To develop the complete history of arithmetic and geometry would be a
task quite beyond the limits of this paper, and of the writer's
knowledge. In arithmetic we were able to observe a stage in which
spontaneous behavior led to the invention of number names and methods of
counting. Then, by certain speculative and "play" impulses, there arose
elementary arithmetical problems which began to be of interest in
themselves. Geometry here also comes into consideration, and, in
connection with positional number symbols, begin those interactions
between arithmetic and geometry that result in the forms of our
contemporary mathematics. The complex quantities represented by number
symbols are no longer merely the necessary results of analyzing
commercial relations or practical measurements, and geometry is no
longer directly based upon the intuitively given line, point, and plane.
If number relations are to be expressed in terms of empirical spatial
positions, it is necessary to construct many imaginary surfaces, as is
done by Riemann in his theory of functions, a construction representing
the type of imagination which Poincare has called the intuitional in
contradistinction to the logical (_Value of Science_, Ch. I). And
geometry has not only been led to the construction of many non-Euclidian
spaces, but has even, with Peano and his school, been freed from the
bonds of any necessary spatial interpretation whatsoever.
To trace in concrete detail the attainment of modern refinements of
number theory would likewise exhibit nothing new in the building up of
mathematical intelligence. We should find, here, a process carried out
without thought of the consequences, there, an analogy suggesting an
operation that might lead us beyond a difficulty that had blocked
progress; here, a play interest leading to a combination of symbols out
of which a new idea has sprung; there, a painstaking and methodical
effort to overcome a difficulty recognized from the start. It is rather
for us now to ask what it is that has been attained by these means, to
inquire finally what are those things called "number" and "line" in the
broad sense in which the terms are now used.
In so far as the cardinal number at least is concerned, the answer
generally accepted by Dedekind, Peano, Russell, and such writers is
this: the number is a "class of similar classes" (Whitehead and Russell,
_Prin. Math._, Vol. II, p. 4). To the interpretation of this answer, Mr.
Russell, the most self-consciously philosophical of these
mathematicians, has devoted his full dialectic skill. The definition has
at least the merit of being free from certain arbitrary psychologizing
that has vitiated many earlier attempts at the problem. Mr. Russell
claims for it "(1) that the formal properties which we expect cardinal
numbers to have result from it; (2) that unless we adopt this definition
or some more complicated and practically equivalent definition, it is
necessary to regard the cardinal number of a class as indefinable"
(_loc. cit._, p. 4). That the definition's terms, however, are not
without obscurity appears in Mr. Russell's struggles with the zigzag
theory, the no-class theory, etc., and finally in his taking refuge in
the theory of "logical types" (_loc. cit._, Vol. III, Part V. E.),
whereby the contradiction that subverted Frege and drove Mr. Russell
from the standpoint of the _Principles of Mathematics_ is finally
overcome.
The second of Mr. Russell's claims for his definition adds nothing to
the first, for it merely asserts that unless we adopt some definition of
the cardinal number from which its formal properties result, number is
undefined. Any such definition would be, _ipso facto_, a practical
equivalent of the first. We need only consider whether or not the
formal properties of numbers clearly follow from this definition.
Mr. Russell's own experience makes us hesitate. When he first adopted
this definition from Frege, he was led to make the inference that the
class of all possible classes might furnish a type for a greatest
cardinal number. But this led to nothing but paradox and contradiction.
The obvious conclusion was that something was wrong with the concept of
class, and the obvious way out was to deny the possibility of any such
all-inclusive class. Just why there should be such limitation, except
that it enables one to escape the contradiction, is not clear from Mr.
Russell's analysis (cf. Brown, "The Logic of Mr. Russell," _Journ. of
Phil., Psych., and Sci. Meth._, Vol. VIII, No. 4, pp. 85-89).
Furthermore, to pass to the theory of types on this ground is to give up
the value of the first claim for the definition (quoted above), since
the formal properties of numbers now merely follow from the definition
because the terms of the definition are reinterpreted from the
properties of number, so that these properties will follow from it. The
definition has become circular.
