Men of Mathematics (68 page)

An honor which pleased him more than any he had ever received was the last, as he lay on his deathbed: he was elected the first foreign member of the National Academy of Sciences of the United States, which was founded during the Civil War. This honor was in recognition of his work in quaternions, principally, which for some unfathomable reason stirred American mathematicians of the time (there were only one or two in existence, Benjamin Peirce of Harvard being the chief) more profoundly than had any other British mathematics since Newton's
Principia.
The early popularity of quaternions in the United States is somewhat of a mystery. Possibly the turgid eloquence of the
Lectures on Quaternions
captivated the taste of a young and vigorous nation which had yet to outgrow its morbid addiction to senatorial oratory and Fourth of July verbal fireworks.

*  *  *

Quaternions has too long a history for the whole story to be told here. Even Gauss with his anticipation of
1817
was not the first in the field; Euler preceded him with an isolated result which is most simply interpreted in terms of quaternions. The origin of quaternions may go back even farther than this, for Augustus de Morgan once half-jokingly offered to trace their history for Hamilton from the ancient Hindus to Queen Victoria. However, we need glance here only at the lion's share in the invention and consider briefly what inspired Hamilton.

The British school of algebraists, as will be seen in the chapter on Boole, put common algebra on its own feet during the first half of the nineteenth century. Anticipating the currently accepted procedure in developing any branch of mathematics carefully and rigorously they founded algebra
postulationally.
Before this, the various kinds of “numbers”—fractions, negatives, irrationals—which enter mathematics when it is
assumed
that all algebraic equations have roots, had been allowed to function on precisely the same footing as the common positive integers which were so staled by custom that all mathematicians believed them to be “natural” and in some vague sense completely understood—they are not, even today, as will be seen when the work of Georg Cantor is discussed. This naîve faith in the self-consistency of a system founded on the blind, formal juggling of
mathematical symbols may have been sublime but it was also slightly idiotic. The climax of this credulity was reached in the notorious
principle of permanence of form,
which stated in effect that a set of rules which yield consistent results for one kind of numbers—say the positive integers—will continue to yield consistency when applied to any other kind—say the imaginaries—even when no interpretation of the results is evident. It does not seem surprising that this faith in the integrity of meaningless symbols frequently led to absurdity.

The British school changed all this, although they were unable to take the final step and
prove
that their postulates for common algebra will never lead to a contradiction. That step was taken only in our own generation by the German workers in the foundations of mathematics. In this connection it must be kept in mind that algebra deals only with
finite
processes; when
infinite
processes enter, as for example in summing an infinite series, we are thrust out of algebra into another domain. This is emphasized because the usual elementary text labelled “Algebra” contains a great deal—infinite geometric progressions, for instance—that is
not
algebra in the modern meaning of the word.

The nature of what Hamilton did in his creation of quaternions will show up more clearly against the background of a set of postulates (taken from L. E. Dickson's
Algebras and Their Arithmetics,
Chicago,
1923)
for common algebra or, as it is technically called, a
field
(English writers sometimes use
corpus
as the equivalent of the German
Körper
or French
corps).

“A field
F
is a system consisting of a set
S
of elements
a, b, c, . . .
and two operations, called addition and multiplication which may be performed upon any two (equal or distinct) elements
a
and
b
of
S,
taken in that order, to produce uniquely determined elements
a
⊕
b
and
a
⊙
b
of
S,
such that postulates I-V are satisfied. For simplicity we shall write
a + b
for
a
⊕
b,
and
ab
for
a
⊙
b,
and call them the
sum
and
product,
respectively, of
a
and
b.
Moreover, elements of
S
will be called elements of
F.

“I. If
a
and
b
are any two elements of
F, a + b
and
ab
are uniquely determined elements of
F,
and

b + a = a + b, ba = ab.

“II. If
a, b, c
are any three elements of
F,

(a + b)
+
c = a + (b + c), (ab)c = a(bc), a(b + c) = ab + ac.

“III. There exist in
F
two distinct elements, denoted by 0, 1, such that if
a
is any element of
F
,
a
+ 0 =
a
,
a
1 =
a
(whence 0 +
a
=
a
, 1
a
=
a
, by I).

“IV. Whatever be the element
a
of F, there exists in Fan element
x
such that
a
+
x
= 0 (whence
x
+ a =
0 by I).

“V. Whatever be the element
a
(distinct from 0) of
F
, there exists in
F
an element
y
such that
ay
= 1 (whence =
ya
, by I).”

