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7 4 T H E FOUNDATIONS OF MATHEMATICS. [ N o v. ,
he supplements this theorem by considering the case when p = 2. The main results may be stated as follows : If the Sylow subgroups of order 2m(m > 1) contained in any group G are either cyclic or contain a cyclic subgroup of order 2 m~l which includes only two invariant operators under one of these Sylow subgroups, then the number of operators of order 2 in G is of the form 1 + 4k. When this condition is not satisfied the number of these operators is always of the form 3 4- 4k. When m = 1, G contains an invariant subgroup which is com- posed of all its operators of odd order, and the number of the subgroups of order 2 may have either of the two forms 1 + 4k, 3 + 4k. This is the only case where the form of the number of the subgroups of order 2 is not determined by the form of this number in a Sylow subgroup. G. A. M I L L E R , Secretary of the Section.
T H E F O U N D A T I O N S O F M A T H E M A T I C S. *
The Principles of Mathematics. By BERTRAND BUSSELL. Volume I. Cambridge. The University Press, 1903. xxix
+ 534 pp.
Essai sur les Fondements de la Géométrie. Par BERTRAND RUSSELL. Traduction par A. CADENAT, revue et annotée par l'auteur et par L. COUTURAT. Paris, Gauthier-Villars,
- x + 2 7 4 p p.
- The Problem. — Pure mathematics has always been con- ceived in the minds of its votaries and by the world at large to be a science which makes up for whatever it lacks in human interest, and in the stimulus of close contact with the infinite variety of nature, by the sureness, the absolute accuracy, of its methods and results. Yet what has been accepted as sure and accurate in one generation has frequently required funda- mental revision in the next. Euclid and his pupils could doubtless have complained of the lack of rigor and logical pre- cision in his predecessors just as forcibly as some modern pupils of Weierstrass berate their scientific ancestors and companions.
- We may also refer our readers to the review by L. Couturat, Bulletin des Sciences Mathématiques, vol. 28, pp. 129-147 (1904). So large is the work of Russell that Coufcurat's review and our own supplement rather than overlap one another.
1 9 0 4. ] THE FOUNDATIONS OF MATHEMATICS. 75
Euler, finding confusion in the theory of the infinite and infini- tesimal, proceeded to explain away the difficulties, that others might be free from the prevailing errors. We cannot accept his reasoning to-day. At the beginning of the last century the state of infinite series was lamentable and Cauchy's memoir on the subject is said to have impressed itself on Laplace to such an extent that he postponed publishing his Mécanique Céleste until he became so hopeless of righting things that he gave up trying to do it. The righting has been accomplished in the pres- ent generation by Poincaré. Yet we very much doubt whether Laplace, before hearing of Cauchy's treatment, would have for a moment granted any possible inaccuracy in his own methods. Somewhat later Dirichlet treated the problem of determining a harmonic function from its boundary values and so careful a mathematician as H. Weber extended the method to the discussion of the equation A F + X F = 0 without any ap- parent qualms as to error. Nevertheless, now-a-days, the theo- retical importance and the practical use of the principles of Dirichletand Thomson are completely obscured for many by the too great emphasis laid upon the errors in the original demon- stration of the principles. We notice that the advance toward our present rigor has been made step by step by great men who, however, were no greater — one might almost say no more careful — than their fellows working in apparent unconsciousness of the impending trouble and perhaps even incredulous at first as to its reality. When will this revision stop? And whereunto will it finally lead? This is the problem of the ultimate foundation of math- ematics. In attempting an answer one can learn only hesi- tancy from the past. The delicacy of the question is such that even the greatest mathematicians and philosophers of to-day have made what seem to be substantial slips of judgment and have shown on occasions an astounding ignorance of the essence of the problem which they were discussing. At times this has been due to the inevitable failings of individual intuition in dealing with matters that are still unsettled ; but all too frequently it has been the result of a wholly unpardonable dis- regard of the work already accomplished by others. Even when guarding as much as may be against this latter sin, those who approach the depths of the subject upon which Russell has so courageously entered may well expect to hear the warning :
Procul, o procul este profani!
