Chapter VI: Part 6

The ideas of causality and of the conservation of matter and energy are not derived from experience in the narrow sense of the word, not from sense-experience, but from experience in the wider sense of the word, i. e. from sense-experience arranged with the assistance of formal laws. We should not forget that mere sense-experience exists only in our abstract thought. In reality all sense-experience is relational, it is inseparable from its form. Form and the laws of form are not something purely mental which is transferred to the world of reality, form is something real, it is objective, it is a quality of the facts and the thought-forms of mind are a part and a product of the formal structure of the universe.

The ideas of causation and of the conservation of matter and energy are not prior to experience, they are a part of experience, i. e. experience in the wider sense. They are not part of the sense-experience, but the results of our experiments with purely formal thought-operations, and being the vital instrument or organ of cognition they are the condition of all methodical experience.

IV. WHY IS MR. MILL’S PROPOSITION UNTENABLE?

In practical life we all expect that 2 × 2 will under all circumstances make 4, and not 5. We reject Mr. Mill’s idea that there may be even a possibility of the latter. Is our expectancy really due to _a posteriori_ experience which having been repeated so often in the lives of our ancestors is now so firmly rooted in our minds that we imagine it to be necessary and _a priori_? No! certainly not. The experiences of our race in the struggle for life has produced our ability of thought, but it has nothing to do with the apriority of the products of formal thought-operations. A statement of formal thought such as “twice two makes four,” cannot be compared to statements of sense-experience such as that sugar has a sweet taste. There may be a moment in which the taste of sugar will be bitter to our tongue. This is quite possible. But to say that twice two might in the future or in any other world, as Mr. Mill maintains, make five is nonsensical, and the possibility of this assumption cannot be placed in one line with the possibilities of extraordinary and exceptional sense-experiences.

What does “twice two makes four” mean? Two means 1 + 1, and twice two means 1 + 1 plus 1 + 1. This sum is called “four”; and what we call five is 4 + 1. To maintain that the operation 2 × 2 might produce the result 5, is to admit conditions which have been excluded. Pure forms are not like animals which multiply; they are and remain such as they have been posited. When we put two amœbas into a glass and then add two other amœbas, it is quite possible that in the mean time one of the first set has divided into two. In that case we would have five amœbas. But the operation 2 × 2 cannot breed any additional units, so as to produce a greater number than the sum of 1 + 1 + 1 + 1. Nor can we let any of these units disappear into naught, so as to produce the result 3, without committing some inconsistency in our thought-operations, for the products of our thought-operations are rigid and must remain such as they have been posited. They are not animals blessed with fertility but pure forms and nothing but pure forms.

How could Mr. John Stuart Mill overlook so palpable a contrast as that between formal knowledge and sense-experience? He was apparently prejudiced against the term “a priori,” which as we freely confess is a very poor and inadequate expression. Mr. Mill himself states the cause of his prejudice in his autobiography. He says:

“There is not any idea, feeling, or power, in the human mind,
which, in order to account for it, requires that its origin
should be referred to any other source than experience.”

Mr. Mill was justly exasperated against anything _a priori_, for in his time, it had become customary among certain philosophers to classify all pet superstitions which could not be proved by experience as _a priori_. Mr. Mill continues:

“Whatever may be the practical value of a true philosophy
of these matters, it is hardly possible to exaggerate the
mischiefs of a false one. The notion that truths external
to the mind may be known by intuition or consciousness,
independently of observation and experience, is, I am
persuaded, in these times the great intellectual support of
false doctrines and bad institutions. By the aid of this theory
every inveterate belief and every intense feeling, of which
the origin is not remembered, is enabled to dispense with the
obligation of justifying itself by reason, and is erected into
its own all-sufficient voucher and justification. There never
was such an instrument devised for consecrating all deep-seated
prejudices.”

We appreciate the cause of Mr. Mill’s prejudice, but we cannot agree with him. And Mr. Mill is mistaken when he imagines that a rejection of apriority will abolish false doctrines and superstitions. On the contrary. The recognition of absolute necessity based upon the universality of formal thought will alone be a safe basis of science through which we can reject _prima facie_ the wrong pretensions of superstitions and pseudo-science. If we assume with Mr. Mill that all formal knowledge partakes of the nature of sense-experience, that there is no difference between the two, no general judgment would be allowable and the idea of necessity would be inadmissible. These consequences are accepted by Mill. In that case science would lose its value and philosophy would be without foundation. The most absurd superstition could not be rejected off-hand on the ground of being contrary to that which through logic and other formal sciences has been found to be necessary and a condemnation of any superstition on the part of science would be mere arrogance. Pseudo-science would have the same right with true science.

It is obvious that without being obliged to consider the apriority of formal laws as innate, we need not accept the consequence of Mr. Mill’s philosophy. We can and we do retain the idea of necessity and we consider it as the corner-stone of all science.

