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Intuition And Demonstration

142. THE main positions occupied by those who defend the Metaphysical Method, and by those who believe in the possibility of Metempirics, are the evidences of a source of knowledge which is antecedent to and independent of Experience, and of a kind of knowledge which transcends Experience. We must have a higher organ, it is said, because we have the higher knowledge. That organ is Intuition, that knowledge is Necessary Truth.

All that has been written in the preceding pages would be either set aside as erroneous, or disregarded as irrelevant, if these two positions were left in the possession of our antagonists.

143. The ancient doctrine of Innate Ideas having been relinquished, or modified till it became ineffectual, the doctrine of Intellectual Intuition was put forward in its place.. The most precise form this doctrine assumed was that given it by Jacobi, when he affirmed that over and above the intuitions of sensible objects, we had a special organ of rational intuition for the perception of supra-sensibles. He admitted that its special intuitions are given in the overflow of feeling (in überschwcinglichen Gefuhle), but declared these to be nevertheless truly objective. It was this organ which Schelling christened the Intellectual Intuition, and to it assigned the principle of Demonstration and final ground of Certitude.

144. Jacobi was misled, I think, by an imperfect appreciation of the nature of Demonstration. ' He said that the conviction gained through demonstration is a certainty at second hand ; it rests on comparison, and can never be perfectly secure. " If, then, every opinion is Faith, which does not issue from Reason, so must conviction from rational grounds issue from Faith, and owe its force to Faith alone." There is an equivoque here. Conviction is assuredly a feeling, and Reason has only force in proportion to the feeling involved. But although the certainty of a demonstration may be reached by a comparison of feelings, and is thus second-hand, what is usually understood by Faith is not this comparison of feelings, not the reduction of inferences to sensations, but the reliance on unverified inferences. Granting that Feeling is the common basis of sensible and rational inference, we cannot admit that any unverified inferences are to be accepted as objective truths. My conviction that the object before me is an apple, and my conviction that the riots in Ireland are parts of a "providential scheme," may be equally true expressions of my state of feeling, — I who have these convictions cannot doubt that I have them, — but one or both may be absolutely false expressions of the objective realities ; and their truth or false-hood can only be demonstrated by the reduction of what is inferential in each to its correspondent sensibles. In the case of the apple such a reduction may be easy. In the case of the providential scheme it is impossible, simply because the providential scheme is a conception framed out of data which never were and never could be sensible. And herein is displayed the futility of this pretended organ. It professes to deal with supra-sensibles, yet these can only be thought of under sensible forms. Nor let it be urged that precisely this is the course followed with respect to extra-sensibles. The only test there admitted, namely, the reduction of the extra-sensibles to the sensible standard, is the very test which the theologian and metempiricist reject. For if that test be admitted, it brings the Supra-sensible within the range of Experience, and thus Religion and Metaphysics become amenable to the Method of Science; a Method which, by excluding whatever cannot be verified, at once sets aside a mass of speculations declared to be unverifiable, and a mass of dogmas declared to be absurd.

145. No reader firmly persuaded that the mind of man is endowed with the power of apprehending the Supra-sensible can be expected to relinquish that belief, coerced by the arguments here advanced ; it is with him a question of Faith, and cannot be shaken by Logic. But in opposition we may say to him : " The existence of such a power requires proof, and when proven it can only serve to construct a system of conceptions which have no analogy, or point of intersection, with the conceptions constructed out of sensible experiences ; this being so, whatever range it may have it must be excluded from all theories having reference to the sensible world."

Our Method does not exclude mystery from the Universe, it only excludes it from Science, and assigns it to the region of the Metempirical,"whose margin fades forever and forever as we move." The doctrine of Intellectual Intuition is not only disputable, it is futile. But while rejecting its pretensions we may with advantage accept and interpret the facts it improperly classifies, and admit the existence of Experiential Intuition. This we will here consider.