The real difficulty lies in the concept of the class. Dogmatic realism
is prone to find here an entity for which, as it is obviously not a
physical thing, a home must be provided in some region of "being." Hence
arises the realm of subsistence, as for Plato the world of facts
duplicated itself in a world of ideas. But the subsistent realm of the
mathematician is even more astounding than the ideal realm of Plato, for
the latter world is a prototype of the world of things, while the world
of the mathematician is peopled by all sorts of entities that never were
on land or sea. The transfinite numbers of Cantor have, without doubt,
a definite mathematical meaning, but they have no known representatives
in the world of things, nor in the imagination of man, and in spite of
the efforts of philosophers it may even be doubted whether an entity
correlative to the mathematical infinite has ever been or can ever be
specified.
Mr. Russell now teaches that "classes are merely symbolic" (_Sci. Meth.
in Phil._, p. 208), but this expression still needs elucidation. It
does, to be sure, avoid the earlier difficulty of admitting "new and
mysterious metaphysical entities" (_loc. cit._, p. 204), but the
"feeling of oddity" that accompanies it seems not without significance.
What can be meant by a merely symbolic class of similar classes
themselves merely symbolical? I do not know, unless it is that we are to
throw overboard the effort aimed at arbitrary and creative definition
and proceed in simple inductive and interpretative fashion. With classes
as entities abandoned, we are left, until we have passed to a new point
of view as to arithmetical entities, in the position of the intelligent
ignoramus who defined a stock market operation as buying what you can't
get with money you never had, and selling what you never owned for more
than it was ever worth.
The situation seems to be that we are now face to face with new
generalizations. Just as number symbols arose to denote operations gone
through in counting things when attention is diverted from the
particular characteristics of the things counted, and remained a symbol
for those operations with things, so now we are becoming self-conscious
of the character of the operations we have been performing and are
developing new symbols to express possible operations with operations.
The infinity of the number series expresses the fact that it is possible
to continue the enumerating process indefinitely, and when we are asked
by certain mathematicians to practise ourselves in such thoughts as that
for infinite series a proper part can be the equal of the whole, where
equality is defined through the establishment of one-one correspondence,
we are really merely informed that among the group of symbols used to
denote the concrete steps of an ever open counting process are groups of
symbols that can be used to indicate operations that are of the same
type as the given one in so far as the characteristic of being an open
series is concerned. If there were anywhere an infinity of things to
count, an unintelligible supposition, it would by no means be true that
any selection of things from that series would be the equivalent of all
things in the series, except in so far as equivalence meant that they
could be arranged in the same type of series as that from which they
were drawn.
Similarly the mathematical conception of the continuum is nothing but a
formulation of the manner in which the cuts of a line or the numbers of
a continuous series must be chosen so that there shall remain no
possible cut or number of which the choice is not indicated.
Correspondence is reached between elements of such series when the
corresponding elements can be reached by an identical process. It seems
to me, however, a mistake to _identify_ the number continuum with the
linear continuum, for the latter must include the irrational numbers,
whereas the irrational number can never represent a spatial position in
a series. For example, the sqrt{2} is by nature a decimal involving an
infinite, i.e., an ever increasing, number of digits to express it and,
by virtue of the infinity of these digits, they can never be looked upon
as all given. It is then truly a number, for it expresses a genuine
numerical operation, but it is not a position, for it cannot be a
determinate magnitude but merely a quantity approaching a determinate
magnitude as closely as one may please. That is, without its complete
expression, which would be analogous to the self-contradictory task of
finding a greatest cardinal number, there can be no cut in the line
which is symbolized by it. But the operations of translating algebraic
expressions into geometrical ones and vice versa (operations which are
so important in physical investigations) are facilitated by the notion
of a one to one correspondence between number and space.
When we pass to the transfinite numbers, we have nothing in the Alephs
but the symbols of certain groupings of operations expressible in
ordinary number series. And the many forms of numbers are all simply the
result of recognizing value in naming definite groups of operations of a
lower level, which may itself be a complication of processes indicated
by the simple numerical signs. To create such symbols is by no means
illegitimate and no paradox results in any forms as long as we remember
that our numbers are not things but are signs of operations that may be
performed directly upon things or upon other operations.
