Volume 2 — THE MONIST. (continued 5)

Negative numbers always presuppose a contrast and such conditions arise naturally wherever the fundamental ideas imply two opposite directions, for instance past and future in time, credit and debit in business, etc. It is a matter of course that there are in reality as little either positive or negative numbers, as there are positive or negative colors or sounds.

Concerning the necessity idea, Ehrenfels says: “Nobody will consider it as possible that five plus seven will in some cases make any other number than twelve. We are confident that the same addition will under all circumstances yield the same sum.” Ehrenfels grants the psychical certitude of this but not the mathematical, and thinks that on this point there is a difference of opinion allowable. Here we disagree from Ehrenfels and refer the reader to former articles on kindred subjects in The Monist, especially the article on “The Origin of Thought-forms,” Vol. II, p. 111. We must bear in mind that in mathematics we are moving in a realm of pure forms and the statement 7 + 5 = 12 is, as the Germans express it, eindeutig bestimmt, i. e. it is determined exhaustively in one and the only one possible way. The numbers 7 and 5 being rigid, their sum and their product will also be rigid.

This difference of opinion may be contingent upon a difference of the conception of the a priori. Ehrenfels defines as “a priori” such judgments which having come into our possession, are readily accepted without proof. We follow Grassmann in rejecting the acceptance of anything without proof, including the idea of mathematical axioms. The a priori in our terminology becomes identical with that which pertains to formal thought: and it would make no difference whether the instance presented is as simple as 1 + 1 = 2 or extremely complex as are the differential calculus and logarithms. Accordingly we disagree also from Ehrenfels when he finds even in such additions as for instance 825 + 217 = 1042 vestiges of an a posteriori character. The employment of the logarithms accordingly appears to Ehrenfels also aposterioristic because the fruits of other peoples’ labors are utilised!

Concerning John Stuart Mill’s view of the subject, Ehrenfels says that “it is still deeply entangled in the errors of that conception which it so bitterly opposes, viz. in the formalism of the old purely a priori conception. For only he who adheres to the view that all mathematics are deduced from a few axioms can think of attributing to those axioms the highest degree of plausibility which is assumed for them on the ground of comprehensive deduction.” We agree with Ehrenfels’s objection to Mill, but we cannot agree with his view that mathematics derives any elements from a posteriori elements, although we grant that quite new departments are created simply by a different employment of certain functions. Accordingly mathematics cannot be derived from a few axioms only but is the products of certain functions.

Ehrenfels calls attention to the fact that the mathematician operating with symbols often forgets entirely what he has to think of in connection with these symbols. “This is not strange,” he adds, “for thoughtless word-combinations present analogous instances, yet it is strange that the result almost without exception comes out right; es stimmt!” We object here; operations with mathematical symbols are not thoughtless combinations, at least, they are not meaningless. They are operations not with things, but with symbols representing certain relations among things. When gamblers play with chips representing real money, they need not think during the game of the value represented by a chip, and yet when the account is made, the result attained with the assistance of the chips will come out right. There is no reason to wonder at it. Chips like mathematical symbols might in a certain sense be called thoughtless, for certainly they do not think; but they are not thoughtless in the sense that they are meaningless, that nothing is thought by them.

Ehrenfels apparently sees a problem where there is none and this is closely connected with another point. He looks upon the mathematician’s inability of thinking out in every respect the objective meaning of mathematical symbols as a shortcoming of man’s intellect. While it appears that we cannot think anything by many mathematical symbols (for instance by a⁰ = 1) except the symbol itself, the enormous success of mathematical thought is evidence that they must have some definite meaning although it is to be excogitated only by those beings who will transgress the average intelligence of to-day, the first germs of whose existence are the mathematical geniuses of the present generation. It appears to us that undoubtedly every mathematical symbol has a definite meaning, representing the result of some function. That the result will sometimes be unattainable or unrealisable, that especially all operations with zero make the whole calculation indefinite (which naturally arises from the nature of zero) does not alter the truth of this proposition in the least.

