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 THE FOUNDATIONS OF 
 MATHEMATICS 
 
 A CONTRIBUTION TO 
 
 THE PHILOSOPHY OF GEOMETRY 
 
 BY 
 
 DR. PAUL CARUS 
 
 o 6in^ dtl ytoyfierpeL- — PLATO. 
 
 CHICAGO 
 THE OPEN COURT I'UBLISHIXG CO. 
 
 LONDON AGENTS 
 
 KECAN PAUL, TRENCH, TrObNER & CO.. LTD. 
 
 IQOS
 
 COPYRIGHT BY 
 
 THE OPEN COURT PUB. CO. 
 1908
 
 URL 
 
 TABLE OF CONTENTS. 
 
 THE SEARCH FOR THE FOUNDATIONS OF GEOM- 
 ETRY: HISTORICAL SKETCH. 
 
 PACE 
 
 Axioms and the Axiom of Parallels i 
 
 Metageometry 5 
 
 Precursors 7 
 
 Gauss T I 
 
 Riemann 15 
 
 Lobatchevsky 20 
 
 Bolyai 22 
 
 Later Geometricians 24 
 
 Grassmann 27 
 
 Euclid Still Unimpaired 31 
 
 THE PHILOSOPHICAL BASIS OF MATHEMATICS. 
 
 The Philosophical Problem 35 
 
 Transcendentalism and Empiricism 3<^ 
 
 The A Priori and the Purely Formal 40 
 
 Anyness and its Universality 46 
 
 Apriority of Different Decrees 49 
 
 Space as a Spread of Motion 56 
 
 Uniqueness of Pure Space 61 
 
 Mathematical Space and Physiological Space 63 
 
 Homf)^,'encity of Space Due to Ab.straction 66 
 
 Even Boundaries as Standards of MeasuremcTit 6g 
 
 The Straight Line Indispensable T2. 
 
 The Superreal 7^' 
 
 Discrete Units and the Continuum 7S 
 
 ^L\THEMATICS AND METAGEOMETRY. 
 
 Different Geometrical Systcm.s X2 
 
 Tridimensionality S4 
 
 Three a Concept of r.ninulary 8<H
 
 IV THE FOUNDATIONS OF MATHEMATICS. 
 
 PAGE 
 
 Space of Four Dimensions 90 
 
 The Apparent A.rbitrariness of the A Priori 96 
 
 Definiteness of Construction 99 
 
 One Space, But Various Systems of Space Measurement.. . 104 
 Fictitious Spaces and the Apriority of All Space Measure- 
 ment 109 
 
 Infinitude 116 
 
 Geometry Remains A Priori 119 
 
 Sense-Experience and Space. . 122 
 
 The Teaching of Mathematics 127 
 
 EPILOGUE 132 
 
 INDEX 139
 
 THE SEARCH FOR THE FOUNDATIONS 
 OF GEOMETRY: HISTORICAL SKETCH. 
 
 AXIOMS AND THE AXIOM OF PARALLELS. 
 
 MATHEMATICS as commonly taught in our 
 schools is based upon axioms. These axioms 
 SO called are a few simple formulas which the be- 
 ginner must take on trust. 
 
 Axioms are defined to be self-evident propo- 
 sitions, and are claimed to be neither demonstrable 
 nor in need of demonstration. They are statements 
 which are said to command the assent of every 
 one who comprehends their meaning. 
 
 The word axiom^ means "honor, reputation, 
 high rank, authority," and is used by Aristotle, 
 almost in the modern sense of the term, as "a self- 
 evident highest principle," or "a truth so obvious 
 as to be in no need of proof." It is derived from 
 the verl) a^Lovv, "to deem worth}-, to think fit, to 
 maintain," and is cognate with afto?, "worth" or 
 "worthy." 
 
 Euclid cloes not use the term "axiom." He 
 starts with Definitions," which describe tlie mean- 
 ings of point, line, surface, j>Iane. angle, etc. He
 
 2 THE FOUNDATIONS OF MATHEMATICS. 
 
 then proposes Postulates'' in which he takes for 
 granted that we can draw straight Hnes from any 
 point to any other point, and that we can prolong 
 any straight line in a straight direction. Finally, 
 he adds what he calls Common Notions* which em- 
 body some general principles of logic (of pure rea- 
 son) specially needed in geometry, such as that 
 things which are equal to the same thing are equal 
 to one another; that if equals be added to equals, 
 the wholes are equal, etc. 
 
 I need not mention here perhaps, since it is a 
 fact of no consequence, that the readings of the 
 several manuscripts vary, and that some proposi- 
 tions (e. g., that all right angles are equal to one 
 another) are now missing, now counted among the 
 postulates, and now adduced as common notions. 
 
 The commentators of Euclid who did not under- 
 stand the difference between Postulates and Com- 
 mon Notions, spoke of both as axioms, and even 
 to-day the term Common Notion is mostly so trans- 
 lated. 
 
 In our modern editions of Euclid we find a 
 statement concerning parallel lines added to either 
 the Postulates or Common Notions. Originally it 
 appeared in Proposition 29 where it is needed to 
 prop up the argument that would prove the equality 
 of alternate angles in case a third straight line falls 
 upon parallel straight lines. It is there enunciated 
 as follows: 
 
 "But those straight Hnes which, with another straight 
 
 ^ aiTirttxara, * Koi.val ivvoiai.
 
 HISTORICAL SKETCH. 3 
 
 line fallinjT upon them, make the interior angles on the same 
 side less than two right angles, do meet if continually pro- 
 duced." 
 
 Now this is exactly a point that calls for proof. 
 Proof was then, as ever since it has remained, alto- 
 gether lacking-. So the proposition was formulated 
 dogmatically thus : 
 
 "If a straight line meet two straight lines, so as to make 
 the two interior angles on the same side of it taken together 
 less than two right angles, these straight lines being con- 
 tinually produced, shall at length meet upon that side on 
 which are the angles which are less than two right angles." 
 
 And this proposition has been transferred by 
 the editors of Euclid to the introductory portion of 
 the book where it now appears either as the fifth 
 Postulate or the eleventh, twelfth, or thirteenth 
 Common Notion. The latter is obviously the less 
 appropriate place, for the idea of ])ara]lelism is 
 assuredly not a Common Notion ; it is not a rule 
 of pure reason such as would be an essential con- 
 dition of all thinking, reasoning, or logical argu- 
 ment. Anrl if we do not give it a place of its own, 
 it should either be classed among the postulates, or 
 recast so as to 1)ecome a pure definition, it is usu- 
 ally referred to as "the axiom of i)arallels." 
 
 It seems to me that no one can read ihc axiom of 
 jxirallels as it stands in Furlid without receiving 
 the impression that {\\v statcni'-nl was affixed by a 
 later redactor. Even in Proposition 2(), the original 
 place of its insertion, it comes in as an afterthought ; 
 and if Euclid himself had considered the diffirultv
 
 4 THE FOUNDATIONS OF MATHEMATICS. 
 
 of the parallel axiom, so called, he would have placed 
 it among the postulates in the first edition of his 
 book, or formulated it as a definition.'^ 
 
 Though the axiom of parallels must be an inter- 
 polation, it is of classical origin, for it was known 
 even to Proclus (410-485 A. D.), the oldest com- 
 mentator of Euclid. 
 
 By an. irony of fate, the doctrine of the parallel 
 axiom has become more closely associated with 
 Euclid's name than anything he has actually writ- 
 ten, and when we now speak of Euclidean geometry 
 we mean a system based upon that determination of 
 parallelism. 
 
 We may state here at once that all the attempts 
 made to derive the axiom of parallels from pure 
 reason were necessarily futile, for no one can prove 
 the absolute straightness of lines, or the evenness of 
 space, by logical argument. Therefore these con- 
 cepts, including the theory concerning parallels, 
 cannot be derived from pure reason ; they are not 
 Common Notions and possess a character of their 
 own. But the statement seemed thus to hang in the 
 air, and there appeared the possibility of a geom- 
 etry, and Qven of several geometries, in whose do- 
 mains the parallel axiom would not hold good. This 
 large field has been called metageometry, hyper- 
 
 ^ For Professor Halsted's ingenious interpretation of the origin 
 of the parallel theorem see The Monist, Vol. IV, No. 4, p. 487. He 
 believes that Euclid anticipated metageometry, but it is not probable 
 that the man who wrote the argument in Proposition 29 had the fifth 
 Postulate before him. He would have referred to it or stated it at 
 least approximately in the same words. But the argument in Propo- 
 sition 29 differs considerably from the parallel axiom itself.
 
 HISTORICAL SKETCH. 5 
 
 geometry, or pangeometry, and may be regarded 
 as due to a generalization of the space-conception 
 involving what might be called a metaphysics of 
 mathematics. 
 
 METAGEOMETRY. 
 
 Mathematics is a most conservative science. Its 
 system is so rigid and all the details of geometrical 
 demonstration are so complete, that the science was 
 comrnonly regarded as a model of perfection. Thus 
 the philosophy of mathematics remained undevel- 
 oped almost two thousand years. Not that there 
 were not great mathematicians, giants of thought, 
 men like the Bernoullis. Leibnitz and Newton, Euler. 
 and others, worthy to be named in one breath with 
 Archimedes, Pythagoras and Euclid, but they ab- 
 stained from entering into ])hilosophical specula- 
 tions, and the very idea of a ])angeometry remained 
 foreign to them. They may privately have reflected 
 on the subject, but they dicl not give utterance to 
 their thoughts, at least they left no records of them 
 to posterity. 
 
 It would be wrong, however, to assume that the 
 mathematicians of former ages were not conscious 
 of the flifficulty. They always felt that there was 
 a flaw in the luiclidean foundation of geometry, 
 but they were satisfied to sui)ply any need of basic 
 l)rincii)le'^ in the shape of axioms, and it has become 
 (|uite customary ( I might almost say ortbodox) to 
 say that mathematics is based upon axioms. In fact, 
 l)eople enjoyed the idea that mathematics, the most
 
 b THE FOUNDATIONS OF MATHEMATICS. 
 
 lucid of all the sciences, was at bottom as mysterious 
 as the most mystical dogmas of religious faith. 
 
 Metageometry has occupied a peculiar position 
 among mathematicians as well as with the public at 
 large. The mystic hailed the idea of "7Z-dimensional 
 spaces," of "space curvature" and of other concep- 
 tions of which we can form expressions in abstract 
 terms but which elude all our attempts to render 
 them concretely present to our intelligence. He 
 relished the idea that by such conceptions mathe- 
 matics gave promise to justify all his speculations 
 and to give ample room for a multitude of notions 
 that otherwise would be doomed to irrationality. 
 In a word, metageometry has always proved attrac- 
 tive to erratic minds. Among the professional math- 
 ematicians, however, those who were averse to phil- 
 osophical speculation looked upon it with deep dis- 
 trust, and therefore either avoided it altogether or 
 rewarded its labors with bitter sarcasm. Prominent 
 mathematicians did not care to risk their reputation, 
 and consequently many valuable thoughts remained 
 unpublished. Even Gauss did not care to speak 
 out boldly, but communicated his thoughts to his 
 most intimate friends under the seal of secrecy, not 
 unlike a religious teacher who fears the odor of 
 heresy. He did not mean to suppress his thoughts, 
 but he did not want to bring them before the public 
 unless in mature shape. A letter to Taurinus con- 
 cludes with the remark: 
 
 "Of a man who has proved himself a thinking mathe- 
 matician, T fear not that he will misunderstand what I say,
 
 HISTORICAL SKETCH. 7 
 
 but under all circumstances you have to regard it merely as 
 a private communication of which in no wise public use, or 
 one that may lead to it, is to be made. Perhaps I shall pub- 
 lish them myself in the future if I should gain more leisure 
 than my circumstances at present permit. 
 
 "C. F. Gauss. 
 "GoETTiNGEx, 8. November, 1824." 
 
 But Gauss never did publish anything upon this 
 topic although the seeds of his thought thereupon 
 fell upon fertile ground and bore rich fruit in the 
 works of his disciples, foremost in those of Riemann. 
 
 PRECURSORS. 
 
 The first attempt at improvement in the matter 
 of parallelism was made by Nasir Eddin ( 1201- 
 1274) whose work on Euclid was printed in Arabic 
 in 1594 in Rome. His labors were noticed by John 
 W'allis who in 165 1 in a Latin translation com- 
 municated Nasir Eddin's exposition of the fifth Pos- 
 tulate to the mathematicians of the University of 
 Oxford, and then propounded his own views in a 
 lecture deHvered on July 11, 1663. Nasir Eddin 
 takes his stand upon the postulate that two straight 
 lines which cut a third straight line, the one at 
 right angles, the other at some other angle, will 
 converge on the side where the angle is actite and 
 diverge where it is obtuse. Wallis, in his endeavor 
 to prove this postulate, starts willi llu- ;mxilinrv 
 theorem : 
 
 "If a limited straight line which lies upon an un- 
 limited straight line be prolonged in a straight direction,
 
 8 THE FOUNDATIONS OF MATHEMATICS. 
 
 its prolongation will fall upon the unlimited straight 
 line." 
 
 There is no need of entering into the details of 
 his proof of this auxiliary theorem. We may call 
 his theorem the proposition of the straight line and 
 may grant to him that he proves the straightness of 
 the straight line. In his further argument Wallis 
 shows the close connection of the problem of paral- 
 lels with the notion of similitude. 
 
 Girolamo Saccheri, a learned Jesuit of the seven- 
 teenth century, attacked the problem in a new way. 
 Saccheri was born September 5, 1667, at San Remo. 
 Having received a good education, he became a 
 member of the Jesuit order March 24, 1685, and 
 served as a teacher of grammar at the Jesuit College 
 di Brera, in Milan, his mathematical colleague be- 
 ing Tommaso Ceva (a brother of the more famous 
 Giovanni Ceva). Later on he became Professor of 
 Philosophy and Polemic Theology at Turin and in 
 1697 at Pavia. He died in the College di Brera 
 October 25, 1733. 
 
 Saccheri saw the close connection of parallelism 
 with the right angle, and in his work on Euclid^ he 
 examines three possibilities. Taking a quadrilateral 
 ABCD with the angles at A and B right angles 
 and the sides AC and BD equal, the angles at C and 
 D are without difficulty shown to be equal each to 
 the other. They are moreover right angles or else 
 they are either obtuse or acute. He undertakes to 
 
 ^ Euclidcs ab omni naevo vindicatus; sive conatus geometricus 
 quo stabiliimtur prima ipsa iiniversae geometriae principia. Auctore 
 Hieronymo Saccherio Societatis Jesu. Mediolani, 1773.
 
 HISTORICAL SKETCH. 
 
 prove the absurdity of these two latter suppositions 
 so as to leave as the only solution the sole possibility 
 left, viz., that they must be right angles. But he 
 finds difficulty in pointing out the contradiction to 
 which these assumptions may lead and thus he opens 
 a path on which Lobatchevsky (1793-1856) and 
 Bolyai (1802-1860) followed, reaching a new view 
 which makes three geometries possible, viz., the 
 geometries of (i) the acute angle, (2) the obtuse 
 angle, and (3) the right angle, the latter being the 
 Euclidean geometry, in which the theorem of paral- 
 lels holds. 
 
 B 
 
 D 
 
 While Saccheri seeks the solution of the problem 
 through the notion of the right angle, the German 
 mathematician Lambert starts from the notion of 
 the angle-sum of the triangle. 
 
 Johann Heinrich Lambert was born August 26, 
 1728, in Miihlhauscn. a city which at that time was 
 a part of Switzerland. He died in 1777. His The- 
 ory of flic Parallel Lines, written in 1766, was not 
 ])ublished till 1786, nine years after his death, by 
 Bernoulli and Ilindcnburg in tlie Mai^acin fi'ir die 
 rcine iiiid auii^cwaiidte MatJicuiatik. 
 
 Lambert ])oints out that there are three possi- 
 bilities: the sum of the angles of a triangle may be
 
 lO THE FOUNDATIONS OF MATHEMATICS. 
 
 exactly equal to, more than, or less than i8o degrees. 
 The first will make the triangle a figure in a plane, 
 the second renders it spherical, and the third pro- 
 duces a geometry on the surface of an imaginary 
 sphere. As to the last hypothesis Lambert said not 
 without humor :'^ 
 
 "This result^ possesses something attractive which easily 
 suggests the wish that the third hypothesis might be true." 
 
 He then adds:^ 
 
 "But I do not wish it in spite of these advantages, be- 
 cause there would be innumerable other inconveniences. 
 The trigonometrical tables would become infinitely more 
 complicated, and the similitude as well as proportionality of 
 figures would cease altogether. No figure could be repre- 
 sented except in its own absolute size ; and astronomy would 
 be in a bad plight, etc.*' 
 
 Lobatchevsky's geometry is an elaboration of 
 Lambert's third hypothesis, and it has been called 
 "imaginary geometry" because its trigonometric 
 formulas are those of the spherical triangle if its 
 sides are imaginary, or, as Wolfgang Bolyai has 
 shown, if the radius of the sphere is assumed to be 
 imaginary =(V — i)r. 
 
 France has contributed least to the literature on 
 the subject. Augustus De Morgan records the fol- 
 lowing story concerning the efforts of her greatest 
 mathematician to solve the Euclidean problem. La- 
 
 ^ P. 351, last line in the Magazin fiir die rcinc uiid angewandte 
 Mathematik, 1786. 
 
 ' Lambert refers to the proposition that the mooted angle might 
 be less than 90 degrees. 
 
 '/fttrf., p. 352.
 
 HISTORICAL SKETCH. II 
 
 grange, he says, composed at the close of his Hfe 
 a discourse on parallel lines. He began to read it 
 in the Academy but suddenly stopped short and 
 said: "II faut que j'y songe encore." With these 
 words he pocketed his papers and never recurred 
 to the subject. 
 
 Legendre's treatment of the subject appears in 
 the third edition of his elements of Euclid, but he 
 omitted it from later editions as too difficult for be- 
 ginners. Like Lambert he takes his stand upon the 
 notion of the sum of the angles of a triangle, and 
 like Wallis he relies upon the idea of similitude, 
 saying that "the length of the units of measurement 
 is indifferent for proving the theorems in ques- 
 tion."^" 
 
 GAUSS. 
 
 A new epoch begins \\ith Gauss, or rather with 
 his ingenious disciple Riemann. While Gauss was 
 rather timid about speaking openly on the subject, 
 he did not wish his ideas to be lost to posterity. In 
 a letter to Schumacher dated May 17, 1831, he 
 said: 
 
 "I have bc'S^iin to jot clown somcthinc^ of my own medi- 
 tations, which are partly older than forty years, but which 
 I have never written out, beinj^ oblijj^ed therefore to excogi- 
 tate many things three or four times over. I do not wish 
 them to pass away with me." 
 
 The notes to whicli Gauss here refers have not 
 been found among his posthumous papers, and it 
 
 ^'' Mcmnirrs dc I'Acadcmic dcs Sciences de I'liislilut de Prance. 
 Vol. XII, 1833.
 
 12 THE FOUNDATIONS OF MATHEMATICS. 
 
 therefore seems probable that they are lost, and our 
 knowledge of his thoughts remains limited to the 
 comments that are scattered through his corres- 
 pondence with mathematical friends. 
 
 Gauss wrote to Bessel (1784-1846) January 27, 
 1829: 
 
 "T have also in my leisure hours frequently reflected upon 
 another problem, now of nearly forty years' standing. I 
 refer to the foundations of geometry. I do not know 
 whether I have ever mentioned to you my views on this 
 matter. J\Iy meditations here also have taken more definite 
 shape, and my conviction that we cannot thoroughly demon- 
 strate geometry a priori is, if possible, more strongly con- 
 firmed than ever. But it will take a long time for me to 
 bring myself to the point of working out and making public 
 my very extensiz'e investigations on this subject, and pos- 
 sibly this will not be done during my life, inasmuch as I 
 stand in dread of the clamors of the Boeotians, which would 
 be certain to arise, if I shotild ever give full expression to 
 my views. It is curious that /;/ addition to the celebrated 
 flaw in Euclid's Geometry, which mathematicians have hith- 
 erto endeavored in vain to patch and never will succeed, 
 there is still another blotch in its fabric to which, so far as 
 I know, attention has never yet been called and which it will 
 by no means be easy, if at all possible, to remove. This is 
 the definition of a plane as a surface in which a straight 
 line joining any tivo points lies wholly in that plane. This 
 definition contains more than is requisite to the determina- 
 tion of a surface, and tacitly involves a theorem which is in 
 need of prior proof." 
 
 Bessel in his answer to Gauss makes a distinc- 
 tion between Euclidean geometry as practical and 
 metageometry (the one that does not depend upon
 
 HISTORICAL SKETCH. I3 
 
 the theorem of parallel lines) as true geometry. 
 He writes under the date of February lo, 1829: 
 
 "I should regard it as a great misfortune if you were to 
 allow yourself to be deterred by the 'clamors of the Boeo- 
 tians' from explaining your views of geometry. From what 
 Lambert has said and Schweikart orally communicated, it 
 has become clear to me that our geometry is incomplete and 
 stands in need of a correction which is hypothetical and 
 which vanishes when the sum of the angles of a plane tri- 
 angle is equal to 180°. This would be the true geometry 
 and the Euclidean the practical, at least for figures on the 
 earth." 
 
 In another letter to Bessel, April 9, 1830, Gauss 
 sums up his views as follows : 
 
 "The ease with which you have assimilated my notions of 
 geometry has been a source of genuine delight to me, espe- 
 cially as so few possess a natural bent for them. I am pro- 
 foundly convinced that the theory of space occupies an en- 
 tirely different position with regard to our knowledge a 
 priori from that ot the theory of numbers (Grossenlehre) ; 
 that perfect conviction of the necessity and therefore the 
 absolute truth which is characteristic of the latter is totally 
 wanting to our knowledge of the former. We must confess 
 in all humility that a number is solely a product of our mind. 
 •Space, on the other hand, possesses also a reality outside of 
 our mind, the laws of which we cannot fully prescribe a 
 priori." 
 
 Another letter of Gauss may l)e quoted here in 
 fidl. It is a reply to Taurinus and contains an ap- 
 preciation of his essay on the Parallel Lines. Gauss 
 writes ivom r,r,iiino;-cn, \ov. S, i<Sj4: 
 
 "Your esteemed communication of October 30th, with
 
 14 THE FOUNDATIONS OF MATHEMATICS. 
 
 the accompanying little essay, I have read with considerable 
 pleasure, the more so as I usually find no trace whatever of 
 real geometrical talent in the majority of the people who 
 ofifer new contributions to the so-called theory of parallel 
 lines. 
 
 "With regard to your effort, I have nothing (or not 
 much) more to say, except that it is incomplete. Your pres- 
 entation of the demonstration that the sum of the three an- 
 gles of a plane triangle cannot be greater than i8o°, does 
 indeed leave something to be desired in point of geometrical 
 precision. But this could be supplied, and there is no doubt 
 that the impossibility in question admits of the most rigorous 
 demonstration. But the case is quite different with the 
 second part, viz., that the sum of the angles cannot be 
 smaller than i8o° ; this is the real difBculty, the rock on 
 which all endeavors are wrecked. I surmise that you have 
 not employed yourself long with this subject. I have pon- 
 dered it for more than thirty years, and I do not believe 
 that any one could have concerned himself more exhaus- 
 tively with this second part than I, although I have not 
 published anything on this subject. The assumption that 
 the sum of the three angles is smaller than i8o° leads to a 
 new geometry entirely dififerent from our Euclidean, — a 
 geometry which is throughout consistent with itself, and 
 which I have elaborated in a manner entirely satisfactory 
 to myself, so that I can solve every problem in it with the 
 exception of the determining of a constant, which is not 
 a priori obtainable. The larger this constant is taken, the 
 nearer we approach tlie Euclidean geometry, and an infin- 
 itely large value will make the two coincident. The propo- 
 sitions of this geometry appear partly paradoxical and ab- 
 surd to the uninitiated, but on closer and calmer considera- 
 tion it will be found that thev contain in them absolutelv 
 nothing that is impossible. Thus, the three angles of a 
 triangle, for example, can be made as small as we will, 
 provided the sides can be taken large enough ; whilst the
 
 HISTORICAL SKETCH. 15 
 
 area of a triangle, however great the sides may be taken, 
 can never exceed a definite Hmit, nay, can never once reach 
 it. All my endeavors to discover contradictions or incon- 
 sistencies in this non-Euclidean geometry have been in vain, 
 and the only thing in it that conflicts with our reason is the 
 fact that if it were true there would necessarily exist in space 
 a linear magnitude quite determinate in itself, yet unknown 
 to us. But I opine that, despite the empty word-wisdom of 
 the metaphysicians, in reality we know little or nothing of 
 the true nature of space, so much so that we are not at liberty 
 to characterize as absolutely impossible things that strike us 
 as unnatural. If the non-Euclidean geometry were the true 
 geometry, and the constant in a certain ratio to such mag- 
 nitudes as lie within the reach of our measurements on the 
 earth and in the heavens, it could be determined a posteriori. 
 I have, therefore, in jest frequently expressed the desire that 
 the Euclidean geometry should not be the true geometry, 
 because in that event we should have an absolute measure 
 a priori." 
 