1 9 0 4. ] THE FOUNDATIONS OF MATHEMATICS. 7 7
bolic logic such a simplification of notation as to relieve it of its unwieldiness and to allow its development into a powerful instrument without which one can hardly hope to get the best results in the treacherous though treasure-laden fields of the foundations of mathematics. Poincaré, to be sure, in his re- view of Hubert's Foundations of Geometry * spurns this pasig- raphy, characterizing it as disastrous in teaching, hurtful to mental development, and deadening for investigators, nipping their originality in the bud. However much we may agree in the first statements (see § 7, page 91), we had best be cautious in accepting such sweeping statements as the last, even from so great an authority — especially in view of the fact that, equipped with this pasigraphy, the Italian investigators, Peano and his pupil Pieri,f with some rights of priority, had given a more fundamental logical J treatment of the subject on
- Translated in BULLETIN, vol. 10, p. 5 (Oct., 1903). f " I principii della geometria di posizione," Memorie della B. Academia delle Scienze di Torino, vol. 48, pp. 1-62. And, " Delia geometria elementare come sistema ipotetico-deduttivo ; Monografia del pun to e moto," ibid., vol. 49, pp. 173-222. t While we appreciate and admire as much as anyone can the beauties of Hubert's famous Grundlagen der Geometrie, we fail to see how the his- torical facts can justify what Poincaré says (I. c , p. 23) : " H e has made the philosophy of mathematics take a long step in advance, comparable to those which were due to Lobachevsky, to Riemann, to Helmholtz, and to L i e. " Poincaré makes the point that Hilbert regards his geometric elements as mere things and on this seems to rest a large part of the praise (I. c. bottom p. 21 and top p. 22). If this be so, it ought to be mentioned as a matter of history that Peano, in 1889, in his Principii di Geometria took precisely this stand, p.
- In 1891-2, Vailati, Bivista, vol. 1, p. 127, vol. 2, p. 71, again formulated the principle in words. By 1897 the Italian school had gone as far beyond this point of view as to make it a postulate that points are classes — thus showing a twofold advance, once in recognizing the presence of a postulate, again in using the word class so as to bring the reasoning into form dependent upon precise logical processes alone. It has also been said that the idea of the in- dependence of the axioms was due to Hilbert. As a matter of fact in 1894, Peano, " S u i fondamenti della geometria," Bivista, vol. 4, pp. 51 et seq., states the problem and, by actually setting up simple systems of elements, proves the independence of certain axioms from certain others. So that by 1899 the idea and method were both five years old at least. Again, in 1889, Peano laid it down as a principle that there should be as few undefined symbols as possible, and he used but few. In 1897-9 Pieri used but two for projective geometry and but two for metric geometry, whereas Hilbert was using a considerable number, seven or eight. (The idea of compatibility seems to have been first stated clearly by Hilbert.) There still remains in the Grundlagen der Geometrie matter enough for the amplest praise. The archimedean axiom, the theorems of Pascal and Desargues, the analysis of segments and area*, and a host of things are treated either for the first time or in a new way, and with consummate skill. We should say that it was in the technique rather than in the philosophy of geometry that Hilbert created an epoch.