From these simple postulates the whole of common algebra follows. A word or two about some of the statements may be helpful to those who have not seen algebra for years. In II, the statement
(a + b)
+
c
=
a + (b + c),
called the
associative law of addition,
says that if
a
and
b
are added, and to this sum is added
c,
the result is the same as if
a
and the sum of
b
and
c
are added. Similarly, with respect to multiplication, for the second statement in II. The third statement in II is called the
distributive law.
In III a “zero” and “unity” are postulated; in IV, the postulated
x
gives the negative of
a;
and the first parenthetical remark in V forbids “division by zero.” The demands in Postulate I are called the
commutative laws of addition and multiplication
respectively.

Such a set of postulates may be regarded as a distillation of experience. Centuries of working with numbers and getting useful results according to the rules of arithmetic—empirically arrived at—suggested most of the rules embodied in these precise postulates, but once the suggestions of experience are understood, the
interpretation
(here common arithmetic) furnished by experience is deliberately suppressed or forgotten, and the
system
defined by the postulates is developed
abstractly,
on its own merits, by common logic plus mathematical tact.

Notice in particular IV, which
postulates the existence
of negatives. We do not attempt to
deduce
the existence of negatives from the behavior of positives. When negative numbers first appeared in experience, as in debits instead of credits, they,
as numbers,
were held in the same abhorrence as “unnatural” monstrosities as were later the “imaginary” numbers
etc., arising from the
formal
solution of equations such as
x
²
+ 1 = 0, x
²
+ 2 = 0, etc. If the reader will glance back at what Gauss did for complex numbers he will appreciate more fully the complete simplicity of the following partial statement of Hamilton's original way of stripping “imaginaries” of their silly, purely imaginary mystery. This simple thing was
one of the steps which led Hamilton to his quaternions, although strictly it has nothing to do with them. It is the
method
and the
point of view
behind this ingenious recasting of the algebra of complex numbers which are of importance for the sequel.

If as usual
i
denotes
, a “complex number” is a number of the type
a + bi,
where
a, b
are “real numbers” or, if preferred, and more generally, elements of the field
F
defined by the above postulates. Instead of regarding
a
+
bi
as one “number,” Hamilton conceived it as an
ordered couple
of “numbers,” and he designated this couple by writing it
(a, b).
He then proceeded to impose definitions of
sum and product
on these couples, as suggested by the
formal
rules of combination sublimated from the experience of algebraists in manipulating complex numbers
as if
the laws of common algebra did in fact hold for them. One advantage of this new way of approaching complex numbers was this: the definitions for sum and product of couples were seen to be
instances
of the general, abstract definitions of sum and product as in a field. Hence, if the consistency of the system defined by the postulates for a field is proved, the like follows, without further proof, for complex numbers and the usual rules by which they are combined. It will be sufficient to state the definitions of sum and product in Hamilton's theory of complex numbers considered as couples
(a, b) (c, d),
etc.

The
sum
of
(a, b)
and
(c, d)
is
(a + b, c
+
d);
their
product
is
(ac – bd, ad + be).
In the last, the minus sign is as in a field; namely, the element
x
postulated in IV is denoted by –
a.
To the 0,
1
of a field correspond here the couples (0, 0), (1, 0). With these definitions it is easily verified that Hamilton's couples satisfy all the stated postulates for a field. But they also accord with the
formal
rules for manipulating complex numbers. Thus, to
(a, b), (c, d)
-correspond respectively
a
+
bi, c
+
di,
and the formal “sum” of these two is
(a + c)
+
i(b + d),
to which corresponds the couple
(a + c, b + d).
Again, formal multiplication of
a
+
bi, c + id
gives
(ac – bd) + i(ad + be),
to which corresponds the couple
(ac – bd, ad + be).
If this sort of thing is new to any reader, it will repay a second inspection, as it is an example of the way in which modern mathematics eliminates mystery. So long as there is a shred of mystery attached to any concept that concept is not mathematical.

Having disposed of complex numbers by
couples,
Hamilton sought to extend his device to ordered
triples
and
quadruples.
Without some
idea of what is sought to be accomplished such an undertaking is of course so vague as to be meaningless. Hamilton's object was to invent an algebra which would do for rotations in space of
three
dimensions what complex numbers, or his couples, do for rotations in space of
two
dimensions, both spaces being Euclidean as in elementary geometry. Now, a complex number
a + bi
can be thought of as representing a
vector,
that is, a line segment having both
length and direction,
as is evident from the diagram, in which the directed segment (indicated by the arrow) represents the vector
OP.