It is necessary to say, in the strict sense of the term, because the same laxity exists in the use of the word Demonstration, as in that of Science, and hence it has lost the distinctive meaning which attaches to it, in its legitimate use, as signifying a mode of reasoning in which the self-evident truths or axioms, with which we start, and every step in the deduction, 'are not only perfectly definite, but incapable of being apprehended differently—if really apprehended, they must be apprehended alike by all and at all times.' It is because this Method of proof exists only in Mathematics, that this alone is denominated the Exact Science, or its branches, the Exact Sciences; Sciences whose Laws or Principles, and the Facts connected with or deduced from them, are irresistible conclusions of thought, in all minds, which conclusions rest upon universally recognized axioms; while the Inexact Sciences, including all except Mathematics, the Sciences in which the Inductive Method prevails, are systems of Laws or Principles, with their related Facts, of the truth of which there is great probability, but of which there is, nevertheless, no complete certainty; whose conclusions are not based upon universally undeniable axioms, or are not themselves irresistible to the human mind.
The superiority of the Deductive Method, both in its mode of advancing to the discovery of new truth and in the precision, clearness, and certainty which accompany its findings, must now easily become apparent. Whether we regard Induction and Deduction as correlative Processes belonging to one Method, each of which has been disproportionately in vogue at different epochs, or as distinctive Methods, having each their own Deductive and Inductive Processes, in either aspect, Induction is only a preparative labor, leading in the more important work of the application of the Law or Principle derived. It is only, indeed, for the purpose of discovering this Law that Observation, Classification, and Induction are undertaken. It has been the triumphant boast of the Inductive Method, that it guarded, by means of these preliminary steps, in the most careful manner, against error in establishing its Laws. To the extent of its capacity it has done so. But we have already seen, that deriving its Principles, as it was obliged to, from less than all the Facts which appertained to the Principles, these must inevitably have been lacking in some particulars; it being impossible to make the whole out of less than all its parts.
The Inductive Method has obtained an importance greatly exaggerated, for the reason that it has been brought into comparison, for the most part, with the Anticipative or Hypothetical, the bastard Deductive Method only, and its superiority over this exhibited in the most detailed manner, while the right application of the Deductive Method, except in Mathematics, has not been considered possible. The reason of this can be made obvious.
The immense superiority of Mathematical Reasoning, as Demonstration is often called, over all other kinds, is universally known and recognized. For in this mode of reasoning there is no room for doubt or uncertainty. It starts from Principles of whose truth there can be no doubt, because it is impossible for the human mind to apprehend them in more than one way, and proceeds by steps, every one of which can likewise be apprehended in only one way. Hence all men arrive inevitably at the same conclusion at the close of the chain of reasoning. It is, therefore, a Method of proof which sets out from a precise, definite, universally established Law or Principle which really contains the conclusion in itself, and which can be developed to the end through a series of necessary and irresistible convincements; while in the Inductive Method we are obliged to start from this or that admitted Fact or Truth assumed after Observation, Classification, and Induction, which may have been rigorously performed, but which, nevertheless, could not, in the nature of the case, prove the Fact or Truth with complete certainty, and which is not, perhaps, universally admitted, and proceed by merely probable inferences drawn from various, diverse, and often uncertain relations, until we reach a conclusion. Such reasoning may be sufficient to incline the mind to a particular conclusion, as against those which tend to any other conclusion, but they are never quite sufficient, as in Demonstrative or true Deductive reasoning, to necessitate the conclusion, and render any other impossible.
A Method of Scientific investigation which proceeds from self-evident truths to necessary results by undeniable steps, would of course be preferable to one which, starting from truths whose precision and certainty might be doubtful, advances by more or less probable inferences to a more or less probable conclusion, did there not exist some powerful cause for a contrary action. A difficulty thus far insurmountable has, indeed, stood in the way of the adoption of the Deductive Method in any department of investigation, save the one already referred to. This Method, we have seen, leads to truth or error accordingly as the Principles or Laws from which it commences its reasoning process are true or false. In the Mathematics, the basic truths, being of a simple character, were arrived at by easy and instinctive mental processes, and the Method achieved in this department great success. But the other domains of human knowledge being more complex, involving more qualities or characteristics than mere Number and Form and Force, which are all that come within the scope of Mathematics, their fundamental bases or truths were not so easily attainable. Hence, when Principles or Ideas which men believed to contain all the fundamentals of a specific domain of thought were adopted as starting points of reasoning, they were generally lacking in some important element, which caused the conclusion to be in some way incorrect. We have seen the historical results of this mode of procedure in what is denominated the Anticipative or Hypothetical Method. The failure of this to secure good results, and the absence of any standard by which to be certain when a Law or Principle was fundamental, exact, and inclusive, when it was a valid basis to reason from, led to the abandonment of the Deductive Method, except in its application to Mathematics, where true starting points were known. The Observation and Classification of Facts was then resorted to, first, in a loose way, in Greece, and afterward, in a more rigorous way, in the world at large, for the purpose of endeavoring to discover, by the only mode considered effective—the examination of Phenomena—the fundamental Principles, which, like those of Mathematics, should include all the essentials of the special domain under consideration. These being discovered, might furnish, it was instinctively felt, starting points from which to work the Deductive Process, with the same success as that which attended its application to Mathematics.
