Men of Mathematics (101 page)

The heart of Dedekind's theory of
irrational
numbers is his concept of the “cut” or “section”
(Schnitt):
a cut separates
all
rational numbers into
two
classes, so that each number in the
first
class is
less than
each number in the
second
class; every such cut which does not “correspond” to a rational number “defines” an irrational number. This bald statement needs elaboration, particularly as even an accurate exposition conceals certain subtle difficulties rooted in the theory of the mathematical infinite, which will reappear when we consider the life of Dedekind's friend Cantor.

Assume
that some rule has been prescribed which separates
all
rational numbers into
two
classes, say an “upper” class and a “lower” class, such that each number in the
lower
class is
less than
every number in the
upper
class. (Such an assumption would not pass unchallenged today by all schools of mathematical philosophy. However, for the moment, it may be regarded as unobjectionable.) On this assumption one of three mutually exclusive situations is possible.

(A) There may be a number in the
lower
class which is
greater
than every other number in that class.

(B) There may be a number in the
upper
class which is
less
than every other number in that class.

(C) 
Neither
of the numbers
(greatest
in [A],
least
in [B]) described in (A), (B) may exist.

The possibility which leads to irrational numbers is (C). For, if (C)
holds, the assumed rule “defines” a definite break or “cut” in the set of all rational numbers. The upper and lower classes strive, as it were, to meet. But in order for the classes to meet the cut must be filled with some “number,” and, by (C), no such filling is possible.

Here we appeal to intuition. All the distances measured from any fixed point along a given straight line “correspond” to “numbers” which “measure” the distances. If the cut is to be left unfilled, we must picture the straight line, which we may conceive of as having been traced out by the
continuous
motion of a point, as now having an unbridgeable gap in it. This violates our intuitive notions, so we say, by definition, that each cut
does define
a number. The number thus defined is not rational, namely it is irrational. To provide a manageable scheme for operating with the
irrationals
thus
defined by cuts
(of the kind [C]) we now consider the
lower class of rationals
in (C) as being equivalent to the irrational which the cut defines.

One example will suffice. The
irrational
square root of 2 is defined by the cut whose upper class contains
all
the positive rational numbers whose squares are greater than 2, and whose lower class contains
all
other
rational
numbers.

If the somewhat elusive concept of cuts is distasteful two remedies may be suggested: devise a definition of irrationals which is less mystical than Dedekind's and fully as usable; follow Kronecker and, denying that irrational numbers exist, reconstruct mathematics without them. In the present state of mathematics some theory of irrationals is convenient. But, from the very nature of an irrational number, it would seem to be necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. The appeal to infinite classes is obvious in Dedekind's definition of a cut. Such classes lead to serious logical difficulties.

It depends upon the individual mathematician's level of sophistication whether he regards these difficulties as relevant or of no consequence for the consistent development of mathematics. The courageous analyst goes boldly ahead, piling one Babel on top of another and trusting that no outraged god of reason will confound him and all his works, while the critical logician, peering cynically at the foundations of his brother's imposing skyscraper, makes a rapid mental calculation predicting the date of collapse. In the meantime all are busy and all seem to be enjoying themselves. But one conclusion appears to be inescapable: without a consistent theory of the mathematical
infinite there is no theory of irrationals; without a theory of irrationals there is no mathematical analysis in any form even remotely resembling what we now have; and finally, without analysis the major part of mathematics—including geometry and most of applied mathematics—as it now exists would cease to exist.

The most important task confronting mathematicians would therefore seem to be the construction of a satisfactory theory of the infinite. Cantor attempted this, with what success will be seen later. As for the Dedekind theory of irrationals, its author seems to have had some qualms, for he hesitated over two years before venturing to publish it. If the reader will glance back at Eudoxus' definition of “same ratio” (Chapter 2) he will see that “infinite difficulties” occur there too, specifically in the phrase “any whatever equimultiples.” Nevertheless some progress has been made since Eudoxus wrote; we are at least beginning to understand the nature of our difficulties.

*  *  *

The other outstanding contribution which Dedekind made to the concept of “number” was in the direction of algebraic numbers. For the nature of the fundamental problem concerned we must refer to what was said in the chapter on Kronecker concerning algebraic number fields and the resolution of algebraic
integers
into their
prime
factors. The crux of the matter is that in
some
such fields resolution into prime factors is
not unique
as it is in common arithmetic; Dedekind restored this highly desirable uniqueness by the invention of what he called
ideals.
An ideal is not a number, but an infinite class of numbers, so again Dedekind overcame his difficulties by taking refuge in the infinite.

The concept of an ideal is not hard to grasp, although there is one twist—
the more inclusive class divides the less inclusive,
as will be explained in a moment—which shocks common sense. However, common sense was made to be shocked; had we nothing less dentable than shock-proof common sense we should be a race of mongoloid imbeciles. An ideal must do at least two things: it must leave common (rational) arithmetic substantially as it is, and it must force the recalcitrant algebraic integers to obey that fundamental law of arithmetic—
unique
decomposition into primes—which they defy.