78 THE FOUNDATIONS OF MATHEMATICS. [Nov.,
which Poincaré was writing than is to be found in the work he was praising so highly. In the fields of arithmetic and algebra, too, Burali-Forti and Padoa, adherents of Peano, had reached a point far beyond the widest view of the chief of the German school that deals with the same subjects.* Further- more, on this one point Poincaré may not be regarded as an authority ; for his own work f in the field should be charac- terized as subjective rather than objective, speculative and suggestive rather than purely logical. J Anyone who is acquainted with the articles presented to the Philosophical Congress at Paris in 1900 by Peano, Burali-Forti, Padoa, and Pieri, cannot be convinced that* these authors had become deadened, and the artificiality of their system is by no means so certain as it might be. Since then, our author, Russell, has simplified and improved the older work of C. S. Peirce on the theory of relations, adapting it to the system of Peano, and has produced a coherent treatment of the great problems underlying mathematics. In view of accomplished facts one inclines more readily to the praise given by Whitehead : " I believe the in- vention of the Peano and Russell symbolism forms an epoch in mathematical reasoning." §
- The Reason. — I t is not hard to detect the reason why mathematics has thus pushed its foundations back until they
- Compare the papers below referred to in the Bibliothèque du Congrès International with No. 2 of Hubert's Mathematical Problems, BULLETIN, vol. 8, p. 447. We may refer also to Padoa, VEnseignement Mathématique, vol. 5, p. 85. See also? 4 of the present review. For our readers, who may be working on the problem No. 2, we may note—what we unfortunately failed to note at the time of translating—namely, that a solution along the lines proposed by Hilbert seems logically impossible. A solution has long since been proposed in the article here referred to. There are those, however, who hold that Padoa has gone so far as to overshoot the mark. Hilbert has again taken up the matter much more searchingly than in 1900. It is to be regretted that his paper which was presented at Heidelberg, August, 1904, is not at hand for comparison. t La science et l'hypothèse, Paris, 1903; and numerous scattered essays. X That Poincaré seems frequently to have in mind the physical rather than the mathematical, the psychological rather than the logical point of view can be seen in several places in his review. On p. 8 he asserts that we know the axioms are non-contradictory "since geometry exists," And on p. 22 he seems to complain that the logical standpoint interests the author to the utter disregard of the psychological. It should be remembered that the first chief aim of the modern researches on the foundations of geometry is to be entirely rid of the psychological element — and this for the very reason that secondly we may decide just what that psychological element must be. This latter problem belongs rather to the philosopher and psychologist than to the mathematician. § American Journal of Mathematics, vol. 24, p. 367 (October, 1902).
8 0 T H E FOUNDATIONS OF MATHEMATICS. [ N o v. ,
looked into, it appears that we constantly use propositions, passing from certain propositions as hypothesis to certain others as conclusion. The laws of implication which govern the rela- tion between hypothesis and conclusion constitute the logical theory or calculus of propositions. Casting about for other principles we come upon classes or sets of objects represented in ordinary speech by common nouns. The development of the interrelations of classes produces the logical calculus of classes. This calculus has a remarkable analogy with the calculus of propositions, but the relation is not quite dual. In the third place we perceive that relations are of the utmost im- portance. Every transformation, every function is a relation. In common language the verb does but express a relation be- tween the subject and object. Thus there appears the necessity for a calculus of relations.* The complete logical calculus, as now used, is a combination of these three types. The whole number of laws of thought or logical premises which seem to be required for establishing the calculus in all the generality necessary for mathematics is small. In addition to these pre- mises there are a certain number of elementary ideas or terms such as implication, and the notions of proposition, class and relation, which must be assumed as known. I t is the dis- cussion of these questions which are of a philosophical rather than mathematical nature, that fills the first Part of RusselPs Principles. We may grant, then, that logic is necessary to mathematics. I t is affirmed to be sufficient This in reality is the remarkable content of the definition given by the author. So immune are we from logical error that the necessity of logic might never force us to a critical examination of its principles ; but the affir- mation of its sufficiency fully justifies and even renders im- perative such an examination. RusselPs entire volume is devoted to establishing this sufficiency. And although the subject is very new and many difficulties philosophical and mathematical are still outstanding, there can be little doubt that to an unexpectedly large extent the author is successful in his attempt and that in these Principles he has given a per-
- Peano and his immediate followers overlook the importance of this sub- ject— so busy are they with other important questions. It is one of the lasting services of Russell, following very closely on the work done twenty years earlier by C. S. Peirce, to have recognized the necessity of this addition to Peano's system and to have supplied the deficiency. See his articles in the Revue de Mathématiques, vol. 7, nos. 2 and following (1901-1902).