We have to make one additional remark. The peculiarity of mathematics that we do not throughout our operations think out the meaning of the symbols is not a shortcoming of our intelligence, but the strength of mathematical science. The advantage of all the formal sciences and especially of mathematics consists in this that we need not think out every detail, but that we can, through the assistance of mathematical symbols, perform the most intricate operations with machine-like exactness. The economy of thought produced in this way is not a deficiency of man’s mind, but a virtue.

Prof. H. Höffding (in No. 4) insists upon the causal law as being indispensable in psychology. There are some people and among others his colleague Professor Kroman who regard moral motives as an exception. “Yet,” says Professor Höffding, “should the decisive moment of a decision not be determined by the causal law, the will could never be determined through a reflection on the possible effects of the action and thus every reason would be missing to attribute to man any responsibility.”

E. Grosse expatiates on the proposition to apply the comparative method of ethnology to æsthetics. Ferd. Rosenberger proposes the following programme: “Knowledge is power; activity based upon such power is the cause of happiness. Therefore with the increase of knowledge, there is an increase of happiness, successful activity however is impossible without virtue. Therefore we conclude that an increase of happiness will be accompanied with an increase of virtue.” A. Marty in this his ninth article blames Steinthal for having misrepresented the eighteenth century theories of the origin of language.

M. Offner reviews Dr. Charles Richet’s reports of his telepathic experiments, but the reviewer cannot assent to Richet’s opinion “that these facts possess a strange coincidence and that they are, probably, the result of a relation and not of pure chance.” (Leipsic: O. R. Reisland.)

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PHILOSOPHISCHE MONATSHEFTE. Vol. XXVIII. Nos. 1 and 2.

CONTENTS:

Zum Begriff der unbewussten Vorstellung. By E. v. Hartmann.

Ueber das Gebet. Ein religionsphilosophisches Fragment. Sendschreiben an Herrn E. Renan in Paris. By M. J. Monrad.

Werke zur Philosophie des socialen Lebens und der Geschichte. Erster Artikel (H. Spencer, Sociologie, Bd. III). By F. Tönnies.

Recensionen: H. Münsterberg, Beiträge zur experimentellen Psychologie No. 3. Neue Grundlegung der Psychophysik. By Th. Ziehen. W. Enoch Der Begriff der Wahrnehmung. By P. Natorp. Ch. Bénard. L’esthétique d’Aristote et de ses successeurs. By A. Döring.

Litteraturbericht.

The well-known philosopher Edward von Hartmann defines his position with reference to the idea of an unconscious representation. Granting that there are no unconscious sensations, perceptions, conceptions or memories, because feeling either is conscious or not at all, he introduces the idea of unconscious representations again as the most adequate determination. He says, “Either we must renounce all speaking and thinking of non-sensual objects or we must be satisfied with using figurative expressions.”

M. J. Monrad, a Norwegian, argues, in the second article against M. E. Renan’s theory of prayer, whom he had visited some years ago in Paris, that prayer has after all an effect upon the objective world and it is not limited to a merely subjective and psychological influence. Monrad presupposes a belief in God, prayer bringing the individual in unison with God, changes the will of the individual into a co-ordinate willing of God and thus renders the individual a co-worker of God. This, however, should not be conceived to take place by magic and in contradiction to nature, but through nature, man using the laws of nature.

F. Tönnies of Kiel gives an exposition of Mr. Spencer’s social views which are, briefly expressed, “the final victory of society over the state.” Professor Tönnies answers that “we all want a higher civilisation, but the development of a higher civilisation is not conditioned by the final victory of society over the state. On the contrary, it may be said that it depends upon a victory of the state over society in so far as public rights will supersede private rights.... The truth is that state and society are contingent, the one upon the other and also limiting each other.” (Berlin: Dr. R. Salinger.)

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FOOTNOTES:

Vol. II. April, 1892. No. 3.