mentally supply, he soon passed from this noumenalism to a
“universal physical phenomenology.” It retains some relics of
Fechner’s influence; first, the theory of identity, according to
which the difference between the physical and psychical is not a
dualism, but everything is at once both; and secondly, the substitution
of mathematical dependence for physical causality, except
that, whereas Fechner only denied causality between physical
and psychical, Mach rejects the entire distinction between
causality and dependence, on the ground that “the law of
causality simply asserts that the phenomena of Nature are
dependent on one another.” He comes near to Hume’s substitution
of succession of phenomena for real causality. He holds,
like Hume, that nothing is real except our sensations and complexes
of sensory elements; that the ego is not a definite, unalterable,
sharply bounded unity, but its continuity alone is important;
and that we know no real causes at all, much less real causes of
our sensations; or, as he expresses it, bodies do not produce
sensations, but complexes of sensations form bodies. If he has
any originality, it consists in substituting for the association of
ideas the “economy of thinking,” by which he means that all
theoretical conceptions of physics, such as atoms, molecules,
energy, &c., are mere helps to facilitate our consideration of
things. But he limits this power of mind beyond sensations
to mere ideas, and like Hume, and also like Lange, holds at last
that, though we may form ideas beyond sensations or phenomena,
we cannot know things. If we ask how Mach arrived at
this scepticism, which is contained in his well-known scientific
work Die Mechanik in ihrer Entwickelung (1883; ed. 1908) as
well as in his psychological work on the Analysis of Sensations
(Beiträge zur Analyse der Empfindungen, 1886), we find two main
causes, both psychological and epistemological; namely, his
views on sense and on inference. In the first place, he displays
in its most naked form the common but unproved idealistic
paradox of a sense of sensations, according to which touch
apprehends not pressure but a sensation of pressure, sight
apprehends not colour but a sensation of colour, and there is no
difference between the sensory operation and the sensible
object apprehended by any sense, even within the sentient
organism. Hence, according to him, sensations are not apprehensions
of sensible objects (e.g. pressures felt) from which we
infer similar objects beyond sense (e.g. similar pressures of outside
things), but are the actual elements out of which everything
known is made; as if sensations were like chemical elements.
Within the limits of these supposed sensory elements he accords
more than many psychologists do to sense; because, following
the nativists, Johannes Müller and Hering, he includes sensations
of time and space, which, however, are not to be regarded as
“pure intuitions” in the style of Kant. But here again he
identifies time and space with the sensations of them (Zeitempfindungen
and Raumempfindungen). On the assumption,
then, that time and space are not objects, but systems, of sensations,
he concludes that a body in time and space is “a relatively
constant sum of touch-and-light-sensations, joined to the same
time-and-space-sensations,” that each man’s own body is included
in his sensations, and that to explain sensations by motions
would only be to explain one set of sensations from another. In
short, sensations are elements and bodies complexes of these
elements. Secondly, his theory of inference contains the admission
that we infer beyond sensations: he remarks that the space
of the geometer is beyond space-sensations, and the time of the
physicist does not coincide with time-sensations, because it
uses measurements such as the rotation of the earth and the
vibrations of the pendulum. But by inference beyond sense he
does not mean a process of concluding from sensible things to
similar things, e.g. from tangible pressures to other similar
pressures in the external world. Inference, according to him,
is merely mental completion of sensations; and this mental
completion has two characteristics: it only forms ideas, and it
proceeds by an “economy of thought.” In the course of his
learned studies on the history of mechanics he became deeply
impressed with Galileo’s appeals to simplicity as a test of truth,
and converted what is at best only one characteristic of thinking
into its essence. According to him, whatever inferences we
make, certain or uncertain, are mere economies of thought,
adapting ideas to sensations, and filling out the gaps of experience
by ideas; whatever we infer, whether bodies, or molecules, or
atoms, or space of more than three dimensions, are all without
distinction equally provisional conceptions, things of thought;
and “bodies or things are compendious mental symbols for
groups of sensations symbols which do not exist outside
thought.” Moreover, he applies the same scepticism to cause
and effect. “In Nature,” says he, “there is no cause and no
effect.” He thinks that repetitions of similar conjunctions
occur in Nature, the connexion of cause and effect only in abstraction.
He refers to Hume as recognizing no causality but only
a customary and habitual succession, but adds that Kant rightly
recognizes that mere observation cannot teach the necessity of
the conjunction. But in reality his theory is neither Hume’s
theory of association nor Kant’s of an a priori notion of understanding
under which a given case is subsumed. He thinks that
there is a notion of understanding (Verstandesbegriff), under
which every new experience is subsumed, but that it has been
developed by former experience, instinctively, and by the
development of the race, as part of the economy of thinking.
“Cause and effect are therefore,” he concludes, “thought-things
of economical function (Gedankendinge von ökonomischer
Function).” His philosophy, therefore, is that all known things
are sensations and complexes of sensory elements, supplemented
by an economy of thinking which cannot carry us beyond ideas
to real things, or beyond relations of dependency to real causes.
It is important to understand that Mach had developed this economical view of thought in 1872, more than ten years before the appearance of his work on the history of mechanics as he tells us in the preface, where he adds that at a later date similar views were expressed by Kirchhoff in his Vorlesungen über mathematische Physik (1874). Kirchhoff asserted that Kirchhoff. the whole object of mechanics is “to describe the motions occurring in Nature completely in the simplest manner.” This view involves the denial of force as a cause, and the assertion that all we know about force is that the acceleration of one mass depends on that of another, as in mathematics a function depends on a variable; and that even Newton’s third law of motion is merely a description of the fact that two material points determine in one another, without reciprocally causing, opposite accelerations. It is evident that Kirchhoff’s descriptive is the same as Mach’s economical view. “When I say,” says Mach, “that a body A exerts a force on a body B, I mean that B, on coming into contraposition with A, is immediately affected by a certain acceleration with respect to A.” In a word, Mach and Kirchhoff agree that force is not a cause, convert Newtonian reciprocal action into mere interdependency, and, in old terminology, reduce mechanics from a natural philosophy of causes to a natural history of mere facts. Now, Mach applies these preconceived opinions to “mechanics in its development,” with the result that, though he shows much skill in mathematical mechanics, he misrepresents its development precisely at the critical point of the discovery of Newton’s third law of motion.
The true order of discovery, however, was as follows:—
(a) Sir Christopher Wren made many experiments before the Royal Society, which were afterwards repeated in a corrected form by Sir Isaac Newton in the Principia, experimentally proving that bodies of ascertained comparative weights, when suspended and impelled against one another, forced one another back by impressing on one another opposite changes of velocity inversely as their weights and therefore masses; that is, by impressing on one another equal and opposite changes of momentum.
(b) Wallis showed that such bodies reduce one another to a joint mass with a common velocity equal to their joint momentum divided by their joint weights or masses. This result is easily deducible also from Wren’s discovery. If m and m′ are the masses, u and v ′ their initial velocities, and V the common velocity, then m(v−V)＝m′(V−v ′), therefore mv + m′v ′＝(m + m′)V, and hence (mv + m′v ′)/(m + m′)＝V.
(c) Wren and Huygens further proved that the law of equal action and reaction, already experimentally established by the former, is deducible from the conservation of the velocity of the common centre of gravity, which is the same as the common velocity of the
bodies, that is, deducible from the fact that their common centre