1 9 0 4. ] THE FOUNDATIONS OF MATHEMATICS. 8 1
manent set to the future philosophy of the questions which he handles.
- Some Notions. — Owing to the wide-spread diversity of usage in the meaning of such fundamental notions as postulates, axioms, undefined symbols, definitions, consistency, independence (of postulates), irreducibility (of undefined symbols), complete- ness (of systems of postulates and undefined symbols), we think it best to enter upon some slight exposition * of these matters instead of taking up the critical discussion of some of the more abstruse problems which are treated by the author and which could scarcely be appreciated before such exposition. Axiom is a word which has so long been used in so many vague ways that its use in pure mathematics had probably best be abandoned. The familiar definition : An axiom is a self-evident truth, means, if it means anything, that the propo- sition which we call an axiom has been approved of by us in the light of our experience and intuition. In this sense pure mathematics has no axioms : for mathematics is a formal subject over which formal and not material implication reigns.f The proper word to use for those statements which we posit would seem to be postulate. What self-evident truths can there be concerning objects which are not dependent on any definite interpretation but are merely marks to be operated upon in accordance with the rules of formal logic? Postulates, how- ever, may be laid down at will so long as they are not contra- dictory. I t is the postulates which give the objects their intellectual though not physical existence. Indeed before we can apply to the physical world any of the systems of logical geometry, for instance, we have the one great axiom : This system fits nature sufficiently for our purposes. To postulate such a statement would avail us naught. We must carefully consider the totality of our experience and decide whether the statement seems to represent a truth.
Definition is a term which has long been used by philoso- phers to stand for a process of analysis and exemplification
*See also E. V. Huutington on " S e t s of independent postulates for the algebra of logic," Transactions, vol. 5, p. 288 (July, 1904). f i n regard to logic on which mathematics rests, we should incline to use the word axiom (if indeed we do not prefer to hold to premise) and not pos- tulate. For here we are dealing with the actual (mental) world and not with a system of marks. The basis of rationality must go deeper than a mere set of marks and postulates. I t is foundation of everything and must be more real than anything else.
1904.] THE FOUNDATIONS OF MATHEMATICS. 83
ber, zero, and successor. These we could connect, as Peano and Padoa have done, by a system of postulates, and thus we should have a definition of number through postulates. In order to prove the non-contradictoriness of our postulates and indefinables, that is, the existence of our elements, we should have to set up some system which afforded one interpretation of the indefinables and of the postulates. As this must be done by going back to the laws of thought we finally get very near to where we started in the other sort of definition. The definition remains, however, slightly more general : for the integers thus defined are not merely one set of elements but any set which satisfies the postulates. Or (3) we may use the principle of abstraction on which Russell places a great deal of emphasis. We may say that two classes of objects, no matter what objects they be, have the same number when there exists a one-to-one relation between their elements.* Thus number becomes the common property of all similar classes, and is their only com- mon property. The class of numbers becomes the class of all similar classes. Owing to RusselPs development of the theory of relations this definition becomes also merely nominal and as it seems to be the most fundamental and philosophic it may be accepted as the best thus far given.
Although the use of postulates other than the premises of logic, and the use of undefined symbols other than those of logic seem needless and to be avoided in pure mathematics, the usage is so common that we may go on to say a few words concerning consistency, independence, irreducibility, and completeness — especially as these ideas are somewhat usable in the founda- tions of logic. To show the consistency of the system of pos- tulates and undefined symbols it is evidently futile to attempt to develop the consequences of the postulates until no contra- diction is reached (this method of stating the thing is sufficient to show wherein lies the futility) : for the most that can be ac- complished in this way is to see that up to a certain point no contradiction has been reached. The method of proof consists merely in finding some system of entities known to exist and affording a possible interpretation of the undefined symbols and postulates. To make the proof really fundamental for the system of integers it seems necessary to go quite out of the
- Russell shows, Principles, p. 113, that this idea is not dependent on the general concept number, nor even on the concept unity. Two classes which can be placed in one-to-one correspondence are called similar.