V. THE MEANING OF “NECESSARY.”

We have to be on our guard lest we introduce some mystical element into the idea of necessity. There is nothing mystical about necessity. Necessary means that a certain operation, if it is exactly the same operation as another one, will produce exactly the same result. When we posit two units twice, we shall have the same result as when we posit one unit four times; and we call this result four. We shall reach the same product whenever we repeat the same operation. Knowing that we shall always reach the same result, we can, _a priori_ (or beforehand) and with certainty, determine the result of certain operations after we have mentally gone through the same operations for all possible cases and under any imaginable conditions. That a perfect apriority with an unfailing certainty is possible only in the domain of formal thought is natural. The reason is that we know our thought-forms exhaustively. They contain nothing but that which has been predicated about them. Our sense-experience however is always piecemeal and never exhaustive.

Comprehension is actually a tracing of form relations and a formulating them in concise statements. The scientist’s work is based upon the methods of measuring and counting (i. e. the methods of formal sciences) and he undertakes to show that certain phenomena, different in some respects, are the same in other respects, that their sameness can be stated in a comprehensive and exact formula. In this way he marks out their determining factors in terms of formal thought (for instance in numbers), so that we can compute them and predict them according to their determining factors, so that we can know, according to their conditions, that they will be always as they are.

The importance of formal thought is paramount in science and the problem about the necessity which attaches to the laws of formal thought is the fundamental problem of philosophy.

There are many philosophers, still, troubling themselves to solve the problem in the fashion of Schopenhauer or of Mill or looking upon the problem as insolvable. We do not doubt that the solution here presented is the only possible solution which as soon as it is understood will find a general acceptance.

Must it be added that the solution of this fundamental problem does not involve the ready solutions of all other minor problems? Oh no! We all know that the solution of one problem is only a stepping-stone for attacking other problems. The possibilities of progress are as unlimited as the scope of cognition. Light on this general subject gives us hope that we shall succeed in throwing light also upon other subjects which are still shrouded to the philosophical inquirer in impenetrable darkness.

VI. MODERN LOGIC.

The problem of modern logic is at bottom no other problem than that of formal thought and of the origin of thought-forms. Professor Dewey in the excellent essay which appears as the leading article of this number says: “Any book of logic will tell us what this conception of thought is: that thought is a faculty or an entity existing in the mind apart from facts and that it has its own fixed forms with which facts have nothing to do—except in so far as they pass under the yoke.” This is the old conception of thought, rightly criticised by Professor Dewey, for, closely considered, it turns out to be dualistic. However, as soon as a proposition is recognised to contain or to rest upon dualism, it becomes a problem. The problem of modern logic is, How can we arrive at a monistic conception of logic, how can we rid ourselves of the dualism on the one side of facts not yet rationalised by the method of thought-forms and on the other side of mind with its empty thought-forms assimilating facts to its own nature.

Our solution of the difficulty has been proposed, in the sense outlined by Professor Dewey, in “Fundamental Problems.” Professor Dewey, according to our opinion, is right when he says there is no such a thing as transcendental thought, or pure thought, thought by itself. And there is no such a thing either as fact, crude irrational chaotic fact. The world of fact, indeed, is a cosmos and no chaos; there never was a chaos and never will be a chaos, for the laws of form are an essential and the most characteristic feature of the world.

Can there be any question how the order found in thought-forms originates in a world in which the inorganic and unfeeling mineral crystallises in a regular shape according to strict mathematical laws, i. e. the laws of form? A world in which the plant grows not otherwise than according to strict mathematical laws building up roots and stem and leaves and petals and stamens and all other organs obedient to a certain plan which will vary according to circumstances, but throughout consistent with the principles of formal laws? Can there be any question that in this world of cosmic harmony thought-forms will develop in feeling beings as a microcosm exhibiting the same regularity and conformity to law as do in this world all other things animate and inanimate? Our pure, i. e. merely formal, thought is an abstraction which serves the purpose of comprehension. And so is the concept “matter,” being that which produces sense-impressions. There are no such ghosts as pure matter or pure thoughts in reality.

Modern logic, so far as we conceive it to be right, is by no means an overthrow of the old formal Logic, generally called Aristotelian. It is simply an amendment made in order to exclude an erroneous interpretation. And so is modern mathematics not so much a revolution as an extension of the old Euclidian system. It is a revolution only against a certain unclear conception of mathematics according to which this science is said to rest upon axioms, these axioms representing absolute truth—unprovable, incomprehensible, and mysterious.[21]

The main truth of monism is that reality forms one indivisible whole and all our concepts are mere abstractions representing certain parts or certain features of the whole. As soon as we try to think of any of them as things in themselves we become involved in inextricable contradictions. It appears as if the formal sciences contained some truths which were absolute and independent of actual reality. But let any one think of any number, of 2 or 3, and he will soon find that conceived as absolute beings they are meaningless and unthinkable.

This is not to say that numbers are phantoms, but that conceived as absolute beings they are phantoms. Numbers and all formal concepts represent something real, they represent pure forms. And form is as much a part and feature of reality as is matter and energy.

