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senses in such a case, for that there sues should be as free from the possible may be deviations from perfect figure intermixture of error as its original infinitely more minute than they are grounds. And we may now therefore able to apprehend. The subjects of enquire, in what manner the same cermathematical reasoning, then, to tainty is obtained in the steps of mawhich it is essential that they should thematical demonstration. It is recorrespond in the most absolute truth marked by Mr Locke, that, « in to their mathematical description, demonstrative knowledge, there must either do not, and cannot subsist in be in every step of reasoning inmaterial nature; or if they do, cannot, tuitive certainty." By intuition is as so existing, be known to the mind; meant the perception of truth by simand cannot, therefore, afford the ple inspection of the subject-as the grounds of its reasoning.
truth of the proposition that things Where, then, have they their exist- equal to the same thing are equal to ence? In the mind only, which, by one another is said to be perceived by assigning to them their essential pro- intuition. It is discerned as soon as perties, gives them existence_or, as the subject is presented to the underit may be truly said, creates them. standing. Now it will be found, that The subjects, then, of all mathemati- all the steps employed in the course of cal reasonings are intellectual concep- mathematical demonstration have such tions merely; and therefore are what intuitive certainty. For the proofs they are conceived. It is of no con- resorted to are either the axioms of sequence that these conceptions are the science, or its definitions. With not possible to be realized in matter. respect to the axioms of mathematical Intellect here frames its own subjects science, they are precisely such truths of thought, and is therefore at liberty as have just been described they are to assign their properties, without re- propositions which, the moment they gard to any laws, except those to are presented to the mind, are diswhich it is itself subjected ; and it can cerned to be necessarily true. They be no objection to the constitution are self-evident truths, or truths of inwhich it ascribes to its subjects, that tuition. With respect to the definiit is found to be in contradiction with tions, these serve to describe the subthe constitution of matter, so long as jects of reasoning by their essential it is not self-contradictory. In this properties; and the occasion of referscience, then, whatever its subjects are ing to them is simply this, that the conceived by the mind, that they are reasoning is brought to a point when -the point is without extension-the the next step of proof depends on that line is without breadth--the tangent essential property assigned in the detouches the circle but in a single point finition. The reference is merely an -the radii of the circle, declared to appeal to that original constitution of be equal, cannot vary by the differ- the subject, which was established by ence of one indivisible atom of matter. the mind as the basis of the science.
But if these subjects are, and can Thus, whether the proof be by referbe nothing else than precisely what ence to the axioms or to the definitions they are conceived by the mind, there of the science, the certainty is perfect: are then two most important grounds since in one case, it rests on a selfof certainty obtained to all its subse- evident or intuitive truth ;-in the quent proceedings—first, that the rela other, on a fact given in the very contions apprehended by the mind, as sub- stitution of the subject of reasoning. sisting in these subjects, do subsist in We may here remark, that the subthe most perfect degree, absolutely jects of mathematical reasoning existand unexceptionably ;-the other ing in thought merely, the truths ascerthat the knowledge which the mind tained respecting them cannot in their possesses of the primitive constitution intellectual purity and rigorous exactof the subjects of its reasoning is a ness be transferred to any thing existperfect knowledge, without omission ing out of thought. Nevertheless, the and without error.
deductions of this reasoning are apSuch then are the grounds of cer- plicable so far to material subjects, as tainty in the subjects themselves of these material subjects approximate to mathematical enquiry.
