THIRD PAPER
In any classification of our intellectual domain which it is possible to make on the basis of Principles now known to the Scientific world at large, the most fundamental characteristic should be, the distinctive separation of those departments of thought in which Certainty is now attainable, from those in which only varying degrees of Probability exist, and the clear exhibition of that which is positive and demonstrable knowledge, in the strict sense of the term, as distinguished from that which is liable to be more or less fallible. Although the precise point at which, in some cases, the proofs of Probable Reasoning cease to be as convincing as those of Demonstration cannot be readily apprehended, yet the essential nature of the two methods of proof is radically and inherently different, and is marked by the most distinctive results. In the latter case, we have always accuracy, precision, and certainty, beyond the possibility of doubt; in the former, always the conviction that, how strong soever the array of evidence may seem to be, in favor of a particular inference, there still remains a possibility that the conclusion may be modified or vitiated by the subsequent advancement of knowledge.
The Generalizations which respectively affirm that all the angles of a triangle are equal to two right angles, or that the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides, rest upon an entirely different basis of proof from those upon which the Generalizations rest which respectively assert that water is composed of certain chemical constituents combined in certain proportions, or that the nerves are the instruments of sensation and of motion. The former are irresistible conclusions of the human mind, because, from the nature of the intellect, they cannot be conceived of as being otherwise. The Laws of Thought are such, that we are unable to think a triangle whose angles will not be equal to two right angles, or a right-angled one, the square of whose hypothenuse will not be equal to the squares of the other two sides. So long, therefore, as man is constituted as he now is—unless the human organization becomes radically changed, these geometrical Laws cannot be conceived as being otherwise than as they are. All men must apprehend them alike if they apprehend them at all. So long as man lives and thinks they remain unalterable verities, about which there can be no shadow of doubt, no possibility of error.
The doctrine that water is composed of certain definite chemical constituents in certain definite proportions, or the theory that the nerves are the instruments of sensation and of motion, rests upon no such foundation. Whenever water has been analyzed, it has yielded the same separate elements in the same proportions; and whenever these elements are put together in the same quantitative ratio they have produced water; so that the conviction is proximately established in the minds of all that water is invariably the product of these elements in certain proportions. But this proof does not establish the generalization as inevitably true, nor show that it is impossible for it to be otherwise. It is possible, in the nature of things, for us to conceive that the fluid which we call water may be produced from other constituents than oxygen or hydrogen, or that such a fluid may even now exist undiscovered, the product of elements altogether unknown.
So in regard to the nerves. Observation and experiment have established to the general satisfaction, that they are the instruments of sensation and motion; but we are not absolutely sure that this is the fact, nor can we know that a human being may not be born in whom no trace of nerves can be detected, and who will nevertheless experience sensation and exhibit motion. We may be as well satisfied, for all practical purposes, of the nature of water and of the office of the nerves as of the nature of a triangle; but the character of the evidence, on which the convincement is based, is essentially different; being, in the one case, incontrovertible and infallible; and, in the other, indecisive and possibly fallacious.
This repetition of that which has been substantially stated before, brings us to the final consideration of the distinctive nature of different departments of Thought, as indicated by the Methods of Proof which respectively prevail in them; and hence as embodying either exact and definite Knowledge, or only varying degrees of Probability. We have already seen that in at least one sphere of intellectual activity we are able to start from the most basic and fundamental conceptions, from axiomatic truths so patent and universal that they cannot even be conceived of as being otherwise than as they are, and to proceed from them, by equally irresistible Inferences, to conclusions which are, from the nature of the human mind, inevitable. It is in the Mathematics, in which the Deductive Method is rightly operative, that this kind of Proof—Demonstration in the strict sense of the term—prevails. The various branches of Mathematics have therefore been appropriately denominated the Exact Sciences, in contradistinction from those domains of Thought whose Laws or Principles are liable to be somewhat indefinite or uncertain; hence, called the Inexact Sciences.
Exact Science—in its largest sense, that which extends to all domains in which the proper Deductive Method has been or may hereafter be rightly employed—is therefore a system or series of truths relating to the whole Universe, or to some department of it, consecutively and necessarily resulting from, and dependent upon, each other, in a definite chain or series; and resting primarily upon some fundamental truth or truths so simple and self-evident, that, when clearly stated, all men must, by the natural constitution of the human mind, perceive them and recognize them as true. Demonstration is the pointing out of the definite links in the chain or series by which we go from fundamental truths, clearly perceived and irresistible, up to the particular truth in question.
Thus far in the history of Science, Mathematics, as a whole, has ranked as the only Exact Science; being the only department of intellectual activity, all of whose Laws or Principles are established on a basis of undeniable certainty. If, however, theories of Cosmogony and considerations of Cosmography be excluded from the field of Astronomy, this Science consists almost wholly of the application of the Laws of Mathematics to the movements of the celestial bodies. Restricting Astronomy proper to this domain, where, as a Science, it strictly belongs, and setting aside its merely descriptive and conjectural features, as hardly an integral part of the Science itself, we have another Exact Science in addition to Mathematics.
Of still another domain, that of Physics, Professor Silliman says, 'all its phenomena are dependent on a limited number of general laws ... which may be represented by numbers and algebraic symbols; and these condensed formulæ enable us to conduct investigations with the certainty and precision of pure Mathematics.'
The various branches of Physics have not hitherto been ranked as Exact Sciences, because, as in Astronomy, unsubstantiated theories and doubtful generalizations, incapable of Mathematical Proof, have mingled with their Demonstrated Laws and Phenomena, as a component part of the Science itself. It has consequently exhibited an ambiguous or problematical aspect, incompatible with the rigorous requirements of Exact Science. Even in Professor Silliman's admirable work, formulæ are given as Laws, which, however correct, have yet no foundation in axiomatic truth; while Inferences are drawn from them which are by no means capable of Demonstration. Strictly speaking, however, only those Laws which do rest upon a Demonstrable basis and the Phenomena derived from them come within the scope of the Science of Physics. So far as these prevail, this department of investigation is entitled to the Mathematical character accorded to it by Professor Silliman, and ranks as an Exact Science.