146. Demonstration is the showing to Sense or Intuition, in other words, the reduction of Inference to its corresponding sensations, either directly through Sense, or indirectly through Intuition.

If I wish to demonstrate that three objects added to three others will form a group numerically equivalent to another group, named six, this can be done by a direct appeal to Sense, placing the groups side by side ; or, by an indirect appeal through Intuition, the ratio symbolized in 3 + 3 = 6 being intuited with a certainty equal to that which accompanied the vision of the groups. For this intuition to be possible, the sensible experiences must have preceded it ; but once formed, the sensible experiences pass into symbols and are intuited. Just as Algebra in virtue of its generality can effect operations which are difficult to Arithmetic, and operations which are impossible to Arithmetic, so Intuition can detect relations which' are obscure to Sense, and relations inaccessible to Sense. Thus although it it easy to see that three objects placed beside three others form a group equivalent to a group of six, the acutest eye would fail to detect at a glance that sixty objects placed beside sixty others were equivalent to a group of one hundred and twenty ; but where Sense is bewildered by the multiplicity of objects, In-tuition sees at a glance the equivalence of their ratios. We may therefore define Intuition as Mental Vision, or as the Perception of Relations.* It is differenced from Sensation on the one hand in that it sees objects not only as they affect Sense, but also in their relations to each other, and sees these present as constituent elements of the group so that the intuition of an object includes a much wider range of Experience than a perception of the object. From Conception, on the other hand, it is differenced by its restriction to definite particular objects and relations, always therefore reproducing the forms of sensible experiences ; whereas Conception never does this, being in its nature analytical, general, abstract.

It is often impossible to demonstrate (to Sense) what it is impossible to doubt when intuited. Thus after proving that the area of a spherical triangle depends on the sum of its angles, we cannot exhibit to Sense that any two spherical triangles which have their sides and angles equal, each to each, have equal areas.; because if they are symmetrical angles they can no more be made to coincide than a right and left hand can get into the same glove.* But the intuition of equality is perfect in this case.

147. This perception of ratios or intuition of equality, that two things equal to a third must be equal to each other, is in constant requisition. If I have a vacant space, and a box which I wish to place in it, my sensible perception of the relation between the boundaries of the space and those of the box is too imperfect for guidance ; I cannot see the equality, but I can measure with a foot-rule both the space and the box, and finding that each contains this measure the same number of times, I conclude that the box will not go into the space ; or, if the space contains the measure and something over, I conclude the box will go into it.

148. Intuition under its ideal aspect is Judgment. Demonstration is the exhibition of the grounds. We call judgment, 1°, intuitive, when the relations seem to embody experiences which are not specified or cannot now be specified, although originally they were capable of being so; and, 2°, discursive, when many or all of the experiences are or can be specified. The conclusion which is seen so rapidly that its premises are but faintly or not at all recognized is said to be seen intuitively : it is an organized judgment. Its rapidity and certainty, together with our reliance on all spontaneous actions, have led to the notion, that Intuition is a source of peculiar validity. But Intuition is ideal vision, and is no less liable to error than sensible vision. It also has its illusions, and needs the control of Verification. In the perception of an object we are unconscious of the many evanescent muscular . feelings by which its distance is estimated and its shape inferred. These relations are intuited; and because the judgments are so rapid, and so inevitable, we regard the perception of distance and the shape of the object as given in an immediate apprehension. Analysis, however, discloses that the evanescent processes of which we are unconscious must have taken place; and in the early days of Experience the processes took place slowly, consciously. All our other intuitions are organized experiences, groups of neural processes which originally were isolated. They are to the mind what automatic actions are to the body. Their mechanism is concealed because their action is so easy and so rapid. Among the automatic actions there are tricks of Habit, peculiar to the individual, tricks peculiar to his family, and tricks peculiar to his race ; these are all perfectly irresistible, al-though often serving no purpose, and representing no vital necessity. Among our intuitions there are likewise tricks of Thought and Feeling, i. e. some personal prejudices, or traditions of the family, sect, nation ; and these are irresistible even when Reason sees them to be absurd. We have to be on our guard against illusory Perception, we must be equally on our guard against illusory Intuition. In both cases the illusion arises from accepting what is only inferred as if it were really seen.