the truth of the intelleetual concerAll that is further necessary is, that tion. In their nearest approximation the steps of reasoning which it pur. they are indeed absolutely divided from that exact intellectual truth; but Let us now pass to Inductive Reason. to the perception of our senses they ing. The object of inductive reasonmay approach so undistinguishably ing is, to obtain knowledge of things near, that, as far as regards all our having absolute and independent expractical purposes, they may be con- istence, their relations, and their laws ceived to coincide with it. Hence we of mutual action. The means are obare able to apply mathematical rea servation of the facts falling under our sonings to the various material sub- notice, with reasonings grounded on jects of scientific observation and of those facts—these reasonings in like art; the difference, in that applica- manner consist of intuitive perception, from the absolute truth, being tions of relations the grounds of cersuch as either is not perceptible by us, tainty are the confidence of the mind or such as does not affect our use. in its own intuitions, and the belief of
The subjects then of all mathemati- constancy and unity in the order of cal enquiry are given in those simple nature. primary relations which are proposed The process of the mind in such and established in its outset. It then inductive reasoning will be best unbecomes the object of the science to derstood by considering particular educe by reasoning the other relations cases of knowledge thus acquired from which these primary oncs necessarily the study of material nature. involve; proceeding from one discovery The first occasion to the great Har. to another, in endless combinations, car vey, of conceiving the circulation of ried on step by step from these few and the blood, was the anatomical obsersimple elements. The labour of the vation of certain valves, that are placmost powerful minds, through a long ed at intervals in those vessels, which successon of time, is unable to exhaust are now known to carry the blood tothe relations involved in the constitu- wards the heart, the veins. The tion of the subject that was at first so structure and situation of these valves simply determined. “I have no is such, that they will permit any fluid doubt," says Dr Reid, “ that after all contained in these vessels to flow in the investigations of mathematicians, one direction, and not in the other : of the simple figure of the circle, it that direction is towards the heart. contains many properties, which are He argued that the blood did flow yet undiscovered." So fruitful to in- along them in that direction. It foltellect is every element of intellectual lowed that the other vessels, the arteconception. The thought of a mo ries, in which no such valves were ment produces a subject for the study found, must serve to carry it from the of ages.
heart over the body. This, then, was In mathematical reasoning, there- at first, an hypothesis, grounded on a fore, what is attained is perfect cer. single observation. Innumerable obsertainty ; the subject of reasoning being vations, of various kinds, made during relations known absolutely to the many years, all coinciding with, and mind, which has conceived them; and explained by this opinion, and none the process of reasoning being always contradicting it, enabled the illustrious such a comparison of one subject with discoverer to establish his theory with another, that the new relation result- irresistible evidence. Here, then, was, ing is perceived by intuition ; the pur- in the first place, a fact observed—a pose being from those fixed and known hypothesis framed consonant to that relations to ascertain unknown rela. fact-and numberless subsequent ob. tions which they involve. And the servations found coinciding with the whole may be thus summed up. The hypothesis. The comparison of these subjects of reasoning are known rela- observations with the hypothesis, and tions—the steps of reasoning intui- the perception of their coincidence, is tions the end other relations—the what in this instance is to be underground of certainty intuition of rela- stood by inductive reasoning. tions among subjects certainly known As an example of the same pro-to doubt its reasonings, the mind cess, far more removed, as it might must either doubt its own intuitions, seem, from absolute observation, which have carried it on at every step may be mentioned the discovery by -or it must doubt its understanding Copernicus of the true orbits of the of its own conceptions, from which planets. “When Copernicus," says originally it set out,
Maclaurin, « considered the form, disposition, and motions of the planet. when other later theories resting on ary system, as they were in his time their own grounds were found to agree represented according to the scheme with it-it was then considered as inof Ptolemy, he found the whole void ductively demonstrated ; yet all this of order, symmetry, and proportion; weight of inductive evidence may be like a picture, as he expresses himself, summed up in this, that each observamade up of parts copied from different tion, severally taken and put in appooriginals, which, not fitting each other, sition, as Locke would say, with this should rather represent a monster than theory, is perceived to coincide with a man. He therefore perused the it, and none to contradict it. The writings of ancient philosophers, to evidence then of inductive reasoning see whether any more rational account is merely an aggregate of simple perhad ever been proposed of the motionsceptions of coincidence. of the heavens. The first hint to this It may thus be understood what is effect that he gained was from Cicero, meant by inductive reasoning. Apwho, in his Academical Questions, pearances are first observed, which is acquaints us, that Nicetas, a Syracu- all the knowledge directly conveyed san, had taught that the earth turns to us of any existence. These apround on its axis, which, to a specta. pearances are to be accounted for, and tor on the earth's surface, makes the the hypothesis which explains them whole heavens appear daily to revolve. reduces them into knowledge. But Afterwards, from Plutarch he found, that hypothesis must, in the first inthat Philolaus had taught that the stance, be matter of conjecture merely, earth moved annually round the sun. and the comparison of various obserHe immediately perceived that, by vations with it, and the perception of allowing these two motions, all the their coincidence, is the inductive perplexity, disorder, and confusion reasoning which confirms it. which had troubled him, in the celes. So far what we have seen is the distial motions, vanished ; and that, in- covery by induction of absolute facts stead of them, a simple regular dispo- of physical nature, which are so situatsition of the whole, and a harmony of ed as to be out of the reach of our the motions appeared, worthy of the knowledge by direct observation, and great Author of the world.”
