[plt-scheme] Natural numbers
On Wed, 2009-03-11 at 22:25 -0400, Stephen Bloch wrote:
. . .
> > In maths, the term "the natural numbers" refers specifically to the
> > set of positive integers (see for example [1]), and not to any set
. . .
> Ah, maybe that's it: I went through graduate school surrounded by
> logicians and computer scientists. :-)
This sparked an entertaining couple of hours for me. I found the
following two line-ups:
Natural numbers starting at 1:
o Eric W.Weisstein, "CRC Concise Encyclopedia of Mathematics",
o H.Benke, F.Bachmann, K.Fladt and W.Suess, "Fundamentals of
Mathematics, Volume I, Foundations of Mathematics/The Real Number
System and Algebra",
o Raymond L. Wilder, "Introduction to The Foundations of Mathematics",
o Giuseppe Peano, "The principles of arithmetic, presented by a new
method",
o Mark Dugopolski, "Elementary and Intermediate Algebra", second
edition.
Natural numbers starting at 0:
o Gottlob Frege, "The concept of number",
o Patrick Suppes, "Axiomatic Set Theory",
o Willard van Orman Quine, "Set Theory and its Logic",
o Frederic B. Fitch, "Elements of Combinatory Logic",
o David Gries and Fred B.Schneider, "A Logical Approach to Discrete
Math".
There does indeed seem to be a pattern here.
I teach elementary algebra out of Dugopolski and I find the definition
of the naturals as the positive integers and the "whole numbers" as the
naturals + {0} inconvenient; students keep asking "why".
References for the curious:
H.Benke, F.Bachmann, K.Fladt and W.Suess, "Fundamentals of Mathematics,
Volume I, Foundations of Mathematics/The Real Number System and
Algebra", (translated by S.H.Gould), MIT Press, 1974, pp. 93-94: "...
natural numbers (in the present section they are simply called
numbers) ... system of axioms: I. 1 is a number ...."
Mark Dugopolski, "Elementary and Intermediate Algebra", second edition,
McGraw-Hill, 2006, p. 2: "... the set of counting numbers or natural
numbers ... {1,2,3,...}".
Frederic B. Fitch, "Elements of Combinatory Logic", Yale University
Press, 1974, p. 83: "By the natural numbers we mean the non-negative
finite integers 0, 1, 2, 3, 4, and so on."
Gottlob Frege, "The concept of number", in "Philosophu of
Mathematics/Selected Readings", Paul Benacerraf & Hilary Putnam,
Cambridge University Press, 1983, p. 147: "1 immediately follows 0 in
the series of natural numbers."
David Gries and Fred B.Schneider, "A Logical Approach to Discrete Math",
Springer-Verlag, 1993, p. 227: "Definition. The set of natural numbers
N, expressed in terms of 0 and a function S (for successor), ... ."
Giuseppe Peano, "The principles of arithmetic, presented by a new
method", 1889, in Jean van Heijenoort, "From Frege to Goedel, A Source
Book in Mathematical Logic, 1879-1931", Harvard University Press, 1967,
p. 94: "Axioms ... 1. 1 <- N." ("<-" for "is an element of")
Willard van Orman Quine, "Set Theory and its Logic", Harvard University
Press, 1969, p. 74: "By numbers in this chapter I shall mean just the
natural numbers: 0 and the positive integers."
Patrick Suppes, "Axiomatic Set Theory", Dover Publications, 1972, p.
135: "Our first task is to prove Peano's five axioms ... Theorem 17. 0
is a natural number."
Eric W.Weisstein, "CRC Concise Encyclopedia of Mathematics", Chapman &
Hall/CRC, 1999, p. 1219 (entry: Natural Number): "A positive integer 1,
2, 3, ... (Sloane's A000027). ... Unfortunately, 0 is sometimes also
included in the list of 'natural' numbers (Bourbaki 1968, Halmos 1974),
and there seems to be no general agreement about whether to include it."
Raymond L. Wilder, "Introduction to The Foundations of Mathematics",
John Wiley and Sons, 1965, p. 46: "...integers 1, 2, 3, ... (these we
shall call the natural numbers), ...". Footnote: "Some authors include
0 among the natural numbers."
-- Bill Wood