For example, let us consider such a symbol as sqrt{-5}. -5 signifies
the totality of a counting process carried on in an opposite sense from
that denoted by +5. To take the square root is to symbolize a number,
the totality of an operation, such that when the operation denoted by
multiplying it by itself is performed the result is 5. Consequently the
sqrt{-5} is merely the symbol of these processes combined in such a
way that the whole operation is to be considered as opposite in some
sense to that denoted by sqrt{5}. Hence, an easy method for the
representation of such imaginaries is based on the principle of analytic
geometry and a system of co-ordinates.
The nature of this last generalization of mathematics is well shown by
Mr. Whitehead in his monumental _Universal Algebra_. The work begins
with the definition of a calculus as "The art of manipulating
substitutive signs according to fixed rules, and the deduction therefrom
of true propositions" (_loc. cit._, p. 4). The deduction itself is
really a manipulation according to rules, and the truth consists
essentially in the results being actually derived from the premises
according to rule. Following Stout, substitutive signs are characterized
thus: "a word is an instrument for thinking about the meaning which it
expresses; a substitutive sign is a means of not thinking about the
meaning which it symbolizes." Mathematical symbols have, then, become
substitutive signs. But this is only possible because they were at an
early stage of their history expressive signs, and the laws which
connected them were derived from the relations of the things for which
they stood. First it became possible to forget the things in their
concreteness, and now they have become mere terms for the relations that
had been generalized between them. Consequently, the things forgotten
and the terms treated as mere elements of a relational complex, it is
possible to state such relational complexes with the utmost freedom. But
this does not mean that mathematics can be created in a purely arbitrary
fashion. The mark of its origin is upon it in the need of exhibiting
some existing situation through which the non-contradictory character of
its postulates can be verified. The real advantage of the generalization
is that of all generalizations in science, namely, that by looking away
from practical applications (as appears in a historical survey) results
are frequently obtained that would never have been attained if our labor
had been consciously limited merely to those problems where the
advantages of a solution were obvious. So the most fantastic forms of
mathematics, which themselves seem to bear no relation to actual
phenomena, just because the relations involved in them are the relations
that have been derived from dealing with an actual world, may contribute
to the solutions of problems in other forms of calculus, or even to the
creation of new forms of mathematics. And these new forms may stand in a
more intimate connection with aspects of the real world than the
original mathematics.
In 1836-39 there appeared in the _Gelehrte Schriften der Universitaet
Kasan_, Lobatchewsky's epoch-making "New Elements of Geometry, with a
Complete Theory of Parallels." After proving that "if a straight line
falling on two other straight lines make the alternate angles equal to
one another, the two straight lines shall be parallel to one another,"
Euclid, finding himself unable to prove that in every other case they
were not parallel, assumed it in an axiom. But it had never seemed
obvious. Lobatchewsky's system amounted merely to developing a geometry
on the basis of the contradictory axiom, that through a point outside a
line an indefinite number of lines can be drawn, no one of which shall
cut a given line in that plane. In 1832-33, similar results were
attained by Johann Bolyai in an appendix to his father's "_Tentamen
juventutem studiosam in elementa matheseosos purae ... introducendi_"
entitled "The Science of Absolute Space." In 1824 the dissertation of
Riemann, under Gauss, introduced the idea of an _n_-ply extended
magnitude, or a study of _n_-dimensional manifolds and a new road was
opened for mathematical intelligence.
At first this new knowledge suggested all sorts of metaphysical
hypotheses. If it is possible to build geometries of _n_-dimensions or
geometries in which the axiom of parallels is no longer true, why may it
not be that the space in which we make our measurements and on which we
base our mechanics is some one of these "non-Euclidian" spaces? And
indeed many experiments were conducted in search of some clue that this
might be the case. Such experiments in relation to "curved spaces"
seemed particularly alluring, but all have turned out to be fruitless in
results. Failure leads to investigation of the causes of failure. If our
space had been some one of these spaces how would it have been possible
for us to know this fact? The traditional definition of a straight line
has never been satisfactory from a physical point of view. To define it
as the shortest distance between two points is to introduce the idea of
distance, and the idea of distance itself has no meaning without the
idea of straight line, and so the definition moves in a vicious circle.
On the metaphysical side, Lotze (_Metaphysik_, p. 249) and others (Merz,
_History of European Thought in the Nineteenth Century_, Vol. II, p.
716) criticized these attempts, on the whole justly, but the best
interpretation of the situation has been given by Poincare.
Two lines of thought now lead to a recasting of our conceptions of the
fundamental notions of geometry. On the one hand, that very
investigation of postulates that had led to the discovery of the
apparently strange non-Euclidian geometries was easily continued to an
investigation of the simplest basis on which a geometry could be
founded. Then by reaction it was continued with similar methods in
dealing with algebra, and other forms of analysis, with the result that
conceptions of mathematical entities have gradually emerged that
represent a new stage of abstraction in the evolution of mathematics,
soon to be discussed as the dominating conceptions in contemporary
thought. On the other hand, there also developed the problem of the
relations of these geometrical worlds to one another, which has been
primarily significant in helping to clear up the relations of
mathematics in its "pure" and "applied" forms.
Geometry passed through a stage of abstraction like that examined in
connection with arithmetic. Beginning with the discovery of
non-Euclidian geometry, it has been becoming more and more evident that
a line need not be a name for an aspect of a physical object such as the
ridge-pole line of a house and the like, nor even for the more abstract
mechanical characteristic of direction of movement;--although the
persistency with which intuitionally minded geometers have sought to
adapt such illustrations to their needs has somewhat obscured this fact.
However, even a cursory examination of a modern treatise on geometry
makes clear what has taken place. For example, Professor Hilbert begins
his _Grundlagen der Geometrie_, not with definition of points, lines,
and planes, but with the assumption of three different systems of things
(Dinge) of which the first, called points, are denoted A, B, C, etc.,
second, called straight lines (Gerade), are denoted a, b, c, etc., and
the third, called planes, are denoted by [Greek: alpha], [Greek: beta],
[Greek: gamma], etc. The relations between these things then receive
"genaue und vollstaendige Beschreibung" through the axioms of the
geometry. And the fact that these "things" are called points, lines, and
planes is not to give to them any of the connotations ordinarily
associated with these words further than are determined by the axiom
groups that follow. Indeed, other geometers are even more explicit on
this point. Thus for Peano (_I Principii di Geometria_, 1889) the line
is a mere class of entities, the relations amongst which are no longer
concrete relations but types of relations. The plane is a class of
classes of entities, etc. And an almost unlimited number of examples,
about which the theorems of the geometry will express truths, can be
exhibited, not one of which has any close resemblance to spatial facts
in the ordinary sense.
Philosophers, it seems to me, have been slow to recognize the
significance of the step involved in this last phase of mathematical
thought. We have been so schooled in an arbitrary distinction between
relations and concepts, that while long familiar with general ideas of
concepts, we are not familiar with generalized ideas of relations. Yet
this is exactly what mathematics is everywhere presenting. A transition
has been made from relations to types of relations, so that instead of
speaking in terms of quantitative, spatial and temporal relations,
mathematicians can now talk in terms of symmetrical, asymmetrical,
transitive, intransitive relational types and the like. These present,
however, nothing but the empirical character that is common to such
relations as that of father and son; debtor and creditor; master and
servant; a is to the left of b, b of c; c of d; a is older than b, b
than c, c than d, etc. Hence this is not abandonment of experience but a
generalization of it, which results in a calculus potentially applicable
not only to it but also to other subject-matter of thought. Indeed, if
it were not for the possibility of this generalization, the almost
unlimited applicability of diagrams, so useful in the classroom, to
illustrate everything from the nature of reality to the categorical
imperative, as well as to the more technical usages of the psychological
and social sciences, would not be understandable.
It would be a paradox, however, if starting out from processes of
counting and measuring, generalizations had been attained that no longer
had significance for counting or measuring, and the non-Euclidian
hyper-dimensional geometries seem at first to present this paradox. But,
as the outcome of our second line of thought proves, this is not the
case. The investigation of the relations of different geometrical
systems to each other has shown (cf. Brown, "The Work of H. Poincare,"
_Journ. of Phil., Psy., and Sci. Meth._, Vol. XI, No. 9, p. 229) that
these different systems have a correspondence with one another so that
for any theorem stated in one of them there is a corresponding theorem
that can be stated in another. In other words, given any factual
situation that can be stated in Euclidian geometry, the aspect treated
as a straight line in the Euclidian exposition will be treated as a
curve in the non-Euclidian, and a situation treated as three-dimensional
by Euclid's methods can be treated as of any number of dimensions when
the proper fundamental element is chosen, and vice versa, although of
course the element will not be the line or plane in our empirical usage
of the term. This is what Poincare means by saying that our geometry is
a free choice, but not arbitrary (_The Value of Science_, Pt. III, Ch.
X, Sec. 3), for there are many limitations imposed by fact upon the
choice, and usually there is some clear indication of convenience as to
the system chosen, based on the fundamental ideal of simplicity.
It is evident, then, that geometry and arithmetic have been drawing
closer together, and that to-day the distinction between them is
somewhat hard to maintain. The older arithmetic had limited itself
largely to the study of the relations involved in serial orders as
suggested by counting, whereas geometry had concerned itself primarily
with the relations of groups of such series to each other when the
series, or groups of series, are represented as lines or planes. But
partly by interaction in analytic geometry, and partly in the
generalization of their own methods, both have come to recognize the
fundamental character of the relations involved in their thought, and
arithmetic, through the complex number and the algebraic unknown
quantities, has come to consider more complex serial types, while
geometry has approached the analysis of its series through interaction
with number theory. For both, the content of their entities and the
relations involved have been brought to a minimum. And this is true even
of such apparently essentially intuitional fields as projective
geometry, where entities can be substituted for directional lines and
the axioms be turned into relational postulates governing their
configurations.
Nevertheless, geometry like arithmetic, has remained true to the need
that gave it initial impulse. As in the beginning it was only a method
of dealing with a concrete situation, so in the end it is nothing but
such a method, although, as in the case of arithmetic, from ever closer
contact with the situation in question, it has been led, by refinements
that thoughtful and continual contact bring, to dissect that situation
and give heed to aspects of it which were undreamed of at the initial
moment. In a sense, then, there are no such things as mathematical
entities, as scholastic realism would conceive them. And yet,
mathematics is not dealing with unrealities, for it is everywhere
concerned with real rational types and systems where such types may be
exemplified. Or we can say in a purely practical way that mathematical
entities are constituted by their relations, but this phrase cannot here
be interpreted in the Hegelian ontological sense in which it has played
so great and so pernicious a part in contemporary philosophy. Such
metaphysical interpretation and its consequences are the basis of
paradoxical absolutisms, such as that arrived at by Professor Royce
(_World and the Individual_, Vol. II, Supplementary Essay). The peculiar
character of abstract or pure mathematics seems to be that its own
operations on a lower level constitute material which serves for the
subject-matter with which its later investigations deal. But mathematics
is, after all, not fundamentally different from the other sciences. The
concepts of all sciences alike constitute a special language peculiarly
adapted for dealing with certain experience adjustments, and the
differences in the development of the different sciences merely express
different degrees of success with which such languages have been
formulated with respect to making it possible to predict concerning not
yet realized situations. Some sciences are still seeking their terms and
fundamental concepts, others are formulating their first "grammar," and
mathematics, still inadequate, yearly gains both in vocabulary and
flexibility.
But if we are to conceive mathematical entities as mere terminal points
in a relational system, it is necessary that we should become clear as
to just what is meant by relation, and what is the connection between
relations and quantities. Modern thought has shown a strong tendency to
insist, somewhat arbitrarily, on the "internal" or "external" character
of relations. The former emphasis has been primarily associated with
idealistic ontology, and has often brought with it complex dialectic
questions as to the identity of an individual thing in passing from one
relational situation to another. The latter insistence has meant
primarily that things do not change with changing relations to other
things. It has, however, often implied the independent existence, in
some curiously metaphysical state, of relations that are not relating
anything, and is hardly less paradoxical than the older view. In the
field of physical phenomena, it seems to triumph, while the facts of
social life, on the other hand, lend some countenance to the view of the
"internalists." Like many such discussions, the best way around them is
to forget their arguments, and turn to a fresh and independent
investigation of the facts in question.