8 4 T H E FOUNDATIONS OF MATHEMATICS. [ N o v. ,
field of mathematics into the domain of logic* The method of showing the independence is merely to set up for each postu- late one existent system of elements in which there are possi- ble interpretations of our undefined symbols and which satisfies all the other postulates but not the particular one in question. This shows the independence of that one. If one of the un- defined symbols used in the statement of the postulates can be given a nominal definition in terms of the others the sys- tem of indefinables is redundant. I t was Padoa f who first made effective use of this idea. To show the irreducibility of the indefinables relative to the system of postulates it is necessary to set up a system of elements which satisfies all the postulates, which affords an interpretation of the undefined symbols, and which continues to satisfy these conditions when one of the undefined symbols is suitably altered : this must be done for each. The problem is quite similar to that of the in- dependence of the postulates and is not difficult to solve in case the number of undefined symbols is small. All this difficulty is avoided in dealing with the different branches of mathematics when Russell's point of v i e w — n o new indefinables, no new postulates — is taken.
Huntington $ seems to have been the first to bring to effec- tive use the idea of completeness. The problem is to show that if there are two sets M and Mf^ of objects each § of w^7 hich satis- fies the postulates and affords interpretations of the indefinables, then the two sets of objects may be brought into one-to-one correspondence in such a way as to preserve the interpretation of the symbols. With the statement of this last idea we have arrived at the limit of present ideas concerning the interrela- tions of the notions at the base of mathematics as defined by postulates.
•*See references given in footnote under § 2, p. 77. The consistency is far more important than the independence, irreducihility, or completeness : for these are merely a matter of elegance, whereas that determines whether or not all our reasoning upon the system in question is void. t Bibliothèque e t c " Essai d'une théorie algébrique des nombres entiers, précédé d'une introduction logique à une théorie deductive quelconque." This remarkable essay should be read by every one. We may note that Padoa uses * transformateurs ' but introduces no theory of relations. In this respect Russell has introduced improvements. % Transactions, vol. 3, pp. 264-282 (1902). See also Veblen, Transactions, vol. 5, p. 346 (1904). § Serious mistakes, resulting in definitions of no essential content, have been made by forgetting that the relations whicji connect the elements must be in correspondence, in addition to any correspondence between the ele- ments. See also footnote under § 6, p. 87.
8 6 THE FOUNDATIONS OF MATHEMATICS. [ N o v , ,
and as evidence that at last we have a principle of addition dis- tinctly above the plane of counting on one's fingers. In like manner the definition of multiplication is such as to be free from the laws of commutation and association of the factors and to apply equally to finite or infinite products of finite or infinite cardinals. Again a vindication of the school-child who rightly cannot see why it should make any difference whether he puts down four rows of three marks or three rows of four. The discussion of the meaning of quantity and magnitude in Part I I I. and its connection with number we will not pause to consider, but we pass directly to the theory of order as de- veloped in Part I V. The treatment of this subject is greatly simplified by the theory of relations. Order is shown to be an asymmetric transitive relation, an essential property of serial relations. I t is clearly pointed out and it is important to notice that when a set of objects is given the relation is not necessarily included ; whereas when the relation is given the field in which it operates must also be given. If recourse is had to the prin- ciple of abstraction the ordinal integer appears as " the common property of classes of serial relations which generate ordinally similar series." As the cardinals are classes of similar classes, so the ordinals are classes of like relations. The principle of induction is intimately associated with the system of ordinals rather than with the system of cardinals although for finite numbers the distinction is not so great as for infinites. We may say that the finite ordinal is that which can be reached by induction from 1. It appears that those who generate their system of numbers by a relation of succession or by counting — that is, by successive acts of attention — must in reality be coming at something which resembles ordinals much more nearly than cardinals. The difference between the theory of infinite cardinals and infinite ordinals brings to light the im- portant fact that in mathematics we have two kinds of infinite : the cardinal, which has the property of being similar to a part of itself, and the ordinal, which cannot be reached by induction from 1.* The discussion naturally brings up the old question of extensive and intensive definition. The definition of an
*This would seem to render invalid the contention of Poincaré in his La science et l'hypothèse to the effect that the principle of induction is the essence of the infinite. We have seen that it is the essence of the finite. The difficulty seems to be that Poincaré has in mind the definition of the infinite as a growing variable. If this be so, the apparent contradiction resolves itself into a mere difference of definitions.
