Astronomy and Physics, viewed in the light in which they are here presented, are rather special branches of Mathematics, than distinct Sciences. But as we often speak of Geometry as a separate Science, although it is in reality only a division of the Mathematical domain, and is so classed by Comte; so there is a sense in which both Astronomy and Physics, as herein defined, may be regarded as individual Sciences, and in that character they will be considered in this paper.
We have, then, three domains in which the true Deductive Method is active; in which we can start from universally recognized Truths and proceed, by irresistible Inferences, to ulterior Principles and Facts. In three Sciences, in Mathematics as commonly defined and understood, in Astronomy and Physics as herein circumscribed, we are able to establish starting points of thought with Mathematical certainty, and to deduce from them all the Phenomena of their respective realms.
Within the scope of these three Sciences, therefore, our information is clearly defined, positive, and indisputable. The conclusions to which we are led by their Principles can no more be gainsayed than human existence can be doubted. While time shall last, while mankind shall endure, while the human Mind is constructed on its present basis; while, in fine, there is a possibility for the exercise of Thought in any way conceivable to the existing Mentality of the universe, the Laws of Mathematics, of Astronomy, and of Physics can be apprehended in no way different from that in which they are now apprehended. There is no conceivable possibility that subsequent investigations will show them to be erroneous or defective. They stand upon a foundation of Proof as unalterable as the fiat of Fate or the decrees of the Almighty, which can neither be shaken nor destroyed.
It is between these three Mathematical Sciences, on the one side, and all other domains of intellectual investigation on the other, that a line of distinct demarcation must be drawn, in any Classification of our so-called Knowledge, in accordance with any method of classification known to the scientific world at large. Not that the Laws or Principles which lie at the base of all other departments of the universe are not as stable, as definite, and as infallible as those which inhere in the Sciences which have been specially indicated. But that, as yet, the endeavor to apprehend fundamental Principles, in other spheres than these, has been attended with only partial success; and hence, the ability to establish a Mathematical or Demonstrable basis for other regions of Thought is yet wanting, so far as is commonly known.
When, therefore, we emerge from the domains of Mathematics, Astronomy, and Physics, we are leaving the field of positive assurance, of undeniable truth, and entering the realms where opinion, conjecture, and variable degrees of certainty prevail. The Facts of Observation may be, indeed, as plain here as elsewhere and as firmly established. But the conclusions drawn from them, the Scientific Principles assumed to be established, may be erroneous or defective, and the power of prevision, the great test of Scientific accuracy, is proportionally wanting. Derived, as we have hitherto seen these conclusions to be, from Phenomena, on the supposition that a given range of Observation will secure all the essential Principles which appertain to the whole of the Phenomena included in the range, we can never be entirely sure that our basis of Facts is sufficient for our purpose, and hence the possibility of error always exists.
It is not to be understood, therefore, that first or observational Facts are not rightly to be known in other departments of investigation than Mathematics, Astronomy, and Physics; but that Laws, Principles, or Generalizations which relate Facts and serve as instruments for penetrating into the deeper arcana of Nature, cannot be precisely, accurately, and certainly known, in their relations and belongings, until we are able to establish their connection with the lowest, most fundamental, and self-evident truths, and in this manner become competent to advance step by step from undeniable first truths to those equally undeniable. In Mathematics, in Astronomy, and in Physics, we are able to do this. We know the Laws or Principles of these Sciences, therefore, so far as we have developed the Sciences themselves. We know the relations of the various Laws within the range of each Science, and the relations of the different Sciences with each other. We can advance, within their boundaries, from the simplest and most positive verities, such as the whole is equal to all its parts—a self-evident truth, which it is impossible to conceive as being otherwise than as here stated—up to the most intricate ulterior Facts of the universe, by Inferences which are as irresistible to the mind as the axioms with which we started. In no other domains of Thought can this be done by any methods now in vogue. In no other realms, therefore, are complete precision and infallibility attainable. It is this which constitutes the peculiar character of these three Sciences, and distinguishes them radically from all others.
The whole body of our authoritative and irrevocably determinate intellectual acquisitions lies, therefore, at the present time, so far as is commonly known, within the range of Mathematics, Astronomy, and Physics. These are in strictness the only Sciences which we possess; and the only domains in which knowledge, in the proper sense of the term, is attainable. In passing their boundaries, we leave the regions of positive certitude, and come into the domain where Conjecture, varying from the strongest presumption to mere plausibility, is the highest proof. Laws or Principles are yet undiscovered there, and in their place we find Generalizations—Suppositive or Proximate Laws—which are in process of proof, or already established by such evidence as the Inductive Method can array, and which carry the conviction of their correctness with varying degrees of force, to larger or smaller classes of investigators.
These three branches of knowledge are unquestionably entitled to the designation of Positive Sciences; and to no others can it with justice be accorded. To apply the name of Science to domains in which real knowledge is not attainable, is, in some sense, an abuse of terms. To denominate Positive Sciences, domains which are not strictly Scientific, and in which positive certainty, in reference to Principles and ulterior Facts, cannot be attained, is still more incongruous. Comte's arrangement of the schedule of the Positive Sciences, in which domains where Demonstrable knowledge prevails are placed upon a common basis with those in which it does not, was probably owing to the want of a clear perception on his part of the essential difference of the nature of proof by the true Deductive Method and of proof by the Inductive Method, of the actual Certainty of the one and the merely proximate Certainty of the other.
If such were the case, his want of discrimination was rather due to an overestimate of Inductive proof than to an undervaluation of Mathematical Demonstration. That Mathematics, Astronomy, and Physics were more perfect Sciences than the others in point of precision, he distinctly affirms, pointing out that 'the relative perfection of the different Sciences consists in the degree of precision of Knowledge,' that this degree of precision is in accordance with the extent to which Mathematical analysis can be applied to the given domain, and that to the above-mentioned Sciences only is its application possible. Notwithstanding this apprehension of the different degrees of precision or exactitude attainable in the various Scientific realms, he does not seem to have sufficiently understood that there was also a vast difference in the nature of the evidence which went to prove the truth of the supposed Principles and ulterior Facts of the various departments of Thought, and hence variable degrees of Certainty in regard to the positive bases of the Principles themselves. He thus falls into the same error which it was one of the main purposes of his Scientific labors to correct—commingling problematical theories with Demonstrable Truths, as equally entitled to belief—and ranks Sociology, including La Morale, afterward called a distinct Science, with Mathematics, Astronomy, and Physics, as domains in which our reasonings, in the present state of Knowledge, can be equally reliable.
It is barely possible that the purpose and design of Comte's Classification had, unconsciously, much to do with its really unscientific and incongruous character. The aim which he had in view was to construct a Sociology or Science of Society which should be a guide in the establishment of a new Government, a new Political Economy, a new Religion, a new Social Life, a new Order of Things, in fine, to take the place of the decrepit institutions, governmental, ecclesiastical, and social, which he thought were fast approaching their period of dissolution. The Generalization which had exhibited to him, that the Laws and Phenomena of the various departments of investigation were dependent on each other in a graduated scale, and had thus enabled him to establish the Hierarchy of the Sciences, showed him that Sociology, including as it does the Principles and Phenomena of the other domains which he regards as Positive Sciences, must be based upon them.