 Schweikart, a contemporary of Gauss, may in- 
 cidentally be mentioned as havinj^ worked out a 
 geometry that would 1)C' independent of the Euclid- 
 ean axiom. He called it astral geometry.^^ 
 
 RIKMANN. 
 
 Gauss's ideas fell upon good soil in his disciple 
 Riemann fi(S26-i866) whose Ilabilitali^n Lecture 
 on "The IIy])otheses which Constitute the Bases of 
 Geometry" inaugurates a new epocli in the history 
 of the i)hilosophy of mathematics. 
 
 Riemann states the situation as follows. I quote 
 
 "Die Thcorie der ParallcUimen, nehsi dem Vorschlag Hirer Vcr- 
 banimng aus der Geometric. Lcipsic :iik1 Jena, 1807.
 
 l6 THE FOUNDATIONS OF MATHEMATICS. 
 
 from Clifford's almost too literal translation (first 
 published in Nature, 1873) • 
 
 "It is known that geometry assumes, as things given, 
 both the notion of space and the first principles of construc- 
 tions in space. She gives definitions of them which are 
 merely nominal, while the true determinations appear in the 
 form of axioms. The relation of these assumptions remains 
 consequently in darkness ; we neither perceive whether and 
 how far their connection is necessary, nor, a priori, whether 
 it is possible. 
 
 "From Euclid to Legendre (to name the most famous of 
 modern reforming geometers) this darkness was cleared up 
 neither by mathematicians nor by such philosophers as con- 
 cerned themselves with it." 
 
 Riemann arrives at a conclusion which is nega- 
 tive. He says : 
 
 "The propositions of geometry cannot be derived from 
 general notions of magnitude, but the properties which dis- 
 tinguish space from other conceivable triply extended mag- 
 nitudes are only to be deduced from experience." 
 
 In the attempt at discovering the simplest mat- 
 ters of fact from which the measure-relations of 
 space may be determined, Riemann declares that — 
 
 "Like all matters of fact, they are not necessary, but 
 only of empirical certainty ; they are hypotheses." 
 
 Being a mathematician, Riemann is naturally 
 bent on deductive reasoning, and in trying to find 
 a foothold in the emptiness of pure abstraction he 
 starts with general notions. He argues that posi- 
 tion must be determined by measuring quantities, 
 and this necessitates the assumption that length of
 
 HISTORICAL SKETCH. jy 
 
 lines is independent of position. Then he starts with 
 the notion of manifoldness, which he undertakes to 
 speciaHze. This speciaHzation, however, may be 
 done in various ways. It may be continuous, as is 
 geometrical space, or consist of discrete units, as 
 do arithmetical numbers. \\'e may construct mani- 
 foldnesses of one, two, three, or n dimensions, and 
 the elements of which a system is constructed may 
 be functions which undergo an infinitesimal dis- 
 placement expressible by dx. Thus spaces become 
 possible in which the directest linear functions (an- 
 alogous to the straight lines of Euclid) cease to be 
 straight and suffer a continuous deflection which 
 may be positive or negative, increasing or decreas- 
 ing. 
 
 Riemann argues that the simplest case will be, 
 if the differential line-element ds is the square root 
 of an always positive integral homogeneous function 
 of the second order of the (|uantities d.v in wliich 
 the coefficients are continuous functions of the quan- 
 tities X, viz., ds =\/tdx-, but it is one instance only 
 of a whole class of possibilities. He says: 
 
 "Manifoldnesses in which, as in the plane and in space, 
 the hne-element may be reduced to the form \/'fdy-\ are 
 therefore only a particular case of the manifoldnesses to he 
 here investip;-ated : they recjuire a special name, anri therefore 
 these manifoldnesses in which the square of the line-element 
 may be expressed as the sum of the squares of complete dif- 
 ferentials I will call Hat." 
 
 The Eticlidean plane is the best-known instance 
 of flat space being a manifold of a zero curvature.
 
 l8 THE FOUNDATIONS OF MATHEMATICS. 
 
 Flat or even space has also been called by the 
 new-fangled word homaloidal,^^ which recommends 
 itself as a technical term in distinction from the 
 popular meaning of even and fiat. 
 
 In applying his determination of the general 
 notion of a manifold to actual space, Riemann ex- 
 presses its properties thus : 
 
 "In the extension of space-construction to the infinitely 
 great, we must distinguish between tmboundedness and in- 
 finite extent ; the former belongs to the extent-relations, the 
 latter to the measure relations. That space is an unbounded 
 threefold manifoldness, is an assumption which is developed 
 by every conception of the outer world ; according to which 
 every instant the region of real perception is completed and 
 the possible positions of a sought object are constructed, and 
 which by these applications is forever confirming itself. The 
 unboimdedness of space possesses in this way a greater em- 
 pirical certainty than any external experience. But its in- 
 finite extent by no means follows from this ; on the other 
 hand, if we assume independence of bodies from position, 
 and therefore ascribe to space constant curvature, it must 
 necessarily be finite, provided this curvature has ever so 
 small a positive value. If we prolong all the geodetics 
 starting in a given surface-element, we should obtain an 
 unbounded surface of constant curvature, i. e., a surface 
 which in a Hat manifoldness of three dimensions would take 
 the form of a sphere, and consequently be finite." 
 
 It is obvious from these quotations that Rie- 
 mann is a disciple of Kant. He is inspired by his 
 teacher Gauss and by Herbart. But while he starts 
 a transcendentalist, employing mainly the method 
 of deductive reasoning, he arrives at results which 
 
 "From the Greek o/xaXos, level.
 
 HISTORICAL SKETCH. I9 
 
 would Stamp him an empiricist of the school of Mill. 
 He concludes that the nature of real space, which is 
 only one instance among many possibilities, must 
 be determined a posteriori. The problem of tri- 
 dimensionality and homaloidality are questions 
 which must be decided by experience, and while 
 upon the whole he seems inclined to grant that 
 Euclidean geometry is the most practical for a solu- 
 tion of the coarsest investigations, he is inclined 
 to believe that real space is non-Euclidean. Though 
 the deviation from the Euclidean standard can onlv 
 be slight, there is a possibility of determining it by 
 exact measurement and observation. 
 
 Riemann has succeeded in impressing his view 
 upon meta-geometricians down to the present day. 
 They have built higher and introduced new ideas, 
 yet the cornerstone of metageometry remained the 
 same. It will therefore be found recommendable 
 in a discussion of the problem to begin with a criti- 
 cism of his Habilitation Lecture. 
 
 It is regrettable that Riemann was not allowed to 
 work out his philosophy of mathematics. He died 
 at the premature age of forty, but the work which 
 he pursued with so much success had already been 
 taken up by two others, Lobatchevsky and Bolyai, 
 who, each in his own way, actually contrived a 
 geometry indej)endent of the theorem of parallels. 
 
 It is perhaps no accident that the two independ- 
 ent and almost simultaneous inventors of a non- 
 luiclidean geometry arc original, not to say way- 
 ward, characters living on the outskirts of Euro-
 
 20 THE FOUNDATIONS OF MATHEMATICS. 
 
 pean civilization, the one a Russian, the other a 
 Magyar. 
 
 LOBATCHEVSKY. 
 
 Nicolai Ivanovich Lobatchevsky^^ was born Oc- 
 tober 22 (Nov. 2 of our calendar), 1793, in the 
 town of Makariev, about 40 miles above Nijni Nov- 
 gorod On the Volga. His father was an architect 
 who died in 1797, leaving behind a widow and two 
 small sons in poverty. At the gymnasium Loba- 
 tchevsky was noted for obstinacy and disobedience, 
 and he escaped expulsion only through the protec- 
 tion of his mathematical teacher, Professor Bartels, 
 w^ho even then recognized the extraordinary talents 
 of the boy. Lobatchevsky graduated with distinc- 
 tion and became in his further career professor of 
 mathematics and in 1827 Rector of the University 
 of Kasan. Two books of his offered for official 
 publication were rejected by the paternal govern- 
 ment of Russia, and the manuscripts may be con- 
 sidered as lost for good. Of his several essays on 
 the theories of parallel lines we mention only the 
 one which made him famous throughout the whole 
 mathematical world. Geometrical Researches on the 
 Theory of Parallels, published by the University of 
 Kasan in 1835.^* 
 
 Lobatchevsky divides all lines, which in a plane 
 go out from a point A with reference to a given 
 
 " The name is spelled differently according to the different 
 methods of transcribing Russian characters. 
 
 " For further details see Prof. G. B. Halsted's article "Loba- 
 chevski" in The Open Court, 1898, pp. 411 ff.
 
 HISTORICAL SKETCH. 
 
 21 
 
 Straight line BC in the same plane, into two classes 
 — cutting and not cutting. In progressing from the 
 not-cutting lines, such as EA and GA, to the cutting 
 lines, such as FA, we must come upon one HA 
 that is the boundary between the two classes ; and it 
 is this which he calls the parallel line. He desig- 
 nates the parallel angle on the perpendicular {p^ 
 AD, dropped from Aupon BC) by n. If X[{p) <T/< tt 
 (viz., 90 degrees) we shall have on the other side 
 of p another angle DAK=n(/)) parallel to DB, 
 so that on this assumption we 
 must make a distinction of 
 sides in parallelisui, and we 
 must allow two parallels, one 
 on the one and one on the 
 other side. If U(p) = y2Tr 
 we have only intersecting 
 lines and one parallel ; but if 
 II(/)) <^7r we have two i)ar- 
 allel lines as boundaries be- 
 tween the intersecting and 
 non-intersecting lines. 
 
 We need not further dex'clop Lobatchevsky's 
 idea. Among other things, he proves that "if in 
 any rectilinear triangle the sum of the three angles 
 is equal to two right angles, so is this also the case 
 for cver\ nilur irianglc," that is to say, each in- 
 stance is a saiii])k' of the whole, and if one case is 
 established, the nature of the whole system to which 
 it belongs is determinerl. 
 
 The inij)ortance of Lobalchevsky's discovery
 
 22 THE FOUNDATIONS OF MATHEMATICS. 
 
 consists in the fact that the assumption of a geom- 
 etry from which the parallel axiom is rejected, does 
 not lead to self-contradictions but to the conception 
 of a general geometry of which the Euclidean is one 
 possibility. This general geometry was later on 
 most appropriately called by Lobatchevsky "Pan- 
 geometry." 
 
 BOLYAI. 
 
 John (or, as the Hungarians say, Janos) Bolyai 
 imbibed the love of mathematics in his father's 
 house. He was the son of Wolfgang (or Farkas) 
 Bolyai, a fellow student of Gauss at Gottingen when 
 the latter was nineteen years old. Farkas was pro- 
 fessor of mathematics at Maros Vasarhely and 
 wrote a two-volume book on the elements of mathe- 
 matics^'^ and in it he incidentally mentions his vain 
 attempts at proving the axiom of parallels. His 
 book was only partly completed when his son Janos 
 wrote him of his discovery of a mathematics of pure 
 space. He said: 
 
 "As soon as I have put it into order, I intend to write 
 and if possible to publish a work on parallels. At this 
 moment, it is not yet finished, but the way which I have fol- 
 lowed promises me with certainty the attainment of my aim, 
 if it is at all attainable. It is not yet attained, but I have 
 discovered such magnificent things that I am myself aston- 
 ished at the result. It would forever be a pity, if they were 
 lost. When you see them, my father, you yourself will con- 
 
 ° Tentamcn juventutem studiosam in elementa matheos etc. iii- 
 troducendi, printed in Maros Vasarhely. By Farkas Bolyai. Part I. 
 Maros Vasarhely, 1832. It contains the essay by Janos Bolyai as an 
 Appendix.
 
 HISTORICAL SKETCH. 23 
 
 cede it. Now I cannot say more, only so much that frotn 
 nothing I have created another wholly new world. All that 
 I have hitherto sent you compares to it as a house of cards 
 to a castle."^® 
 
 Janos being convinced of the futility of proving 
 Euclid's axiom, constructed a geometry of absolute 
 space which would be independent of the axiom of 
 parallels. And he succeeded. He called it the Sci- 
 ence Absolute of Space}"^ an essay of twenty-four 
 pages which Bolyai's father incorporated in the 
 first volume of his Tent amen as an appendix. 
 
 Bolyai was a thorough Magyar. He was wont 
 to dress in high boots, short wide Hungarian trou- 
 sers, and a white jacket. He loved the violin and 
 was a good shot. \\'hile serving as an officer in 
 the Austrian army, Janos was known for his hot 
 temper, which finally forced him to resign his com- 
 mission as a captain, and we learn from Professor 
 Halsted that for some provocation he was chal- 
 lenged by thirteen cavalry officers at once. Janos 
 calmly accepted and proposed to fight them all, one 
 after the other, on condition that he be permitted 
 after each duel to play a piece on his violin. We 
 know not the nature of these duels nor the construc- 
 tion of the pistols. 1)iit the fact remains assured that 
 he came out unhurt. As for the rest of the report 
 that "he came out victor from the thirteen duels, 
 
 " See Halsted's introduction to the English translation of Bo- 
 lyai's Science Absolute of Space, p. .xxvii. 
 
 '^''Appendix scicniiam spalii ahsnJutc veram cxhihcns: a rcritate 
 aut falsitatc axiotnalis XI. Luclidci (a priori hand tinquam deci- 
 dettda) independcntem; Adjecta ad casum falsitatis quadratura cir- 
 cuit geomelrica.
 
 24 THE FOUNDATIONS OF MATHEMATICS. 
 
 leaving his thirteen adversaries on the square," 
 we may he permitted to express a mild but deep- 
 seated doubt. 
 
 Janos Bolyai starts with straight lines in the 
 same plane, which may or may not cut each other. 
 Now there are two possibilities: there may be a 
 system in which straight lines can be drawn which 
 do not cut one another, and another in which they all 
 cut one another. The former, the Euclidean he 
 calls t, the latter S. "All theorems," he says, 
 "which are not expressly asserted for % or for S 
 are enunciated absolutely, that is, they are true 
 whether t or S is reality."^^ The system S can 
 be established without axioms and is actualized in 
 spherical trigonometry, (ibid. p. 21). Now S can 
 be changed to , viz., plane geometry, by reducing 
 the constant i to its limit (where the sect y^o) 
 which is practically the same as the construction 
 of a circle with r ^ co , thus changing its periphery 
 into a straight line. 
 
 LATER GEOMETRICIANS. 
 
 The labors of Lobatchevsky and Bolyai are sig- 
 nificant in so far as they prove beyond the shadow 
 of a doubt that a construction of geometries other 
 than Euclidean is possible and that it involves us 
 in no absurdities or contradictions. This upset the 
 traditional trust in Euclidean geometry as absolute 
 truth, and it opened at the same time a vista of new 
 
 " See Halsted's translation, p. 14.
 
 HISTORICAL SKETCH. 25 
 
 problems, foremost among which was the question 
 as to the mutual relation of these three different 
 geometries. 
 
 It was Cayley who proposed- an answer which 
 was further elaborated by Felix Klein. These two 
 ingenious mathematicians succeeded in deriving by 
 projection all three systems from one common ab- 
 original form called by Klein Grundgebild or the 
 Absolute. In addition to the three geometries hith- 
 erto known to mathematicians, Klein added a fourth 
 one which he calls elliptic}^ 
 
 Thus we may now regard all the different ge- 
 ometries as three species of one and the same genus 
 and we have at least the satisfaction of knowing that 
 there is terra Urma at the bottom of our mathemat- 
 ics, though it lies deeper than was formerly sup- 
 posed. 
 
 Prof. Simon Newcomb of Johns Hopkins Uni- 
 versity, although not familiar with Klein's essays, 
 worked along the same line and arrived at similar 
 results in his article on "Elementarv Theorems Re- 
 
 ml 
 
 lating to the Geometry of a Space of Three Dimen- 
 sions and of Uniform Positive Curvature in the 
 Fourth Dimension.^'' 
 
 In the meantime the problem of geometry be- 
 came interesting to outsiders also, for the theorem 
 of parallel lines is a problem of space. A most 
 
 " "Uebcr die .so(?enanntc nicht-ciiklidischc Geometric" in Math. 
 Annalen, 4, 6 (1871-1872). Vorlcsungcn iihcr niclit-culclidischc Cco- 
 tnctrie, Gottingen, 1893. 
 
 "Crelle's Journal fur die reinc und anficivandtc Mathcmatik. 
 1877.
 
 26 THE FOUNDATIONS OF MATHEMATICS. 
 
 excellent treatment of the subject came from the 
 pen of the great naturalist Helmholtz who wrote 
 two essays that are interesting even to outsiders 
 because written in a most popular style.^^ 
 
 A collection of all the materials from Euclid to 
 Gauss, compiled by Paul Stackel and Friedrich 
 Engel under the title Die Theorie der Parallellinien 
 von Euklid bis auf Gauss, eine Urkundensaininlimg 
 sur Vorgeschichte dcr nicht-euklidischen Geometric, 
 is perhaps the most useful and important publica- 
 tion in this line of thought, a book which has become 
 indispensable to the student of metageometry and 
 its history. 
 
 A store of information may be derived from 
 Bertrand A. W. Russell's essay on the Foundations 
 of Geometry. He divides the history of metageom- 
 etry into three periods: The synthetic, consisting 
 of suggestions made by Legendre and Gauss; the 
 metrical, inaugurated by Riemann and character- 
 ized by Lobatchevsky and Bolyai; and the projec- 
 tive, represented by Cayley and Klein, who reduce 
 metrical properties to projection and thus show that 
 Euclidean and non-Euclidean systems may result 
 from "the absolute." 
 
 Among American writers no one has contrib- 
 uted more to the interests of metageometry than the 
 indefatigable Dr. George Bruce Halsted.^^ He has 
 
 ^ "Ueber die thatsachlichen Grundlagen der Geometrie," in Wis- 
 scnschaftl. Ahh.., 1866, Vol. H., p. 610 ff., and "Ueber die Thatsachen, 
 die der Geometrie zu Grunde liegen," ibid., 1868, p. 618 ff. 
 
 '"From among his various publications we mention only his 
 translations : Geometrical Researches on the Theory of Parallels by 
 Nicholaus Lobatchezvsky. Translated from the Original. And The
 
 HISTORICAL SKETCH. 2J 
 
 not only translated Bolyai and Lobatchevsky. but 
 in numerous articles and lectures advanced his own 
 theories toward the solution of the problem. 
 
 Prof. B. J. Delboeuf and Prof. H. Poincare have 
 expressed their conceptions as to the nature of the 
 bases of mathematics, in articles contributed to The 
 Monisf.^^ The latter treats the subject from a purely 
 mathematical standpoint, while Dr. Ernst Mach in 
 his little book Space mid Geometry,^* in the chapter 
 "On Physiological, as Distinguished from Geomet- 
 rical, Space," attacks the problem in a very original 
 manner and takes into consideration mainly the nat- 
 ural growth of space conception. His exposition 
 might be called "the physics of geometry." 
 
 GRASSMANN. 
 
 I cannot conclude this short sketch of the historv 
 of metageometry without paying a tribute to the 
 memory of Hermann Grassmann of Stettin, a math- 
 ematician of first degree whose highly important 
 results in this line of work have only of late found 
 
 Science Absolute of Space, Independent of the Truth or Falsity of 
 huclid's Axiom XI. (zi.'hich can never be decided a priori). By John 
 Bolyai. Translated from the Latin, both pubhslicd in Austin, Texas, 
 the translator's former place of residence. I-iirther, \vc refer the 
 reader to Halsted's bibliography of the literature on hyperspace and 
 non-Euclidean geometry in the American Journal of Mathematics, 
 Vol. L, pp. 261-276, 384, 385, and Vol. IL, pp. 65-70. 
 
 "They are as follows: "Are the Dimensions of the Physical 
 World Absolute?" by Prof. B. J. Delba-uf. The Monist, January. 
 1894; "Oil the Foundations of Geometry," by Prof. U. Pcjincare, 
 The Monist. October, 1898; also "Relations Between Experimental 
 and Mathematical Physics," The Monist, July, 1902. 
 
 ''Chicago: Open Court Pub. Co., 1906. This chapter also ap- 
 peared in The Monist for Ai)rii, 1901.
 
 28 THE FOUNDATIONS OF MATHEMATICS. 
 
 the recognition which they so fully deserve. I do 
 not hesitate to say that Hermann Grassmann's Li- 
 neare AusdehnimgsleJire is the best work on the 
 philosophical foundation of mathematics from the 
 standpoint of a mathematician. 
 
 Grassmann establishes first the idea of mathe- 
 matics as the science of pure form. He shows that 
 the mathematician starts from definitions and then 
 proceeds to show how the product of thought may 
 originate either by the single act of creation, or by 
 the double act of positing and combining. The 
 former is the continuous form, or magnitude, in the 
 narrower sense of the term, the latter the discrete 
 form or the method of combination. He distin- 
 guishes between intensive and extensive magnitude 
 and chooses as the best example of the latter the 
 sect^^ or limited straight line laid down in some 
 definite direction. Hence the name of the new sci- 
 ence, "theory of linear extension." 
 
 Grassmann constructs linear formations of 
 which systems of one, two, three, and n degrees 
 are possible. The Euclidean plane is a system of 
 second degree, and space a system of third degree. 
 He thus generalizes the idea of mathematics, and 
 having created a science of pure form, points out 
 that geometry is one of its applications which origi- 
 nates under definite conditions. 
 
 Grassmann made the straight line the basis of 
 
 ^ Grassmann's term is Strecke, a word connected with the Anglo- 
 Saxon "Stretch," being that portion of a Hne that stretches between 
 two points. The translation "sect," was suggested by Prof. G. B. 
 Halsted.
 
 HISTORICAL SKETCH. 29 
 
 his geometrical definitions. He defines the plane 
 as the totality of parallels which cut a straight line 
 and space as the totalit}^ of parallels which cut the 
 plane. Here is the limit to geometrical construction, 
 but abstract thought knows of no bounds. Having 
 generalized our mathematical notions as systems 
 of first, second, and third degree, we can continue 
 in the numeral series and construct systems of four, 
 five, and still higher degrees. Further, we can de- 
 termine any plane by any three points, given in the 
 figures Xi, Xo, X:>, not lying in a straight line. If the 
 equation between these three figures be homogene- 
 ous, the totality of all points that correspond to it will 
 be a system of second degree. If this homogeneous 
 equation is of the first grade, this system of second 
 degree will be simple, viz., of a straight line; but 
 if the equation be of a higher grade, we shall have 
 curves for which not all llic laws of plane geometry 
 hold good. The same considerations lead to a dis- 
 tinction between homaloidal space and non-Euclid- 
 ean systems.^* 
 
 Being professor at a German gymnasium and 
 not a universitw Grassniann's 1)ook remained neg- 
 lected and ihc newness of liis methods i)revented 
 superficial readers from ajjpreciating the sweeping 
 significance of bis i:)rr)positions. Since there was 
 no call whatexer for llie book, the ])ul)lishers re- 
 turned tbe whole edition to tlie ])aper mill, and the 
 complimentary copies which tlie author had sent 
 
 "Sec Grassniann's Ausdehunu^slrhrr. 1S44, Anhanj? I. pii. 273- 
 274.
 
 30 THE FOUNDATIONS OF MATHEMATICS. 
 
 out to his friends are perhaps the sole portion that 
 was saved from the general doom. 
 
 Grassmann, disappointed in his mathematical 
 labors, had in the meantime turned to other studies 
 and gained the honorary doctorate of the Univer- 
 sity of Tubingen in recognition of his meritorious 
 work on the St. Petersburg Sankrit Dictionary, 
 when Victor Schlegel called attention to the simi- 
 larity of Hamilton's theory of vectors to Grass- 
 mann's concept oiStrccke, both being limited straight 
 lines of definite direction. Suddenly a demand for 
 Grassmann's book was created in the market; but 
 alas! no copy could be had, and the publishers 
 deemed it advisable to reprint the destroyed edition 
 of 1844. The satisfaction of this late recognition 
 was the last joy that brightened the eve of Grass- 
 mann's life. He wrote the introduction and an ap- 
 pendix to the second edition of his Lincare Aus- 
 dehminzdehre, but died while the forms of his book 
 were on the press. 
 
 At the present day the literature on metageo- 
 metrical subjects has grown to such an extent that 
 we do not venture to enter into further details. We 
 will only mention the appearance of Professor 
 Schoute's work on more-dimensional geometry^^ 
 which promises to be the elaboration of the pan- 
 geometrical ideal. 
 