this was the first point of discussion Now it is to be observed, that this proposed. There is, however, another theory of Copernicus was, in the first great result of the application of ininstance, a simple hypothesis. It ac- ductive reasoning to the observation counted for the known appearances of of natural existence, that is, the disthe heavens ; but those known appear- covery of the general laws by which ances did not require it. For, in the nature is governed. But when philosystem generally adopted at the time, sophy speaks of the discovery of genethat of Ptolemy, according to the na ral laws, it is not to be understood as tural suggestion as it may seem of hu- if it presumed to ascertain the power man iinagination, and which placed which determines action, or the true the earth in the centre of the world, connexions in nature, which are not there was nothing inconsistent with discernible by us all that is meant the then ascertained facts of the pla by such discovery is the perception of netary motions. The choice was harmonies in the system of being by therefore between two hypotheses; the intellectual extension of the law of and the reason for the preference of one appearance to many, or, as it is that of Copernicus was, not that it ex commonly expressed, by resolving plained what was left unexplained by particular into more general facts. the other, but that one made the struc. Thus the knowledge of the true places ture of our system perplexed and in- of the bodies of our planetary system, tricate, and the other reduced its mo- and of the true orbits which the planets tions to simple and beautiful order. by their revolutions describe, are But when this hypothesis bad by its merely facts. When Newton showed beautiful simplicity recommended its that the course of those motions agreed probability, then was science excited with what must take place if they to find observations concurring with were determined by the same cause by this hypothesis ; and when numberless which unsupported bodies fall, assigrfacts were ascertained which this hy- ing gravitation as the principle which pothesis would explain and no other regulated the descent of heavy bodies and the motion of planets, he gave no In this resolution of various appearexplanation of the cause, for the cause ances into universal laws, it is imporin both instances is equally unknown tant to observe what is and what is he merely resolved the two unlike not effected. There is no discovery facts of the fall of bodies, and the de- of absolute causation, which is always flexion of the planets' course from a hidden from our sight-what is obright line, into one common fact or tained is the extended recognition of a law, namely, that the bodies of matter single fact under a great diversity of tend towards one another. This more appearances, viz. that matter tends general law might be said to be a har- towards matter. We had occasion to mony found in nature connecting the show in what manner a particular fact events which take place on the surface was ascertained, namely, by the proof our globe and the motions of the posal of a well-conceived hypothesis, planets in one system. The mere and then by inductive reasoning jus. discovery of the same tendency of tifying and confirming it. In the matter to matter, existing alike here same manner, in the extension of a and in the mutual relations of those single fact through such variety of apvast worlds, is itself a wide embracing pearances, or its generalization, as it harmony; but that effects so unlike in is called, there is the same process appearance fall under one law is a pursued. First is the bold conjecture harmony of another kind, a harmony of philosophy that this is indeed the of simplicity in the design by which common law which prevails through the world is ordered. Even when all these appearances, and next in within the limits of our own planet order is that inductive reasoning which philosophy examines and analyses the confirms and establishes the hypothesis. various appearances and effects which But if we ask in what that inductive fall under this common law of gravita- reasoning consists, it is found as betion, tracing the operation of this one fore that it consists in nothing more principle in the agency that has given than an aggregate of intellectual per. to the earth its solidity, in the law ceptions of coincidence. It consists in appointed to the ocean alike to lift up the collection of innumerable minute its tides and to retain its waters in and partial facts-of appearances their bed, when it finds that by the among the subjects—in the compari. same power the vapours are carried up son of them with the hypothesis, and and sustained in the floating clouds, the perception of their coincidence. and that by this the upward growth The particular manner in which these of all vegetation is determined, with innumerable confirming observations numberless other effects in which the are pursued and ascertained does not operation of this one principle is more need to be considered in the present or less disguised in its appearances- enquiry. It is when they are ascerwhen we see how many and various tained that they become, with respect these results are, how much of the to that hypothesis, the matter of whole of the complex system of our inductive reasoning. knowledge and of our world they It needs only further to be obsercomprehend, we do not indeed obtain ved, that every general law thus asany knowledge of causation, but we certained, becomes a means of further resolve a vast diversity of facts under discovery ; because the philosopher that one common fact, whatever its expects other effects yet unexplained cause may be, that matter tends to to fall under it, and strong in this dismatter. But in merely thus reducing covery, he goes on to observe and to these diverse effects under a common explore, that he may comprehend law, we cannot but feel that we have under the same knowledge what yet greatly enlarged the comprehension remains unknown. His process being of our own intelligence; and with merely this-to observe individual respect to our knowledge of the uni. facts, to generalize them, and then to verse, we have acquired insight into advance again to the observation of the harmony of design in its constitu- further effects, to bring them under his tion when we have seen one single known general law ; but here too his principle employed, through so wide induction is still the same, being an extent of created existence, to pro never more than the comparison of duce effects so various from one one thing with another, and the peranother.