149. I will select examples of illusory intuition not from Theology or Ethics, where some intuitions which are demonstrable fallacies are often appealed to as final arbiters, but from Science, e. g. the. once common, now exploded, induction of "Nature's horror of a vacuum," and the more common and still popular induction of weight being inherent in bodies two judgments which had become so organized that they passed for intuitions. The first has long been recognized to be a fiction ; the second, which seems like direct experience, is an illusion. When daily and hourly familiarity showed that bodies had weight, and that no alteration in their condition affected this weight, but that whether solid, liquid, or aeriform, the balance proved them to preserve this quality throughout these changes, experiment seemed to guarantee the intuitive judgment; and the most sceptical regarded weight as an absolute quality belonging to the very nature of bodies, since it was a quality which did not alter under changing conditions. Now as a judgment expressing the facts of experience this intuition of weight was exact ; but as an inference respecting the absolute quality inherent in bodies, the intuition was illusory, that is to say, it was an induction, not a real intuition. It was proved to be illusory when Newton showed that gravity was a relation dependent on the position of bodies. The weight of a body was unaffected by any change in the condition of form, structure, or combination of that body, simply because these conditions were not co-operant factors ; the phenomenon did not express them, did not depend on them, therefore it could not vary with their variations. No sooner were the real factors of gravity detected than weight was found to vary with them, and thus, like all other qualities, was seen to be variable and relative. The illusion consisted in inferring that what was true of bodies under all changes which had been investigated, would be equally true of bodies under all changes whatever, and that no investigation of other relations would disclose a variation in weight. But this inference needed verification; and it needed it all the more because when men observed that bodies did not vary under certain varying conditions, they ought to have suspected that this constancy was an indication of the observed conditions not being factors, since the real factors could not vary without a corresponding variation in the phenomenon.

150. The reader has doubtless noticed with surprise and misgiving that in the foregoing passage the word Intuition has a wider range than usual, wider indeed than my own definition of it as the perception of relations. Not only does it there represent Judgment, but even Induction. My purpose was to fix attention on the possible illusions of Intuition, because so many writers regard it with a sort of superstitious reverence, as if coming from a supra-natural source and further I wished to insist on the essential uniformity in all psychical processes. Intuition is beholding: Anschauung, the Germans call it. We have sensible intuitions, rational intuitions, and moral intuitions, each of which is liable to the same possibility of illusion. Our intuitions of Space, Time, Motion, Quantity, etc., are constructed out of sensible experiences which lie so far back in the dim past that the subtlest analysis is tasked to detect their elements, and therefore many philosophers regard these intuitions as anterior to all Experience, being original endowments of the organism. The sense in which this is acceptable is expounded in § 23. A similar remark applies to our rational intuitions, such as Substance, Cause, Equality, etc., which, in the gratuitous and restricted meaning of the word Experience (that of sensible affection), never could have been experienced. In a less degree there is a similar difficulty with respect to such moral intuitions as Freedom, Responsibility, Duty, etc.

151. The validity of all these intuitions depends on their reduction to identical propositions ; in other words, whether the relations are what we see them to be. The possibility of error lies in the possibility of our supposing that we see what we only infer. Intuition must there-fore be distinguished from Induction as vision is from Inference. Intuition is the clear vision of relations ; Induction is the inference that the phenomena now seen in this particular case, or these few cases, will be equally visible in many, or all, cases resembling these. That the angles at the base of an isosceles triangle are equal, or that the square of 5 is 25, are intuitions, and admit of no doubt when the relations are clearly seen. We then know that the relations are what they are seen to _be : for we have before us all the elements expressed by the propositions. But "all crows are black" is an induction : it is an inference that whenever a bird is found presenting the general characters classified under crow, it will also present this one character of a black plumage. The uncertainty of this inference lies in our not having before us all the generating conditions, and therefore we cannot know that there are not birds possessing all the other characters of the crow, and with these a white or gray plumage. We cannot reduce our proposition to an identical proposition ; although if we choose to throw it into that form and say "all black crows are and must every-where be black," it has the same irresistible certainty that belongs to our propositions about angles and squares.