 "" Mehrdimensionale Gcomctrie von Dr. P. H. Schoute, Professor 
 der Math, an d. Reichs-Universitat zii Groningen, Holland. Leipsic. 
 Goschen. So far only the first volume, which treats of linear space, 
 has appeared.
 
 HISTORICAL SKETCH. 3 1 
 
 EUCLID STILL UNIMPAIRED. 
 
 Having briefly examined tlie chief innovations 
 of modern times in the field of elementary geometry, 
 it ought to be pointed out that in spite of the well- 
 deserved fame of the metageometricians from Wal- 
 lis to Halsted, Euclid's claim to classicism remains 
 unshaken. The metageometrical movement is not 
 a revolution against Euclid's authority but an at- 
 tempt at widening our mathematical horizon. Let 
 us hear what Halsted, one of the boldest and most 
 iconoclastic among the champions of metageometry 
 of the present day, has to say of Euclid. Halsted 
 begins the Introduction to his English translation 
 of Bolyai's Science .Ibsoliifc of Space with a terse 
 description of the history of Euclid's great book 
 The Elements of Geometry, the rediscovery of which 
 is not the least factor that initiated a new epoch in 
 the development of Europe which may be called the 
 era of inventions, of discoveries, and of the appre- 
 ciation as well as growth of science. Halsted says : 
 
 "The immortal Elements of EucUd was already in dim 
 antiquity a classic, regarded as al)solutely perfect, valid 
 without restriction. 
 
 "Elementary geometry was for two thousand years as 
 stationary, as fixed, as peculiarly Greek as the Partlu-non. 
 r)n this foundatif)n piin- science rose in Arcliinudi's. in 
 Apollonius, in Pappus; struggled in Theon. in I lypatia ; 
 declined in Proclus ; fell into the long decadence of the Dark- 
 Ages. 
 
 "The book that monkish Kurope could no longer under-
 
 32 THE FOUNDATIONS OF MATHEMATICS. 
 
 stand was then taught in Arabic by Saracen and Moor in 
 the Universities of Bagdad and Cordova. 
 
 "To bring the Hght, after weary, stupid centuries, to 
 Western Christendom, an EngHshman, Adelhard of Bath, 
 journeys, to learn Arabic, through Asia Minor, through 
 Egypt, back to Spain. Disguised as a Mohammedan stu- 
 dent, he got into Cordova about 1120, obtained a Moorish 
 copy of Euclid's Elements, and made a translation from the 
 Arabic into Latin. 
 
 "The first printed edition of Euclid, published in Venice 
 in 1482, was a Latin version from the Arabic. The trans- 
 lation into Latin from the Greek, made by Zamberti from a 
 manuscript of Theon's revision, was first published at Ven- 
 ice in 1505. 
 
 "Twenty-eight years later appeared the editio princeps in 
 Greek, published at Basle in 1533 by John Hervagius, 
 edited by Simon Grynaeus. This was for a century and 
 three-quarters the only printed Greek text of all the books, 
 and from it the first English translation (1570) was made 
 by 'Henricus Billingsley,' afterward Sir Henry Billingsley, 
 Lord Mayor of London in 1591. 
 
 "And even to-day, 1895. in the vast system of examina- 
 tions carried out by the British Government, by Oxford, 
 and by Cambridge, no proof of a theorem in geometry will 
 be accepted which infringes Euclid's sequence of propo- 
 sitions. 
 
 "Nor is the work unworthy of this extraordinary im- 
 mortality. 
 
 "Says Clifl^ord: 'This book has been for nearly twenty- 
 two centuries the encouragement and guide of that scientific 
 thought which is one thing with the progress of man from 
 a worse to a better state. 
 
 " 'The encouragement : for it contained a body of knowl- 
 edge that was really known and could be relied on. 
 
 " 'The guide ; for the aim of every student of every sub-
 
 HISTORICAL SKETCH. 33 
 
 ject was to bring his knowledge of that subject into a form 
 as perfect as that which geometry had attained.' " 
 
 Euclid's Elements of Geometry is not counted 
 among the books of divine revelation, but trul}- it 
 deserves to be held in religious veneration. There 
 is a real sanctity in mathematical truth which is 
 not sufhciently appreciated, and certainly if truth, 
 helpfulness, and directness and simplicity of presen- 
 tation, give a title to rank as divinely inspired litera- 
 ture, Euclid's great work should be counted among 
 the canonical books of mankind. 
 
 * * * 
 
 Is there any need of warning our readers that 
 the foregoing sketch of the history of metageometry 
 is both brief and popular? We have purposely 
 avoided the discussion of technical details, limiting 
 our exposition to the most essential points and try- 
 ing to show them in a light that will render them 
 interesting even to the non-mathematical reader. 
 It is meant to serve as an introduction to the real 
 matter in hand, viz., an examination of the founda- 
 tions upon which geometrical truth is to be ration- 
 ally justified. 
 
 The author has purposely introduced what 
 might be called a biographical element in these ex- 
 positions of a subject which is commonly regarded 
 as dry and abstruse, and endeavored to give some- 
 thing of the lives of the men who have struggled 
 and labored in this line of thought and have sacri- 
 ficed their time and energy on the altar of one of the
 
 34 THE FOUNDATIONS OF MATHEMATICS. 
 
 noblest aspirations of man, the delineation of a 
 philosophy of mathematics. He hopes thereby to 
 relieve the dryness of the subject and to create an 
 interest in the labor of these pioneers of intellectual 
 progress.
 
 THE PHILOSOPHICAL BASIS OF ^lATHE- 
 
 MATICS. 
 
 THE PHILOSOPHICAL PROBLEM. 
 
 HAVING thus reviewed the history of non- 
 EucHdean geometry, which, rightly consid- 
 ered, is but a search for the philosophy of mathe- 
 matics, I now turn to the problem itself and, in the 
 conviction that I can offer some hints which con- 
 tain its solution, I will formulate my own views in 
 as popular language as would seem compatible with 
 exactness. Not being a mathematician by profes- 
 sion I have only one excuse to offer, which is this: 
 that I have more and more come to the conclusion 
 that the problem is not mathematical but philo- 
 sophical ; and I hope that those who are competent 
 to judge will correct me where I am mistaken. 
 
 The problem of the philosophical foundation of 
 mathematics is closely connected with the topics 
 of Kant's Critique of Pure Reason. It is the old 
 quarrel between Empiricism and Transcendental- 
 ism. Hence our method of dealing with it will nat- 
 urally be philosophical, not typically mathematical. 
 
 The proper solution can be attained only by 
 analysing the fundamental concepts of mathematics
 
 36 THE FOUNDATIONS OF MATHEMATICS. 
 
 and by tracing them to their origin. Thus alone can 
 we know their nature as w^ell as the field of their 
 applicability. 
 
 We shall see that the data of mathematics are 
 not without their premises; they are not, as the 
 Germans say, voraiissetBun gslos ] and though math- 
 ematics is built up from nothing, the mathematician 
 does not start with nothing. He uses mental im- 
 plements, and it is they that give character to his 
 science. 
 
 Obviously the theorem of parallel lines is one 
 instance only of a difficulty that betrays itself every- 
 where in various forms; it is not the disease of 
 geometry, but a symptom of the disease. The the- 
 orem that the sum of the angles in a triangle is 
 equal to 180 degrees; the ideas of the evenness or 
 homaloidality of space, of the rectangularity of the 
 square, and more remotely even the irrationality of 
 TT and of e, are all interconnected. It is not the 
 author's intention to show their interconnection, 
 nor to prove their interdependence. That task is 
 the work of the mathematician. The present in- 
 vestigation shall be limited to the philosophical side 
 of the problem for the sake of determining the na- 
 ture of our notions of evenness, which determines 
 both parallelism and rectangularity. 
 
 At the bottom of the difficulty there lurks the old 
 problem of apriority, proposed by Kant and decided 
 by him in a way which promised to give to mathe- 
 matics a solid foundation in the realm of transcen- 
 dental thought. And yet the transcendental method
 
 THE PHILOSOPHICAL BASIS. 37 
 
 finally sent geometry away from home in search of 
 a new domicile in the wide domain of empiricism. 
 
 Riemann, a disciple of Kant, is a transcendental- 
 ist. He starts with general notions and his arguments 
 are deductive, leading him from the abstract down 
 to concrete instances; but when stepping from the 
 ethereal height of the absolute into the region of 
 definite space-relations, he fails to find the necessary 
 connection that characterizes all a priori reasoning ; 
 and so he swerves into the domain of the a posteriori 
 and declares that the nature of the specific features 
 of space must be determined by experience. 
 
 The very idea seems strange to those who have 
 been reared in traditions of the old school. An un- 
 sophisticated man, when he speaks of a straight 
 line, means that straightness is implied thereby; 
 and if he is told that space may be such as to render 
 all straightest lines crooked, he will naturally be 
 "bewildered. If his metageometrical friend, with 
 much learnedness and in sober earnest, tells him that 
 when he sends out a ray as a straight line in a for- 
 ward direction it will imperceptibly deviate and 
 finally turn back upon his occiput, he will naturally 
 become suspicious of the mental soundness of his 
 company. Would not many of us dismiss such ideas 
 with a shrug if there were not geniuses of the very 
 first rank who subscribe to the same? So in all 
 modesty we ha\-e to defer our judgment until com- 
 petent study and mature reflection have enabled us 
 to understand the difficulty which they encounter 
 and then judge their solution. One thing is sure,
 
 38 THE FOUNDATIONS OF MATHEMATICS. 
 
 however: if there is anything wrong with meta- 
 geometry, the fault Hes not in its mathematical 
 portion but must be sought for in its philosophical 
 foundation, and it is this problem to which the 
 present treatise is devoted. 
 
 While we propose to attack the problem as a 
 philosophical question, we hope that the solution 
 will prove acceptable to mathematicians. 
 
 TRANSCENDENTALISM AND EMPIRICISM. 
 
 In philosophy we have the old contrast between 
 the empiricist and transcendentalist school. The 
 former derive everything from experience, the latter 
 insist that experience depends upon notions not de- 
 rived from experience, called transcendental, and 
 these notions are a priori. The former found their 
 representative thinkers in Locke, Hume, and John 
 Stuart Mill, the latter was perfected by Kant. Kanf 
 establishes the existence of notions of the a priori 
 on a solid basis asserting their universality and 
 necessity, but he no longer identified the a priori 
 with innate ideas. He granted that much to em- 
 piricism, stating that all knowledge begins with ex- 
 perience and that experience rouses in our mind 
 the a priori which is characteristic of mind. Mill 
 went so far as to deny altogether necessity and uni- 
 versality, claiming that on some other planet 2X2 
 might be 5. French positivism, represented by 
 Comte and Littre, follows the lead of Mill and thus 
 they end in agnosticism, and the same result was
 
 THE PHILOSOPHICAL BASIS. 39 
 
 reached in England on grounds somewhat different 
 by Herbert Spencer. 
 
 The way which we propose to take may be char- 
 acterized as the New Positivism. We take our stand 
 upon the facts of experience and estabhsh upon tiie 
 systematized formal features of our experience a 
 new conception of the a priori, recognizing the uni- 
 versality and necessity of formal laws but rejecting 
 Kant's transcendental idealism. The a priori is 
 not deducible from the sensorv elements of our sen- 
 sations, but we trace it in the formal features of 
 experience. It is the result of abstraction and sys- 
 tematization. Thus we establish a method of dealing 
 with experience (commonly called Pure Reason) 
 which is possessed of universal validity, implying 
 logical necessity. 
 
 The New Positivism is a further development of 
 philosophic thought which combines the merits of 
 both schools, the Transcendentalists and Empiri- 
 cists, in a higher unity, discarding at the same time 
 their aberrations. In this way it becomes possible 
 to gain a firm basis upon the secure ground of facts, 
 according to the principle of positivism, and yet to 
 preserve a method established by a study of the 
 purely formal, which will not end in nescience (the 
 ideal of agnosticism) 1)Ut justify science, and thus 
 establishes the philosophy of science.^ 
 
 'We have treated the pliilosonhic.-il problem of tlie a fi'iori at 
 full length in a discussion of Kant s Prolegomena. See the author's 
 Kant's Prolegomena, edited in English, with an essay on Kant's Phi- 
 losophy and other .Supplementary Material for the study of Kant, 
 pp. 167-240. Cf. Fundamental Prnblems,. the chapters "Form and
 
 40 THE FOUNDATIONS OF MATHEMATICS. 
 
 It is from this standpoint of the philosophy of 
 science that we propose to investigate the problem 
 of the foundation of geometry. 
 
 THE A PRIORI AND THE PURELY FORMAL. 
 
 The bulk of our knowledge is from experience, 
 i. e., we know things after having become acquainted 
 with them . Our knowledge of things is a posteriori. 
 If we want to know whether sugar is sweet, we must 
 taste it. If we had not done so, and if no one had 
 tasted it, we could not know it. However, there is 
 another kind of knowledge which we do not find out 
 by experience, but by reflection. If I want to know 
 how much is 3X3, or {a-\-hY or the angles in a 
 regular polygon, I must compute the answer in 
 my own mind. I need make no experiments but 
 must perform the calculation in my own thoughts. 
 This knowledge which is the result of pure thought 
 is a priori; \nz., it is generally applicable and holds 
 good even before we tried it. \\'hen we begin to 
 make experiments, we presuppose that all our a 
 priori arguments, logic, arithmetic, and mathemat- 
 ics, will hold good. 
 
 Kant declared that the law of causation is of the 
 same nature as arithmetical and logical truths, and 
 that, accordingly, it will have to be regarded as 
 a priori. Before we make experiments, we know 
 that every cause has its effects, and wherever there 
 
 Formal Thought," pp. 26-60, and "The Old and the New Mathemat- 
 ics," pp. 61-73; and Primer of Philosophy, pp. 51-103.
 
 THE PHILOSOPHICAL BASIS. 4I 
 
 is an effect we look for its cause. Causation is not 
 proved by, but justified through, experience. 
 
 The doctrine of the a priori has been much mis- 
 interpreted, especially in England. Kant calls that 
 which transcends or goes beyond experience in the 
 sense that it is the condition of experience "tran- 
 scendental," and comes to the conclusion that the 
 a priori is transcendental. Our a priori notions 
 are not derived from experience but are products 
 of pure reflection and they constitute the conditions 
 of experience. By experience Kant understands 
 sense-impressions, and the sense-impressions of the 
 outer world (which of course are a posteriori) are 
 reduced to system by our transcendental notions; 
 and thus knowledge is the product of the a priori 
 and the a posteriori. 
 
 A sense-impression becomes a perception by be- 
 ing regarded as the effect of a cause. The idea of 
 causation is a transcendental notion. A\'ithout it 
 experience would be impossil)le. An astronomer 
 measures angles and determines the distance of the 
 moon and of the sun. Experience furnishes the 
 data, they are a posteriori: l)ut his mathematical 
 methods, the number system, and all arithmetical 
 functions are a priori. He knows them before he 
 collects the details of his investigation; and in so 
 far as they are the condition without which his 
 sense - impressions could not be transformed into 
 knowledge, they are called transcendental. 
 
 Note here Kant's use of the word transcendental 
 which denotes the clearest and most reliable knowl-
 
 42 THE FOUNDATIONS OF MATHEMATICS. 
 
 edge in our economy of thought, pure logic, arith- 
 metic, geometry, etc. But transcendental is fre- 
 quently (though erroneously) identified with "tran- 
 scendent," which denotes that which transcends our 
 knowledge and accordingly means "unknowable." 
 Whatever is transcendental is, in Kantian terminol- 
 ogy, never transcendent. 
 
 That much will suffice for an explanation of the 
 historical meaning of the word transcendental. We 
 must now explain the nature of the a priori and its 
 source. 
 
 The a priori is identical with the purely formal 
 which originates in our mind by abstraction. When 
 we limit our attention to the purely relational, drop- 
 ping all other features out of sight, we produce a 
 field of abstraction in which we can construct purely 
 formal combinations, such as numbers, or the ideas 
 of types and species. Thus we create a world of 
 pure thought which has the advantage of being 
 applicable to any purely formal consideration of 
 conditions, and we work out systems of numbers 
 which, when counting, we can use as standards of 
 reference for our experiences in practical life. 
 
 But if the sciences of pure form are built upon 
 an abstraction from which all concrete features are 
 omitted, are they not empty and useless verbiage? 
 
 Empty they are, that is true enough, but for all 
 that they are of paramount significance, because 
 they introduce us into the sanctum sanctissimum of 
 the world, the intrinsic necessity of relations, and 
 thus they become the key to all the riddles of the
 
 THE PHILOSOPHICAL BASIS. 43 
 
 universe. They are in need of being supplemented 
 by observation, by experience, by experiment; but 
 while the mind of the investigator builds up purely 
 formal systeuLS of reference (such as numbers) and 
 purely formal space-relations (such as geometry), 
 the essential features of facts (of the objective 
 world) are in their turn, too, purely formal, and 
 they make things such as they are. The suchness 
 of the world is purely formal, and its suchness alone 
 is of importance. 
 
 In studying the processes of nature we watch 
 transformations, and all we can do is to trace the 
 changes of form. Matter and energy are words 
 which in their abstract significance have little value; 
 tliey merely denote actuality in general, the one of 
 being, the other of doing. \Miat interests us most 
 are the forms of matter and energy, how they 
 change by transformation ; and it is obvious that 
 the famous law of the conservation of matter and 
 energy is merely the reverse of the truth that cau- 
 sation is transformation. Tn its elements which in 
 their totality are called matter and energy, the ele- 
 ments of existence remain the same, but the forms in 
 which they combine cliange. The sum-total of the 
 mass anrl the sum-total of the forces of the world 
 ran be neither increased nor diminished; they re- 
 main the same to-day that they have ahvavs been 
 and as they w ill remain forever. 
 
 All a pnslcr'wri cognition is concrete and par- 
 ticular, while all a priori cognition is abstract and 
 general. 'I'he concrete is (at least in its relation
 
 44 THE FOUNDATIONS OF MATHEMATICS. 
 
 to the thinking subject) incidental, casual, and indi- 
 vidual, but the abstract is universal and can be used 
 as a general rule under which all special cases may 
 be subsumed. 
 
 The a priori is a mental construction, or, as Kant 
 says, it is ideal, viz., it consists of the stuff that 
 ideas are made of, it is mind-made. While we grant 
 that the purely formal is ideal we insist that it is 
 made in the domain of abstract thought, and its 
 fundamental notions have been abstracted from ex- 
 perience by concentrating our attention upon the 
 purely formal. It is, not directly but indirectly and 
 ultimately, derived from experience. It is not de- 
 rived from sense-experience but from a considera- 
 tion of the relational (the purely formal) of ex- 
 perience. Thus it is a subjective reconstruction of 
 certain objective features of experience and this 
 reconstruction is made in such a way as to drop 
 every thing incidental and particular and retain 
 only the general and essential features ; and we gain 
 the unspeakable advantage of creating rules or for- 
 mulas which, though abstract and mind-made, apply 
 to any case that can be classified in the same cate- 
 gory. 
 
 Kant made the mistake of identifying the term 
 "ideal" with "subjective," and thus his transcen- 
 dental idealism was warped by the conclusion that 
 our purely formal laws were not objective, but were 
 imposed by our mind upon the objective world. Our 
 mind (Kant said) is so constituted as to interpret 
 all facts of experience in terms of form, as appear-
 
 THE PHILOSOPHICAL BASIS. 45 
 
 ing in space and time, and as being subject to the law 
 of cause and effect; but what things are in them- 
 selves we cannot know. We object to Kant's sub- 
 jectivizing the purely formal and look upon form 
 as an essential and inalienable feature of objective 
 existence. The thinking subject is to other thinking 
 subjects an object moving about in the objective 
 world . E\'en when contemplating our own exist- 
 ence we must grant (to speak with Schopenhauer) 
 that our bodily actualization is our own object; i. e., 
 we (each one of us as a real living creature) are as 
 much objects as are all the other objects in the 
 world. It is the objectified part of our self that in 
 its inner experience abstracts from sense-experience 
 the interrelational features of things, such as rioht 
 and left, top and bottom, shape and figure and struc- 
 ture, succession, connection, etc. The formal ad- 
 heres to the object and not to the subject, and every 
 object (as soon as it develops in the natural way of 
 evolution first into a feeling creature and then into 
 a thinking being) will be a1)le to build up a /^riori 
 from the abstract notion of form in general the sev- 
 eral systems of formal thought: arithmetic, geom- 
 etry, algebra, logic, and the conceptions of time, 
 space, and causalitv. 
 
 Accordingly, all formal tliought, although we 
 grant its idcdity, is fashioned from niatcrinls ab- 
 stracted from the oljjective world, and it is tlu-reforc 
 a matter of course that they are applicable to the 
 objectix'c world. Tliey belong to the object and, 
 when we thinking subjects beget them from our own
 
 46 THE FOUNDATIONS OF MATHEMATICS. 
 
 minds, we are able to do so only because we are ob- 
 jects that live and move and have our being in the 
 objective world. 
 
 ANYNESS AND ITS UNIVERSALITY. 
 
 We know that facts are incidental and hap- 
 hazard, and appear to be arbitrary; but we must not 
 rest satisfied with single incidents. We must gather 
 enough single cases to make abstractions. Abstrac- 
 tions are products of the mind; they are subjective; 
 but they have been derived from experience, and 
 they are built up of elements that have objective 
 significance. 
 
 The most important abstractions ever made by 
 man are those that are purely relational. Every- 
 thing from which the sensory element is entirely 
 omitted, where the material is disregarded, is called 
 "pure form," and the relational being a considera- 
 tion neither of matter nor of force or energy, but 
 of number, of position, of shape, of size, of form, 
 of relation, is called "the purely formal." The no- 
 tion of the purely formal has been gained by ab- 
 straction, viz., by abstracting, i. e., singling out and 
 retaining, the formal, and by thinking away, by can- 
 celling, by omitting, by leaving out, all the features 
 which have anything to do with the concrete sensory 
 element of experience. 
 
 And what is the result ? 
 
 We retain the formal element alone which is 
 void of all concreteness, void of all materialitv, void
 
 THE PHILOSOPHICAL BASIS. 47 
 
 of all particularity. It is a mere nothing and a 
 non-entity. It is emptiness. But one thing is left, 
 — position or relation. Actuality is replaced by 
 mere potentiality, viz., the possible conditions of 
 any kind of being that is possessed of form. 
 
 The word "any" denotes a simple idea, and yet 
 it contains a good deal of thought. Mathematics 
 builds up its constructions to suit any condition. 
 "Any" implies universality, and universality in- 
 cludes necessity in the Kantian sense of the term. 
 
 In every concrete instance of an experience the 
 subject-matter is the main thing with which we are 
 concerned; but the purely formal aspect is after all 
 the essential feature, because form determines the 
 character of things, and thus the formal (on account 
 of its anyness) is the key to their comprehension. 
 
 The rise of man above the animal is due to his 
 ability to utilize the purely formal, as it revealed 
 itself to him especially in types for classifying things, 
 as genera and species, in tracing transformations 
 which present themselves as effects of causes and re- 
 ducing them to shapes of measurable relations. The 
 abstraction of the formal is made through the in- 
 strumentality of language and the result is reason, 
 — the faculty of abstract thought. Man can see the 
 universal in the particular; in the single experiences 
 he can trace the laws that are generally applicable 
 to cases of the same class; he observes some in- 
 stances and can describe them in a general formula so 
 as to cover an\- other instance of the same kind, and 
 thus he becomes master of the situation; he learns
 
 48 THE FOUNDATIONS OF MATHEMATICS. 
 
 to separate in thought the essential from the acci- 
 dental, and so instead of remaining the prey of cir- 
 cumstance he gains the power to adapt circum- 
 stances to himself. 
 
 Form pervades all nature as an essential constit- 
 uent thereof. If form were not an objective feature 
 of the w^orld in which we live, formal thought would 
 never without a miracle, or, at least, not without 
 the mystery of mysticism, have originated accord- 
 ing to natural law, and man could never have arisen. 
 But form being an objective feature of all existence, 
 it impresses itself in such a way upon living crea- 
 tures that rational beings will naturally develop 
 among animals whose organs of speech are per- 
 fected as soon as social conditions produce that de- 
 mand for communication that will result in the crea- 
 tion of language. 
 
 The marvelous advantages of reason dawned 
 upon man like a revelation from on high, for he did 
 not invent reason, he discovered it; and the senti- 
 ment that its blessings came to him from above, 
 from heaven, from that power which sways the des- 
 tiny of the whole universe, from the gods or from 
 God, is as natural as it is true. The anthropoid did 
 not seek reason : reason came to him and so he be- 
 came man. Man became man by the grace of God, 
 by gradually imbibing the Logos that was with God 
 in the beginning ; and in the dawn of human evolu- 
 tion we can plainly see the landmark of mathemat- 
 ics, for the first grand step in the development of
 
 THE PHILOSOPHICAL BASIS. 49 
 
 man as distinguished from the transitional forms of 
 the anthropoid is the abihty to count. 
 