1 9 0 4. ] T H E FOUNDATIONS OF MATHEMATICS. 8 7
object is said to be extensive when the object is given by the enumeration of its parts ; it is said to be intensive when the object is characterized by its properties. In the treatment of these questions and of transfinite cardinals and ordinals there is much which is instructive for the mathematician and the phi- losopher. The author points out with his customary frank desire to state no more than the truth that there still remain difficulties to solve. Thanks to his lucid and modest presenta- tion there is no reason why he should not find adherents who will take up the work and attempt the solution in a spirit of hearty cooperation. There is a school of creationists who, when they find that certain infinite processes lead to no rational limit nor yet to a number which becomes infinite, postulate the existence of a limit and thus obtain the irrational numbers. The author does not consider an ipse dixit like this to be a sufficiently good theorem of existence. He therefore considers infinite sets of rationals and by means of them he forms a set of things which he calls real numbers. A real number is neither a rational nor an irrational ; it is a certain infinite set of rationals. The real numbers thus defined are shown to satisfy the notion of a con- tinuum. According to the method followed, the continuum appears as an idea wholly ordinal in nature. With the aids thus prepared the author is able to give a very satisfactory account of the philosophy of the infinite and of the continuous. His treatment of the paradoxes of Zeno shows that the argu- ments of the ancient philosopher are by no means so far from right as might be imagined and that the contradictions are more apparent than real.
- Geometry and Mechanics. — A short study of the proper- ties of multiple series leads to a point from which it may be seen that : Geometry is the study of series of two or more di- mensions.* In this manner the necessity of new postulates and new indefinables is avoided. The procedure is evidently reason- able. Mathematical geometry has long since been divested of all spatial relations between its elements. The above definition
- As the serial relation is emphasized rather than its domain (see discus- sion of order given above) the author avoids a definition which is null and which makes dimensious impossible. Compare discussion of completeness and footnote, __ 4. For a fuller discussion of this important point we may refer to " T h e so-called foundations of geometry," by the present reviewer, in the Archiv der Mathematik und Phyxik, vol. 6, pp. 104-122 (1903). Toward the end of the discussion a change, which may cause some confusion, is made to the point of view of physical geometry. The first part, however, deals solely with purely mathematical geometry.
