Science "Abstract" or "Selective."—Mathematical science (which is the "pure" science par excellence) deals not—as is generally supposed—with "things," but with certain selected aspects of things. For example, for purposes of arithmetic, every leaf on a tree is an "unit" (i.e. all are "identical"); but, in point of fact, there exist no two leaves that are alike, as Leibniz, long ago, pointed out. Again, for geometrical purposes two fields may be regarded as of like area; but no two fields are, or ever have been, so.
Thus mathematics—where scientific method is seen at its purest—proceeds by deliberately disregarding individuality; it regards the differences between individuals as non-essential, and irrelevant to its purpose.
Economy of Thought.—And mathematical science is justified in acting in this way. This method, highly abstract as it is—in fact, just because it is highly abstract—leads to invaluable results. It's justification is that it is economical of thought; disregarding all irrelevant considerations, it is able, by using a short-cut, to reach its goal. Did the mathematician have to take into consideration all the manifold and complex aspects of each concrete "thing" (whether it be leaf, or field, or lever, or what not) with which he deals, he would never be able to cut his way through the jungle. His method of abstraction carries him at once to his goal.
Mach on the "Mechanical View."—Mach's criticism of the mechanical view of nature proceeded upon similar lines. He termed that view "analogical," by which he meant that mechanical "laws of nature" serve us as formal patterns to which the processes of nature may (for convenience sake) be represented as conforming. A clear account, though not a complete account, of all physical processes may be given in terms of mechanical "law."
And in fact it remains a question, Mach observed, "whether the mechanical view of things, instead of being the profoundest, is not in point of fact, the shallowest of all."[52]
Science not Invalid but Incomplete.—This line of criticism of scientific method—i.e. that it deals with abstractions and analogies rather than with things, for the sake of economy and convenience of thought—does not deprive science of validity, but only invalidates that superficial dogmatism which had crept into so many investigations. A critical estimate of scientific methods makes it evident how much and how little we have the right to expect from them. They will enable us to give a simple description of phenomena as they are seen when reduced to their simplest terms of matter and motion; but of ultimate and final causes they will tell us nothing.
"The system of conceptions by which the exact sciences try to describe the phenomena of nature ... is symbolic, a kind of shorthand, unconsciously invented and perfected for the sake of convenience and for practical use ... the leading principle is that of Economy of Thought" (Merz, Vol. III, p. 579).
Boutroux.—This criticism of the mechanical method of dealing with reality was seconded by Boutroux's criticism of the principle of Natural Law. Émile Boutroux (1845-1918)—Professor at the Sorbonne—in two important treatises, examines with great minuteness this aspect of the scientific method. In the earlier of these works, The Contingency of the Laws of Nature (1879) he suggests that these laws only give, so to speak, the habits which things display. They constitute, as it were, "the bed in which the stream of occurrence flows, which the stream itself had hollowed out, although its course has come to be determined by this bed" (Höffding, Modern Philosophers, p. 101).
In his Natural Law in Science and Philosophy (1895), Boutroux lays it down that the laws of nature, as science describes them, may indeed represent, but are by no means identical with, the laws of nature as they really are. The laws of science are true, not absolutely but relatively, i.e. are not elements in, but symbols of, reality. The notion that everything is "determined" (i.e. the opposite of "contingent"), though absolutely indispensable to the mechanical theory, is nevertheless a way of looking at things rather than a faithful picture of reality—a way in which we see things rather than the way things exist in themselves.
As Boutroux himself puts it in his final chapter: "That which we call the 'laws of nature' is the sum total of the methods we have discovered for adapting things to the mind, and subjecting them to be moulded by the will."