 Man's distinctive characteristic remains, even 
 to-day, reason, the faculty of purely formal thought ; 
 and the characteristic of reason is its general appli- 
 cation. All its verdicts are universal and involve 
 apriority or beforehand knowledge so that man can 
 foresee events and adapt means to ends. 
 
 APRIORITY OF DIFFERENT DEGREES. 
 
 Kant has pointed out the kinship of all purely 
 formal notions. The validity of mathematics and 
 logic assures us of the validity of the categories 
 including the conception of causation; and yet ge- 
 ometry cannot be derived from pure reason alone, 
 hut contains an additional element which imparts 
 to its fundamental conceptions an arbitrary appear- 
 ance if we attempt to treat its deductions as rigidly 
 a priori. Why should there be straight lines at all? 
 Why is it possible that by quartering the circle we 
 should have right angles with all their peculiarities? 
 All these and similar notions can not be sul)sumed 
 under a general formula of pure reason from which 
 we could derive it with logical necessity. 
 
 When dealing with lines we observe their exten- 
 tion in one direction, when dealing with planes we 
 have two dimensions, when measuring solids we 
 have three. Whv can we not rr)ntinue and construct 
 bodies that extend in t'our (Hmensions? 'Hie hmit 
 set us by space as it jK)sitively presents itself to us
 
 50 THE FOUNDATIONS OF MATHEMATICS. 
 
 seems arbitrary, and while transcendental truths 
 are undeniable and obvious, the fundamental no- 
 tions of geometry seem as stubborn as the facts of 
 our concrete existence. Space, generally granted 
 to be elbow-room for motion in all directions, after 
 all appears to be a definite magnitude as much as 
 a stone wall which shuts us in like a prison, allowing 
 us to proceed in such a way only as is permissible 
 by those co-ordinates and no more. We can by 
 no resort break through this limitation. Verily we 
 might more easily shatter a rock that impedes our 
 progress than break into the fourth dimension. The 
 boundary line is inexorable in its adamantine rigor. 
 
 Considering all these unquestionable statements, 
 is there not a great probability that space is a con- 
 crete fact as positive as the existence of material 
 things, and not a mere form, not a mere potentiality 
 of a general nature ? Certainly Euclidean geometry 
 contains some such arbitrary elements as we should 
 expect to meet in the realm of the a posteriori. No 
 wonder that Gauss expressed "the desire that the 
 Euclidean geometry should not be the true geom- 
 etry," because "in that event we should have an ab- 
 solute measure a priori.''' 
 
 Are we thus driven to the conclusion that our 
 space-conception is not a priori; and if, indeed, it is 
 not a priori, it must be a posteriori ! What else can 
 it be? Tertium non datitr. 
 
 If we enter more deeply into the nature of the 
 a priori, we shall learn that there are different kinds
 
 THE PHILOSOPHICAL BASIS. 5I 
 
 of apriority, and there is a difference between the 
 logical a priori and the geometrical a priori. 
 
 Kant never investigated the source of the a pri- 
 ori. He discovered it in the mind and seemed satis- 
 fied with the notion that it is the nature of the mind 
 to be possessed of time and space and the categories. 
 He w^ent no further. He never asked, how did mind 
 originate? 
 
 Had Kant inquired into the origin of mind, he 
 would have found that the a priori is woven into 
 the texture of mind by the uniformities of experi- 
 ence. The uniformities of experience teach us the 
 laws of form, and the purely formal applies not to 
 one case only but to any case of the same kind, and 
 so it involves "anyness," that is to say. it is (/ priori. 
 
 Mind is the product of memory, and we may 
 briertv describe its oriijin as follows: 
 
 Contact with the outer world produces impres- 
 sions in sentient substance. The traces of these 
 impressions arc preserved (a condition which is 
 called "memory") and they can be revived (which 
 state is called "recollection"). Sense-impressions 
 are different in kind and leave dift'erent traces, but 
 those which are the same in kind, or similar, leave 
 traces the forms of which are the same or similar; 
 and sense-impressions of the same kind are regis- 
 tered in the traces haxing the same form. As a note 
 of a definite pitch makes chords of the same i)itch 
 vibrate while it passes all others by; so new sense- 
 impressions revive those traces only into wliich they 
 fit. and therebv announce themselves as beiuLr the
 
 52 THE FOUNDATIONS OF MATHEMATICS. 
 
 same in kind. Thus all sense-impressions are sys- 
 tematized according to their forms, and the result 
 is an orderly arrangement of memories which is 
 called "mind."2 
 
 Thus mind develops through uniformities in sen- 
 sation according to the laws of form. Whenever 
 a new sense-perception registers itself mechanically 
 and automatically in the trace to which it belongs, 
 the event is tantamount to a logical judgment which 
 declares that the object represented by the sense- 
 impression belongs to the same class of objects 
 ^^•hich produced the memory traces with which it is 
 registered. 
 
 If we abstract the interrelation of all memory- 
 traces, omitting their contents, we have a pure sys- 
 tem of genera and species, or the a priori idea of 
 "classes and subclasses." 
 
 The a priori, though mind-made, is constructed 
 of chips taken from the objective world, but our 
 several a priori notions are by no means of one and 
 the same nature and rigidity. On the contrary, 
 there are different degrees of apriority. The emp- 
 tiest forms of pure thought are the categories, and 
 the most rigid truths are the logical theorems, which 
 can be represented diagrammatically so as to be a de- 
 nt onst ratio ad oculos. 
 
 If all bs are B and if /Sis a b, then yS is a B. If 
 all dogs are quadrupeds and if all terriers are dogs, 
 then terriers are quadrupeds. It is the most rigid 
 
 ' For a more detailed exposition see the author's Soul of Man ; 
 also his Whence and Whither.
 
 THE PHILOSOPHICAL BASIS. 53 
 
 kind of argument, and its statements are practically 
 tautologies. 
 
 The case is different with causation. The class 
 of abstract notions of which causation is an instance 
 is much more complicated. No one doubts that 
 everv effect must have had its cause, but one of the 
 keenest thinkers was in deep earnest when he 
 doubted the possibility of proving this obvious state- 
 ment. And Kant, seeing its kinship with geometry 
 and algebra, accepted it as a pviori and treated it 
 as being on equal terms with mathematical axioms. 
 Yet there is an additional element in the formula 
 of causation which somehow disguises its a priori 
 origin, and the reason is that it is not as rigidly 
 a priori as are the norms of pure logic. 
 
 What is this additional element that somehow 
 savors of the a posteriori? 
 
 If we contemplate the interrelation of genera 
 and species and su1)species, we find that the cate- 
 gories with \\hich we operate are at rest. They 
 stand before us like a well arranged cabinet with 
 several di\isions and drawers, and these drawers 
 have sulxlivisions and in these subdivisions we keep 
 boxes. The cabinet is our (/ priori system of classi- 
 fication and we store in it our a posteriori impres- 
 sions. If a thing is in box /8, we seek for it in 
 drawer b which is a sul)division of the dei)artment B. 
 
 How different is causation! While in logic 
 everything is al rest, causation is not conceivable 
 without mot inn. The norms of pure reason are 
 static, the law of cause and effect is dynamic; and
 
 54 THE FOUNDATIONS OF MATHEMATICS. 
 
 thus we have in the conception of cause and effect 
 an additional element which is mobility. 
 
 Causation is the law of transformation. We have 
 a definite system of interrelated items in which we 
 observe a change of place. The original situation 
 and all detailed circumstances are the conditions; 
 the motion that produces the change is the cause; 
 the result or new arrangement of the parts of the 
 whole system is the effect. Thus it appears that cau- 
 sation is only another version of the law of the con- 
 servation of matter and energy. The concrete items 
 of the whole remain, in their constitutional elements, 
 the same. No energy is lost; no particle of matter 
 is annihilated; and the change that takes place is 
 mere transformation.^ 
 
 The law of causation is otherwise in the same 
 predicament as the norms of logic. It can never be 
 satisfactorily proved by experience. Experience 
 justifies the a priori and verifies its tenets in single 
 instances which prove true, but single instances can 
 never demonstrate the universal and necessary va- 
 lidity of any a priori statement. 
 
 The logical a priori is rigidly a priori; it is the 
 a priori of pure reason. But there is another kind 
 of a priori which admits the use of that other ab- 
 stract notion, mobility, and mobility as much as 
 form is part and parcel of the thinking mind. Our 
 conception of caiise and efifect is just as ideal as our 
 conception of genera and species. It is just as much 
 
 * See the author's Ursachc, Grund unci Zzvcck. Dresden, 1883; 
 also his Fundamental Problems.
 
 THE PHILOSOPHICAL BASIS. 55 
 
 mind-made as they are, and its intrinsic necessity 
 and universal validity are the same. Its apriority 
 cannot be doubted ; but it is not rigidly a priori, and 
 we will call it purely a priori. 
 
 We may classify all a priori notions under two 
 headings and both ar-t transcendental (viz., con- 
 ditions of knowledge in their special fields) : one is 
 the a priori of being, the other of doing. The rigid 
 a priori is passive anyness, the less rigid a priori 
 is active anyness. Geometry belongs to the latter. 
 Its fundamental concept of space is a product of ac- 
 tive apriority; and thus we cannot derive its laws 
 from pure logic alone. 
 
 The main difficulty of the parallel theorem and 
 the straight line consists in our space-conception 
 which is not derived from rationality in general, 
 but results from our contemplation of motion. Our 
 space-conception accordingly is not an idea of pure 
 reason, but the product of pure activity. 
 
 Kant felt the difference and distinguished be- 
 tween pure reason and pure intuition or Anschau- 
 iiHi^. He did not expressly say so, but his treatment 
 suggests the idea that we ought to distinguish be- 
 tween two different kinds of a priori. Transcen- 
 dental logic, and with it all common notions of 
 Euclid, are mere applications of the law of consist- 
 ency; they are "rigidly a priori."' lUit our ])urc 
 space-conception presupposes, in addition to j)urc 
 reason, our own .activity, the potentiality of moving 
 about in any kind of a field, and thus it admits 
 another factor which cannot be derived from pure
 
 56 THE FOUNDATIONS OF MATHEMATICS. 
 
 reason alone. Hence all attempts at proving the 
 theorem on rigidly a priori grounds have proved 
 failures. 
 
 SPACE AS A SPREAD OF MOTION. 
 
 Mathematicians mean to start from nothingness, 
 so they think away everything, but they retain their 
 own mentality. Though even their mind is stripped 
 of all particular notions, they retain their principles 
 of reasoning and the privilege of moving about, 
 and from these two sources geometry can be con- 
 structed. 
 
 The idea of causation goes one step further: it 
 admits the notions of matter and energy, emptied 
 of all particularity, in their form of pure'generali- 
 zations. It is still a priori, but considerably more 
 complicated than pure reason. 
 
 The field in which the geometrician starts is pure 
 nothingness ; but we shall learn later on that noth- 
 ingness is possessed of positive qualifications. We 
 must therefore be on our guard, and we had better 
 inquire into the nature and origin of our nothing- 
 ness. 
 
 The geometrician cancels in thought all positive 
 existence except his own mental activity and starts 
 moving about as a mere nothing. In other words, 
 we establish by abstraction a domain of monotonous 
 sameness, which possesses the advantage of "any- 
 ness," i. e., an absence of particularity involving 
 universal validity. In this field of motion we pro- 
 ceed to produce geometrical constructions.
 
 THE PHILOSOPHICx\L BASIS. 57 
 
 The geometrician's activity is pure motion, 
 which means that it is mere progression ; the ideas 
 of a force exerted in moving and also of resistance 
 to be overcome are absokitely excluded. 
 
 We start moving, but whither? Before us are 
 infinite possibilities of direction. The inexhaus- 
 tibility of chances is part of the indifference as 
 to definiteness of determining the mode of motion 
 (be it straight or curved). Let us start at once 
 in all possible directions which are infinite, (a propo- 
 sition which, in a way, is realized by the light), 
 and having proceeded an infinitesimal way from the 
 starting-point A to the points B, Bi,, B2, B3, B4, 
 . . . . B ; we continue to move in infinite directions 
 at each of these stations, reaching from B the points 
 C, Ci, C2, Cj, C4, . . . . C.^. From Bi we would switch 
 
 off to the points C\\ Cf , Cf , Cf Cl\ etc. until 
 
 we reach from B^ the points C^", Cf», Cf«, Cf*, 
 . . . -C^", thus exhausting all the points which clus- 
 ter around every Bi, B2, B:-, B4 B^. Thus, by 
 
 moving after the fashion of the light, spreading 
 again and again from each new ])oint in all direc- 
 tions, in a medium that offers no resistance what- 
 ever, we obtain a uniform spread of light whose 
 intensity in every j)oint is in liie inverse square of 
 its distance from its source. Every lighted s])ot be- 
 comes a center of its own from which light travels 
 on in all directions. But among these infinite direc- 
 tions there are rays, A, B, C, Ai, Bi. d,. . . ., 
 
 A2, B2, C2 etc.. th.'it is lo '^ay. lines of motion 
 
 that pursue the original direct ion and are j^aths of
 
 58 THE FOUNDATIONS OF MATHEMATICS. 
 
 maximum intensity. Each of these rays, thus ide- 
 ally constructed, is a representation of the straight 
 line which being the shortest path between the start- 
 ing-point A and any other point, is the climax of 
 directness : it is the upper limit of effectiveness and 
 its final boundary, a non plus ultra. It is a maxi- 
 mum because there is no loss of efficacy. The 
 straight line represents a climax of economy, viz., 
 the greatest intensity on the shortest path that is 
 reached among infinite possibilities of progression 
 by uniformly following up all. In every ray the 
 maximum of intensity is attained by a minimum 
 of progression. 
 
 Our construction of motion in all directions after 
 the fashion of light is practically pure space; but 
 to avoid the forestalling of further implications we 
 will call it simply the spread of motion in all direc- 
 tions. 
 
 The path of highest intensity in a spread of 
 motion in all directions corresponds to the ray in an 
 ideal conception of a spread of light, and it is equiv- 
 alent to the straight line in geometry. 
 
 We purposely modify our reference to light in 
 our construction of straight lines, for we are well 
 aware of the fact that the notion of a ray of light 
 as a straight line is an ideal which describes the 
 progression of light only as it appears, not as it is. 
 The physicist represents light as rays only when 
 measuring its effects in reflection, etc., but when 
 considering the nature of light, he looks upon rays 
 as transversal oscillations of the ether. The notion
 
 THE PHILOSOPHICAL BASIS. 59 
 
 of light as rays is at bottom as much an a priori 
 construction as is Newton's formula of gravitation. 
 
 The construction of space as a spread of motion 
 in all directions after the analogy of light is a sum- 
 mary creation of the scope of motion, and we call it 
 "ideal space." Everything that moves about, if it 
 develops into a thinking subject, when it forms the 
 abstract idea of mobility, will inevitably create out 
 of the data of its own existence the ideal "scope of 
 motion," which is space. 
 
 When the geometrician starts to construct his 
 figures, drawing lines and determining the position 
 of points, etc., he tacitly presupposes the existence 
 of a spread of motion, such as we have described. 
 Motility is part of his equipment, and motility pre- 
 supposes a field of motion, \'iz., space. 
 
 Space is the possibility of motion, and by ideally 
 moving about in all possible directions the number 
 of which is inexhaustible, we construct our notion 
 of pure space. 1 f we speak of space wc mean this 
 construction of our mobility. It is an a priori con- 
 struction and is as unique as logic or arithmetic. 
 There is but one space, and all spaces are but por- 
 tions of this one construction. The prol)lem of tri- 
 dimensionality will be considered later on. Here 
 we insist only on the objective validity of our a 
 priori construction, wliich is the same as the ob- 
 jective validity of all our a priori constructions — 
 of iDgic and arithmetic and causality, and it rests 
 upon the same foundation. Our mathematical space 
 omits all particularity and serves our purpose of
 
 6o THE FOUNDATIONS OF MATHEMATICS. 
 
 universal application: it is founded on "anyness," 
 and thus, within the limits of its abstraction, it holds 
 good everywhere and under all conditions. 
 
 There is no need to find out by experience in the 
 domain of the a posteriori whether pure space is 
 curved. Anvness has no particular qualities; we 
 create this anyness by abstraction, and it is a matter 
 of course that in the field of our abstraction, space 
 will be the same throughout, unless by another act 
 of our creative imagination we appropriate partic- 
 ular qualities to different regions of space. 
 
 The fabric of which the purely formal is woven 
 is an absence of concreteness. It is (so far as mat- 
 ter is concerned) nothing. Yet this airy nothing 
 is a pretty tough material, just on account of its 
 indifferent "any"-ness. Being void of particularity, 
 it is universal ; it is the same throughout, and if we 
 proceed to build our air-castles in the domain of 
 anyness, we shall find that considering the absence 
 of all particularity the same construction will be the 
 same, w^herever and whenever it mav be conceived. 
 
 Professor Clifford says :^ ''We assume that two 
 lengths which are equal to the same length are 
 equal to each other." But there is no "assumption" 
 about it. The atmosphere in which our mathemat- 
 ical creations are begotten is sameness. Therefore 
 the same construction is the same wherever and 
 whenever it may be made. We consider form only; 
 we think away all other concrete properties, both 
 
 * Loc. cit., p. 53.
 
 THE PHILOSOPHICAL BASIS. 6l 
 
 of matter and energy, mass, weight, intensity, and 
 qualities of any kind. 
 
 UNIQUENESS OF PURE SPACE. 
 
 Our thought-forms, constructed in the realm of 
 empty abstraction, serve as models or as systems of 
 reference for any of our observations in the real 
 world of sense-experience. The laws of form are 
 as well illustrated in our models as in real things, 
 and can be derived from either; but the models of 
 our thought-forms are always ready at hand while 
 the real things are mostly inaccessible. The any- 
 ness of pure form explains the parallelism that ob- 
 tains between our models and actual experience, 
 which was puzzling to Kant. And truly at first 
 sight it is mystifying that a pure thought-construc- 
 tion can reveal to us some of the most im])ortant 
 and deepest secrets of objective nature ; but the sim- 
 ple solution of the mystery consists in this, that the 
 actions of nature are determined by the same con- 
 ditions of possible motions wilh which pure thought 
 is confronted in its efforts to construct its models. 
 Here as well as there we have consistency, that is 
 to say, a thing done is uiii(|ucly determined, and, 
 in pure thought as well as in reality, it is such as it 
 has been made l)v construction . 
 
 Our constructions are made in anyness and apply 
 to all possible instances of tlic kind; and thus we 
 may as well define si)ace as the potentiality of meas- 
 uring, which presU]iposes moving about. Mobility
 
 62 THE FOUNDATIONS OF MATHEMATICS. 
 
 granted, we can construct space as the scope of our 
 motion in anyness. Of course we must bear in mind 
 that our motion is in thou£>-ht only and we have 
 dropped all notions of particularity so as to leave 
 an utter absence of force and resistance. The motor 
 element, qua energy, is not taken into consideration, 
 but we contemplate only the products of progres- 
 sion. 
 
 Since in the realm of pure form, thus created 
 by abstraction, we move in a domain void of par- 
 ticularity, it is not an assumption (as Riemann 
 declares in his famous inaugural dissertation), but 
 a matter of course which follows with logical ne- 
 cessity, that lines are independent of position; they 
 are the same anywhere. 
 
 In actual space, position is by no means a negli- 
 gible quantity. A real pyramid consisting of actual 
 material is possessed of dififerent qualities according 
 to position, and the line AB, representing a path 
 from the top of a mountain to the valley is very 
 different from the line BA, which is the path from 
 the valley to the top of the mountain. In Euclidean 
 geometry AB=BA. 
 
 Riemann attempts to identify the mathematical 
 space of a triple manifold with actual space and ex- 
 pects a proof from experience, but, properly speak- 
 ing, they are radically different. In real space po- 
 sition is not a negligible factor, and would necessi- 
 tate a fourth co-ordinate which has a definite rela- 
 tion to the plumb-line; and this fourth co-ordinate 
 (which we may call a fourth dimension) suffers a
 
 THE PHILOSOPHICAL BASIS. 63 
 
 constant modification of increase in inverse propor- 
 tion to the scjuare of the distance from the center of 
 this planet of ours. It is rectiHnear, yet all the plumb- 
 lines are converging toward an inaccessible center; 
 accordingly, they are by no means of equal value in 
 their different parts. How different is mathemat- 
 ical space ! It is homogeneous throughout. And it 
 is so because we made it so by abstraction. 
 
 Pure form is a feature which is by no means 
 a mere nonentity. Having emptied existence of all 
 concrete actuality, and having thought away every- 
 thing, we are confronted by an absolute vacancy — 
 a zero of existence: but the zero has positive char- 
 acteristics and there is this peculiarity about the 
 zero that it is the mother of infinitude. The thought 
 is so true in mathematics that it is trite. Let any 
 number be divided by nought, the result is the in- 
 finitely great ; and let nought be divided by any 
 number, the result is the infinitely small. In think- 
 ing away everything concrete we retain with our 
 nothingness potentiality. Potentiality is the empire 
 of purely formal constructions, in the dim back- 
 ground of which lurks the phantom of infinitude. 
 
 MATHEMATICAL SPACE AND I' 1 1 YSIOLOGICAL SPACE. 
 
 If we admit to our conception of space the qual- 
 ities of bodies such as mass, our conception of real 
 space will become more comj)licaled still. What 
 we gain in concrete definitencss we lose from uni- 
 x'ersality. and we can return tn the general ap])li
 
 64 THE FOUNDATIONS OF MATHEMATICS. 
 
 cability of a priori conditions only by dropping 
 all concrete features and limiting our geometrical 
 constructions to the abstract domain of pure form. 
 
 Mathematical space with its straight lines, 
 planes, and right angles is an ideal construction. 
 It exists in our mind only just as much as do logic 
 and arithmetic. In the external world there are 
 no numbers, no mathematical lines, no logarithms, 
 no sines, tangents, nor secants. The same is true 
 of all the formal sciences. There are no genera and 
 species, no syllogisms, neither inductions nor deduc- 
 tions, running about in the world, but only concrete 
 individuals and a concatenation of events. There 
 are no laws that govern the motions of stars or mol- 
 ecules ; yet there are things acting in a definite way, 
 and their actions depend on changes in relational 
 conditions which can be expressed in formulas. All 
 the generalized notions of the formal sciences are 
 mental contrivances which comprise relational fea- 
 tures in general rules. The formulas as such are 
 purely ideal, but the relational features which they 
 describe are o1)jectively real. 
 
 Thus, the space-conception of the mathematician 
 is an ideal construction ; but the ideal has objective 
 significance. Ideal and subjective are by no means 
 synonyms. With the help of an ideal space-con- 
 ception we can acquire knowledge concerning the 
 real space of the objective world. Here the New- 
 tonian law may be cited as a conspicuous example. 
 
 How can the thinking subject know a priori any- 
 thing about the object? Simply because the subject
 
 THE PHILOSOPHICAL BASIS. 05 
 
 is an object moving about among other objects. 
 Mobility is a qualification of the object, and I, the 
 thinking subject, become conscious of the general 
 rules of motion only, because I also am an object 
 endowed with mobility. My ''scope of motion" can- 
 not be derived from the abstract idea of myself as 
 a thinking subject, but is the product of a conside- 
 ration of my mobility, generalized from my activities 
 as an object by omitting all particularities. 
 
 Mathematical space is a priori in the Kantian 
 sense. However, it is not ready-made in our mind, 
 it is not an innate idea, but the product of much 
 toil and careful thought. Nor will its construction 
 be possible, except at a maturer age after a long 
 development. 
 
 Physiological space is the direct and unsophisti- 
 cated space-conception of our senses. It originates 
 through experience, and is, in its way, a truer pic- 
 ture of actual or physical space than mathematical 
 space. The latter is more general, the former more 
 concrete. In physiological space position is not in- 
 different, for high and low, right and left, and up 
 and down are of great importance. Geometrically 
 congruent figures produce (as Mach has shown) 
 remarkably different impressions if they i)resent 
 themselves to the eyes in different positions. 
 
 In a geometrical plane the figures can be shoved 
 about without suffering a change of form. If they 
 are flopped, their inner relations remain the same, 
 as, e.g., helices of opposite directions arc, mathe- 
 matically considered, congruent, while in actual life
 
 66 THE FOUNDATIONS OF MATHEMATICS. 
 
 they would always remain mere symmetrical coun- 
 terparts. So the right and the left hands considered 
 as mere mathematical bodies are congruent, while 
 in reality neither can take the place of the other. 
 A glove which we may treat as a two-dimensional 
 thing can be turned inside out, but we would need 
 a fourth dimension to flop the hand, a three-dimen- 
 sional body, into its inverted counterpart. So long- 
 as we have no fourth dimension, the latter being a 
 mere logical fiction, this cannot be done. Yet mathe- 
 matically considered, the two hands are congruent. 
 Why? Not because they are actually of the same 
 shape, but because in our mathematics the quali- 
 fications of position are excluded ; the relational 
 alone counts, and the relational is the same in both 
 cases. 
 