1 9 0 4. ] T H E FOUNDATIONS OF MATHEMATICS. 8 9
(the material points) with all time, that is, a, b, c = R(aQ, b 09 c 0 , t), where a, 6, c are the coordinates of the material points. This relation R is so chosen as to allow for the impossibility of gen- erating or destroying matter. The relation is also chosen so that if the relation between matter and time is known at two instants it is known at every instant. In this way is stated the causality in the universe. This seems very far off from the real world. I t must delight the hearts of philosophers who believe in a pure idealism. I t is found that arithmetic may be handled adequately with no help save from logic. This does not surprise us. Then geometry is put in the same cate- gory. Modern mathematicians have so accustomed us to look on merely the logical side of the subject that we are not troubled. Finally comes dynamics. Why not thermodynamics, electro- dynamics, biodynamics, anything we please? There is no reason why not. There is in reality no place to stop, save when we have become tired of pure logic, if once we include geometry. As a matter of fact all our concepts whether of space, or matter, or electricity, or life, are but idealizations more or less well-defined, and, if we insist on subjecting the world to purely logical explanation, they all belong in the same class. Upon this matter we may best quote Russell who, amid all his refinements, keeps a clear idea of their proper place in the system of all knowledge. He says : The laws of motion, like the axiom of parallels in regard to space, may be viewed either as parts of a definition of a class of possible material universes, or as empirically verified assertions concerning the actual mate- rial universe. But in no way can they be taken as à priori truths necessarily applicable to any possible material wTorld. The à priori truths involved in dynamics are only those of logic ; as a system of deductive reasoning, dynamics requires nothing further, while as a science of what exists, it requires experiment and observation. Those who have admitted a sim- ilar conclusion in geometry are not likely to question it here ; but it is important to establish separately every instance of the principle that knowledge as to what exists is never derivable from general philosophical considerations, but is always and wholly empirical. *
- I t would be interesting to discuss in how far this attitude is really in accord or out of accord with the apparently very different view of Poincaré (La science et l'hypothèse) that the question whether the parallel axiom is true or not true is devoid of sense owing to the fact that it is merely a con- venient method of correlating experience and a convenlion can have neither truth nor falsity.
9 0 THE FOUNDATIONS OF MATHEMATICS. [ N o v. ,
- Some Conclusions. —There is one conclusion in logic which suggests itself almost inevitably at this point. For there are a considerable number of systems of logic current at present. Different authors have treated the subject differently — each choosing the system of indefinables and laws of thought which seemed best to him at the time. Now it is by no means true that these various systems of logic have been proved coexten- sive or even not mutually contradictory. If it should appear that they cannot be brought into harmonious relation one with another there will be some instructive, if bewildering, conclu- sions to draw. And as we have such complex entities as infinity and continua with which to deal it might not be regarded as surprising if some points were found to stand out permanently, so that logicians will permanently disagree. In fact at present there seems to be a grave logical difficulty in our logical system as developed by Russell. This trouble had been felt by Frege and a solution had been proposed by him ; but it does not seem entirely satisfactory. * In view of the outstanding diffi- culties and the possible divergence of systems of logic held by equally good authorities, we come to the conclusion that it is dangerous to accept the naïve point of view of those who claim that a certain piece of reasoning depends on the operation of logic alone but who fail to state what those operations are and to use all the means possible to avoid the intrusion of extraneous ideas. They may not fall into error, but they are merely fol- lowing in the footsteps of those who " knew " what infinity and continuity were.
From the pedagogic point of view we may also draw some conclusions. I t is hardly necessary to trouble the student with the commutative and associative laws in multiplica- tion of integers or with elaborate deductions of a number system before he is readily able to appreciate the needlessness of the former and the relation which the latter bears to the theory of finite and infinite cardinals and ordinals, the ideas of compactness and continuity, and the two kinds of infinity.
- I n a long appendix, Eussell gives a detailed exposition of the important work of Frege, which culminated in the Grundgesetze der Arithmetik, and he discusses this troublesome contradiction again from a different standpoint. I t is this contradiction which Hubert had in mind in his Heidelberg address referred to under __ 2. He, therefore, attempts to recast the principles of logic and of arithmetic in such a manner as to render them sufficient for mathe- matical reasoning. We certainly hope that he has succeeded in doing so to the satisfaction of both mathematicians and philosophers.
