A distinguished thinker, Stephen Pearl Andrews, from whose writings some of these suggestions concerning Methods have been borrowed, points out three sources of confusion in the minds even of the learned themselves, in connection with this subject. First, in the verbal point of view, the terms Induction and Deduction are applied in a way directly the opposite of that which their Etymology would indicate: In-duction is used for the De-rivation of a Law from Facts, and De-duction for the Intro-duction of new Facts under the Law. Secondly, the two terms Inductive and Deductive, which are alone usually spoken of, are not enough to designate all the processes involved in the several Scientific Methods; and, thirdly, these terms are sometimes used to denote Processes merely, and sometimes to designate Methods which are merely characterized by the predominance of one or the other of these Processes.
This intricate subject of Methods may be better understood after a statement of the following considerations. Induction, as a Process, occurs whenever Facts are used as an instrument by which to discover a Principle or Law of Nature. The Principle is derived from, or, as Scientists have chosen to conceive it, induced upon the Facts. Deduction, as a Process, occurs whenever a Principle or Law of Nature is used as an instrument by which to discover Facts. The new Facts are ranged under, or, as it is conceived, deduced from the Principle.
Each, of these Processes occurs in every Scientific Method; but different Methods are characterised by that one of these two Processes which is put first or takes the lead in the given Method. Thus, in both Methods which are included in the one generally called the Deductive, the main Process was Deduction, there being no perceptible Induction from Collected Facts in the proper Hypothetical or Anticipative Method, while in the true Deductive Method, as applied to Mathematics, the Inductive stage is so short and so slight that it is performed instinctively by all people and the Deductive stage at once reached. The other branch of the Hypothetical Method, that used by Aristotle and the Greek Scientists, was, as we have seen, in reality a first and imperfect attempt to use the Inductive Method. In this Method itself, on the other hand, the main Process is the Induction or derivation of a Principle or Law from accumulated Facts, while Deduction, or the bringing in of new Facts under the Law, is a subordinate or Secondary Process.
In reality, there is but one Method, having several stages or Processes, which Processes, preponderating at different epochs, have not been clearly apprehended as necessary complements of each other, and have, hence, been regarded as different Methods. In one phase of the Anticipative or Hypothetical stage,—the assumption of basic Principles as points to reason from,—the Observation and Collection of Facts, and the Induction therefrom, were processes so imperfectly performed, that they appeared to have no existence; in another phase, that employed by Aristotle, these Processes were apparent, but still imperfectly conducted, and hence, in both cases, the Law or Principle employed for the Deductive Process was liable to be defective, and therefore insufficient as a guide to the acquisition of certain knowledge. In the Inductive stage or Method, on the other hand, the Processes thus defectively employed in the former stage, the Hypothetical, are preëminently and disproportionately active, while the Deductive Process is given a very inferior position. The establishment of the just, reciprocal activity of these two Processes in intellectual investigation would secure the perfection of the one true Scientific Method.
The Inductive Method—preserving the term Method to avoid confusion—in which the mode of procedure from Facts to Principles predominates, and which is hence sometimes called the Empirical, or the Experimental, or the Positive, or the à posteriori Method, is that which now prevails in the world, which is extolled as if it were the only legitimate Method, and the only possible route to Scientific Discovery. That the just claims of the Inductive Method are very great is universally admitted, but let us not stultify ourselves by assuming a position in its defence which is in direct violation of the teachings of the Method itself,—namely, the assumption of a theory which is not verified by Facts. That the Inductive Method is vastly superior to the Anticipative or Hypothetical one, is abundantly proved; but that it is the only correct path to Scientific truth, that it is the best path to Scientific truth which will ever be known, or that in a rightly balanced Method it would be the main Process, is an averment for which there is no warrant. On the contrary, a very cursory examination of the Inductive Method will show defects which render it unavailable as the sole or the chief guide in Scientific inquiry.
The leading characteristic of the Inductive Method, that for which it is mainly admired, is its cautious, laborious, oftentimes tedious Observation and Collection of the Facts of Experience, and their careful Classification as a basis for the derivation of a Principle or Law applicable to the Phenomena grouped together. By this means, it is said, we secure precision and certainty, by which is intended, not only the certainty of that which is already observed and classified, but also the certainty of that which is deduced from the Law or Principle derived from these known Facts. It is just here, however, that the Inductive Method is lacking. Experience may testify a thousand, ten thousand, any indefinite number of times, to the repetition of the same Phenomena, and yet we can have no certainty of the recurrence of the same Phenomena, in the future, in the same way. All the Facts of Observation and Experience for thousands of years went to convince men that the earth was at rest and the sun and stars passing around it. A larger Experience showed them their error. How shall we know that our Observation has at any time included all the Facts necessary to establish a Law? The history of Science, even under the guidance of the Inductive Method, is a history of Principles announced as firmly established, which a little later were found to be defective and had to be adjusted to the advanced stage of human Experience. The very nature of the Inductive Method indicates its inadequacy for the largest and most important purposes of Science. It gives certainty, where it does give it, only up to the point of the present, it can never afford complete certainty for the future. The logical and rigid testimony of this Method can never be more than this;—Observation and Experience show that such has been the uniform operation of Nature in this particular so far as can be discovered, and in all probability it will always continue to be such. High Probability, amounting, it may be, at times, to an assurance of certainty, is the strongest proof which this Method can, from its very nature, produce. To establish a Principle or Law for a certainty beyond any possibility of doubt by the Inductive Method, it is essential that we should know that we are in possession of every Fact in the universe which has any relation to the given Principle, or rather that we should know that there are no Facts in the universe at variance with it. To know this, it is necessary to be in possession of all the Facts in the universe, since the Inductive Method has no mode of discovering when it has sifted out of the immense mass of Facts all those which exist in connection with any given Law. As we shall never be in possession of all the Facts of the universe, we shall never be able, by the Inductive Method, to possess certainty in respect to the future operations of Nature; and thus we discover the insufficiency of this Method as a perfect guide to the acquisition of knowledge.
