BOOKS.

THE REALM OF PURE THOUGHT.* TIM distinction between the workaday world that we actually—so far as consciousness is a guide—live in, and the worlds, real beyond all tricks of the imagination, that mathe- maticians and philosophers construct out of the stuff that thoughts are made of, comes out very vividly in the remark- able book which is noticed, with unavoidable inadequacy, in

the present review. Any attempt to reproduce in language to be nnderstanded of the general reader the methods of analysis, or even the conclusions arrived at, in the region of pure thought must necessarily be inadequate in the extreme, while the actual severity of the reasoning in this high region makes perusal as exhausting, and perhaps as delightful, as high mountaineering. The worlds of Mr. Russell are not the world we live in, and an excursion into them makes our flat land of struggle, sorrow, joy, and survival seem unprofitable enough. But the air on these bleak peaks is too rarefied, the absence of heaven and hell as well as earth too certain, to make a long sojourn possible. The sights and sounds of daily life are a necessity of intellectual existence. Even Wordsworth, who more than any other poet carried poetry into the region of pure thought, knew this :-

"And, as I mounted up the hill, The music in my heart I bore, Long after it was heard no more."

Despite the extreme severity of the reasoning in Mr. Bertrand Russell's work on The Principles of Mathematics, his real gift of style, his strong sense of humour—note the remark in his powerful attack on Lotze's arguments against the existence of absolute space : "Points do not assign, positions to:each other, as though they were each other's pew-openers "- make his book a pleasure to read. Dealing with abstruse questions of extraordinary complexity, he states his case in terse, plain English, introduces very few technical terms, coins practically no words, and makes the lay mind finally understand that the strange jargon in which Mr. Herbert Spencer habitually indulges is an abuse and not a necessity of science. We should say that Mr. Russell has an inherited place in literature or statesmanship waiting for him if he will condescend to come down to common day.

Mr. Russell, however, needs all the lucidity that he ean command in his present book, for he has come definitely out as an opponent of much current philosophic and mathematical thought. A man who flings down a formal challenge on the one hand to Kant and his school, and on the other hand attacks with something approaching ferocity the mathematical doctrine of the" infinitesimal," cannot afford, if he is to be taken seriously, to wrap his arguments in a nebulous mist ef words. There is no nebulosity about Mr. Russell. He is as downright as his strongest opponent could wish, and to the unimpassioned onlooker—for there is passion in these serene realms, as there is also in the realm of transcendental ethics, witness the controversies of Sidgwick, Green, Spencer, and Martineau— his blows go home. So far as may be, we must notice some of Mr. Russell's more notable demonstrations. He is largely concerned to attack the Kantian assertion that mathematical reasoning is not strictly formal, "but always uses intuitions, —i.e., the a priori knowledge of space and time." In order to

do this it was necessary to evolve a living, as opposed to an arid, philosophy of mathematics. It is, we think, irrefutably shown that all mathematics are reducible to certain funda- mental notions of logic. This demonstration—which if sound finally wrecks the Kantian position—is due to that develop- ment of symbolic logic by Professor Peano and others which Mr. Russell regards as "one of the greatest discoveries of our age." His own work in developing symbolic logic

• The Prineiplos of Mathamaties. By Bertrand Russell, DLL, late Fellow of Trinity College, Cambridge. VoL L Cambridge c at the Univeraity Pima. Ens. 6d, net.] has been of very real importance. In Part L of this book "the apparatus of general logical notions with which mathematics operate" is reviewed, and the practical identity of mathematics with symbolic logic established. We are then shown that, "mathematically, a number is nothing but a class of similar classes " (where a class is a unit in being or a numerical conjunction of such units), and the basis of an elaborate theory of arithmetic is evolved. This is followed by the mathematical distinction between finite and infinite. This distinction enables the author to deal with arithmetic as merely a development of a special branch of general logic. The theory of finite numbers is rigidly developed from certain fundamental notions and propositions. Mr. Russell thus bases all mathematics on "a certain body of indefinable entities and indemonstrable propositions." It is this basis which concerns the philosopher, and the philosopher is challenged to produce by inspection (which, as is shrewdly pointed out, is really the method of philosophic argument) any other fundamental indefinable set of entities commonly called numbers in substitution for the entities produced. It is shown in elaboration that no philosophical argument could overthrow the theory of cardinal numbers set forth.

The question of "infinite wholes" is next discussed,—a necessary antecedent of any discussion of infinite space. The conclusion arrived at appears to be that there are infinite wholes consisting of an aggregate with an infinite number of terms, and that the existence of such an aggregate pre- supposes the existence of an infinite number of indivisible finite parts. This latter conclusion (as, indeed, Mr. Russell suggests) may perhaps be questioned. It is doubtful if it is supportable by physical experiment or by that "common sense" which Mr. Russell considers as of weight in the question of the reality of the universe. The entire independence of numbers and quantity (which, it seems, is not a pure mathematical notion at all) is next demonstrated. The problem of infinity —the ultimate difficulty in mathematical ideas—is really one that concerns order and not quantity. An infinite number is one that does not obey the law of mathematical induction. Thus the problems of continuity, the infinite, and the infinitesimal belong to the theories of number and order. Therefore "the whole philosophy of space and time depends upon the view we take of order." All order is shown abso- lutely to depend upon transitive asymmetrical relations. These terms are difficult to grasp, but Mr. Russell aptly illustrates them from human relationship. The relation brother is " transitive" ; son-in-law is " asymmetrical "; descendant is both " transitive " and "asymmetrical." "Order," in the mathematical sense, is of the "descendant" type. We are next led to a theory of open series (that is, a series without, or without an arbitrary, beginning) in a form that does not presuppose numbers. It is then demonstrated that distance is not a notion which is essential to series. From this basis Mr. Russell proceeds to attack the philosophic difficulties involved in the ideas of infinity and con. tinuity. The Kantian idea that continuity has an essential reference to space and time is demolished by the fact that con- tinuity depends entirely on order. Moreover, "the mathe- matical treatment of continuity rests wholly upon the doctrine of limits." The Hegelian assertion that the notion of con- tinuity is incapable of analysis is, of course, condemned. Cantor's definition of continuity obtained by analysis (a triumph of intellectual force) is purely ordinal. From the mathematical conception of continuity we pass to that of infmity,—a subject of which philosophy treats without any precise definition, and therefore without any valuable results. It is shown that the so-called infinitesimal calculus has nothing to do with the in- finitesimal, and that infinity and continuity are both abso- lutely independent of it. Infinity is then defined to be