 Mathematical space being an ideal construction, 
 it is a matter of course that all mathematical prob- 
 lems must be settled by a priori operations of pure 
 thought, and cannot be decided by external experi- 
 ment or by reference to a posteriori information. 
 
 HOMOGENEITY OF SPACE DUE TO ABSTRACTION. 
 
 When moving about, we change our place and 
 pass by different objects. These objects too are 
 moving; and thus our scope of motion tallies so 
 exactly with theirs that one can be used for the com- 
 putation of the other. All scopes of motion are pos- 
 sessed of the same anyness. 
 
 Space as we find it in experience is best defined
 
 THE PHILOSOPHICAL DASIS. 67 
 
 as the juxtaposition of things. If there is need of 
 distinguishing it from our ideal space-conception 
 which is the scope of our mobihty, we may call the 
 former pure objective space, the latter pure sub- 
 jective space, but, our subjective ideas being rooted 
 in our mobility, which is a constitutional feature 
 of our objective existence, for many practical pur- 
 poses the two are the same. 
 
 But though pure space, whether its conception 
 be established objectively or subjectively, must be 
 accepted as the same, are we not driven to the con- 
 clusion that there are after all two different kinds 
 of space: mathematical space, which is ideal, and 
 physiological space, which is real? And if they are 
 different, must we not assume them to be indepen- 
 dent of each other ^ \Miat is their mutual relation? 
 
 The two spaces, the ideal construction of mathe- 
 matical space and the reconstruction in our senses 
 of the juxtaposition of things surrounding us, are 
 different solely because they have been 1)uilt up upon 
 two different planes of abstraction ; physiological 
 space includes, and mathematical space excludes, 
 the sensory data of juxtaposition. T"*hysiological 
 space admits concrete facts, — man's own upright 
 position, gravity, perspective, etc. Matliematical 
 space is purely formal, and to lay its foundation we 
 have dug down to the bed-rock of our formal knowl- 
 edge, which is "anyness," Mathematical space is 
 a priori, albeit the a priori of motion. 
 
 At present it is sufficient to state that the homo- 
 geneity of a mathematical space is its anyness, and
 
 68 THE FOUNDATIONS OF MATHEMATICS. 
 
 its anyness is due to our construction of it in the 
 domain of pure form, involving universality and 
 excluding everything concrete and particular. 
 
 The idea of homogeneity in our space-conception 
 is the tacit condition for the theorems of similarity 
 and proportion, and also of free mobility without 
 change, viz., that figures can be shifted about with- 
 out suffering distortion either by shrinkage or by 
 expanse. The principle of homogeneity being ad- 
 mitted, we can shove figures about on any surface 
 the curvature of which is either constant or zero. 
 This produces either the non-Euclidean geometries 
 of spherical, pseudo-spherical, and elliptic surfaces, 
 or the plane geometry of Euclid — all of them a 
 priori constructions made without reference to real- 
 
 ity. 
 
 Our a priori constructions serve an important 
 purpose. We use them as systems of reference. 
 We construct a priori a number system, making a 
 simple progression through a series of units which 
 we denominate from the starting-point o, as i, 2, 
 3, 4, 5, 6, etc. These numbers are purely ideal con- 
 structions, but with their help we can count and 
 measure and weigh the several objects of reality 
 that confront experience ; and in all cases we fall 
 back upon our ideal number system, saying, the 
 table has four legs ; it is two and a half feet high, it 
 weighs fifty pounds, etc. We call these modes of 
 determination quantitative. 
 
 The element of quantitative measurement is the 
 ideal construction of units, all of which are assumed
 
 THE PHILOSOPHICAL BASIS. 69 
 
 to be discrete and equivalent. The equivalence of 
 numbers as much as the homogeneity of space, is 
 due to abstraction. In reality equivalent units do 
 not exist any more than different parts of real space 
 may be regarded as homogeneous. Both construc- 
 tions have been made to create a domain of anyness, 
 for the purpose of standards of reference. 
 
 EVEN BOUNDARIES AS STANDARDS OF MEASUREMENT. 
 
 Standards of reference are useful only when 
 they are unique, and thus we cannot use any path 
 of our spread of motion in all directions, but must 
 select one that admits of no equivocation. The only 
 line that possesses this quality is the ray, viz., the 
 straight line or the path of greatest intensity. 
 
 The straight line is one instance only of a whole 
 class of similar constructions which with one name 
 may be called "even boundaries," and by even I 
 mean congruent with itself. They remain the same 
 in any position and no change originates however 
 they may be turned. 
 
 Clifford, starting from objective space, con- 
 structs the plane by polishing three surfaces, A, B, 
 and C, until they fit one another, which means until 
 they are congruent.^ Tlis proposition leads to the 
 same result as ours, but the essential ihing is not 
 so much fas Clifford has it ) that the three ])lanes 
 are congruent, each to the two others, but that each 
 plane is congruent with its own inversion. Thus, 
 
 ' Common Sense of the Exact Sciences, Applcton & Co., p. 66.
 
 70 THE FOUNDATIONS OF MATHEMATICS. 
 
 under all conditions, each one is congruent with 
 itself. Each plane partitions the whole infinite space 
 into two congruent halves. 
 
 Having divided space so as to make the boun- 
 dary surface congruent with itself (viz., a plane), 
 we now divide the plane (we will call it P) in the 
 same way, — a process best exemplified in the fold- 
 ing of a sheet of paper stretched flat on the table. 
 The crease represents a boundary congruent with 
 itself. In contrast to curved lines, which cannot be 
 flopped or shoved or turned without involving a 
 change in our construction, we speak of a straight 
 line as an even boundary. 
 
 A circle can be flopped upon itself, but it is not 
 an even boundary congruent with itself, because 
 the inside contents and the outside surroundings 
 are dififerent. 
 
 If we take a plane, represented by a piece of 
 paper that has been evenly divided by the crease 
 AB, and divide it again crosswise, say in the point 
 O, by another crease CD, into two equal parts, M^e 
 establish in the four angles round O a new kind 
 of even boundary. 
 
 The bipartition results in a division of each half 
 plane into two portions which again are congruent 
 the one to the other ; and the line in the crease CD, 
 constituting, together with the first crease, AB, two 
 angles, is (like the straight line and the plane) 
 nothing more nor less than an even boundary con- 
 struction. The right angle originates by the pro-
 
 THE PHILOSOPHICAL BASIS. 71 
 
 cess of halving the straight Une conceived as an 
 angle. 
 
 Let us now consider the significance of even 
 boundaries. 
 
 A point being a mere locus in space, has no ex- 
 tension whatever: it is congruent with itself on 
 account of its want of any discriminating parts. If 
 it rotates in any direction, it makes no difference. 
 
 There is no mystery about a point's being con- 
 gruent with itself in any position. It results from 
 our conception of a point in agreement with the ab- 
 straction we have made; but when we are con- 
 fronted witli lines or surfaces that are congruent 
 with themselves we believe ourselves nonplussed; 
 yet the mystery of a straight line is not greater than 
 that of a point. 
 
 A line which when flopped or turned in its direc- 
 tion remains congruent with itself is called straight, 
 and a surface which when flopped or turned round 
 on itself remains congruent with itself is called 
 plane or flat. 
 
 The straight lines and the flat surfaces are, 
 among all possible boundaries, of special imi)or- 
 tance, for a similar reason that the abstraction of 
 ])ure form is so useful. In the domain of pure form 
 we get rid of all particularity and thus establish 
 a norm fit for universal application. In geometry 
 straight lines and |)lane surfaces are the climax 
 of simplicity; they are void of any particularity 
 that needs further description, or would complicate 
 the situation, and this absence of complications in
 
 ^22 THE FOUNDATIONS OF MATHEMATICS. 
 
 their construction is their greatest recommendation. 
 The most important point, however, is their quaUty 
 of being unique by being even. It renders them 
 specially available for purposes of reference. 
 
 We can construct a priori different surfaces that 
 are homogeneous, yielding as many different sys- 
 tems of geometry. Euclidean geometry is neither 
 more nor less true than spherical or elliptic geom- 
 etry; all of them are purely formal constructions, 
 they are a priori, being each one on its own premises 
 irrefutable by experience; but plane geometry is 
 more practical for general purposes. 
 
 The question in geometry is not, as some meta- 
 geometricians would have it, "Is objective space 
 flat or curved?" but, 'Ts it possible to make con- 
 structions that shall be unique so as to be service- 
 able as standards of reference?'' The former ques- 
 tion is due to a misconception of the nature of math- 
 ematics ; the latter must be answered in the affirma- 
 tive. All even boundaries are unique and can there- 
 fore be used as standards of reference. 
 
 THE STRAIGHT LINE INDISPENSABLE. 
 
 Straight lines do not exist in reality. How 
 rough are the edges of the straightest rulers, and 
 how rugged are the straightest lines drawn with 
 instruments of precision, if measured by the stan- 
 dard of mathematical straightness ! And if we con- 
 sider the paths of motion, be they of chemical atoms 
 or terrestrial or celestial bodies, we shall always
 
 THE PHILOSOPHICAL BASIS. ^^ 
 
 find them to be curves of high conii)lexity. Never- 
 theless the idea of the straight hne is justified by 
 experience in so far as it helps us to analyze the 
 complex curves into their elementary factors, no 
 one of which is truly straight; but each one of 
 which, when we ^o to the end of our analysis, can 
 be represented as a straight line. Judging from 
 the experience we have of moving bodies, we cannot 
 doubt that if the sun's attraction of the earth (as 
 well as that of all other celestial bodies) could be 
 annihilated, the earth would fly off into space in 
 a straight line. Thus the mud on carriage wheels, 
 when spurting off, and the pebbles that are thrown 
 with a sling, are flying in a tangential direction 
 which would be absolutely straight w^re it not for 
 the interference of the gravity of the earth, which 
 is constantly asserting itself and modifies the 
 straightest line into a curve. 
 
 Our idea of a straight line is suggested to us by 
 experience when we attempt to resolve compound 
 forces into their constituents, but it is not traceable 
 in experience. It is a product of our method of 
 measurement. It is a creation of our own doings, 
 yet it is justified by the success which attends its 
 employment. 
 
 The great question in geometry is not, whether 
 straight lines are real but whether their construc- 
 tion is not an indispensable ref|uisite for any pos- 
 sible system of space measurement, and further, 
 what is the nature of straight lines and planes and 
 right angles; how does their conception originate
 
 74 THE FOUNDATIONS OF MATHEMATICS. 
 
 and why are they of paramount importance in ge- 
 ometry. 
 
 \\> can of course posit that space should be 
 filled up with a medium such as would deflect every 
 ray of light so that straight rays would be impos- 
 sible. For all we know ether may in an extremely 
 slight degree operate in that way. But there w^ould 
 be nothing in that that could dispose of the think- 
 ability of a line absolutely straight in the Euclidean 
 sense with all that the same involves, so that Eu- 
 clidean geometry would not thereby be invalidated. 
 
 Now the fact that the straight line (as a purely 
 mental construction) is possible cannot be denied: 
 we use it and that should be sufficient for all prac- 
 tical purposes. That we can construct curves also 
 does not invalidate the existence of straight lines. 
 
 So again while a geometry based upon the idea 
 of homaloidal space wall remain what it has ever 
 been, the other geometries are not made thereby il- 
 legitimate. Euclid disposes as little of Lobachevsky 
 and Bolyai as they do of Euclid. 
 
 As to the nature of the straight line and all the 
 other notions connected therewith, we shall always 
 be able to determine them as concepts of boundary, 
 either reaching the utmost limit of a certain func- 
 tion, be it of the highest (such as » ) or lowest meas- 
 ure (such as o) ; or dividing a whole into two con- 
 gruent parts. 
 
 The utility of such boundary concepts becomes 
 apparent when we are in need of standards for 
 measurement. An even boundary being the utmost
 
 THE PHILOSOPHICAL BASIS. 75 
 
 limit is unique. There are innumerable curves, but 
 there is only one kind of straight line. Accordingly, 
 if we need a standard for measuring curves, we 
 must naturally fall back upon the straight line and 
 determine its curvature by its deviation from the 
 straight line which represents a zero of curvature. 
 
 The straight line is the simplest of all boundary 
 concepts. Hence its indispensableness. 
 
 If we measure a curvature we resolve the curve 
 into infinitesimal pieces of straight lines, and then 
 determine their change of direction. Thus we use 
 the straight line as a reference in our measurement 
 of curves. The simplest curve is the circle, and its 
 curvature is expressed by the reciprocal of the ra- 
 dius; but the radius is a straight line. It seems 
 that we cannot escape straightness anywhere in 
 geometry; for it is the simplest instrument for meas- 
 uring distance. \\'e may replace metric geometry 
 by projective geometry, but what could projective 
 geometricians do if they had not straight lines for 
 their projections? Without them they would be 
 in a strait indeed! 
 
 But suppose we renounced with Lobatchevsky 
 the conventional method of even boundary concep- 
 tions, especially straightness of line, and were satis- 
 fied with straightest lines, what would be the result? 
 He does not at the same time, surrender either the 
 principle of consistency or tlie assumption of the 
 homogeneity of space, and thus he builds up a ge- 
 ometry independent of the theorem of parallel lines, 
 which would be applical^lc to two systems, the Eu-
 
 /G THE FOUNDATIONS OF MATHEMATICS. 
 
 clidean of straight lines and the non-Eudidean of 
 curved space. But the latter needs the straight line 
 as much as the former and finds its natural limit 
 in a sphere whose radius is infinite and whose curva- 
 ture is zero. He can measure no spheric curvature 
 without the radius, and after all he reaches the 
 straight line in the limit of curvature. Yet it is 
 noteworthy that in the Euclidean system the straight 
 line is definite and rr irrational, while in the non- 
 Euclidean, TT is a definite number according to the 
 measure of curvature and the straight line becomes 
 irrational. 
 
 THE SUPERREAL. 
 
 We said in a former chapter (p. 48), "man did 
 not invent reason, he discovered it," which means 
 that the nature of reason is definite, unalterable, 
 and therefore valid. The same is true of all any- 
 ness of all formal thought, of pure logic, of mathe- 
 matics, and generally of anything that with truth 
 can be stated a priori. Though the norms of any- 
 ness are woven of pure nothingness, the flimsiest 
 material imaginable, they are the factors which de- 
 termine the course of events in the entire sweep 
 of actual existence and in this sense they are real. 
 They are not real in the sense of materiality; they 
 are real only in being efficient and in distinction 
 to the reality of corporeal things we may call them 
 superreal. 
 
 On the one hand it is true that mathematics is
 
 THE PHILOSOPHICAL BASIS. y"] 
 
 a mental construction ; it is purely ideal, which means 
 it is \vo\'en of thought. On the other hand we must 
 grant that the nature of this construction is fore- 
 determined in its minutest detail and in this sense 
 all its theorems must be discovered. A\ e erant 
 that there are no sines, and cosines, no tangents 
 and cotangents, no logarithms, no number tt, nor 
 even lines, in nature, but there are relations in 
 nature which correspond to these notions and sug- 
 gest the invention of symbols for the sake of de- 
 termining them with exactness. These relations 
 possess a normative value. Stones are real in the 
 sense of offering resistance in a special place, but 
 these norms are superreal because they are efficient 
 factors everywhere. 
 
 The reality of mathematics is well set forth in 
 these words of Prof. Cassius Jackson Keyser, of 
 Columbia University: 
 
 "Phrase it as you will, there is a world that is peopled 
 with ideas, ensembles, propositions, relations, and implica- 
 tions, in endless variety and multiplicity, in structure ran- 
 ging- from the very simple to the endlessly intricate and com- 
 plicate. That world is not the product but the object, not 
 the creature but the quarry of thought, the entities compos- 
 ing it — propositions, for example, — being no more identical 
 with thinking them than wine is identical with the drinking 
 of it. Mind or no mind, that world exists as an extra- 
 l)ersonal affair, — pragmatism to llu' contrary notwithstand- 
 ing." 
 
 While llie relational possesses objective signifi- 
 cance, the method of describing it is subjective and 
 of course the symbols arc arbitrarv.
 
 78 THE FOUNDATIONS OF MATHEMATICS. 
 
 In this connection we wish to call attention to a 
 most important point, which is the necessity of cre- 
 ating fixed units for counting. As there are no 
 logarithms in nature, so there are no numbers; 
 there are only objects or things sufficiently equal 
 which for a certain purpose may be considered 
 equivalent, so that we can ignore these differences, 
 and assuming them to be the same, count them. 
 
 DISCRETE UNITS AND THE CONTINUUM. 
 
 Nature is a continuum ; there are no boundaries 
 among things, and all events that happen proceed 
 in an uninterrupted flow of continuous transforma- 
 tions. For the sake of creating order in this flux 
 which would seem to be a chaos to us, we must 
 distinguish and mark off individual objects with 
 definite boundaries. This method may be seen in all 
 branches of knowledge, and is most in evidence in 
 arithmetic. When counting we start in the domain 
 of nothingness and build up the entire structure of 
 arithmetic with the products of our own making. 
 
 We ought to know that whatever we do, we must 
 first of all take a definite stand for ourselves. When 
 we start doing anything, we must have a starting- 
 point, and even though the world may be a constant 
 flux we must for the sake of definiteness regard our 
 starting-point as fixed. It need not be fixed in real- 
 ity, but if it is to serve as a point of reference we 
 must regard it as fixed and look upon all the rest 
 as movable; otherwise the world would be an in-
 
 THE PHILOSOPHICAL BASIS. 79 
 
 determinable tangle. Here we have the first rule 
 of mental activity. There may be no rest in the 
 world vet we must create the fiction of a rest as a 
 Sos jxoL TTov crT(o aud wheuevcr we take any step 
 we must repeat this fictitious process of laying 
 down definite points. 
 
 All the things which are observed around us are 
 compounds of qualities which are only temporarily 
 combined. To call them things as if they were 
 separate beings existing without reference to the 
 rest is a fiction, but it is part of our method of 
 classification, and without this fictitious comprehen- 
 sion of certain groups of qualities under definite 
 names and treating them as units, we could make 
 no headway in this world of constant flux, and all 
 events of life would swim 1)efore our mental eye. 
 
 Our method in arithmetic is similar. We count 
 as if units existed, yet the idea of a unit is a fiction. 
 We count our fingers or the l^eads of an abacus or 
 any other set of things as if they were equal. W^e 
 count the feet which we measure off in a certain 
 line as if each one were cfjuivalent to all the rest. 
 For all we know they may be different, but for our 
 pur])ose of measuring they possess the same signifi- 
 cance. This is neither an hypothesis nor an assump- 
 tion nor a fiction, but a postulate needed for a 
 definite purpose. For our purpose and according 
 to the method emi)]oyed they are the same. We 
 postulate their sameness. We have made them the 
 same, we treat them as equal. Their sameness de-
 
 8o THE FOUNDATIONS OF MATHEMATICS. 
 
 pends upon the conditions from which we start and 
 on the purpose which we have in view. 
 
 There are theorems which are true in arithmetic 
 but which do not hold true in practical life. I will 
 only mention the theorem 2+3+4 = 4+3+2 = 
 4+2+3 etc. In real life the order in which things 
 are pieced together is sometimes very essential, but 
 in pure arithmetic, when we have started in the 
 domain of nothingness and build with the products 
 of our own counting which are ciphers absolutely 
 equivalent to each other, the rule holds good and 
 it will be serviceable for us to know it and to utilize 
 its significance. 
 
 The positing of units which appears to be an 
 indispensable step in the construction of arithmetic 
 is also of great importance in actual psychology 
 and becomes most apparent in the mechanism of 
 vision. 
 
 Consider the fact that the kinematoscope has 
 become possible only through an artificial separation 
 of the successive pictures which are again fused 
 together into a new continuum. The film which 
 passes before the lens consists of a series of little 
 pictures, and each one is singly presented, halting 
 a moment and being separated from the next by a 
 rotating fan which covers it at the moment when it 
 is exchanged for the succeeding picture. If the 
 moving figures on the screen did not consist of a 
 definite number of pictures fused into one by our 
 eye which is incapable of distinguishing their quick 
 succession the whole sight would be blurred and we
 
 THE PHILOSOPHICAL BASIS. 8l 
 
 could see nothing but an indiscriminate and un- 
 analyzable perpetual flux. 
 
 This method of our mind which produces units 
 in a continuum may possess a still deeper signifi- 
 cance, for it may mark the very beginning of the 
 real world. For all we know the formation of the 
 chemical atoms in the evolution of stellar nebulas 
 may be nothing but an analogy to this process. The 
 manifestation of life too begins with the creation 
 of individuals — of definite living creatures which 
 develop dift'erently under dififerent conditions and 
 again the soul becomes possible by the definiteness 
 of single sense-impressions which can be distin- 
 guished as units from others of a dififerent tvpe. 
 
 Thus the contrast between the continuum and 
 the atomic formation appears to be fundamental 
 and gives rise to many problems which have be- 
 come especially troublesome in mathematics. But 
 if we bear in mind that the method, so to speak, of 
 atomic division is indispensable to change a world 
 of continuous flux into a system that can be com- 
 puted and determined wMth at least approximate 
 accuracy, we will be apt to appreciate thai the 
 atomic fiction in arithmetic is an indispensable part 
 of the method by whicii the whole science is created.
 
 MATHEMATICS AND METAGEOMETRY. 
 
 DIFFERENT GEOMETRICAL SYSTEMS. 
 
 STRAIGHTNESS, flatness, and rectangularity 
 are qualities which cannot (like numbers) be 
 determined in purely quantitative terms; but they 
 are determined nevertheless by the conditions under 
 which our constructions must be made. A right 
 angle is not an arbitrary amount of ninety degrees, 
 but a quarter of a circle, and even the nature of 
 angles and degrees is not derivable either from 
 arithmetic or from pure reason. They are not purely 
 quantitative magnitudes. They contain a qualita- 
 tive element which cannot be expressed in numbers 
 alone. A plane is not zero, but a zero of curvature 
 in a boundary between two solids ; and its qualita- 
 tive element is determined, as Kant would express 
 it, by Anschaunng, or as we prefer to say, by pure 
 motility, i. e., it belongs to the domain of the a 
 priori of doing. For Kant's term Anschaunng has 
 the disadvantage of suggesting the passive sense 
 denoted by the word "contemplation,'' while it is 
 important to bear in mind that the thinking subject 
 by its own activities creates the conditions that de- 
 termine the qualities above mentioned.
 
 MATHEMATICS AND METAGEOAIETRY. 
 
 ^3 
 
 Our method of creating by construction the 
 straight Hue, the plane, and the right angle, does 
 not exclude the possibility of other methods of 
 space-measurement, the standards of which would 
 not be even boundaries, such as straight lines, but 
 lines possessed of either a positive curvature like 
 the sphere or a negative curvature rendering their 
 surface pseudo-spherical. 
 
 Spheres are well known and do not stand in 
 need of description. Their curvature which is posi- 
 tive is determined by the reciprocal of their radius. 
 
 A A 
 
 B 
 
 Fig. I. Fig. 2. 
 
 Pseudo-s])heres are surfaces of negative curva- 
 ture, and pseudo-spherical surfaces are saddle- 
 shaped. Only limited pieces can be connectedly 
 rci)resented, and wc reproduce from Hclmholtz,' 
 two instances. If arc ah in figure i revolves round 
 an axis AR, it will describe a concave-convex sur- 
 face like that of the inside of a wedding-ring; and 
 in the same way, if either of the curves of figure 2 
 revolve round their axis of symmetry, it will de- 
 scribe one half of a jiseudosphcrical surface rescm- 
 
 ' Loc. cit., p. 42.
 
 84 THE FOUNDATIONS OF MATHEMATICS. 
 
 bling the shape of a morning-glory whose tapering 
 stem is infinitely prolonged. Helmholtz compares 
 the former to an anchor-ring, the latter to a cham- 
 pagne glass of the old style. 
 
 The sum of the angles of triangles on spheres 
 always exceeds i8o°, and the larger the sphere the 
 more will their triangles resemble the triangle in 
 the plane. On the other hand, the smii of the angles 
 of triangles on the pseudosphere will always be 
 somewhat less than i8o°. If we define the right 
 angle as the fourth part of a whole circuit, it will 
 be seen that analogously the right angle in the 
 plane dififers from the right angles on the sphere as 
 well as the pseudosphere. 
 
 We may add that while in spherical space sev- 
 eral shortest lines are possible, in pseudospherical 
 space w^e can draw one shortest line only. Both sur- 
 faces, however, are homogeneous (i. e., figures can 
 be moved in it without suffering a change in dimen- 
 sions), but the parallel lines which do not meet are 
 impossible in either. 
 
 Wt may further construct surfaces in which 
 changes of place involve either expansion or con- 
 traction, but it is obvious that they would be less 
 serviceable as systems of space-measurement the 
 more irregular they grow. 
 