The famed Inductive Method, like the Anticipative or Hypothetical, furnishes, in truth, only an assumption as a starting point for reasoning in the endeavor to establish other Facts than those already known. The verification of the Law or Principle assumed is, indeed, in the former Method, as complete as it can be, in the nature of the case, while in the latter it is not; but we have just seen that the strongest proof which Observation, Classification, and Induction can give is that of High Probability, on the supposition that a certain number of Facts from which a Law is derived include substantially all that the whole range of Phenomena belonging to the given sphere would represent. Any possible application of the Inductive Method is, therefore, only a nearer or more remote approximation to an Exactitude and Certainty which the Method itself can never fully attain.
The Inductive Method being thus defective as a Scientific guide, in the most important requirement of Science, it is unnecessary to enter into an exposition of minor defects, not the least of which is the slowness with which conclusions must necessarily be arrived at, when they are reached only by the gradual accumulation of Facts and the derivation of a Law from these. A Method or a Process which lacks that which is the very essence of Science—the power of making known, of introducing certainty into investigation, may be an important factor in the true Scientific Method, but cannot constitute the Method itself, or its leading feature. Let it not be understood, however, that in bringing the Inductive Method in Science to the ordeal of a critical examination, it is designed to detract from its just dues or to depreciate its true value. Science is preëminently severe in its probings; and that which, asserts its claim to the highest Scientific position, and affects to be the only guide to exact knowledge, cannot expect anything less than the most rigorous inquiry into the validity of such claim, and the most peremptory insistence upon the production of proper credentials before so lofty a seat be accorded it. If inquiry discovers deficiencies in its character, Science should rejoice that truth is vindicated, and that, by correctly understanding the nature and powers of their present guide, Scientific men may avoid being tempted to consider it as competent to conduct them into regions where the blind must inevitably be leading the blind, and both be in danger of the ditch. If the devotees of the Inductive Method have in their enthusiasm set up claims for it which cannot be substantiated, they must not blame the rigorous hand, which, in the service of Science, unmasks their idol and exhibits its defects, but rather impute to their own deviation from the severity of Scientific truth, the disappointment which they may experience. The question of Method lies at the foundation of all Science. Until it is thoroughly understood, until the exact character of all our Methods or Processes is definitely and rightly apprehended, there can be no full understanding of the true nature of Science, and, hence, no critical and exact line drawn between that which is Science and that which is not.
Our examination of the Methods in use thus far in our past search after knowledge has developed these facts:—that prior to an era which is commonly said to commence with Bacon, the Method of intellectual investigation was mainly by attempting to proceed from Principles to Facts, and that the attempt exhibits three distinct phases: one, in which the Induction stage is so simple and so short as to be instinctively and correctly performed by all people, and the Deductive stage at once reached—this furnishes the Mathematics, the only Science in which hitherto the true Deductive Method has prevailed; a second, in which Principles are assumed to reason from, without any previous effort at Induction, such as existed, being unconsciously made from the supposed Facts or Knowledge which the mind was in possession of; and a third, in which Facts were collected, classified, and Induction therefrom as a basis of further investigation attempted, but in which the Laws or Principles assumed as established by the Facts were not rigorously and accurately derived from Facts; or, in other words, in which the Facts were not strictly used for the purpose of deriving from them just such Laws or Principles only as they actually established, but were wrenched to the attempted support of Laws, Principles, or Ideas more or less fanciful or unrelated to the Facts. These two last phases are included in what is known among Scientists as the Anticipative or Hypothetical Method; while the three phases are commonly undiscriminated and collectively termed the Deductive Method. It was also developed that the results of this period of intellectual activity were fruitless of definite Scientific achievements, except so far as the true Deductive Method had been employed. It was furthermore seen that since Bacon's time, the opposite Method of procedure, namely, from Facts to Principles, has been chiefly in vogue; that under its impulse distinctness and clearness have been brought to pervade those stores of knowledge which were already in our possession, thus fulfilling one of the requisites of a perfect Scientific Method, while, however, the other necessary requirement, that of furnishing a certain guide to future discoveries, has been only proximately attained by it.
It is obvious from this exhibition of the characteristics of the two leading Scientific Methods, or the two leading Processes of the one Method, in whichever light we may choose to view them, that so far from being the best or the only true Method or Process of intellectual investigation, the Inductive is far inferior to the true Deductive Method or Process, in all the essentials of a Scientific guide. The Inductive can give us only a high degree of precision and definiteness, with only proximate certainty for the future as the result of a slow mode of procedure; while the true Deductive Method gives us perfect precision, exactitude, and complete certainty, as the result of a rapid mode. The true Deductive Method—brought into disrepute by being confounded with the Anticipative or Hypothetical, which differs from it only in this, that the Principles from which the latter reasons are true, while those of the former are doubtful—has thus far prevailed in Mathematics alone, and Mathematics is, up to our day, the only recognized Exact Science, the only Science in which Demonstration, in the strict sense of that term, is now possible,—the fruits of the Inductive Method being known as the Inexact Sciences, in which only Probable Reasoning prevails.