P. C.

FOOTNOTES:

[21] Hermann Grassmann, one of the founders of modern mathematics, has called attention to the fact that Euclid had a clearer conception of the fundamental concepts of mathematics than his ill-informed translators and interpreters. Grassmann says in his _Ausdehnungslehre_: “From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free, Euclid incorporated the former among his postulates (αἰτήματα) while he separated the latter as common concepts (κοιναὶ ἐννοιαι)—a proceeding which even on the part of his commentators was no longer understood, and likewise with modern mathematicians, unfortunately for science, has met with little imitation. As a matter of fact, the abstract methods of mathematical science know no axioms at all.”—Quoted from _The Open Court_, Vol. II. No. 77, _A Flaw in the Foundation of Geometry_, by Hermann Grassmann, translated from his _Ausdehnungslehre_ by μκρκ.

BOOK REVIEWS.

BELIEF IN GOD. Its Origin, Nature, and Basis. Being the Winkley Lectures of the Andover Theological Seminary for the Year 1890. By _Jacob Gould Schurmann_. New York: Charles Scribner’s Sons. 1890.

The learned Sage Professor of Philosophy in Cornell University, after tracing the historical origin and development of the belief in God, expresses his conviction that the problem of the modern theist consists in the union of the Aryan and Semitic modes of interpreting existence. We shall then have “a synthesis of the Father of all spirits with the ground of all nature.” This is the hypothesis developed in the course of the lectures delivered by Professor Schurmann last year before the Andover Theological Seminary on the Winkley foundation, that form the contents of the present volume. The theism embodied in that hypothesis is called by the author _anthropocosmic_, because, while it is based on the facts of the universe and those of human nature, the universe must be interpreted in the terms of man, and not man in the terms of the universe. The key to the universe is the self-conscious spirituality which makes us selves and persons. Hence anthropocosmic theism is the doctrine of a Supreme Being “who is ground both of nature and of man, but whose essence is not natural but spiritual.”

Before considering the evidence for this hypothesis, let us see what the author has to say with reference to the logical character of the belief in God. He shows that _agnosticism_—of which he treats under its three significations as referring to the method of knowledge, the object of knowledge, and the subject of knowledge—is not consistent with the insight into nature and the constitution of knowledge gained by the Newtonian method of hypotheses and verifications. Science, as opposed to pure phenomenalism, assumes that what has not originated in the percipient subject is objectively real, and it postulates the universality of law in nature, a postulate which is the expression of a conviction of “the unity and universal inner connection of all reality.” The objective world cannot be understood without reference to our own conscious experience, and as the only reality we know from the inside is a spiritual _ego_, self-conscious spirit must be ascribed to the one ultimate reality whose existence science assumes, as that which will alone satisfy the requirement of unity in the midst of change.

It might be objected here, that the existence in man of a spiritual _ego_ requires proof before that of a universal spirit or world-soul can be inferred from it. The author takes the existence of the _ego_ for granted, a course which is quite allowable from the theistic standpoint, although, in the face of what is now known as to the dissolution of the ego under abnormal conditions of the organism, not scientific. Having made that assumption and inferred the existence of God from that of the human spirit, the author explains the nature of the one by reference to that of the other. He tells us, that the finite spirit is identical, within the limits of its range, with the infinite spirit, because it is an _ego_, and that in the _ego_ we have, not merely a mode of the divine activity, but a part of the divine essence. Such being the case, the author has no difficulty in inferring the attributes of God from the phenomena exhibited by man. Thus God is a God of righteousness because the moral capacity in the human spirit must have its ground in the infinite Spirit. Again universal benevolence or love is the ideal of which human morality is the realisation; hence we must conceive of the Spirit of the universe as a God of love.

We do not think the author’s final conclusion, that “the phenomena both of the universe and of human life require the thinking mind to postulate a Supreme Ground of things which we are entitled to describe as self-conscious Spirit and loving Father,” is warranted by his premises. But we can accept the statement that our knowledge of God must continue to grow with our knowledge of man and nature. Through these alone can we know Him, but the difficulty is to interpret the revelation. Let it be admitted also that the end of nature is the production of man, and that what is referred to by the author as the human spirit is “the organ of that communication of God which is the end of the universe.” This does not in reality throw any light on the nature of God. The utmost that can be said is that, as man is an organism possessing certain functions, the universe, viewed as God, must have an organic existence with functions _corresponding_ to those exhibited by the human organism.