 TRIDIMEXSIONALITY. 
 
 Space is usually regarded as tridimensional, but 
 there are some people who, following Kant, express
 
 MATHEMATICS AND METAGEOMETRY. 85 
 
 themselves with reserve, saying that the mind of 
 man may be buik up in such a way as to conceive 
 of objects in terms of three dimensions. Others 
 think that the actual and real thing that is called 
 space may be quite different from our tridimensional 
 conception of it and may in point of fact be four, 
 or five, or /z-dimensional. 
 
 Let us ask first what "dimension" means. 
 
 Does dimension mean direction ? Obviously not, 
 for we have seen that the possibilities of direction in 
 space are infinite. 
 
 Dimension is only a popular term for co-ordi- 
 nate. In space there are no dimensions laid down, 
 but in a space of infinite directions three co-ordinates 
 are needed to determine from a given point of ref- 
 erence the position of any other point. 
 
 In a former section on "Even Boundaries as 
 Standards of Measurement," we have halved space 
 and produced a plane Pi as an e\en boundary be- 
 tween the two halves; we have halved the plane Pi 
 by turning the plane so upon itself, that like a crease 
 in a folded sheet of paper the straight line AB 
 was produced on the plane. We then halved the 
 straight line, the even boundary between the two 
 half-planes, by again turning the plane U])on itself 
 so that the line AB covered its own prolongation. 
 It is as if our folded sheet of paper were folded a 
 second time upon itself so that the crease would 
 be foldcfl upon itself and rtne part of the same fall 
 exactly upon and cover the other ])art. On opening 
 the sheet we have a second crease crossing the first
 
 86 THE FOUNDATIONS OF MATHEMATICS. 
 
 one making- the perpendicular CD, in the point O, 
 thus producing right angles on the straight line 
 AB, represented in the cross-creases of the twice 
 folded sheet of paper. Here the method of producing 
 even boundaries by halving comes to a natural end. 
 So far our products are the plane, the straight line, 
 the point and as an incidental but valuable by- 
 product, the right angle. 
 
 We may now venture on a synthesis of our ma- 
 terials. We lay two planes, Po and P?., through the 
 two creases at right angles on the original plane 
 Pi, represented by the sheet of paper, and it becomes 
 apparent that the two new planes Po and P3 will 
 intersect at O, producing a line EF common to both 
 planes P2 and P3, and they will bear the same rela- 
 tion to each as each one does to the original plane 
 Pi, that is to say: the whole system is congruent 
 with itself. If we make the planes change places. Pi 
 may as well take the place of P2 and Po of P3 and 
 P3 of P2 or Pi of P3, etc., or vice versa, and all the 
 internal relations would remain absolutely the same. 
 Accordingly we have here in this system of the 
 three planes at right angles (the result of repeated 
 halving), a composition of even boundaries which, 
 as the simplest and least complicated construction 
 of its kind, recomends itself for a standard of meas- 
 urement of the whole spread of motility. 
 
 The most significant feature of our construction 
 consists in this, that we thereby produce a con- 
 venient system of reference for determining every
 
 MATHEMATICS AND METAGEOMETRY. 87 
 
 possible point in co-ordinates of straight lines stand- 
 ing at right angles to the three planes. 
 
 If we start from the ready conception of objec- 
 tive space (the juxtaposition of things) we can 
 refer the several distances to analogous loci in our 
 system of the three planes, mutually perpendicular, 
 each to the others, \\'e cut space in two equal halves 
 by the horizontal plane Pi. We repeat the cutting so 
 as to let the two halves of the first cut in their angu- 
 lar relation to the new cut (in P2) be congruent 
 with each other, a procedure which is possible only 
 if we make use of the even boundary concept with 
 which we have become acquainted. Accordingly, 
 the second cut should stand at right angles on the 
 first cut. The two planes Pi and P2 have one line 
 in common, EF, and any plane placed at right an- 
 gles to EF (in the point O) will again satisfy the 
 demand of dividing space, including the two planes 
 Pi and P2, into two congruent halves. The two 
 new lines, produced by the cut of the third plane 
 P.i through the two former planes Pi and Po. stand 
 both at right angles to EF. Should we continue 
 our method of cutting space at rigjit angles in O 
 on cither of these lines, we would produce a plane 
 coincident with P,, which is to say, that the possi- 
 1)ilities of the system arc exhausted. 
 
 This implies that in any system of pure space 
 three co-ordinates are suflicient for the determina- 
 tion of any place from a given reference point.
 
 88 THE FOUNDATIONS OF MATHEMATICS. 
 
 THREE A CONCEPT OF BOUNDARY. 
 
 The niiml^er three is a concept of boundary as 
 much as the straight line. Under specially compli- 
 cated conditions we might need more than three 
 co-ordinates to calculate the place of a point, but in 
 empty space the number three, the lowest number 
 that is really and truly a number, is sufficient. If 
 space is to be empty space from which the notion 
 of all concrete things is excluded, a kind of model 
 constructed for the purpose of determining juxta- 
 position, three co-ordinates are sufficient, because 
 our system of reference consists of three planes, 
 and we have seen above that there is no possibility 
 of introducing a fourth plane without destroying 
 its character of being congruent with itself, which 
 imparts to it the simplicity and uniqueness that 
 render it available for a standard of measurement. 
 
 Three is a peculiar number which is of great 
 significance. It is the first real number, being the 
 simplest multiplex. One and two and also zero 
 are of course numbers if we consider them as mem- 
 bers of the number-system in its entirety, but singly 
 regarded they are not yet numbers in the full sense 
 of the word. One is the unit, two is a couple or a 
 pair, but three is the smallest amount of a genuine 
 plurality. Savages who can distinguish only be- 
 tween one and two have not yet evolved the notion 
 of number; and the transition to the next higher 
 stage involving the knowledge of "three" passes 
 through a mental condition in which there exists
 
 MATHEMATICS AND METAGEOMETRY. 8g 
 
 only the notion one, two, and plurality of any kind. 
 When the idea of three is once definitely recognized, 
 the naming of all other numbers can follow in rapid 
 succession. 
 
 In this connection we may incidentally call at- 
 tention to the significance of the grammatical dual 
 number as seen in the Semitic and Greek languages. 
 It is a siirviving relic and token of a period during 
 which the unit, the pair, and the uncounted plural- 
 ity constituted the entire gamut of human arith- 
 metic. The dual form of grammatical number by 
 the development of the number-system became re- 
 dundant and cumbersome, being retained only for 
 a while to express the idea of a couple, a pair that 
 naturally belong together. 
 
 Certainly, the origin of the notion three has its 
 germ in the nature of abstract anyness. Nor is 
 it an accident that in order to construct the simplest 
 figure which is a real figure, at least three lines 
 are needed. The importance of the triangle, which 
 becomes most prominent in trigonometry, is due to 
 its being the simplest possible figure wliich accord- 
 ingly possesses the intrinsic worth of economy. 
 
 The number three j)lays also a significant part 
 in logic, and in llie l)ranclu's of ihe a])])lied sciences, 
 and thus we need not l)e astonishe<l at fuuling the 
 very idea, three, held in religious reverence, for the 
 doctrine of llic Trinity has its basis in llie constitu- 
 tion of the nni\erse and can be fully justified by the 
 laws of j)ure form.
 
 90 THE FOUNDATIONS OF MATHEMATICS. 
 
 SPACE OF FOUR DIMENSIONS. 
 
 The several conceptions of space of more than 
 three dimensions are of a purely abstract nature, 
 yet they are by no means vague, but definitely de- 
 termined by the conditions of their construction. 
 Therefore we can determine their properties even 
 in their details w^ith perfect exactness and formu- 
 late in abstract thought the laws of four-, five-, six-, 
 and 7i-dimensional space. The difficulty with which 
 we are beset in constructing 7?-dimensional spaces 
 consists in our inability to make them representable 
 to our senses. Here we are confronted with what 
 may be called the limitations of our mental constitu- 
 tion. These limitations, if such they be, are con- 
 ditioned by the nature of our mode of motion, which, 
 if reduced to a mathematical system, needs three 
 co-ordinates, and this means that our space-con- 
 ception is tridimensional. 
 
 We ourselves are tridimensional; we can meas- 
 ure the space in which we move with three co-ordi- 
 nates, yet we can definitely say that if space were 
 four-dimensional, a body constructed of two fac- 
 tors, so as to have a four-dimensional solidity, would 
 be expressed in the formula: 
 
 We can calculate, compute, excogitate, and de- 
 scribe all the characteristics of four-dimensional 
 space, so long as we remain in the realm of abstract 
 thought and do not venture to make use of our 
 motility and execute our plan in an actualized con-
 
 MATHEMATICS AND METAGEOMETRY. 
 
 91 
 
 struction of motion. From the standpoint of pure 
 logic, there is nothing irrational about the assump- 
 tion ; but as soon as we make an a priori construc- 
 tion of the scope of our motility, we find out the 
 incompatibility of the whole scheme. 
 
 In order to make the idea of a space of more than 
 three dimensions plausible or intelligible, we resort 
 to the relation between two-dimensional beings and 
 tridimensional space. The nature of tridimensional 
 space may be indicated yet not fully represented in 
 
 two-dimensional space. Tf we construct a square 
 upon the line AB one inch long, it will be bounded 
 by four lines each an inch in length. In order to 
 construct upon the square ABCD a cube of the 
 same measure, we must raise the square by one inch 
 into the tliird flimcn^ion in a direction at right 
 angles to its surface, the result being a figure 
 bounded by six surfaces, each of which is a one-inch 
 square. Tf two-dimensional beings who could not 
 rise into the third dimension wished to gain an idea 
 of space of a higher dimensionality and picture in
 
 92 THE FOUNDATIONS OF MATHEMATICS. 
 
 their own two-dimensional mathematics the resuUs 
 of three dimensions, they might push out the square 
 in any direction within their own plane to a distance 
 of one inch, and then connect all the corners of the 
 image of the square in its new position with the 
 corresponding points of the old square. The result 
 would be what is to us tridimensional beings the 
 picture of a cube. 
 
 When we count the plane quadrilateral figures 
 produced by this combination we find that there are 
 six, corresponding to the boundaries of a cube. We 
 must bear in mind that only the original and the new 
 square will be real squares, the four intermediary 
 figures which have originated incidentally through 
 our construction of moving the square to a distance, 
 exhibit a slant and to our two-dimensional beings 
 they appear as distortions of a rectangular relation, 
 which faultiness has been caused by the insuffi- 
 ciency of their methods of representation. More- 
 over all squares count in full and where their sur- 
 faces overlap they count double. 
 
 Two-dimensional beings having made such a 
 construction must however bear in mind that the 
 field covered by the sides, GEFH and BFDH does 
 not take up any room in their own plane, for it is 
 only a picture of the extension which reaches out 
 either above or below their own plane; and if they 
 venture out of this field covered by their construc- 
 tion, they have to remember that it is as empty and 
 unoccupied as the space beyond the boundaries AC 
 and AB.
 
 MATHEMATICS AND METAGEOMETRY. 93 
 
 Now if we tridimensional beings wish to do the 
 same, how shall we proceed? 
 
 We must move a tridimensional body in a rect- 
 ang-ular direction into a new (i. e., the fourth) 
 dimension, and being unable to accomplish this we 
 may represent the operation by mirrors. Having 
 three dimensions we need three mirrors standing 
 at right angles. We know by a priori considera- 
 tions according to the principle of our construction 
 that the boundaries of a four-dimensional body must 
 be solids, i. e. .tridimensional bodies, and while the 
 sides of a cube (algebraically represented by cf) 
 must be six surfaces (i. e., two-dimensional figures, 
 one at each end of the dimensional line) the boun- 
 daries of an analogous four-dimensional body built 
 u]) like the cube and the square on a rectangular 
 jjlan, must be eight solids, i. e., cubes. If we build 
 up three mirrors at right angles and place any 
 object in the intersecting corner we shall see the 
 object not once, but eight times. The body is re- 
 llected below, and the object thus doubled is mir- 
 rored not only on both upright sides but in addition 
 in the corner beyond, appearing in either of the up- 
 right mirrors coincidingly in the same place. Thus 
 tlie total muhiplication of our iridimcnsionnl boun- 
 daries of a four-dinicn^ioiial complex is rendered 
 eightfold. 
 
 We nuist now bear in mind that this representa- 
 tion of a fourth dimension suffers from nil llic faults 
 of the analogous figure of a rube in two-dimensional 
 space. Tlic severed figures are noi eight indepcn-
 
 94 THE FOUNDATIONS OF MATHEMATICS. 
 
 dent bodies but they are mere boundaries and the 
 four dimensional space is conditioned by their inter- 
 relation. It is that unrepresentable something which 
 they enclose, or in other words, of which they are 
 assumed to be boundaries. If we were four-dimen- 
 sional beings we could naturally and easily enter 
 into the mirrored space and transfer tridimensional 
 bodies or parts of them into those other objects re- 
 flected here in the mirrors representing the bound- 
 aries of the four-dimensional object. While thus 
 on the one hand the mirrored pictures would be as 
 real as the original object, they would not take up 
 the space of our three dimensions, and in this re- 
 spect our method of representing the fourth dimen- 
 sion by mirrors would be quite analogous to the 
 cube pictured on a plane surface, for the space to 
 which we (being limited by our tridimensional 
 space-conception) would naturally relegate the seven 
 additional mirrored images is unoccupied, and if 
 we should make the trial, we would find it empty. 
 
 Further experimenting in this line would render 
 constructions of a more complicated character more 
 and more difficult although not quite impossible. 
 Thus we might represent the formula (a+^)^ by 
 placing a wire model of a cube, representing the 
 proportions (a-f /?)■^ in the corner of our three mir- 
 rors, and we would then verify by ocular inspec- 
 tion the truth of the formula 
 
 a*+4a'b+6a^b^+4ab^+b\ 
 
 However, we must bear in mind that all the 
 solids here seen are merely the boundaries of four-
 
 MATHEMATICS AND METAGEOMETRY. 95 
 
 dimensional bodies. All of them with the exception 
 of the ones in the inner corner are scattered around 
 and yet the analogous figures would have to be re- 
 garded as being most intimately interconnected, 
 each set of them forming one four-dimensional com- 
 plex. Their separation is in appearance only, being 
 due to the insufiiciency of our method of presenta- 
 tion. 
 
 We might obviate this fault by parceling our 
 wire cube and instead of using three large mirrors 
 for reflecting the entire cube at once, we might in- 
 sert in its dividing planes double mirrors, i. e., mir- 
 rors which would reflect on the one side the magni- 
 tude a and on the other the magnitude b. In this 
 way we would come somewhat nearer to a faithful 
 representation of the nature of four-dimensional 
 space, but the model being divided up into a num1:)er 
 of mirror-walled rooms, would l)ecome extremely 
 complicated and it would be diflicult for us to bear 
 always in mind that the mirrored spaces count on 
 both sides at once, although they overlap and (tri- 
 dimensionally considered) seem to fall the one into 
 the other, thus presenting to our eyes a real laby- 
 rinth of spaces that exist within each other without 
 interfering with one anotlicr. They thus render 
 new dejilhs visil)le in all three dimensions, and in 
 order to represent tiie whole scheme of a four-dimen- 
 sional comj)lcx in its full completeness, we ought 
 to have three mirrors at right angles placed at 
 everv i)oint in our tridimensional space. The scheme 
 itself is impossible, but the idea will render the
 
 96 THE FOUNDATIONS OF MATHEMATICS. 
 
 nature of four-dimensional space approximately 
 clear. If we were four-dimensional beings we would 
 be possessed of the mirror-eye which in every di- 
 rection could look straightway round every corner 
 of the third dimension. This seems incredible, but it 
 can not be denied that tridimensional space lies 
 open to an inspection from the domain of the fourth 
 dimension, just as every point of a Euclidean plane 
 is open to inspection from above to tridimensional 
 vision. Of course we may demur (as we actually 
 do) to believing in the reality of a space of four 
 dimensions, but that being granted, the inferences 
 can not be doubted. 
 
 THE APPARENT ARBITRARINESS OF THE A PRIORI. 
 
 Since Riemann has generalized the conception of 
 space, the tridimensionality of space seems very 
 arbitrary. 
 
 Why are three co-ordinates sufficient for pure 
 space determinations? The obvious answer is, Be- 
 cause we have three planes in our construction of 
 space-boundaries. We might as well ask, why do 
 the three planes cut the entire space into 8 equal 
 parts? The simple answer is that we have halved 
 space three times, and 2^=8. The reason is prac- 
 tically the same as that for the simpler question, 
 why have we two halves if we divide an apple into 
 two equal parts? 
 
 These answers are simple enough, but there is 
 another aspect of the question which here seems in
 
 MATHEMATICS AXD METAGEOMETRV. 97 
 
 order: ^^'hy not continue the method of halving? 
 And there is no other answer than that it is impos- 
 sible. The two superadded planes P2 and P3 both 
 standing at right angles to the original plane, neces- 
 sarily halve each other, and thus the four right an- 
 gles of each plane P2 and P3 on the center of inter- 
 section, form a complete plane for the same reason 
 that four quarters are one whole. We have in each 
 case four quarters, and ■*/4=i. 
 
 Purely logical arguments (i. e., all modes of 
 reasoning that are rigorously a priori, the a priori 
 of abstract being) break down and we must resort to 
 the methods of the a priori of doing. We cannot 
 understand or grant the argument without admit- 
 ting the conception of space, previously created by 
 a spread of pure motion. Kant would say that we 
 need here the data of reine Anschanuug, and Kant's 
 reine Anschauung is a product of our motility. As 
 soon as we admit that there is an a priori of doing 
 (of free motility) and that our conception of pure 
 space and time (Kant's rciiie AiiscJianung) is its 
 product, we understand that our mathematical con- 
 ceptions cannot be derived from pure logic alone 
 1)iit must finally depend upon our motility, viz., the 
 function that begets our notion of space. 
 
 Tf we divide an a])i)le by a vertical cut through 
 its center, we have two halves. Tf we cut it again 
 by a horizontal cut tlirough its center, we have 
 four fjuarters. Tf we cut it again with a cut that 
 is at right angles to both prior cuts, we have eight 
 eighths. Tt is obviously impossible to insert among
 
 98 THE FOUNDATIONS OF MATHEMATICS. 
 
 these three cuts a fourth cut that would stand at 
 right angles to these others. The fourth cut through 
 the center, if we needs would have to make it, will 
 fall into one of the prior cuts and be a mere repeti- 
 tion of it, producing no new result ; or if we made 
 it slanting, it would not cut all eight parts but only 
 four of them; it would not produce sixteen equal 
 parts, but twelve unequal parts, viz., eight six- 
 teenths plus four quarters. 
 
 If we do not resort to a contemplation of the 
 scope of motion, if we neglect to represent in our 
 imagination the figure of the three planes and rely 
 on pure reason alone (i. e., the rigid a priori), we 
 have no means of refuting the assumption that we 
 ought to be al)le to continue halving the planes by 
 other planes at right angles. Yet is the proposition 
 as inconsistent as to expect that there should be 
 regular pentagons, or hexagons, or triangles, the 
 angles of which are all ninety degrees. 
 
 From the standpoint of pure reason alone we 
 cannot disprove the incompatibility of the idea of a 
 rectangular pentagon. If we insist on constructing 
 by hook or crook a rectangular pentagon, we will 
 succeed, but we must break away from the straight 
 line or the plane. A rectangular pentagon is not 
 absolutely impossible; it is absolutely impossible in 
 the plane ; and if we produce one, it will be twisted. 
 
 Such was the result of Lobatchevsky's and Bol- 
 yai's construction of a system of geometry in which 
 the theorem of parallels does not hold. Their ge- 
 ometries cease to be even ; they are no longer Euclid-
 
 MATHEMATICS AND METAGEOMETRY. QQ 
 
 can and render the even boundary conceptions un- 
 available as standards of measurement. 
 
 If by logic we understand consistency, and if 
 anvthing that is self-contradictory and incompatible 
 with its own nature be called illogical, we would 
 say that it is not the logic of pure reason that ren- 
 ders certain things impossible in our geometric con- 
 structions, but the logic of our scope of motion. The 
 latter introduces a factor which determines the na- 
 ture of geometry, and if this factor is neglected or 
 misunderstood, the fundamental notions of geom- 
 etry must appear arbitrary. 
 
 DEFINITEXESS OF CONSTRUCTION. 
 
 The problems which puzzle some of the meta- 
 physicians of geometry seem to have one common 
 foundation, which is the definiteness of geometrical 
 construction. Geometry starts from empty nothing- 
 ness, and we are confronted with rigid conditions 
 which it does not lie in our power to change. \\'e 
 make a construction, and the result is something 
 new, perhaps something which we have not in- 
 tended, something at which we are surprised. The 
 synthesis is a product of our own making, yet there 
 is an objective element in it over which we have no 
 command, and ihi^ objective element is rigid, un- 
 compromising, an irrefragable necessity, a stubborn 
 fact, immorvable, inflexible, immutable. What is it ? 
 
 Our metageometricians overlook the fact thai 
 their nothingness i^ not an absolute nothing, but
 
 lOO THE FOUNDATIONS OF MATHEMATICS. 
 
 only an absence of concreteness. If they make defi- 
 nite constructions, they must (if they only remain 
 consistent) expect definite results. This definite- 
 ness is the logic that dominates their operations. 
 Sometimes the results seem arbitrary, but they never' 
 are ; for they are necessary, and all questions zvhyf 
 can elicit only answers that turn in a circle and are 
 mere tautologies. 
 
 Why, we may ask, do two straight lines, if they 
 intersect, produce four angles? Perhaps we did 
 not mean to construct angles, but here we have 
 them in spite of ourselves. 
 
 And why is the sum of these four angles equal 
 to 360° ? Why. if two are acute, will the other two 
 be found obtuse? Why, if one angle be a right 
 angle, will all four be right angles? Why will the 
 sum of any two adjacent angles be equal to two 
 right angles? etc. Perhaps we should have pre- 
 ferred three angles only, or four acute angles, but 
 we cannot have them, at least not by this construc- 
 tion. 
 
 We have seen that the tridimensionality of space 
 is arbitrary only if we judge of it as a notion of 
 pure reason, without taking into consideration the 
 method of its construction as a scope of mobility. 
 Tridimensionality is only one instance of apparent 
 arbitrariness among many others of the same kind. 
 
 We cannot enclose a space in a plane by any 
 figure of two straight lines, and we cannot construct 
 a solid of three even surfaces. 
 
 There are only definite forms of polyhedra pos-
 
 MATHEMATICS AND METAGEOMETRY. lOI 
 
 sible, and the surfaces of every one are definitely 
 determined. To the mind tminitiated into the se- 
 crets of mathematics it would seem arbitrary that 
 there are two hexahedra (viz., the cube and the 
 duplicated tetrahedron), while there is no hepta- 
 hedron. And why can we not have an octahedron 
 with quadrilateral surfaces ? Wq might as Avell ask, 
 why is the square not round ! 
 
 Prof. G. B. Halsted says in the Translator's 
 Appendix to his English edition of Lobatchevsky's 
 Theory of Parallels, p. 48: 
 
 "But is it not absurd to speak of space as interfering 
 with anything? If you think so, take a knife and a raw 
 potato and try to cut it into a seven-edged solid." 
 
 Truly Professor Halsted's contention, that the 
 laws of space interfere with our operations, is true. 
 Yet it is not space that squeezes us, but the laws of 
 construction determine the shape of the figures 
 which we make. 
 
 A simple instance that illustrates the way in 
 which space interferes with our plans and move- 
 ments is the impossible demand on the chessboard 
 to start a rook in one corner (Ai) and pass it with 
 the rook motion over all the fields once, but only 
 once, and let it end its journey on ilic opposite corner 
 (HH ). Rightly considered it is not space that inter- 
 feres with our mode of action, but the law of con- 
 sistence riic ])roposition does not contain any- 
 thing illogical ; the words arc f|uitc rational and the 
 sentences grammaticallv correct. Yet is the task
 
 102 
 
 THE FOUNDATIONS OF MATHEMATICS. 
 
 impossible, because we cannot turn to the right and 
 left at once, nor can we be in two places at once, 
 neither can we undo an act once done or for the 
 
 ABCDEFGH 
 
 nonce change the rook into a bishop ; but something 
 of that kind would have to be done, if we start from 
 A I and pass with the rook motion through A2 
 
 and Bi over to B2. In other words: 
 Though the demand is not in con- 
 flict with the logic of abstract being 
 (ir the grammar of thinking, it is 
 ' impossible because it collides with 
 the logic of doing ; the logic of mov- 
 ing about, the a priori of motility. 
 The famous problem of crossing seven bridges 
 leading to the two Konigsberg Isles, is of the same
 
 MATHEMATICS AND METAGEOMETRY. 
 