The point about a more inclusive class dividing a less inclusive refers to the following phenomenon (and its generalization, as stated presently). Consider the fact that 2 divides 4—
arithmetically,
that is,
without remainder.
Instead of this obvious fact, which leads nowhere if followed into algebraic number fields, we replace 2 by the
class
of
all
its integer multiples, . . . , −8, −6, −4, −2, 0, 2, 4, 6, 8, . . . As a matter of convenience we denote this class by (2). In the same way (4) denotes the class of
all
integer multiples of 4. Some of the numbers in (4) are . . ., −16, −12, .−8, −4, 0, 8, 12, 16, . . . It is now obvious that (2) is the more inclusive class; in fact (2) contains
all
the numbers in (4) and in addition (to mention only two) −6 and 6. The fact that (2) contains (4) is symbolized by writing (2) |(4). It can be seen quite easily that if
m, n
are any common whole numbers then
(m)
|(n)
when, and only when, m divides n.

This might suggest that the notion of common arithmetical divisibility be replaced by that of class inclusion as just described. But this replacement would be futile if it failed to preserve the characteristic properties of arithmetical divisibility. That it does so preserve them can be seen in detail, but one instance must suffice. If
m
divides
n,
and
n
divides /, then
m
divides /—for example, 12 divides 24 and 24 divides 72, and 12 does in fact divide 72. Transferred to classes, as above, this becomes: if (
m)|(n)
and
(n)
|(
l
), then (
m)
|(
l
) or, in English, if the class
(m)
contains the class
(n),
and if the class
(n)
contains the class (
l
), then the class
(m)
contains the class (
l
)—which obviously is true. The upshot is that the replacement of numbers by their corresponding classes does what is required when we add the definition of “multiplication”:
(m)
×
(n)
is defined to be the class
(mn);
(2) × (6) = (12). Notice that the last is a definition; it is not meant to follow from the meanings of
(m)
and
(n).

Dedekind's ideals for algebraic numbers are a generalization of what precedes. Following his usual custom Dedekind gave an
abstract
definition, that is, a definition based upon essential properties rather than one contingent upon some particular mode of representing, or picturing, the thing defined.

Consider the set (or class) of
all
algebraic
integers
in a given algebraic number field. In this all-inclusive set will be subsets. A subset is called an
ideal
if it has the two following properties.

A. The
sum
and
difference
of any two integers in the subset are also in the subset.

B. If any integer in the subset be multiplied by any integer in the all-inclusive set, the resulting integer is in the subset.

An ideal is thus an infinite
class
of integers. It will be seen readily
that
(m), (n),
 . . . , previously defined, are ideals according to A, B. As before, if one ideal contains another, the first is said to divide the second.

It can be proved that every ideal is a class of integers all of which are of the form

x
₁
a
₁
+ x
₂
a
₂
+ . . . + s
_n
a
_n
,

where
a
₁
, a
₂
, . . . , a
_n
are fixed
integers of the field of degree
n
concerned, and each of
x
₁
x
₂
, . . . , x
_n
may be any integer whatever in the field. This being so, it is convenient to symbolize an ideal by exhibiting only the fixed integers
a
₁
, a
₂
, . . . , a
_n
,
and this is done by writing
(a
₁
, a
₂
, . . . , a
_n
)
as the symbol of the ideal. The order in which
a
₁
a
₂
,
 . . .,
a
_n
are written in the symbol is immaterial.

“Multiplication” of ideals must now be defined:
the product
of the two ideals
(a
₁
, . . . , a
_n
), (b
₁
 . . . , b
_n
)
is the ideal whose symbol is
(a
₁
b
₁
, . . . , a
1
b
_n
, . . . , a
₁
b
_n
),
in which all possible products
a
₁
b
₁
,
etc., obtained by multiplying an integer in the first symbol by an integer in the second occur. For example, the product of
(a
₁
, a
₂
)
and
(b
₁
, b
₂
)
is
(a
₁
b
₁
, a
₁
b
₂
, a
₂
b
₁
, a
₂
b
₂
).
It is always possible to reduce any such product-symbol (for a field of degree
n)
to a symbol containing at most
n
integers.

One final short remark completes the synopsis of the story. An ideal whose symbol contains
but one
integer, such as (a
₁
), is called a
principal
ideal. Using as before the notation (a
₁
)|(b
₁
) to signify that
(a
₁
)
contains
(b
₁
), we can see without difficulty that
(a
₁
)|(b
₁
) when,
and
only when,
the integer
a
₁
divides
the integer
b
₁
. As before, then, the concept of arithmetical divisibility is here—for algebraic integers—completely equivalent to that of class inclusion.
A prime
ideal is one which is not “divisible by”—included in—any ideal except the all-inclusive ideal which consists of
all
the algebraic integers in the given field. Algebraic integers being now replaced by their corresponding principal ideals, it is proved that a given ideal is a product of prime ideals in one way only, precisely as in the “fundamental theorem of arithmetic” a rational integer is the product of primes in one way only. By the above equivalence of arithmetical divisibility for algebraic integers and class inclusion, the fundamental theorem of arithmetic has been restored to integers in algebraic number fields.