The author’s reasoning in support of the belief in God as cause or ground of the world, and as realising purpose in the world is very ingenious. He affirms that the creational form of the argument from causality is insufficient. It satisfies the devotional needs of a certain class of worshipers, but what the religious, as well as the scientific, consciousness demands is a God “here in the world, not there outside of it or making it.” It has not yet been shown that the universe has had a beginning in time, and the argument in favor of the eternity of matter ends with an assurance of the eternity of spirit alone. Spirit is the eternal reality, and nature its eternal manifestation, the latter being no more separable from the former “than the spoken word from the thought it symbolises.” The causal relation is, however, absolutely necessary for our apprehension of the facts of the universe, and as it cannot be interpreted without contradiction as an action between independent beings, it must be explained as the eternal dependence of the world upon God. This implies that God must be volitional as well as self-conscious; “for without will there could be no activity, no efficient causation, no material universe.” The universe is thus the eternal expression of the divine will. But what is the purpose realised in creation? The activity of the divine will precludes the notion of a blindly working nature. As creation is the eternal self-revelation of God, the supreme and preconceived end of all things must be the glory of God. But man is indispensable for the attainment of God’s glory, and therefore the end of nature as a realised scheme of divine ideas is the production of man. The volitional and teleological arguments as thus stated by the author are consistent with the theory of evolution developed by Darwin, but they may be combatted on other grounds connected with the conditions of the existence of God as one with Nature. With this observation, we must leave Professor Schurmann’s very thoughtful book which, although for the reasons we have stated, not conclusive, presents the theistic argument with great clearness and in its strongest form.

Ω.

JUDAISM AND CHRISTIANITY. A Sketch of the Progress of Thought from Old Testament to New Testament. By _Crawford Howell Toy_. Boston: Little, Brown & Co. 1890.

This work is another contribution to that genuine history which is alone competent to impart any true instruction. In it the author undertakes to give an account of the genesis of Christianity as a child of Judaism.

It seems to be the thesis of the author that those conceptions and beliefs that characterise any form of religion are rather determined by the social evolution than that the social progress and its features are determined by the evolution of ideas among which the religious ideas are specially influential. In his introduction the author sketches out his view of the general laws of social progress as the same are related to religious thought. He notices in history the tendency of ethnic religions to give way to or to develop into universal religions, and argues that Christianity is destined to overcome all its rivals and prevail universally. This kind of a conclusion is a natural one to follow from the theory that the character of thought is determined by social circumstances and progress. But if it be true that the special course of the evolution of thought and its characteristic form at any epoch is determined by causes that are uncontrolled by social conditions, that as between thought and society thought is the masterful factor, then quite another conclusion may ensue. But the dubitable nature of the main thesis of the work does not much detract from the great excellence of the work in general. As a history of the evolution of Jewish religious conceptions and beliefs from the very first until the establishment of Christianity, it is in the highest degree interesting and instructive.

After a discussion of the literature of the Jews and the formation of the canon, the author proceeds to describe in full detail the nature genesis and mutations of the cardinal religious doctrines as they revealed themselves to the Israelite, Jewish, and early Christian insight. The entire body of the data are interpreted in consonance with the modern scientific idea of the organic nature of society. Jesus is regarded as the master spirit that created the Christian Church, and Paul whom many would install as the real author of the same is accordingly given only a second place. Altogether it may be said that Professor Toy has given us a most valuable contribution to religious history and to the scientific interpretation of the same.

ρσλ.

PRONAOS TO HOLY WRIT. Establishing, on Documentary Evidence, the Authorship, Date, Form, and Contents of each of its Books, and the Authenticity of the Pentateuch. By _Isaac M. Wise_. Cincinnati: Robert Clarke & Co. 1891.

Rabbi Isaac Wise, the president of the Hebrew Union College of Cincinnati and the Nestor of Orthodox Judaism in America, presents in the “Pronaos to Holy Writ” a review of the Biblical books with comments as to their authenticity and the times in which they were written. Having read these “books and every word thereof in the original for a term of sixty-six years, i. e. from boyhood up to his seventy-second birthday,” and having “acquainted himself with all the ancient versions and commentaries and a large portion of the modern translations and commentaries of the Bible,” the author is entitled to be heard. Rabbi Wise is a stern monotheist and he declares: “God only did create light out of darkness; man cannot produce truth out of fiction, unless in his self-delusion problematic truth satisfies him. All so-called gems of truth buried under the quicksand of fiction and deception are problematic at best, if not supported by authoritative corroborants.” This is true. All truth depends upon verification. We cannot make truth, but must find it, we must be able to corroborate it, and the corroborants of truth are its authority. Dr. Wise’s idea of a corroborant is different from ours, he says: “No one can speak conscientiously of Bible truth before he knows that the Bible is true, and especially in its historical data.” This seems to indicate that we must have a belief in the truth of the Bible before we investigate it and that moral truths, the ethics, the philosophy of the Bible depend upon its historical data. We cannot go so far with the author of the Pronaos. Dr. Wise says: “The science commonly called Modern Biblical Criticism, actually Negative Criticism which maintains on the strength of unscientific methods that the Pentateuch is not composed of original Mosaic material, no Psalms are Davidian, no Proverbs Solomonic, the historical books are unhistorical, the prophecies were written post festum, there was no revelation, inspiration, or prophecy, must also maintain that the Bible is a compendium of pious or even impious frauds, wilful deceptions, unscrupulous misrepresentations.” Dr. Wise thought it necessary to meet Negative Criticism with the documentary evidence and for this purpose he wrote his Pronaos, which is to be an entrance-hall to the Temple of Biblical Truth. We do not side with the negativism of certain biblical critics, for we believe that historical investigations have proved large portions of the Pentateuch to be Mosaic, several psalms to be Davidian, and the historical books to contain as much history as many old historical books contain. We believe that they have to be judged and searched and commented just as much and in the same spirit of scientific inquiry as our philologists treat Herodotus or Livy. But the value of the Bible, in our opinion, does not depend upon the acceptance or rejection of these or those historical data; nor is it necessary to consider the Hebrew prophets as special revelations of God, in contradistinction to the divine revelation in nature and the history of mankind in general. It may be true enough that the orthodox God-idea of Monotheism depends upon the belief in special revelation and prophecies, and it is also true that most of the Biblical criticism has been destructive and negative. But there is a way possible between both standpoints which may be called positive criticism. This positive criticism attempts to understand the very life and meaning of the old religion, it attempts to comprehend the belief of the orthodox and construe it in the terms of science—i. e. of rational and clear thought. Religion is certainly not a mere fraud or a vain illusion, it is an ideal which developed naturally out of certain needs of man and the conditions of society. That religious ideas, especially the idea of God as the cosmic power which represents the moral authority, are no mere fictions, is proved by their survival, and those who believe in evolution should not be blind to the fact that there is something good, something true, something well adapted to surroundings in religion. To find these elements of truth and goodness which constitute the life of religion is not mere negative criticism, but positive criticism, and it is not at all necessary for those who aspire in this direction, to believe in any historical data, or in special revelations, or in prophecies, or in the personality of God, but simply to trust in truth. Truth is the only way of salvation even though it may shatter the most sacred idols of a venerable orthodoxy.