 103 
 
 kind. Near the mouth of the Pregel River there 
 is an island called Kneiphof, and the situation of 
 the seven bridges is shown as in the adjoined dia- 
 
 THE SEVEN BRIDGES OF KONIGSBERG. 
 
 gram. A discussion arose as to whether it was pos- 
 sible to cross all the bridges in a single promenade 
 
 EULER S DIAGRAM. 
 
 without crossing any one a second time. Finally 
 Euler solved the problem in a memoir presented to
 
 I04 THE FOUNDATIONS OF MATHEMATICS. 
 
 the Academy of Sciences of St. Petersburg in 1736, 
 pointing- out why the task could not be done. He 
 reproduced the situation in a diagram and proved 
 that if the number of hues meeting at the point K 
 (representing the island Kneiphof as K) were even 
 the task w^as possible, but if the number is odd it 
 can not be accomplished. 
 
 The squaring of the circle is similarly an im- 
 possibility. 
 
 We cannot venture on self-contradictory enter- 
 prises without being defeated, and if the relation 
 of the circumference to the diameter is an infinite 
 transcendent series, being 
 
 i"^ — J- — 3-rs — 7-1-9 — ll-ri3 — 15-ri7 — 19-r21 — 23-t-''*' 
 
 we cannot expect to square the circle. 
 
 If we compute the series, tt becomes 3. 141 59265 
 . . . . , figures which seem as arbitrary as the most 
 whimsical fancy. 
 
 It does not seem less strange that 6=2.71828; 
 and yet it is as little arbitrary as the equation 3X4 
 =12. 
 
 The definiteness of our mathematical construc- 
 tions and arithmetical computations is based upon 
 the inexorable law of determinism, and everything 
 is fixed by the mode of its construction. 
 
 ONE SPACE, BUT VARIOUS SYSTEMS OF SPACE- 
 MEASUREMENT. 
 
 Riemann has generalized the idea of space and 
 would thus justify us in speaking of "spaces." The
 
 MATHEMATICS AND METAGEOMETRY. IO5 
 
 common notion of space, which agrees best with 
 that of Euchdean geometry, has been degraded 
 into a mere species of space, one possible instance 
 among many other possibiHties. And its very legit- 
 imacy has been doubted, for it has come to be looked 
 upon in some quarters as only a popular (not to 
 say vulgar and commonplace ) notion, a mere work- 
 ing hypothesis, infested with many arbitrary con- 
 ditions of which the ideal conception of absolute 
 space should be free. How much more interesting 
 and aristocratic are curved space, the dainty two- 
 dimensional space, and above all the four-dimen- 
 sional space with its magic powers! 
 
 The new space-conception seems bewildering. 
 Some of these new spaces are constructions that are 
 not concretely representablc, 1)ut only abstractly 
 thinkable ; yet they allow us to indulge in ingenious 
 dreams. Think only of two-dimensional creatures, 
 and how limited they are! They can have no con- 
 ception of a third dimension! Then think of four- 
 dimensional beings ; how superior they must be to 
 us poor tridimensional bodies! As we can take a 
 figure situated within a circle through the third di- 
 mension and put it down again outside tlic circle 
 without crossing the circumference, so four-dimen- 
 sional beings could take tridimensional things en- 
 cased in a tridimensional box from their hiding- 
 place anrl |)nt them bark on some other spot on the 
 outside. They could easily help themselves to all 
 the money in the steal-lined safes of our banks, and 
 they could perform the most difficult obstetrical
 
 I06 THE FOUNDATIONS OF MATHEMATICS. 
 
 feats without resorting to the dangerous Csesarean 
 operation. 
 
 Curved space is not less interesting. Just as 
 light may pass through a medium that offers such 
 a resistance as will involve a continuous displace- 
 ment of the rays, so in curved space the lines of 
 greatest intensity would be subject to a continuous 
 modification. The beings of curved space may be 
 assumed to have no conception of truly straight 
 lines. They must deem it quite natural that if they 
 walk on in the straightest possible manner they will 
 finally but unfailingly come back to the same spot. 
 Their world-space is not as vague and mystical as 
 ours: it is not infinite, hazy at a distance, vague 
 and without end, but definite, well rounded off, and 
 perfect. Presumably their lives have the same ad- 
 vantap'es moving in boundless circles, while our 
 progression in straight lines hangs between two 
 infinitudes — the past and the future! 
 
 All these considerations are very interesting be- 
 cause they open new vistas to imaginative specula- 
 tors and inventors, and we cannot deny that the 
 generalization of our space-conception has proved 
 helpful by throwing new light upon geometrical 
 problems and widening the horizon of our mathe- 
 matical knowledge. 
 
 Nevertheless after a mature deliberation of Rie- 
 mann's proposition, I have come to the conclusion 
 that it leads us off in a wrong direction, and in 
 contrast to his conception of space as being one 
 instance among many possibilities, I would insist
 
 MATHEMATICS AND METAGEOMETRY. 10/ 
 
 Upon the uniqueness of space. Space is the possi- 
 bility of motion in all directions, and mathematical 
 space is the ideal construction of our scope of motion 
 in all directions. 
 
 The homogeneity of space is due to our abstrac- 
 tion which omits all particularities, and its homa- 
 loidality means only that straight lines are possible 
 not in the real world, but in mathematical thouglit, 
 and will serve us as standards of measurement. 
 
 Curved space, so called, is a more complicated 
 construction of space-measurement to which some 
 additional feature of a particular nature has been 
 admitted, and in which we waive the advantages of 
 even boundaries as means of measurement. 
 
 Space, the actual scope of motion, remains dif- 
 ferent from all systems of space-measurement, be 
 they homaloidal or curved, and should not be sub- 
 sumed with them under one and the same category. 
 
 Riemann's several space - conceptions are not 
 spaces in the proper sense of the word, but systems 
 of space-measurement. It is true that space is a 
 tridimensional manifold, and a plane a two-dimen- 
 sional manifold, and we can think of other systems 
 of ;i-manifo]dness; but for that reason all these dif- 
 ferent manifolds do not become spaces. Man is a 
 mammal having two ])rehensiles (his hrmd^) ; the 
 elephant is a mammal with one prehensile (his 
 trunk) ; tailless monkeys like the pavian have four; 
 and tailcfl monkeys have five prehensiles. Ts there 
 any logic in extending the denomination man to all 
 these animals, and should we define the clc])hant
 
 I08 THE FOUNDATIONS OF MATHEMATICS. 
 
 as a man with one prehensile, the pavian as a man 
 with four prehensiles and tailed monkeys as men 
 with five prehensiles ? Our zoologists would at once 
 protest and denounce it as an illogical misuse of 
 names. 
 
 Space is a manifold, but not every manifold is 
 a space. 
 
 Of course every one has a right to define the 
 terms he uses, and obviously my protest simply re- 
 jects Riemann's use of a word, but I claim that his 
 identification of "space" with "manifold" is the 
 source of inextricable confusion. 
 
 It is well known that all colors can be reduced 
 to three primary colors, yellow, red, and blue, and 
 thus we can determine any possible tint by three 
 co-ordinates, and color just as much as mathemat- 
 ical space is a threefold, viz., a system in which 
 three co-ordinates are needed for the determination 
 of any thing. But because color is a threefold, no 
 one would assume that color is space. 
 
 Riemann's manifolds are systems of measure- 
 ment, and the system of three co-ordinates on 
 three intersecting planes is an a priori or purely 
 formal and ideal construction invented to calculate 
 space. We can invent other more complicated sys- 
 tems of measurement, with curved lines and with 
 more than three or less than three co-ordinates. 
 We can even employ them for space-measurement, 
 although they are rather awkward and unservice- 
 able; but these systems of measurement are not 
 "spaces," and if they are called so, they are spaces
 
 MATHEMATICS AND METAGEOMETRY. IO9 
 
 by courtesy only. By a metaphorical extension we 
 allow the idea of system of space-measurement to 
 stand for space itself. It is a brilliant idea and quite 
 as ingenious as the invention of animal fables in 
 \Ahich our quadruped fellow-beings are endowed 
 with speech and treated as human beings. But such 
 poetical licences, in which facts are stretched and 
 the meaning of terms is slightly modified, is possible 
 only if instead of the old-fashioned straight rules of 
 logic we grant a slight curvature to our syllogisms. 
 
 FICTITIOUS SPACES AND THE APRIORITY OF ALL 
 SYSTEMS OF SPACE-MEASUREMENT. 
 
 Mathematical space, so called, is strictly speak- 
 ing no space at all, but the mental construction of 
 a manifold, being a tridimensional system of space- 
 measurement invented for the determination of ac- 
 tual space. 
 
 Neither can a manifold of two dimensions be 
 called a space. It is .-i mere ])oundarv in space, it 
 is no reality, but a conce])t, a construction of pure 
 thought. 
 
 h\u"lher, the manifold of fonr dimensions is a 
 .system of measurement .-ippliral)k' to an\- reality 
 for the determination of which fonr co-ordinates 
 are needed. It is ap])licabk' to real s])ar(^ if there 
 is connected with it in addition lo ilic llircc ])lanes 
 at right angles another condition of a constant na- 
 ture, such as gravity. 
 
 At any rate, we must denv the applicaliilil \' of a
 
 no THE FOUNDATIONS OF MATHEMATICS. 
 
 system of four dimensions to empty space void of 
 any such particularity. The idea of space being 
 four-dimensional is chimerical if the word space is 
 used in the common acceptance of the term as 
 juxtaposition or as the scope of motion. So long- 
 as four quarters make one whole, and four right 
 angles make one entire circumference, and so long 
 as the contents of a sphere which covers the entire 
 scope of motion round its center equals sttP, there 
 is no sense in entertaining the idea that empty space 
 might be four-dimensional. 
 
 But the argument is made and sustained by 
 Helmholtz that as two-dimensional beings perceive 
 two dimensions only and are unable to think how 
 a third dimension is at all possible, so we tridimen- 
 sional beings cannot represent in thought the possi- 
 bility of a fourth dimension. Helmholtz, speaking 
 of beings of only two dimensions living on the sur- 
 face of a solid body, says: 
 
 "If such beings worked out a geometry, they would of 
 course assign only two dimensions to their space. They 
 would ascertain that a point in moving describes a line, and 
 that a line in moving describes a surface. But they could 
 as little represent to themselves what further spatial con- 
 struction would be generated by a surface moving out of 
 itself, as w^e can represent what should be generated by a 
 solid moving out of the space we know. By the much- 
 abused expression 'to represent' or 'to be able to think how 
 something happens' I understand — and I do not see how 
 anything else can be understood by it without loss of all 
 meaning — the power of imagining the whole series of sen- 
 sible impressions that would be had in such a case. Now,
 
 MATHEMATICS AND METAGEOMETRY. Ill 
 
 as no sensible impression is known relating to such an un- 
 heard-of event, as the movement to a fourth dimension 
 would be to us, or as a movement to our third dimension 
 would be to the inhabitants of a surface, such a 'represen- 
 tation' is as impossible as the 'representation' of colors 
 would be to one born blind, if a description of them in gen- 
 eral terms could be given to him. 
 
 "Our surface-beings would also be able to draw shortest 
 lines in their superficial space. These would not necessarily 
 be straight lines in our sense, but what are technically called 
 geodetic lines of the surface on which they live ; lines such 
 as are described by a tense thread laid along the surface, 
 and which can slide u])()n it freely.". . . . 
 
 "Now, if beings of this kind lived on an infinite plane, 
 their geometry would be exactly the same as our planimetry. 
 They would affirm that only one straight line is possible 
 between two points ; that through a third point lying with- 
 out this line only one line can be drawn parallel to it : that 
 the ends of a straight line never meet though it is produced 
 to infinity, and so on.". . . . 
 
 "But intelligent beings of the kind supposed might also 
 live on the surface of a sphere. Their shortest or straightest 
 line between two points would then be an arc of the great 
 circle passing through them.". . . . 
 
 "Of parallel lines the sphere-dwellers would know noth- 
 ing. They would maintain that any two straightest lines, 
 sufficiently produced, must finally cut not in one only but in 
 two points. The sum of the angles of a triangle would be 
 always greater than two right angles, increasing as the sur- 
 face of the triangle grew greater. They could thus have 
 no conception of geometrical similarity between greater and 
 smaller figures of the same kind, ior with them a greater 
 triangle must have diflferent angles from a smaller oue. 
 Their space would be unlimited, but would be found to be 
 finite or at least represented as such. 
 
 "It is clear, then, that such beings must set uj) a very
 
 112 THE FOUNDATIONS OF MATHEMATICS. 
 
 / 
 
 different system of geometrical axioms from that of the 
 inhabitants of a plane, or from ours with our space of three 
 dimensions, though the logical powers of all were the same." 
 
 I deny what Helmholtz implicitly assumes that 
 sensiljje impressions enter into the fabric of our 
 concepts of purely formal relations. We have the 
 idea of a surface as a boundary between solids, but 
 surfaces do not exist in reality. All real objects 
 are solid, and our idea of surface is a mere fiction 
 of abstract reasoning. Two dimensional things are 
 unreal, we have never seen any, and yet we form 
 the notion of surfaces, and lines, and points, and 
 pure space, etc. There is no straight line in ex- 
 istence, hence it can produce no sense-impression, 
 and yet we have the notion of a straight line. The 
 straight lines on paper are incorrect pictures of the 
 true straight lines which are purely ideal construc- 
 tions. Our a priori constructions are not a product 
 of our sense-impressions, but are independent of 
 sense or anything sensed. 
 
 It is of course to be granted that in order to 
 have any conception, we must have first of all sen- 
 sation, and we can gain an idea of pure form only 
 by abstraction. But having gained a fund of ab- 
 stract notions, we can generalize them and modify 
 them; we can use them as a child uses its building 
 blocks, we can make constructions of pure thought 
 unrealizable in the concrete world of actuality. Some 
 of such constructions cannot be represented in con- 
 crete form, but they are not for that reason un- 
 thinkable. Even if we grant that two-dimensional
 
 MATHEMATICS AND METAGEOMETRY. IT3 
 
 beings were possible, we would have no reason to 
 assume that two-dimensional beings could not con- 
 struct a tridimensional space-conception. 
 
 Two-dimensional beings could not be possessed 
 of a material body, because their absolute flatness 
 substantially reduces their shape to nothingness. 
 But if they existed, they would be limited to move- 
 ments in two directions and thus must be expected 
 to be incredulous as to the possibility of jumping 
 out of their flat existence and returnino- into it 
 through a third dimension. Having never moved 
 in a third dimension, they could speak of it as the 
 blind might discuss colors ; in their flat minds they 
 could have no true conception of its significance and 
 would be unable to clearly picture it in their imagi- 
 nation ; but for all their limitations, they could verv 
 well develop the abstract idea of tridimensional 
 space and therefrom derive all particulars of its 
 laws and conditions and possibilities in a similar 
 way as we can acquire the notion of a space of four 
 dimensions. 
 
 Helmholtz continues: 
 
 "But let us proceed still farther. 
 
 "Let ns think of reasoning;- heinq-s existin<T on the surface 
 of an eg:j2:-shaped body. Shortest lines could be drawn be- 
 tween three points of such a surface and a triangle con- 
 structed. Rut if the attempt were marie to construct con- 
 S^ruent triang^les at flifferent parts of the surface, it would 
 be found that two trianjijles. with three pairs of equal sides, 
 would not have their anj^^les equal." 
 
 Tf there were two-dimensional beings lix'ino- on
 
 114 THE FOUNDATIONS OF MATHEMATICS. 
 
 an egg-shell, they would most likely have to deter- 
 mine the place of their habitat by experience just 
 as much as we tridimensional beings living on a 
 flattened sphere have to map out our world by meas- 
 urements made a posteriori and based upon a priori 
 systems of measurement. 
 
 If the several systems of space-measurement 
 were not a priori constructions, how could Helm- 
 holtz who does not belong to the class of two-dimen- 
 sional beings tell us w^hat their notions must be 
 like? 
 
 I claim that if there were surface beings on a 
 sphere or on an egg-shell, they would have the 
 same a priori notions as we have; they would be 
 able to construct straight lines, even though they 
 were constrained to move incurves only; they would 
 be able to define the nature of a space of three di- 
 mensions and would probably locate in the third 
 dimension their gods and the abode of spirits. I 
 insist that not sense experience, but a priori con- 
 siderations, teach us the notions of straight lines. 
 
 The truth is that we tridimensional beings ac- 
 tually do live on a sphere, and we cannot get away 
 from it. W'hat is the highest flight of an aeronaut 
 and the deepest descent into a mine if measured by 
 the radius of the earth? If w^e made an exact imi- 
 tation of our planet, a yard in diameter, it would 
 be like a polished ball, and the highest elevations 
 w^ould be less than a grain of sand ; they would not 
 be noticeable were it not for a difference in color 
 and material.
 
 MATHEMATICS AND METAGEOMETRY. II5 
 
 \\'hen we become conscious of the nature of our 
 habitation, we do not construct a priori conceptions 
 accordingly, but feel limited to a narrow surface 
 and behold with wonder the infinitude of space be- 
 yond. We can very well construct other a priori 
 notions which would be adapted to one, or two, or 
 four-dimensional worlds, or to spaces of positive 
 or of negative curvatures, for all these constructions 
 are ideal ; they are mind-made and we select from 
 them the one that would best serve our purpose of 
 space-measurement. 
 
 The claim is made that if we were four-dimen- 
 sional beings, our present three-dimensional world 
 would appear to us as flat and shallow, as the plane 
 is to us in our present tridimensional predicament. 
 That statement is true, because it is conditioned by 
 an *'if." And what pretty romances have been 
 built upon it ! I remind my reader only of the in- 
 genious story Flatland, JVriffcn by a Square, and 
 portions of Wilhelm Busclrs charming tale Ed- 
 zvard's Dream- \ but the wDrth of conditional truths 
 depends u])on the assum])ti()n upon which they arc 
 made contingent, and the argument is easy enough 
 that if things were different, they would not be what 
 they are. If T had wings, I could fly; if 1 had gills 
 T could live under water; if T were a magician I 
 could wr)rk miracle^. 
 
 ^The open Louri, Vol. VIII. p. 4266 ct scq.
 
 Il6 THE FOUNDATIONS OF MATHEMATICS. 
 
 INFINITUDE. 
 
 The notion is rife at present that infinitude is 
 self-contradictory and impossible. But that notion 
 originates from the error that space is a thing, an 
 objective and concrete reality, if not actually mate- 
 rial, yet consisting of some substance or essence. 
 It is true that infinite things cannot exist, for things 
 are always concrete and limited; but space is pure 
 potentiality of concrete existence. Pure space is 
 'materially considered nothing. That this pure space 
 (this apparent nothing) possesses some very defi- 
 nite positive qualities is a truth which at first sight 
 may seem strange, but on closer inspection is quite 
 natural and will be conceded by every one who com- 
 prehends the paramount significance of the doctrine 
 of pure form. 
 
 Space being pure form of extension, it must be 
 infinite, and infinite means that however far we go, 
 in whatever direction we choose, we can go farther, 
 and will never reach an end. Time is just as infinite 
 as space. Our sun will set and the present day will 
 pass away, but time will not stop. We can go back- 
 ward to the beginning, and we must ask what was 
 before the beginning. Yet suppose we could fill the 
 blank with some hypothesis or another, mytholog- 
 ical or metaphysical, we would not come to an ab- 
 solute beginning. The same is true as to the end. 
 And if the universe broke to pieces, time would 
 continue, for even the duration in which the world 
 would lie in ruins would be measurable.
 
 MATHEMATICS AND METAGEOMETRY. II7 
 
 Not only is space as a totality infinite, but in 
 every part of space we have infinite directions. 
 
 What does it mean that space has infinite direc- 
 tions ? If you lay down a direction by drawing a line 
 from a g•i^^en point, and continue to lay down other 
 directions, there is no way of exhausting- your pos- 
 sibilities. Light travels in all directions at once; 
 but "all directions" means that the whole extent of 
 the surroundings of a source of light is agitated, 
 and if we attempt to gather in the whole by picking 
 up every single direction of it, we stand before a 
 task that cannot be finished. 
 
 In the same way any line, though it be of definite 
 length, can suffer infinite division, and the fraction 
 V;. is quite definite while the same amount if ex- 
 pressed in decimals as 0.333 can never ])e com- 
 pleted. Light actually travels in all directions, 
 which is a definite and concrete process, but if we 
 try to lay them down one 1w one we find that we can 
 as little exhaust their numl)er as we can come to 
 an end in divisibility or as we can reach the bound- 
 ary of space, or as we can come to an ultimate 
 number in counting. Tn other words, realitv is ac- 
 tual and definite but oin- mode of measuring it or 
 reducing il to formulas admits of a more or less ap- 
 ])roximate treatment onlv. being the fund inn of an 
 infinite progress in some direction or other. There 
 is an objective raison d'clrr ^ov llic conce])tion of 
 the infinite, biil our formulation of it is subjective, 
 and the puzzling feature of it originates from treat- 
 ing the subjective feature as an objective fact.
 
 Il8 THE FOUNDATIONS OF MATHEMATICS. 
 
 These considerations indicate that infinitude 
 does not appertain to the thing, but to our method 
 of viewing the thing. Things are always concrete 
 and definite, but the relational of things admits of 
 a progressive treatment. Space is not a thing, but 
 the relational feature of things. If we say that 
 space is infinite, we mean that a point may move in- 
 cessantly and will never reach the end where its 
 progress would be stopped. 
 
 There is a phrase current that the finite cannot 
 comprehend the infinite. Man is supposed to be 
 finite, and the infinite is identified with God or the 
 Unknowable, or anything that surpasses the com- 
 prehension of the average intellect. The saying is 
 based upon the prejudicial conception of the infinite 
 as a realized actuality, while the infinite is not a 
 concrete thing, but a series, a process, an aspect, 
 or the plan of action that is carried on without stop- 
 ping and shall not, as a matter of principle, be cut 
 short. Accordingly, the infinite (though in its com- 
 pleteness unactualizable) is neither mysterious nor 
 incomprehensible, and though mathematicians be 
 finite, they may very successfully employ the infinite 
 in their calculations. 
 
 I do not say that the idea of infinitude presents 
 no difficulties, but I do denv that it is a self-contra- 
 dictorv notion and that if space must be conceived 
 to be infinite, mathematics will sink into mysticism.
 
 MATHEMATICS AND METAGEOMETRY. II9 
 
 GEOMETRY REMAINS A PRIORI. 
 
 Those of our readers who have closely followed 
 our arguments will now understand how in one im- 
 portant point we cannot accept ]Mr. B. A. W. Rus- 
 sell's statement as to the main result of the meta- 
 geometrical inquisition. He says: 
 
 "There is thus a complete divorce between geometry and 
 the study of actual space. Geometry does not give us cer- 
 tain knowledge as to w^hat exists. That peculiar position 
 which geometry formerly appeared to occupy, as an a priori 
 science giving knowledge of something actual, now appears 
 to have been erroneous. It points out a whole series of pos- 
 sibilities, each of which contains a whole system of con- 
 nected propositions ; but it throws no more light upon the 
 nature of our space than arithmetic throws upon the popu- 
 lation of Great Britain. Thus the plan of attack suggested 
 by non-Euclidean geometry enables us to capture the last 
 stronghold of those who attempt, from logical or a priori 
 considerations, to deduce the nature of what exists. The 
 conclusion suggested is, that no existential proposition can 
 be deduced from one which is not existential. But to prove 
 such a conclusion would demand a treatise upon all branches 
 of philosophy."^ 
 
 It is a matter of course that the single facts as to 
 the population of Great Britain must be suj)plicd 
 by counting, and in the same way the measurements 
 of angles and actual distances must be taken by a 
 posteriori transactions; but having ascertained 
 some lines and angles, we can (assuming our data 
 to be correct) calculate other items with absolute 
 
 " Tn the new volumes nf tJtr Encyclof^crdia Britannica, Vol. 
 XXVIII, of the complete work, s. v. Geometry, Non-Euclidean, p. 
 674.
 
 IJO THE FOUNDATIONS OF MATHEMATICS. 
 
 exactness by purely a priori argument. There is 
 no need (as Mr. Russell puts it) ''from logical or 
 a priori considerations to deduce the nature of what 
 exists," — which seems to mean, to determine special 
 features of concrete instances. No one ever as- 
 sumed that the nature of particular cases, the qual- 
 ities of material things, or sense-affecting proper- 
 ties, could be determined by a priori considerations. 
 The real question is, whether or not the theorems 
 of space relations and, generally, purely formal con- 
 ceptions, such as are developed a priori in geom- 
 etry and kindred formal sciences, will hold good in 
 actual experience. In other words, can we assume 
 that form is an objective quality, which would im- 
 ply that the constitution of the actual world must be 
 the same as the constitution of our purely a priori 
 sciences? We answer this latter question in the 
 affirmative. 
 