9 2 T H E FOUNDATIONS OF MATHEMATICS. [ N o v. ,
existence of 1, from the fact that zero is a unit-class (the null- class being its only member). By an evident induction we get all finite numbers. From the class of finite cardinals follows first the existence of the smallest of the infinite cardinals, and second, by considering them in the order of magnitude, the ex- istence of ordinals and the smallest of the infinite ordinals. We may go on to obtain the rationals, compact enumerable series, continuous series. From the last we may see the ex- istence of complex numbers, of the class of euclidean spaces, of projective spaces, of hyperbolic spaces, and of spaces with va- rious metrical properties. Finally we may prove the existence of the class of dynamical worlds. Throughout this process no entities are employed but such as are definable in terms of the fundamental logical constants. Thus the chain of definitions and existence-theorems is complete, and the purely logical nature of mathematics is established throughout. This is as far as we are conducted. But we are promised a second volume — may it be soon forthcoming — written with the collaboration of Whitehead. Herein will be contained actual chains of deduction leading from the premises of logic through arithmetic to geometry. Herein will also be found various original developments in which the notations of Peano and Russell have been found useful. For those who wish sooner to get at the Peano-Russell point of view in the matter, we append a bibliography, which while very incomplete may still be found useful in tracing the development of the ideas : (1) Arithmetices principia nova methodo exposita, Turin, Bocca Frères, 1889. (2) I principii di geometria logicamente esposti, Turin, 1889. These two works by Peano are the starting point of the whole movement. They were written in the days when a careful ex- planation and translation of the symbolic method was in vogue and form a good starting point for study. The Formulaire de Mathématiques , edited by Peano, is rather hard to begin on. The Bivista di Matematica, now the Revue de Mathématiques, also edited by Peano, furnishes much easy and instructive read- ing matter. Logica matematica by Burali-Forti in the series of Manuali Hoepli may serve as a textbook. Omitting important memoirs by Burali-Forti on arithmetic and by Pieri on geom- etry which we have quoted in footnotes, we cite again (3) Bibliothèque du congrès international de philosophie, volume 3 (1901).
1904.] NOTES. 93
The articles by Peano, Burali-Forti, Padoa, and Pieri show the point at which the Italian school had arrived in 1900. I t is since that time that most of Russell's technical work has ap- peared. For the present state of the science, we would note a memoir by Whitehead : (4) " O n cardinal numbers," American Journal of Mathemat- ics, volume 24 (1902), pages 367-394; and a still more recent paper by Burali-Forti, " Sulla teoria generale delle grandezze e dei numeri," Atti délia B. Accademia delle Scienze di Torino, volume 39, (January, 1904). E D W I N B I D W E L L W I L S O N. Y A L E U N I V E R S I T Y , July, 1904.
NOTES.
T H E closing (October) number of volume 26 of the Ameri- can Journal of Mathematics contains the following papers : " In- variants of a system of linear partial differential equations, and the theory of congruences of rays," by E. J. WILCZYNSKI ; " On elements connected each to each by one or the other of two reciprocal relations/' by C. DE POLIGNAC.
T H E opening (October) number of volume 6 of the Annals of Mathematics contains the following papers : " On the sub- groups of an abelian group," by G. A. M I L L E R ; " Note on the continued product of the operators of any group of finite order," by W. B. F I T E ; " Reduction of an elliptic integral to Legen- dre's normal form," by N. R. WILSON ; " The necessary and sufficient condition under which two linear homogeneous differ- ential equations have integrals in common," by A. B. P I E R C E ; " A general method of evaluating determinants," by G. M A O LOSKIE ; " Application of groups to a complex problem in arrangements," by L. E. DICKSON ; " On functions defined by an infinite series of analytic functions of a complex variable," by M. B. PORTER.
A T the Cambridge meeting of the British association for the advancement of science (cf. October B U L L E T I N , page 28), Pro- fessor A. R. FORSYTH presided over the subsection of pure mathematics, whose programme included the following papers : " A fragment of elementary mathematics," " Geometry of the complex variable," by Professor F. M O R L E Y ; " Peano's