The contents of the book show that the standpoint of the author does not blind him to the finer traits of the natural development of his religion. So, for instance, Solomon’s rationalism is excellently contrasted with the spirit which manifested itself in the Judges as well as the Prophets. The author of the Judges was an outspoken theocratic democrat. “He literally pours out his abhorrence of the monarchical anti-theocratic institution in narrating the story of the first usurper Abimelech, the son of Gideon.... Entirely different are the language and tendency of the two appendices, evidently written by another author, who evinces his animosity to the democratic form of government by saying four times: ‘In those days there was no king in Israel,’ to which he adds twice ‘every man did what seemed right in his sight’” (p. 46). “The Solomonic ethics is a commentary on the Mosaic ethics, as by reason understood.... Man’s knowledge of ethical doctrine is identical with his knowledge of God’s moral attributes, and all moral obligation has its root in the Mosaic God-idea....” According to Solomon “wisdom based upon and rooted in the fear of Jehovah with the revealed material before them was all-sufficient, without any further special oracles of any prophets. This peculiar rationalism brought upon him the ire of prophets and rabbis” (p. 111).

Some reviewers of Dr. Wise’s book will probably find fault with him that he has taken little if any account of the results of modern biblical investigations. And this is a grievous fault in our times where it seems to be essential for a scholar and author to have read the very latest things published on a subject while an acquaintance with the views of the classical old authorities is considered unnecessary. It appears that Dr. Wise did not intend to present his views or criticisms of and his answers to the latest biblical investigations. It may even be that he is not familiar with many of them. Granting this to be a fault of his book it is, nevertheless, refreshing to us to find an author who has actually read and is excellently familiar with all the old sources of the subject he is writing upon.

κρς.

THE FOUNDATIONS OF GEOMETRY. By _Edward T. Dixon_. Cambridge (Eng.): Deighton, Bell & Co. 1891.

This work is divided into three parts, the first containing such doctrines of psychology and logic as the author deems sound and useful for his purposes, the second exhibiting the author’s “subjective theory of geometry deduced from the two fundamental concepts _position_ and _direction_,” and the third “on the applicability of the foregoing subjective geometry to the geometry of material space.”

In his preface the author expresses his desire that those who criticise his work shall “consider categorically” certain questions relating to his theory of definition, to the definitions and axioms prescribed by him, to his proofs of propositions and to the “objective applications” of his three axioms.

Geometry may be studied for two distinct purposes, neither of which necessarily involves the other. Unless the aim is mainly the discipline of the logical faculty, it is plainly a poor method of study to pore over the definitions, axioms, postulates, theorems, problems, and demonstrations of Euclid or any similar text-book. Practical resources and geometrical information can be acquired much better and more rapidly by a course of mechanical drawing with here and there a more or less loose explanation of the grounds and reasons that warrant the geometrical doctrines, than by means of the Euclidian course. Under such a method of instruction the student would rarely feel any real doubt as to the truth of his geometrical knowledge.

But where the paramount aim is the training of the reason the Euclidian rigor is all important. Hence the perfection of that method by the discovery and certification of the ultimate grounds on which, and the principles by which, it may be unfolded systematically and in necessary and sufficient sequence without presumption or fallacy, is an object of the most momentous concern to science, to philosophy, and to culture in general. For it is well known that however good an account elementary geometry may give of its superstructure the reports given of its foundations are all very far from satisfactory.

Repeated and strenuous efforts have been made, and by the most competent of our race, to discover and certify the true state of the case in respect to the geometrical foundations, in order that the whole edifice of that science shall display throughout the same thorough-going necessity and sufficiency that distinguishes it in general.