In all these reasonings, however, originally impressed upon our minds there is never more attained than pro- in their constitution, and which are bability upon probability, infinitely confirmed by the constant tenor of our accumulated. There can be no de experience. Upon the supposition of monstration. The belief that is forced such a constancy, it is evident that all upon the mind is indeed irresistible. reliance upon our knowledge must be But the conviction yet remains, that founded, since, if these laws could under the weight of all this evidence vary, the whole structure of our there may yet by possibility be error. knowledge would be immediately It is still remembered that “no hypo- overthrown. Upon this belief, then, thesis, how numerous soever the facts rests our assurance that what we have may be with which it tallies, can com ascertained is invariably true. The pletely exclude the possibility of ex. other element of our reasoning, nameceptions or limitations, hitherto undis- ly, our belief of unity in the design of covered." It is still considered, that nature, may require to be more fully our belief is grounded merely on the considered. This is the foundation concurrence of numberless appearances of our reasoning by analogy, when, coinciding with one Hypothesis ; but observing like appearances, the mind that the absolute truth of the Hypo is irresistibly impelled to ascribe them thesis, or the connexion in nature in to like causes an implanted tendency which it subsists, is placed out of the which all experience confirms. It is reach of our discovery.
the ground of that expectation which Such then is the nature of inductive carries us forward continually in phireasoning. We have taken the illus- losopbic investigation, and which is trations which were necessary to ex- the whole spirit of philosophy, the plain it, from the appearances, facts, expectation of reducing more and and laws of the Physical world, be more facts under the laws already cause these are more simply and easily known, so as to enlarge continually stated, and are more generally and the conception of the law itself, and indisputably recognised. But the pro- to comprehend under the dominion cess of inductive reasoning is the same of intellect more and more of the with respect to the mind: all reason- yet uncomprehended facts of the uniing which is employed to establish its verse. Lastly, it is the true ground facts or its laws, from appearances of our belief in our reasoning at all, collected by self-observation or obser- because the sole evidence to us of vation of others, being precisely the truth in our reasonings is their harsame--the aggregate result of many mony ; but this harmony is evidence comparisons, and of intuitive percep of their truth, only because we pretions—upon such comparison made- suppose unity in the design of created of disagreement or coincidence. things, and the harmony we find in
We have seen upon what grounds our own thoughts appears to us to the belief of mathematical reasoning betoken a correspondence between rests, namely, on the confidence of the them, and that believed unity of demind in its perception of connexions sign in the constitution of the world. among its own thoughts. We may If it were possible for us to conceive now consider upon what grounds its diversity of purpose in the scheme of belief in the reasonings of induction is creation, we could never know that established. In all inductive enquiry, what we beheld in one part had any then, the ground of our reasoning is connexion with what we had known a belief of constancy in the appointed in another; we could not find, even order, and of unity in the design of in the consistency of our reasonings, a nature. Our belief of constancy in proof of their truth. It might be a the appointed order of nature, that is proof of error. to say, the determination of our mind . Such then is the nature of inductive to conceive, that whatever it discerns reasoning : a proceeding of the mind, to take place, takes place by establish- not appertaining to science alone, but ed and permanent laws, and will there proper to the human intelligence from fore recur continually in like circum- the beginning of its operations, accomstances, appears to be, as we observed panying it in all its steps, instructing in speaking formerly on the relation those whose knowledge of the worlds of cause and effect, one of those inhe- physical and mental with which we are rent intellectual tendencies which are conversant is rudest and most unform