152. It is important to bear in mind the grounds on which we admit the validity of intuitions, because, as was formerly hinted, there are judgments which have the characters of intuitions (namely, immediate apprehension and irresistible conviction) which are nevertheless illusions, — are spurious intuitions. On the other hand there are inductions which although formulated without a clear vision of the generating conditions, are nevertheless freed from all uncertainty by being enunciated so as to include these, and to exclude all other conditions. The uncertainty lies precisely in the ignorance of whether the cases to which our induction is extended are, or are not, legitimately classed with the cases which furnished the inference; and it would necessarily cease if this assumed homogeneity could be verified ; or if the induction were converted into an identical equation. Thus although we do not know all the conditions which determine the death of animals, the induction that all animals must die is reducible to an identical proposition by the assumed homogeneity of the terms : we know that all animals must die if " all animals include only animals precisely similar in nature to those that have died, and are placed under precisely similar conditions ; and if with this intuition of known terms we exclude all unknown terms, our proposition becomes equally certain with a proposition about angles. Nor is this invalidated by the possibility that in other worlds or in other times there may be animals precisely resembling those known to us which will not die. That is to introduce the very element our proposition has excluded. In a space of two or of four dimensions many geometrical propositions which relate to a space of three dimensions would not be true. Who doubts it ? Who expects that the same results can be the product of different factors ?

153. " Our judgments," says Reid, " are distinguished into intuitive, which are not grounded upon any preceding judgment, and discursive, which are deduced from some preceding judgment by reasoning." In psycho-logical strictness every intuition is grounded upon some preceding experience, and. this may be either simple perception, or a group of complex judgments. The difference between the intuitive and discursive judgments lies in the different degrees of rapidity with which the constituent elements of the groups are apprehended. Suppose I see a glass accidentally swept from the table, I have an in-tuition of the consequences ; this makes me snatch at the glass, to prevent its falling. The judgment and the action are instantaneous; and if I am asked why I exerted myself to catch the glass, I answer that I knew the brittle nature of glass, and saw that if it reached the ground it would be smashed. But these reasons which are furnished by Reflection were not distinctly present to my mind, although they were the organized experiences which determined my act. The proof that they were so is evident in the fact that if a child or savage had witnessed the fall no attempt would have been made to arrest it ; or if instead of a brittle glass, a tin mug had fallen, I should have been impassive.* A discursive judgment is therefore what in its more exact and verifiable form is called a demonstration, namely, a judgment of which the constituent elements are shown instead of being simply felt.

154. Intuition is distinguished from Demonstration as an operation indicated but not performed. By an intuition the ratio of the square root of a to the square root of b may be seen to be identical with the fraction 2/3. To demonstrate this is to perform the operation indicated, and to show that if the value of a is here 4 and the value of b is 9, while the square root of 4 is 2 and that of 9 is 3, the conclusion that a/b = â is the identification of the two expressions, since a is 2 and b is 3. Obviously the correctness of this operation, whether indicated or performed, whether intuited or demonstrated, depends on the correctness of the primary assumption that the values assigned to a and b are 4 and 9.

155. Intuition is of much greater range than Demonstration, because the greater fund of Experience on which we rely is too complex, and drawn too much from the for-gotten past for us to be capable of showing all the successive steps which Demonstration requires. All the great discoveries were seen intuitively long before it was possible to exhibit the correctness of their grounds, and to disentangle the involved data. But we must not on this account place unrestricted confidence in Intuition, for we know but too painfully how many absurd speculations have been propounded on " intuitive grounds." Demonstration is not an instrument of discovery, but a means of control. Intuition is seeing ; Demonstration is showing. What is seen, and what is shown, may be illusory ; they are only proved to be objectively valid when each inference has been reduced to its corresponding sensible.