The author of the work under review is persuaded that he is now able to perform this so desirable service. He avers his belief that the system of geometry he “has set forth in this book is _logically sound_ and that consequently the more it is discussed and criticised, the more firmly will it become established.” He takes his stand upon two fundamental concepts, _position_ and _direction_, which he defines not explicitly but “implicitly.” This leads us to consider his first question and his theory of definition.

The embarrassments that involve the foundations of elementary geometry are mainly, if not wholly, those which involve the general problems of definition. Now a definition is the certification of the purport of a name by means of a statement or a conspiracy of statements necessary and sufficient to that end. But names are constituents absolutely necessary for the formation of any statement, so that the above definition of a definition may be restated thus: A definition is the certification of the purport of one name by means of other names, necessary and sufficient to certify the purport of the one defined. Evidently then, definition can only lead us from name to name in unending process, or to some undefinable name, or to some name that we choose to leave undefined; and the question arises, on what sort of names shall we take our stand as ultimate grounds?

Our author answers this question as follows: “The propounder of a scientific theory is not of course expected to teach his readers to speak, it is only necessary for him to define the terms peculiar to his science, or those to which he wishes to attach peculiar meanings. He may therefore assume that the meanings of all other words are known to his readers.”

He then propounds that “all that is logically required for a definition is one or more assertions with regard to the word to be defined or, its attributes,” provided “they are not demonstrably incompatible with each other.”

Although our author conceives that logical competence requires no more than this for a good definition, he yet goes on to remark, that “if the definition is to form the basis of a deductive science it is further advisable that the assertions should be independent,” and that “where it is required to define a term whose denotation is already known, it is further necessary not only that the assertions should be commonly accepted as true with respect to it, but that they should restrict the meaning of the term exactly to its accepted denotation, neither more nor less, and should do so in the simplest manner that can be devised.”

It is upon this theory of definition that our author requests of his critics a “categorical” answer to his first question, “Do you accept the requirements I have laid down for a logical definition? (If not please state which of them you object to, why you object to it, and what you would propose to substitute for it.)”

Since it is a “categorical” answer that is requested and since also it is the matter of definition that is put in issue, we wish that our author had been more definite and had made his propositions better issuable, for we must protest that we regard ourselves obliged to answer to what we can best conceive to be the author’s true meanings rather than to what he has explicitly said.

We do not conceive that he regards it as _necessary_ to a definition that the defining assertions should be expressed “in the simplest manner that can be devised.” We have also to take his use of the word “restrict” as importing completion as well as limitation, and his use of the word “requirements” as intending conditions that together are sufficient as well as necessary.

If we are right in our understanding of the meanings of our author he contemplates four cases, first, the definition of a name that has no denotation already known and that is not to form the basis of a deductive science, second, the definition of a name that has no denotation already known but which is to form the basis of a deductive science, third, the definition of a name that has a denotation already known but which is not to form the basis of a deductive science, and fourth the definition of a name that has a denotation already known and is to form the basis of a deductive science.

In this fourth case our author deems it requisite for a logical definition that there shall be made one or more assertions about the subject of definition that are not demonstrably incompatible with one another, that are independent of one another, that are commonly accepted as true in respect to the subject defined and that “restrict” the meaning of the name under definition exactly to its accepted denotation.

It seems to us that this last requirement dispenses with the necessity of all the rest. If we have provided an assertion or a set of assertions that do in fact complete and limit the meaning of the subject of definition exactly to its proper denotation that is a definition in full. It implies that the defining assertions are all consistent with one another, and in case any assertion is dependent upon one or more of the rest that is a circumstance wholly immaterial. _Utile per inutile non nocetur._

Again, what is it to be commonly accepted as true? Does logical competence depend on the altering states of our knowledge or on the fluctuations of opinion? Was a whale logically defined as a fish before we learned that it was a mammal?

The third case allows of the application of the same comment as that made upon the fourth. But in the first and second cases the doctrines of the author as well as his suppositions are very notable. He supposes the anomaly of names without any known denotation, by which he may mean those which have no application whatever. In respect to such he propounds that they may be given a logical definition by making one or various consistent assertions as applicable to them or to their attributes.

“The proof of the pudding will be found in the eating,” as our author says. So let us say that a troft may be perceived whenever our attention is excited, and that trofts are of multitudinous variety. Do these assertions constitute a logical definition? It is a prime requisite for a definition that the defining assertion or assertions shall have a meaning, which is the same as to say that names must be employed that are already significant. These significant names must be so used that the intellectual sensibility shall be excited to perceive in a determinate way that which is intended to be defined. In other words, sense and not nonsense must be produced in the mind that considers the definition. Perhaps, however, our author intends such words as electricity, or spirit, or energy.

Because of the considerations above indicated and others we cannot accept the author’s requirements for a logical definition as a whole. Some of them are in some of his cases unnecessary, while taken together they supply no new means whereby to solve the several problems of definition.

The author’s subjective theory of geometry is plainly the outgrowth of his confidence in the solvent power of the concept of direction as a prime datum of geometry.

