather than the "natural numbers." In some formaltheories (for example, the conditions of the famous proof known as Gödel's Theorem--see Prime Family, pp. 35-38) it is important to make a clear distinction between the natural numbers and the counting numbers, but that is too fancy for this project. I generally use whichever term fits the context.

Commonly Seen Species

1

1 One I unity

French une; German ein; Spanish unos; Italian uno; Latin unus; Greek heîs; Papuan body counting, right little finger; classical and medieval finger numerals, left little finger folded at joints, thumb raised

 

There is only the one, but it comes in two guises. One is sometimes a number, but sometimes not; sometimes it is a discriminator.

The word "discriminator" extends the grammatical idea of the article. Articles form the smallest grammatical category in English, comprising only two members: "a" and "the." The main function of an article is to indicate that what follows immediately is a noun and not some other part of speech. This function is sufficiently unnecessary that part of it is dropped in other languages such as Russian, which has no word for "the." Thus: word "discriminator" extends grammatical idea of article.

The broader class of discriminator includes, along with the articles, such words as "this," "that," "such," and, of course, "one." "Pointer" would be a more useful class distinction than "discriminator," since what most discriminators dois point: this child, that elephant, such apples. The other discriminators are those that specifically fail to point: a child, an elephant, one apple.

"One" can also be a nonpointing discriminator, functioning approximately as a more emphatic "a" or "an." The sentences "There'll come a day when I'll find you" and "There'll come one day when I'll find you" mean essentially the same, but "one day" is stronger than "a day." This discriminator function of "one" gradually merges into the number function. If you say "Give me a rose" you mean the same as "Give me one rose," but in the second sentence the discriminator has drifted further toward numberhood. In the other direction, when used as a pronoun "one" has another nonnumerical use, which is distinctly unlike the number function. When I say "One starts counting with the number one," the first "one" does not mean a single individual, but quite the opposite. As a pronoun, "one" means "everyone" --the ultimate nonpointer.

 

FIELD MARKS The number 1 is the least counting number, the...

Details zur Ausgabe

• Verlag: W.H.Freeman & Co Ltd
• Erscheinungsdatum: 2001
• Sprache: Englisch
• ISBN 10: 0716744473
• ISBN 13: 9780716744474
• Einband: Taschenbuch
• Anzahl der Seiten: 388
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar