The controlling factors for the progress of scientific thought are
inventions that lead the scientist into closer contact with his data,
and direct attention to complexities which would otherwise have escaped
observation. This end is best fulfilled by conceiving entities that
under some point of view are practically isolable from the context in
which they occur. Only too often philosophic thought has confused this
practical segregation with ontological separation, and so been obliged
to introduce metaphysical and external relations to bring these entities
together again in a real world, when in reality they have never been
separated from one another and hence not from the real world.
Furthermore, the conceptual model, built on the lines of a calculus of
mathematics, is often considered the truth _par excellence_ after the
analogy of a camera's portrait. Progress in science, however, shows that
these models have to be continually rebuilt. Each seems to lead to
further knowledge that necessitates its reconstruction, so that truth
takes on an ideal value as an ultimate but unattained, if not
unattainable, goal, while existing science becomes reduced to working
hypotheses. From a positivistic point of view, however, the goal is not
only practically unattainable, but it is irrational, for there seems to
be every evidence that it expresses something contrary to the nature of
the real. Yet scientific theory is not wholly arbitrary. We cannot
construe nature as constituted of any sorts of entities that may suit
our whim. And this is because science itself recognizes that its
entities are not really isolated, but are endowed with all sorts of
properties that serve to connect them with other entities. They are only
symbols of critical points of reality which, conceived in a certain way,
make the behavior of the whole intelligible. Indeed, the only
significant sense in which they are true for the scientist is that they
indicate real connections that might otherwise have been overlooked, and
this is only possible from the fact that reality has the characteristics
that they present and that, with their relations, they give an
approximate presentation of what is actually presented just as a
successful portrait painter considers the individuality of the eyes,
nose, mouth, etc., although he does not imply that a face is compounded
of these separate features as a house is built of boards.
The atomic theory, for example, has undoubtedly been of the greatest
service to chemistry, and atoms undoubtedly denote a significant
resting-place in the analysis of the physical world. Yet in the light of
electron theories, it is becoming more and more evident that atoms are
not ultimate particles, and are not even all alike (Becker, "Isostasy
and Radioactivity," _Sci._, Jan. 29, 1915) when they represent a single
substance. Again, while there is as yet no evidence to suggest that the
electron must itself be considered as divisible (unless it be the
distinction between the positive and negative electron), there are
suggestions that electrons may themselves arise and pass away (cf.
Moore, _Origin, and Nature of Life_, p. 39). "A wisely positivistic
mind," writes Enriques (_Problems of Science_, p. 34), "can see in the
atomic hypothesis only a subjective representation,"[34] and, we might
add, "in any other hypothesis." He continues (pp. 34-36): "robbing the
atom of the concrete attributes inherent in its image, we find ourselves
regarding it as a mere symbol. The logical value of the atomic theory
depends, then, upon the establishment of a proper correspondence between
the symbols which it contains and the reality which we are trying to
represent.
"Now, if we go back to the time when the atomic theory was accepted by
modern chemistry, we see that the plain atomic formulae contain only the
representation of the invariable relations in the combination of simple
bodies, in weight and volume; these last being taken in relation to a
well-defined gaseous state.
"But, once introduced into science, the atomic phraseology suggested the
extension of the meaning of the symbols, and the search in reality for
facts in correspondence with its more extended conception.
"The theory advances, urged on, as it were, by its metaphysical
nature, or, if you wish, by the association of ideas which the concrete
image of the atom carries with it.
"Thus for the plain formulae we have substituted, in the chemistry of
carbon compounds, structural formulae, which come to represent, thanks to
the disposition or grouping of atoms in a molecule, structural relations
of the second degree, that is to say, relations inherent in certain
chemical transformations with respect to which some groups of elements
have in some way an invariant character. And here, because the image of
a simple molecule upon a plane does not suffice to explain, for example,
the facts of isomerism, we must resort to the stereo-chemical
representation of Van't Hoff.
"Must we further recall the kinetic theory of gases, the facts explained
by the breaking up of molecules into ions, the hypothesis suggested, for
example, by Van der Waals by the view that an atom has an actual bulk?
Must we point to a physical phenomenon of quite a different class, for
example, to the coloring of the thin film forming the soap-bubbles which
W. Thomson has taken as the measure of the size of a molecule?
"Such a resume of results shows plainly that we cannot help the progress
of science by blocking the path of theory and looking only at its
positive aspects, that is to say, at the collection of facts that it
explains. The value of a theory lies rather in the hypothesis which it
can suggest, by means of the psychological representation of the
symbols.
"We shall not draw from all this the conclusion that the atomic
hypothesis ought to correspond to the extremely subtle sensations of a
being resembling a perfected man. We shall not even reason about the
possibility of those imaginary sensations, in so far as they are
conceived simply as an extension of our own. But we shall repeat, in
regard to the atomic theory, what an illustrious master is said to have
remarked as to the unity of matter: if on first examination a fact seems
possible which contradicts the atomic view of things, there is a strong
probability that such a fact will be disproved by experience.
"Does not such a capacity for adaptation to facts, thus furnishing a
model for them, perhaps denote the _positive_ reality of a theory?"
And the above principles are as true of mathematical concepts as of
chemical. Everywhere it is "capacity of adaptation to facts" that is the
criterion of a branch of mathematics, except, of course, that in
mathematics the facts are not always physical facts. Mathematics has
successfully accomplished a generalization whereby its own methods
furnish the material for higher generalizations. The imaginary number
and the hyper-dimensional or non-Euclidian geometries may be absurd if
measured by the standard of physical reality, but they nevertheless have
something real about them in relation to certain mathematical processes
on a lower level. There is no philosophic paradox about modern
arithmetic or geometry, once it is recognized that they are merely
abstractions of genuine features of simpler and more obviously practical
manipulations that are clearly derived from the dealing of a human
being with genuine realities.
In the light of these considerations, I cannot help feeling that the
frequent attempts of mathematicians with a philosophical turn of mind,
and philosophers who are dipping into mathematics, to derive geometrical
entities from psychological considerations are quite mistaken, and are
but another example of those traditional presuppositions of psychology
which, Professor Dewey has pointed out (_Jour. of Phil., Psy., and Sci.
Meth._, XI, No. 19, p. 508), were "bequeathed by seventeenth-century
philosophy to psychology, instead of originating within psychology" ...
that "were wished upon it by philosophy when it was as yet too immature
to defend itself."
Henri Poincare (_Science and Hypothesis_, Ch. IV, _The Value of
Science_, Ch. IV) and Enriques (_Problems of Science_, Ch. IV, esp.
B--_The Psychological Acquisition of Geometrical Concepts_) furnish two
of the most familiar examples of this sort of philosophizing. Each
isolates special senses, sight, touch, or motion, and tries to show how
a being merely equipped with one or the other of these senses might
arrive at geometrical conceptions which differ, of course, from space as
represented by our familiar Euclidian geometry. Then comes the question
of fusing these different sorts of experience into a single experience
of which geometry may be an intelligible transcription. Enriques finds a
parallel between the historical development and the psycho-genetic
development of the postulates of geometry (_loc. cit._, p. 214 _seq._).
"The three groups of ideas that are connected with the concepts that
serve as the basis for the theory of continuum (_Analysis situs_), of
metrical, and of projective geometry, may be connected, as to their
psychological origin, with three groups of sensations: with the general
tactile-muscular sensations, with those of special touch, and of sight,
respectively." Poincare even evokes ancestral experience to make good
his case (_Sci. and Hyp._, Ch. V, end). "It has often been said that if
individual experience could not create geometry, the same is not true of
ancestral experience. But what does that mean? Is it meant that we could
not experimentally demonstrate Euclid's postulate, but that our
ancestors have been able to do it? Not in the least. It is meant that by
natural selection our mind has _adapted_ itself to the conditions of the
external world, that it has adopted the geometry _most advantageous_ to
the species: or in other words, the _most convenient_."
Now undoubtedly there may be a certain modicum of truth in these
statements. As implied by the last quotation from Poincare, the modern
scientist can hardly doubt that the fact of the adaptation of our
thinking to the world we live in is due to the fact that it is in that
world that we evolved. As is implied by both writers, if one could limit
human contact with the world to a particular form of sense response,
thought about that world would take place in different terms from what
it now does and would presumably be less efficient. But these admissions
do not imply that any light is thrown upon the nature of mathematical
entities by such abstractions. Russell (_Scientific Method in
Philosophy_) is in the curious position of raising arithmetic to a
purely logical status, but playing with geometry and sensation after the
manner of Poincare, to whom he gives somewhat grudging praise on this
account.
The psychological methods upon which all such investigations are based
are open to all sorts of criticisms. Chiefly, the conceptions on which
they are based, even if correct, are only abstractions. There is not the
least evidence for the existence of organisms with a single
differentiated sense organ, nor the least evidence that there ever was
such an organism. Indeed, according to modern accounts of the evolution
of the nervous system (cf. G. H. Parker, _Pop. Sci. Month._, Feb., 1914)
different senses have arisen through a gradual differentiation of a more
general form of stimulus receptor, and consequently, the possibility of
the detachment of special senses is the latter end of the series and not
the first. But, however this may be, the mathematical concepts that we
are studying have only been grasped by a highly developed organism, man,
but they had already begun to be grasped by him in an early stage of his
career before he had analyzed his experience and connected it with
specific sense organs. It may of course be a pleasant exercise, if one
likes that sort of thing, to assume with most psychologists certain
elementary sensations, and then examine the amount of information each
can give in the light of possible mathematical interpretations, but to
do so is not to show that a being so scantily endowed would ever have
acquired a geometry of the type in question, or any geometry at all.
Inferences of the sort are in the same category with those from
hypothetical children, that used to justify all theories of the
pedagogue and psychologist, or from the economic man, that still, I
fear, play too great a part in the world of social science.