Everything of consequence in his essay depends upon the worth of this concept as a geometrical foundation. Considering the disparagement that has been visited upon that concept by numerous writers of good geometrical rank we naturally look for considerations tending to remove the discredit that has befallen that notion. Instead however of this we find the most palpable set of circular definitions. Direction is defined by direction in the most distracting way, thus:

“(_a_) A direction may be conceived to be indicated by naming two points as the direction from one to the other.”

The inaptitude of the term direction for use in geometry is rooted in its ambiguous purport. As commonly used it means at least three distinct but closely associated notions which become confused in thought and expression unless the most solicitous care is taken to distinguish them. When we speak of the direction of one point from another or of the direction from one point to another we mean the straight off-ness or from-ness or to-ness which one bears to the other; in other words a relation of separation and straight mediation. When again we speak of the direction of a motion we intend the indefinite straight sense of its procession, which is not a relation but an attribute of the motion. When still again we speak of the direction of a line we mean its straight _lay_ as compared or as comparable with other actual or possible correlates which is again a relation but not necessarily the same relation as that that obtains between two points.

In all these meanings the notion of straightness is involved, and could we say in lieu of straightness first directness and then direction and holding fast in thought this sense of the word, make a noun of it, so that a direction would intend the same as a straightness and no more, it might obtain a useful geometric term and notion.

To define it we might first define a line thus: A line is a space boundary that is indefinitely long but not otherwise of any extent. Then, a direction is a line such that between the points that bound any assigned parcel of it no copy of said parcel is possible.

But direction purports to our author the second of the meanings above set forth, namely, the indefinite straight sense of the procession of a motion. Definite parcels of a direction thus understood are identical with vectors.

Now the notion of straightness is after the notions of point and line the most fundamental one of geometry and the one which is altogether the most prominent and useful. It is the necessary means for any definition of a vector or of the notion which our author deems so important. As straightness is attributable only to lines and long things which a line may represent it makes no difference whether we define straightness or a straight line, but a masterful performance of this definition is absolutely necessary before the foundations of geometry can be abidingly certified.

Our author defines a straight line thus: “A straight line is a continuous series of points extending from each of them in the same two directions.” What kind of a thing a continuous series of points may be we are not told but as a point is defined to be “a portion of matter so small that for the purpose in hand variations of positions within it may be neglected” we take it that a straight line is a continuous series of particles of matter. The “purpose in hand” in this case must of course be the purpose of geometry.

In defining an angle our author first lays down that “The difference between two directions is called their _inclination_ to one another” and then “The measure of an inclination is called an _angle_.”

Considering that it is the doctrine of the author that every straight line has two contrary directions the measure of whose inclination is an angle of one hundred and eighty degrees, we imagine a northeast southwest line cutting an east west line and wonder if the right hand upper angle is really two angles according to whether or not the directions both pass to the left or both pass to the right or pass one to the left and the other to the right.

Were this an ordinary work we might regard it as due to the author to notice the many excellencies which characterise it, in spite of the defects which we notice. But as our author evidently realises, the eminent dignity of the topic challenges and its singular importance demands unsparing criticism. He who offers to instruct the world on the foundations of geometry draws his sword and throws away the scabbard, and like a doughty champion he will scorn to accept any favor, prizing only such success as he shall take at the point of an efficacy of treatment that conquers all competent and candid criticism.

Stringent as are such terms of contest an author who is a worthy competitor in the field of geometric research can be well content with them in the perception that the very same conditions apply in full force to the comments of his critics.

The author is undoubtedly an able man and a close thinker. He has concentrated his mind upon a work that is worth the energy of a lifetime. But we must confess our judgment to be that in spite of his capacity and evident devotion he has come short of the high result to which he has aspired.

ρσλ.

LES FÊTES DE MONTPELLIER. PROMENADE A TRAVERS LES CHOSES, LES HOMMES ET LES IDEES. By _J. Delbœuf_. Paris: Félix Alcan.

We have here a charming narrative by the well-known Professor at the University of Liège of his visit to the fêtes of Montpellier, undertaken in great measure to make the personal acquaintance of M. Dauriac, the critic in the _Revue Philosophique_ of the author’s work “La matière brute et la matière vivante.” The description given of the fêtes, which marked the sixth centenary of the University of Montpellier, is very entertaining, as is the account of the journey through the South of France; but as M. Delbœuf says that he was more curious to become acquainted with men than with places, what he tells us about the former will be the more interesting.