156. "The method of demonstration in Mathematics," says Hutton, t " is the same with that of drawing conclusions from principles in Logic. Indeed the demonstrations of mathematicians are no other than a series of enthymemes ; everything is concluded by force of syllogism, only omitting the premises which occur either of their own accord or are collected by means of quotation." In other words, it is the exhibition of a necessary connection, or identification of the conclusion with its premises. Mathematical demonstration is the type of exactness be-cause the validity of the premises is never questioned, they are either intuitively evident, or have been rendered irresistible by previous demonstration. When the premises are thus unquestioned the certainty of the result is necessary.

A demonstration is the exhibition of a necessary connection between the proposition to be demonstrated and one or more other propositions which have already been shown to be true, or may be assumed to be so. This assumption will not affect the rigor and consistency of the operation ; but it may be wholly at variance with objective fact. The terms may be absurd, yet the form of the operation correct. The truth of a proposition is not given simply by showing that it is a necessary consequence from some preceding proposition ; that is only showing the logical operation to have been irreproachable ; and an operation may be accurately performed although its premises are inexact. A proposition is objectively true only in as far as it exhibits the equivalence of inference and sensation; and this equivalence may be exhibited directly or indirectly : an inference, or a demonstration, once verified, has all the value of the sensations by which it was verified ; that the square root of 25 is 5, is not less absolutely certain than that 5 is 5.

157. Demonstration is the exhibition of the equivalence of propositions, the presentation of some object or property which is not apparent, through its equivalence with some object or property which is apparent. Since the presentation is thus always mediate, — always by means of something else seen by Intuition to be equivalent, and therefore convertible, the mediation of Intuition may be effected by a succession of equations, or by one. But whether the demonstration depend on a succession of steps, or on only one step, it is always an intuition of equivalence.

158. Had this been clearly apprehended there would perhaps have been less misplaced ingenuity exerted by mathematicians in efforts to demonstrate, geometrically, propositions which are capable only of logical intuition, and for which geometric constructions are superfluous, the intuition being a mental construction. For example the proposition respecting parallel lines : attempts with-out number have been made to demonstrate it, and all attempts have failed ; yet Laplace admits that the "enunciation alone carries along with it the fullest conviction"; why then seek for evidence of what is intuitively evident ? That a geometric proof is impossible, does not disturb the certainty that a line perpendicular to one parallel is perpendicular to the other,— a certainty which belongs to all identical propositions, and which cannot be increased by any geometric exhibition. Mr. De Morgan, indeed, denies that the definition of parallel lines — "lines which are equidistant from one another at every , point "—gets rid of Euclid's postulate ; for he says, "in this case before the name parallel can be allowed to be-long to anything, it must be proved that there are lines such that perpendicular to one is always perpendicular to the other, and that the parts of these perpendiculars intercepted between the two are always equal." Mr. De Morgan thinks that in defining parallel straight lines to be such that any two points in the one are at equal distances from the other there is an assumption without proof, — since it cannot be stated â priority of two straight lines that more than two points of the one shall be at equal distances from the other. I admit that there is an assumption here, but it is the assumption of homogeneity which is fixed in the definition. Two equidistant points suffice,— and to prove that they are equidistant is to prove that A is A. What is meant by parallelism is equidistance ; the two points are prolonged indefinitely, and as according to the assumption of homogeneity the lines are nowhere changed, nowhere cease to be parallel, what was true of the two points remains true of their infinite prolongation. The one act of Intuition by which the relation of two parallel lines, however small, is perceived, is the Intuition of the relation prolonged to infinity by universalizing the terms.

( Originally Published 1874 )

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The Passage From The Abstract To The Concrete

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Intuition And Demonstration

Axioms And Their Validity

Necessary Truths

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