The Inductive Principle, considered either as a Process or a Method, is valuable, therefore, mainly as it furnishes proper starting points for the activity of the Deductive Principle. Thus far in the history of the Natural Sciences it has been the best and safest guide in affording such starting points. But the indications are numerous all about us that the progress of Scientific discovery will ere long bring us to a stage, where the Laws or Principles which underlie every department of the Universe being fully revealed, the function of the Inductive Principle as a guide to fundamental bases, will be at an end, and the Deductive Method once more assume the leadership, opening to us all departments of investigation, with the rapidity, precision, and certainty which characterize Mathematical research and Demonstrative Reasoning.
This desideratum must necessarily result whenever a Unitary Law shall be discovered in Science; whenever the Sciences, and the Phenomena within the different Sciences, shall be basically connected. All the present conditions and tendencies of knowledge indicate that the attainment of this crowning intellectual goal was predestined to our epoch. It has been the grand work of the Inductive Method to arrange Facts under Principles, and these latter as Facts or Truths under a smaller number of Principles, and these in turn under a still smaller number, until all the Phenomena of the different domains of thought which are reckoned as Sciences are included within a few Principles which lie at the foundation of each domain. The connection of these few Principles by a still more fundamental Law, is all that is necessary to the completion of the work of the centuries and the establishment of a Universal or Unitary Science. Already those recognized as leaders in the Scientific world watch expectantly the signs of the times and await the advent of the Grand Discovery which is to usher in a new intellectual era, 'We have reached the point,' says Agassiz, in one of his Atlantic Monthly articles, 'where the results of Science touch the very problem of existence, and all men listen for the solving of that mystery. When it will come, and how, none can say; but this much, at least, is certain, that all our researches are leading up to that question, and mankind will never rest till it is answered.'
'All the Phenomena of Physics,' says Professor Silliman, in his First Principles of Philosophy, 'are dependent on a limited number of general laws, of which they are the necessary consequences. However various and complex may be the phenomena, their laws are few, and distinguished for their exceeding simplicity. All of them 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. As in geometry, all the properties of figures are deduced from a few axioms and definitions; so when the general laws of Physics are known, we may deduce from them, by a series of rigorous reasonings, all the phenomena to which they give rise.'
Auguste Comte, in his elaborate and encyclopædic Course of Positive Philosophy, tells us that 'these three laws [the Law of Inertia, the Law of the Equality of action and reaction, and the Law of the Composition of forces] are the experimental basis of the Science of Mechanics. From them the mind may proceed to the logical construction of the Science, without further reference to the external world. * * * We cannot, however, conceive of any case which is not met by these three laws of Kepler, of Newton, and of Galileo, and their expression is so precise that they can be immediately treated in the form of analytical equations easily obtained.' While also exhibiting the small number of Principles which lie at the foundation of other domains of our intellectual accumulations, Comte remarks: 'The ultimate perfection of the Positive system would be (if such perfection could be hoped for) to represent all phenomena as particular aspects of a single general fact;—such as Gravitation, for instance.'
These are a few specimens of what may be found in the books, pointing out the gradual approach of Scientific investigation to the discovery of a Unitary Law, and the expectation among Scientists of the advent, at some period not far distant, of a new Science, the greatest among Sciences, a true Pantology or Universology. Upon the apprehension of this Law, which must establish the true basis of every domain of thought or activity, and show it to be identical or analogous in the several domains, we shall be placed, in relation to the whole universe, precisely where we now stand in relation to Mathematics, Mechanics, and Physics; that is, the General Law or Laws of every domain of investigation will become known, as we now know those of these Sciences, and, to adopt the words of the French writer, 'from them the mind may proceed to the logical construction of the Science [being now the Science of the whole Universe], without further reference to the external world;' or to use the language of Professor Silliman, 'when the general laws of [the Universe] are known, we may deduce from them, by a series of rigorous reasonings, all the phenomena to which they give rise.' Thus, upon the discovery of a Unitary Law, linking the Sciences together, and showing the identity of their starting points or bases, the Deductive Principle, considered either as a Method or a Process, must once more take the lead, and the Inductive occupy its legitimate position as a subordinate and corroborative auxiliary. Under the guidance of this new adjustment of the Deductive and Inductive Principles, a full, exact, complete, definite, Scientific Classification of our knowledge will become possible, and the true boundaries of every domain of intellectual examination may be critically and clearly established. In the absence of such a Classification, it is only by viewing departments of the Universe with reference to the Method or Process employed in the investigation of their Phenomena, that we are able to estimate their present relations to Science, and to ascertain proximately their Scientific or Unscientific character. We proceed, then, to examine the connection of History, in its present development, with Science, a task to which the foregoing brief and incomplete consideration of the subject of Method has been a necessary preliminary.