The author, with the companions of his tour, could not pass Nancy without stopping to see “the masters in the science of hypnotism” there. An account of what he saw and heard gives the author the opportunity of repeating “That he does not regard forgetfulness on awaking as characteristic of profound hypnosis, and that experience is against the efficacity of criminal suggestion unless the subject is criminally inclined.” The fêtes at Montpellier commenced with a religious service in the Cathedral, during which the Bishop, M. de Cabrières, preached a sermon so liberal in tone, that M. Delbœuf thinks the time is arriving when the church will demonstrate that Moses was the precursor of Darwin. At the University reception which followed, M. Delbœuf sought out among the professors for his friend M. Dauriac, whom he had figured when first he heard from him as small, thin and dark, but now found, in accordance with the usual rule in such cases, that he was tall, robust and fair. In the course of their subsequent conversations the two Professors made mutual confidences, M. Dauriac confessing that his true vocation was music, and that he was preparing a work on the psychology of the musician; while M. Delbœuf informed his friend that he was about to reply to his criticism of “La matière brute et la matière vivante,” and that he would throw the greatest light on the origin, which was still obscure, of life and death. If the genial Liège Professor can do this, he may be the first to reap the benefit referred to in his own words: “The discovery of the cause of death could not fail to assure the immortality of its author and its inspirer, and sooner or later that of humanity at large.” For, according to a medical adage, if the cause of a disease is known it is already conquered.

Montpellier was honored during the fêtes with the presence of Helmholtz, to whom but for national jealousy would have been confided the part of speaking in the name of the foreign universities. Nevertheless he was the true hero of the occasion, and when at the official reception, on the President of the Republic shaking his hand and saying a few gracious words someone feebly hissed, Helmholtz received in response a perfect ovation of applause. M. Delbœuf met with a congenial spirit in the Professor of Zoology, M. Sabatier, who has a laboratory at Cette. Their views on free-will were in sympathy. They agreed in allowing freedom not only to the superior animals, and to inferior animals and plants, but even to so called inorganic matter. M. Sabatier is a Christian and at the same time a convinced transformist; having arrived at his views from religious considerations. He cited M. Dauriac as saying, “The reign of determinism is not in the objective world; its empire extends itself over nature only after having been exercised over thought. There is no other necessity than that of logic or mathematics.” M. Delbœuf is evidently an “indeterminist” by nature. He heartily sympathised with the students in all their demonstrations of freedom, although one of them assumed a somewhat serious character. Dining in the open air with M. Milhaud the author of an article in the _Revue Philosophique_ on non-Euclidian geometry, he was prepared to talk mathematics. The surroundings were too much for him, however, and in recalling the scene he cries, “To the devil with philosophy and mathematics! I cannot recall what we said; in my remembrances, I see only blooming faces, I hear only the indistinct bursts of gaity.” M. Delbœuf’s sympathetic nature is shown in the fact, which he records, that wild animals in confinement soon become familiar with him.

One of the principal objects of the author’s journey was to see M. Gabriel Tarde, “one of the most prolific and original publicists in France, if not in Europe,” who resides at Sarlat. After quoting passages from an article of M. Tarde on Social Darwinism, which appeared in the _Revue Philosophique_, M. Delbœuf remarks that nothing is more attractive and at the same time more fatiguing than the reading of his works. M. Tarde is “the locomotive that carries you to the end of your journey across countries by turns wild, agricultural, industrial, picturesque; but without giving you time to regard and admire.” Referring to M. Tarde’s acute criticisms of Lombroso and his theories, the author says, “It is not that he strikes the pseudo-thinker with formidable blows, but he makes him drop gently to the ground.” The French publicist sees in _imitation_ the source of social life, and he has been long engaged in developing the idea, to the great importance of which M. Delbœuf bears witness; although he objects to the use which M. Tarde makes of terms taken from mathematics, physics, and biology, to express his sociological views. On the question of free-will there was no agreement. Although the latter is a determinist, he believes in penal responsibility, on the ground of personal identity; the diseased person or the madman is no longer himself, in which they differ from the criminal.

We can say nothing of M. Delbœuf’s visit to the canons of the Tarn. Here was captured a lizard which displayed, when compared with a Spanish lizard in captivity with it, as much difference in character as could be found between two men chosen at hazard. The author concludes an amusing description of the habits of the two captives by recommending their history to the politicians and the historians of France and Spain, as likely to throw light on that of the peoples themselves. We leave M. Delbœuf, whose book of seventy-five pages may be said to be as full of interesting matter as an egg is of meat, with quoting his postscript: “On the day that these lines appear (March 1891) the Spanish lizard has finally cast off his savage character and follows in the footsteps of the French. Effect of imitation.”

Ω.

DER POSITIVISMUS VOM TODE AUGUST COMTE’S BIS AUF UNSERE TAGE (1857-1891). By _Hermann Gruber_, S. J. Freiburg im Breisgau: Herder’sche Verlagshandlung. 1891.

This pamphlet of 194 pages is the continuation of another pamphlet on August Comte, the founder of Positivism, which was reviewed in _The Open Court_, No. 134. The author is a Jesuit and it is a matter of course that all the facts he relates, all the doctrines he explains are represented from the standpoint of Roman Catholicism. The booklet is of great importance in so far as we learn through it what an erudite Catholic mind thinks of that recent movement of philosophy which has been called by the collective name Positivism. The method pursued by Hermann Gruber is most recommendable. He states facts and quotes abundantly so as to let the various philosophers speak for themselves. He is economical with the salt of his own opinion, yet he uses it with such a discretion that Roman Catholics can become thoroughly acquainted with infidel views without suffering in their faith.