something which cannot be reached by mathematical in- duction starting from the number 1, and as something which has parts that have the same number of terms as itself, —this last fact being the only solution of the old paradox of Achilles and the tortoise. A system is thus constructed in which the ideas of continuity and infinity involve no definite contradictions, and Mr. Russell proceeds to apply them to space, time, and motion.

Geometry, we are told, is the study of series of two or more dimensions. We are no longer tied down to Euclidean

geometry. Premises other than Euclid's can give us results "empirically indistinguishable, within the limits of observa- tion, from those of the orthodox system Indirectly the increased analysis and knowledge of possibilities, resulting from modern Geometry, has thrown immense light upon our actual space." But the mathematics of the subject do not depend upon any intuitive idea of space. Multiple series are generated and dimensions are defined in purely abstract terms, and we have presented to us an actual logical analysis of continuous space. Mr. Russell deals at con- siderable length with the various kinds of space and the systems of geometry evolved from logical first principles, and supplies us, in a breathing interval, with a crushing display of the errors that occur in the first twenty-six propositions of the First Book of Euclid. The space dis- covered by pure reason may not be our space. "We cannot prove that our actual space must be continuous, but we cannot prove that it is not so, and we can prove that a continuous space would not differ in any discoverable manner from that in which we live." The space of pure reason is composed of points. Now philosophers declare that this is logically impos- sible. Mr. Russell asks:" Is a space composed of points self-con-

tradictory? If this question be answered in the nega- tive, the sole ground for denying that such a space exists in the actual world is removed." He concludes that there is no reason "to deny the ultimate and absolute philosophical validity of a theory of geometry which regards space as composed of points, and not as a mere assemblage of relations between non- spacial terms." Mr. Russell therefore attacks Kant's theory of space. In order to do this successfully it is necessary to ask whether the reasonings of mathematics are in any way different from those of formal logic, and whether there are any contradictions in the notions of time and space. "If these two pillars of the Kantian edifice can be pulled down, we shall have successfully played the part of Samson towards his disciples." The first pillar is already down, and it is shown that since the Kantian contradictions apply to all continuous series, they do not specially invoh a time and space, and are capable of solution. "Thus, although wo discussed no problems specially concerned with what actually exists, we incidentally answered all the arguments usually alleged against the exist- ence of an absolute space. Since common sense affirms this existence, there seems therefore no longer any reason for denying it."

The demolition of Kant's theory of space, and the establishment of the existence of absolute, though possibly subjective, space, lead to the discussion of matter and motion. "It does not follow, merely because there is space, that therefore there are things in it. If we are to believe this, we must believe it on new grounds, or rather on what is called the evidence of the senses." To think about matter we have to conceive of a new indefinable simple relation,—occupation by a class of terms which occupy

• both points and instants. [In this class it is impossible to dis- tinguish, as classes of concepts, between bits of matter and their secondary qualities, such as colour.] The new concept of a material unit has, therefore, to be defined with respect to space and time. If space and time are then replaced by certain continuous dimensional series, and the material unit by a certain correlation between all moments of time and some points of apace as represented in these series, a material universe is constructed. Motion in the universe consists in the correlation of different terms in the two series. As a result of the denial of the infinitesimal "we must entirely reject the notion of a state of motion." Motion consists merely in the occupation of different places at different times subject to continuity. There is no transition from place to place, and no physical existence of velocity and acceleration. Newton's laws of motion can "in no way be taken as a priori truths necessarily applicable to any possible material world." They must be 'viewed "either as parts of a definition of a class of possible material universes, or as empirically verified

assertions concerning the actual material universe Knowledge as to what exists is never derivable from general philosophical considerations, but is always and wholly empirical." It is important to notice that the possibility that there is no such thing as action at a distance is admitted. Mr. Russell asserts finally, and finds in the assertion a. powerful confirmation of the logic on which he has based his book, that absolute motion is essential to dynamics and involves absolute space.

We have endeavoured to indicate, imperfectly and lamely enough we are afraid, the argument round which this really great work is built. 'Unless we are much mistaken, its lucid application and development of the great dis- coveries of Peano and Cantor mark the opening of a new epoch in both philosophic and mathematical thought. There is something wonderful, beyond all the doings of the unearthly beings that people the imagination of the East, in the results of modern thought. On the one hand, it revolutionises our conceptions of, and our doings upon, the tangible earth ; on the other, it evolves out of ideas common to all ratiocinating creatures ground-plans not only approximately for the universe that is, but also for universes that might be, and, for aught that we know, are. With the abolition of mystery new mysteries peer forth. We almost seem to see the choice of creative ground-plans that lay before the Architect of the universe. The universe, according to the Book of Genesis, was created in God's mind before it took actual shape. A shadow of the creative power is delegated to us, for we can within our minds create universes comparable with that of which we form a part.