 We cannot determine by a priori reasoning the 
 population of Great Britain. But we can a pos- 
 teriori count the inhabitants of several towns and 
 districts, and determine the total by addition. The 
 rules of addition, of division, and multiplication can 
 be relied upon for the calculation of objective facts. 
 
 Or to take a geometrical example. When we 
 measure the distance between two observatories 
 and also the angles at which at either end of the 
 line thus laid down the moon appears in a given 
 moment, we can calculate the moon's distance from 
 the earth ; and this is possible only on the assump- 
 tion that the formal relations of objective space
 
 MATHEMATICS AND METAGEOMETRY. 121 
 
 are the same as those of mathematical space. In 
 other words, that our a priori mathematical calcu- 
 lations can be made to throw light upon the nature 
 of space, — the real objective space of the world in 
 which we live. 
 
 * * * 
 
 The result of our investigation is quite conserva- 
 tive. It re-establishes the apriority of mathematical 
 space, yet in doing so it justifies the method of meta- 
 physicians in their constructions of the several non- 
 Euclidean systems. All geometrical systems, Eu- 
 clidean as well as non-Euclidean, are purely ideal 
 constructions. If we make one of them we then 
 and there for that purpose and for the time being, 
 exclude the other systems, but they are all, each 
 one on its own premises, equally true and the ques- 
 tion of preference between them is not one of truth 
 or untruth but of adequacy, of practicability, of use- 
 fulness. 
 
 The question is not: "Is real s])ace that of Eu- 
 clid or of Riemann, of Lobatchevsky or I)olyai?'* 
 for real space is sim])Iy the juxtapositions of things, 
 while our geometries are ideal schemes, mental con- 
 structions of models for si)ace measurement. 'I lie 
 real rjuestion is, "Which system is the most con- 
 venient to determine the juxtaposition of tilings?" 
 
 /I priori considered, all geometries have equal 
 rights, but for all that luiclidean geoniciry, which 
 in the parallel theorem lakes the bull 1)\- the horn, 
 will remain classical forever, for after all the non-
 
 122 THE FOUNDATIONS OF MATHEMATICS. 
 
 Euclidean systems cannot avoid developing the no- 
 tion of the straight line or other even boundaries. 
 Any geometry could, within its own premises, be 
 utilized for a determination of objective space; but 
 we will naturally give the preference to plane ge- 
 ometry, not because it is truer, but because it is 
 simpler and will therefore be more serviceable. 
 
 How an ideal (and apparently purely subjec- 
 tive) construction can give us any information of 
 the objective constitution of things, at least so far as 
 space-relations are concerned, seems mysterious but 
 the problem is solved if we bear in mind the objec- 
 tive nature of the a priori, — a topic which we have 
 elsewhere discussed.* 
 
 SENSE-EXPERIENCE AND SPACE. 
 
 We have learned that sense-experience cannot 
 be used as a source from which we construct our 
 fundamental notions of geometry, yet sense-experi- 
 ence justifies them. 
 
 Experience can verify a priori constructions as, 
 e. g., tridimensionality is verified in Newton's laws: 
 but experience can never refute them, nor can it 
 change them. We may apply any system if we only 
 remain consistent. It is quite indififerent whether 
 we count after the decimal, the binary or the duo- 
 decimal system. The result will be the same. If 
 experience does not tally with our calculations, we 
 have either made a mistake or made a wrong ob- 
 
 * See also the author's exposition of the problem of the a priori 
 in his edition of Kant's Prolegomena, pp. 167-240.
 
 MATHEMATICS AND METAGEOMETRY. I23 
 
 servation. For our a priori conceptions hold good 
 for any conditions, and their theory can be as Httle 
 wrong as reahty can be inconsistent. 
 
 However, some of the most ingenious thinkers 
 and great mathematicians do not conceive of space 
 as mere potentiaHty of existence, which renders it 
 formal and purely a priori, but think of it as a 
 concrete reality, as though it were a big box, pre- 
 sumably round, like an immeasurable sphere. If it 
 were such, space would be (as Riemann says) 
 boundless but not infinite, for we cannot find a 
 boundary on the surface of a sphere, and yet the 
 sphere has a finite surface that can be expressed in 
 definite numbers. 
 
 I should like to know what Riemann would call 
 that something which lies outside of his spherical 
 space. Would the name ''province of the extra- 
 spatial" perhaps be an appropriate term? I do not 
 know how we can rid ourselves of this enormous 
 portion of unutilized outside room. Strange though 
 it may seem, this space-conception of Riemann 
 counts among its advocates mathematicians of first 
 rank, among whom I will here mention only the 
 name of Sir Rol)crt Hall. 
 
 It will be interesting to Jicar a modern thinker 
 who is strongly afifected by metageometrical studies, 
 on the nature of space. Mr. Charles S. Pcircc, an 
 uncommonly keen logician and an original thinker 
 of Uf) mean repute. proj)oses the following llirre 
 alternatives. Me says:
 
 124 THE FOUNDATIONS OF MATHEMATICS. 
 
 "First, space is, as Euclid teaches, both unlimited and 
 immeasurable, so that the infinitely distant parts of any 
 plane seen in perspective appear as a straight line, in which 
 case the sum of the three angles amounts to i8o° ; or, 
 
 "Second, space is immeasurable but limited, so that the 
 infinitely distant parts of any plane seen in perspective ap- 
 pear as in a circle, beyond which all is blackness, and in this 
 case the sum of the three angles of a triangle is less than 
 i8o° by an amount proportional to the area of the tri- 
 angle : or 
 
 "Third, space is unlimited but ^nite (like the surface of 
 a sphere), so that it has no infinitely distant parts; but a 
 finite journey along any straight line would bring one back 
 to his original position, and looking off with an unobstructed 
 view one would see the back of his own head enormously 
 magnified, in which case the sum of the three angles of a 
 triangle exceeds i8o° by an amount proportional to the 
 area. 
 
 "Which of these three hypotheses is true we know not. 
 The largest triangles we can measure are such as have the 
 earth's orbit for base, and the distance of a fixed star for 
 altitude. The angular magnitude resulting from subtracting 
 the sum of the two angles at the base of such a triangle 
 from i8o° is called the star's parallax. The parallaxes of 
 only about forty stars have been measured as yet. Two of 
 them come out negative, that of Arided (a Cycni), a star 
 of magnitude ly^, which is — o."o82. according to C. A. F. 
 Peters, and that of a star of magnitude 7^, known as 
 Piazzi in 422, which is — o."o45 according to R. S. Ball. 
 But these negative parallaxes are undoubtedly to be attrib- 
 uted to errors of observation ; for the probable error of 
 such a determination is about =frO."o75, and it would be 
 strange indeed if we were to be able to see, as it were, more 
 than half way round space, without being able to see stars 
 with larger negative parallaxes. Indeed, the very fact that 
 of all the parallaxes measured only two come out negative
 
 MATHEMATICS AND METAGEOMETRY. 125 
 
 would be a strong arg-ument that the smallest parallaxes 
 really amount to +o."i, were it not for the reflexion that 
 the publication of other negative parallaxes may have been 
 suppressed. I think we may feel confident that the parallax 
 of the furthest star lies somewhere between — o."o5 and 
 -l-o."i5, and within another century our grandchildren will 
 surely know whether the three angles of a triangle are 
 greater or less than i8o°, — that they are exactly that amount 
 is what nobody ever can be justified in concluding. Tt is 
 true that according to the axioms of geometry the sum of 
 the three sides of a triangle is precisely i8o° ; but these 
 axioms are now exploded, and geometers confess that they, 
 as geometers, know not the slightest reason for supposing 
 them to be precisely true. They are expressions of our in- 
 born conception of space, and as such are entitled to credit, 
 so far as their truth could have influenced the formation of 
 the mind. But that affords not the slightest reason for sup- 
 posing them exact." (The Monist. Vol. I, pp. 173-174.) 
 
 Now, let us for argiinienrs sake assume that 
 the measurements of star-parallaxes unequivocally 
 yield results which indicate that the sum of the 
 angles in cosmic triangles is either a trifle more 
 or a trifle less than 180'' ; would we have to conclude 
 that cosmic space is curved, or would we not have 
 to look for some concrete and special cause for the 
 aberration of the light? Tf ihc moon is eclip'^ed 
 while the sun still ap])ears on the liDrizon. it ])r()ves 
 r)nly that the refraction of the solar rays makes the 
 sun appear liighcr than it really stands, if its posi- 
 tion is determined hy a straight line. I»n1 il does not 
 refute the straight line concci)tion of geometry. 
 Measurements of star-parallaxes fit' iliey could no 
 longer he accounted for hy the personal equation
 
 126 THE FOUNDATIONS OF MATHEMATICS. 
 
 of erroneous observation) , may prove that ether can 
 slightly deflect the rays of light, but it will never 
 prove that the straight line of plane geometry is 
 really a cirve. We might as well say that the norms 
 of logic are refuted when we make faulty observa- 
 tions or whenever we are confronted by contradic- 
 tory statements. No one feels called upon, on ac- 
 count of the many lies that are told, to propose a 
 theory on the probable curvature of logic. Yet, 
 seriously speaking, in the province of pure being 
 the theory of a curved logic has the same right to 
 a respectful hearing as the curvature of space in the 
 province of the scope of pure motility. 
 
 Ideal constructions, like the systems of geom- 
 etry, logic, etc., cannot be refuted by facts. Our 
 observation of facts may call attention to the log- 
 ical mistakes we have made, but experience can- 
 not overthrow logic itself or the principles of think- 
 ing. They bear their standard of correctness in 
 themselves which is based upon the same principle 
 of consistency that pervades any system of actual 
 or purely ideal operations. 
 
 But if space is not round, are we not driven to 
 the other horn of the dilemma that space is infinite? 
 
 Perhaps we are. What of it? I see nothing 
 amiss in the idea of infinite space. 
 
 By the by, if objective space were really curved, 
 would not its twist be dominated in all probability 
 by more than one determinant ? Why should it be 
 a curvature in the plane which makes every straight 
 line a circle? Might not the plane in which our
 
 MATHEMATICS AND METAGEOMETRY. 127 
 
 straightest line lies be also possessed of a twist so 
 as to give it the shape of a flat screw, which would 
 change every straightest line into a spiral? But 
 the spiral is as infinite as the straight line. Ob- 
 viously, curved space does not get rid of infinitude ; 
 besides the infinitely small, which would not be 
 thereby eliminated, is not less troublesome than the 
 infinitely great. 
 
 THE TEACHING OF MATHEMATICS. 
 
 As has been pointed out before, Euclid avoided 
 the word axiom, and I believe with Grassmann, that 
 its omission in the Elements is not accidental but 
 the result of well-considered intention. The intro- 
 duction of the term among Euclid's successors is 
 due to a lack of clearness as to the nature of geom- 
 etry and the conditions through which its funda- 
 mental notions originate. 
 
 It may be a flaw in the Euclidean Elements that 
 the construction of the plane is presui)posed, but it 
 does not invalidate the details of his glorious work 
 which will forever remain classical. 
 
 The invention of other geometries can only serve 
 to illustrate the truth that all geometries, the ])lanc 
 geometry of Euclid included, are a priori construc- 
 tions, and were not for obvious reasons Euclid's 
 j)lane geometry preferable, other systems nu"ght as 
 well be employed {ov the ])urpose of space-determi- 
 nation. Neither homaloidality nor curvature be- 
 longs to space; they l)elong to the several .systems of
 
 128 THE FOUNDATIONS OF MATHEMATICS. 
 
 manifolds that can be invented for the determina- 
 tion of the juxtapositions of things, called space. 
 
 If I had to rearrange the preliminary expositions 
 of Euclid, I would state first the Coimnon Notions 
 which embody those general principles of Pure Rea- 
 son and are indispensable for geometry. Then I 
 would propose the Postulates which set forth our 
 OAvn activity (viz., the faculty of construction) and 
 the conditions under which we intend to carry out 
 our operations, viz., the obliteration of all particu- 
 larity, characterizable as "anyness of motion." 
 Thirdly, I would describe the instruments to be 
 employed: the ruler and the pair of compasses; 
 the former being the crease in a plane folded upon 
 itself, and the latter to be conceived as a straight 
 line (a stretched string) one end of which is sta- 
 tionary while the other is movable. And finally I 
 would lay down the Definitions as the most elemen- 
 tary constructions which are to serve as tools and 
 objects for experiment in the further expositions 
 of geometry. There would be no mention of axioms, 
 nor would we have to regard anything as an as- 
 sumption or an hypothesis. 
 
 Professor Hilbert has methodically arranged 
 the principles that underlie mathematics, and the 
 excellency of his work is universally recognized.'' 
 It is a pity, however, that he retains the term 
 "axiom," and we would suggest replacing it by 
 some other appropriate word. "Axiom" with Hil- 
 
 ' The Foundations of Geometry, The Open Court Pub. Co., 
 Chicago, 1902.
 
 MATHEMATICS AND METAGEOMETRY. IZg 
 
 bert does not mean an obvious truth that does not 
 stand in need of proof, but a principle, or rule, viz., 
 a formula describing certain general characteristic 
 conditions. 
 
 Alathematical space is an ideal construction, and 
 as such it is a priori. But its apriority is not as 
 rigid as is the apriority of logic. It presupposes 
 not only the rules of pure reason but also our own 
 activity (viz., pure motility) both being sufficient 
 to create any and all geometrical figures a priori. 
 
 Boundaries that are congruent with themselves 
 being limits that are unique recommend themselves 
 as standards of measurement. Hence the signifi- 
 cance of the straight line, the plane, and the right 
 angle. 
 
 The theorem of parallels is only a side issue of 
 the imi)lications of the straight line. 
 
 The postulate that figures of the same relations 
 are congruent in whatever place they may be, and 
 also that figures can be drawn similar to any figure, 
 is due to our abstraction which creates the condition 
 of anyness. 
 
 The teaching of mathematics, now uttcrlv neg- 
 lected in the public schools and not specially favored 
 in the high schools, should begin early. 1)U( luiclid's 
 method with his ])cdantic pro])ositions and proofs 
 should 1)c replaced by construct inn work. Let chil- 
 dren begin geometry by doing, not by reasoning. 
 Tlie reasoning faculties are not yet sufficient I v de- 
 veloped in a child. Abstract reasoning is tedious, 
 but if it comes in as an incidental aid to construe-
 
 130 THE FOUNDATIONS OF MATHEMATICS. 
 
 tion, it will be welcome. Action is the main-spring 
 of life and the child will be interested so long as 
 there is something to achieve.*^ 
 
 Lines must be divided, perpendiculars dropped, 
 parallel lines drawn, angles measured and trans- 
 ferred, triangles constructed, unknown quantities 
 determined with the help of proportion, the nature 
 of the triangle studied and its internal relations 
 laid down and finally the right-angled triangle com- 
 puted by the rules of trigonometry, etc. All in- 
 struction should consist in giving tasks to be per- 
 formed , not theorems to be proved; and the pupil 
 should find out the theorems merely because he 
 needs them for his construction. 
 
 In the triangle as w'ell as in the circle we should 
 accustom ourselves to using the same names for the 
 same parts.'^ 
 
 Every triangle is ABC. The angle at A is al- 
 ways a, at B y8, at C y. The side opposite A is a, 
 opposite B b, opposite C c. Altitudes (heights) are 
 ha) hbt he. The lines that from A, B, and C pass 
 through the center of gravity to the middle of the 
 opposite sides I propose to call gravitals and would 
 designate them^^.^^,^^. The perpendiculars erected 
 upon the middle of the sides meeting in the center of 
 the circumscribed circle are /«, Z^, Z^. The lines 
 that divide the angles a, /3, y and meet in the center 
 
 " Cp. the author's article "Anticipate the School" {Open Court. 
 1899, p. 747). 
 
 ' Such was the method of my teacher. Prof. Hermann Grass- 
 mann.
 
 MATHEMATICS AND METAGEOMETRY. I3I 
 
 of the inscribed circle I propose to call "dichotoms"^ 
 and would designate them as da, dt, d,. The radius 
 of the circumscribed circle is r, of the inscribed 
 circle p, and the radii of the three ascribed circles 
 are /)«, p^, p,. The point where the three heights 
 meet is H ; where the three gravitals meet, G; where 
 the three dichotoms meet, O.** The stability of 
 designation is very desirable and perhaps indispen- 
 sable for a clear comprehension of these important 
 interrelated parts. 
 
 " From StxoTO/uos. I purposely avoid the term bisector and also 
 the term median, the former because its natural abbreviation b is 
 already appropriated to the side opposite to the point B, and the 
 latter because it lias beeen used to denote sometimes the gravitals 
 and sometimes the dichotoms. It is thus reserved for general use 
 in the sense of any middle lines. 
 
 ' The capital of the Greek p is objectionable, because it cannot be 
 distinguished from the Roman P.
 
 EPILOGUE. 
 
 WHILE matter is eternal and energy is in- 
 destructible, forms change; yet there is a 
 feature in the changing of forms of matter and 
 energy that does not change. It is the norm that 
 determines the nature of all formations, commonly 
 called law or uniformity. 
 
 The term "norm" is preferable to the usual 
 word "law" because the unchanging uniformities 
 of the domain of natural existence that are formu- 
 lated by naturalists into the so-called "laws of na- 
 ture," have little analogy with ordinances properly 
 denoted by the term "law." The "laws of nature" 
 are not acts of legislation ; they are no ukases of a 
 Czar-God, nor are they any decrees of Fate or of 
 any other anthropomorphic supremacy that sways 
 the universe. They are simply the results of a 
 given situation, the inevitable consequents of some 
 event that takes place under definite conditions. 
 They are due to the consistency that prevails in 
 existence. 
 
 There is no compulsion, no tyranny of external 
 oppression. They obtain by the internal necessity 
 of causation. What has been done produces its 
 proper effect, good or evil, intended or not in-
 
 EPILOGUE. 133 
 
 tended, pursuant to a necessity which is not dy- 
 namical and from without. l)ut logical and from 
 within, yet, for all that, none the less inevitable. 
 The basis of every so-called "law of nature" is 
 the norm of formal relations, and if we call it a law 
 of form, we must bear in mind that the term "law'' 
 is used in the sense of uniformity. 
 
 Form (or rather our comprehension of the for- 
 mal and of all that it implies) is the condition that 
 dominates our thinking and constitutes the norm 
 of all sciences. From the same source we derive 
 the principle of consistency which underlies our 
 ideas of sameness, uniformity, rule, etc. This norm 
 is not a concrete fact of existence but the universal 
 feature that permeates both the anyness of our 
 mathematical constructions and the anyness of ob- 
 jective conditions. Its application produces in the 
 realm of mind the a priori, and in the domain of 
 facts the uniformities of events which our scientists 
 reduce to formulas, called laws of nature. On a 
 superficial inspection it is pure nothingness, Imt in 
 fact it is universality, eternality, and r)mnipresence; 
 and it is the factor objectively of tlic world order 
 and subjectively of science, the latter being ninn's 
 capability of reducing the inniinieral)le sense-im- 
 pressions of experience to a nieiho(heal system of 
 knowledge. 
 
 Faust, seeking the ideal of l)eauty, is advised to 
 search for it in the doni;iiti of the elernaj lype^ of 
 existence, wliicli i^ tlie omnipresent Xowhere. the 
 ever-enduring Xever. Mephistopheles calls it the
 
 134 THE FOUNDATIONS OF MATHEMATICS. 
 
 Naught. The norm of being, the foundation of 
 natural law, the principle of thinking, is non-exis- 
 tent to Mephistopheles, but in that nothing (viz., 
 the absence of any concrete materiality, implying a 
 general anyness) from which we weave the fabric 
 of the purely formal sciences is the realm in which 
 Faust finds "the mothers" in whom Goethe personi- 
 fies the Platonic ideas. When Mephistopheles calls 
 it "the nothing," Faust replies: 
 
 "In deinem Nichts hoff' ich das All zu finden." 
 ['Tis in thy Naught I hope to find the AIL] 
 
 And here we find it proper to notice the analogy 
 which mathematics bears to religion. In the his- 
 tory of mathematics we have first the rigid presen- 
 tation of mathematical truth discovered (as it were) 
 by instinct, by a prophetic divination, for practical 
 purposes, in the shape of a dogma as based upon 
 axioms, which is followed by a period of unrest, 
 being the search for a philosophical basis, which 
 finally leads to a higher standpoint from which, 
 though it acknowledges the relativity of the primi- 
 tive dogmatism, consists in a recognition of the 
 eternal verities on which are based all our thinking, 
 and being, and yearning. 
 
 The "Naught" of Mephistopheles may be empty, 
 but it is the rock of ages, it is the divinity of exist- 
 ence, and we might well replace "All" by "God," 
 thus intensifying the meaning of Faust's reply, and 
 say: 
 
 " 'Tis in thy naught T hope to find my God."
 
 EPILOGUE. 135 
 
 The norm of Pure Reason, the factor that shapes 
 the world, the eternal Logos, is omnipresent and 
 eternal. It is God. The laws of nature have not 
 been fashioned by a creator, they are part and parcel 
 of the creator himself, 
 
 Plutarch quotes Plato as saying that God is 
 always geometrizing.^ In other words, the purely 
 formal theorems of mathematics and logic are the 
 thoughts of God. Our thoughts are fleeting, but 
 God's thoughts are eternal and omnipresent veri- 
 ties. They are intrinsically necessary, universal, 
 immutable, and the standard of truth and right. 
 
 Matter is eternal and energy is indestructible, 
 but there is nothing divine in either matter or en- 
 ergy. That which constitutes the divinity of the 
 world is the eternal principle of the laws of ex- 
 istence. That is the creator of the cosmos, the 
 norm of truth, and the standard of right and wrong. 
 If incarnated in living beings, it produces mind, 
 and it continues to be the source of inspiration for 
 aspiring mankind, a refuge of the struggling and 
 storm-tossed sailors on the ocean of life, and the 
 holy of holies of the religious devotee and wor- 
 shiper. 
 
 The norms of logic and of mathematics are 
 uncreate and uncreatable, they arc irrefragable and 
 immutable, and no power on earth or in heaven can 
 change them. We can imagine that the world was 
 made by a great world builder, but we cannot think 
 
 ThitarcVius Convivia. VTIT, 2: Twt J]\iTtov fktyt riv Oeic d«2 ytu- 
 ixirpdv, HaviiiK hunted in vain for llic fanmiis passaRc. I am in- 
 debted for the reference to Professor Ziwet of Ann .\rbor, Midi.
 
 136 THE FOUNDATIONS OF MATHEMATICS. 
 
 that logic or arithmetic or geometry was ever fash- 
 ioned by either man, or ghost, or god. Here is the 
 rock on which the old-fashioned theology and all 
 mythological God - conceptions must founder. If 
 God were a being like man, if he had created the 
 world as an artificer makes a tool, or a potter shapes 
 a vessel, we would have to confess that he is a lim- 
 ited being. He might be infinitely greater and more 
 powerful than man, but he would, as much as man, 
 be subject to the same eternal laws, and he would, 
 as much as human inventors and manufacturers, 
 have to mind the multiplication tables, the theorems 
 of mathematics, and the rules of logic. 
 
 Happily this conception of the deity may fairly 
 well be regarded as antiquated. We know now that 
 God is not a big individual, like his creatures, but 
 that he is God, creator, law, and ultimate norm of 
 everything. He is not personal but superpersonal. 
 The qualities that characterize God are omnipres- 
 ence, eternality, intrinsic necessity, etc., and surely 
 wherever we face eternal verities it is a sign that 
 we are in the presence of God, — not of a mytholog- 
 ical God, but the God of the cosmic order, the God 
 of mathematics and of science, the God of the human 
 soul and its aspirations, the God of will guided by 
 ideals, the God of ethics and of duty. So long as we 
 can trace law in nature, as there is a norm of truth 
 and untruth, and a standard of right and wrong, 
 we need not turn atheists, even though the tradi- 
 tional conception of God is not free from crudities 
 and mythological adornments. It will be by far
 
 EPILOGUE. 137 
 
 preferable to purify our conception of God and re- 
 place the traditional notion which during the un- 
 scientific age of human development served man as 
 a useful surrogate, by a new conception of (lod, 
 that should be higher, and nobler, and better, be- 
 cause truer.
 
 INDEX. 
 
 Absolute, The, 2$. 
 
 Anschauung, 82, 97. 
 
 Anyness, 46fF., 76; Space founded on, 
 60. 
 
 Apollonius, 31. 
 
 A posteriori, 43, 60. 
 
 A priori, 38, 64; and the purely for- 
 mal, 4off; Apparent arbitrariness of 
 the, 96ff; constructions, 112; con- 
 structions. Geometries are, 127; 
 constructions verified by experience, 
 122; Geometry is, i ipff ; is ideal. 
 44; Source of the, 51; The logical, 
 54; The purely, 55; The rigidly, 54, 
 55- 
 
 Apriority of different degrees, 49fT; 
 of mathematical space, 121, 129; of 
 space-measurement, io9ff; Problem 
 of, 36. 
 
 Archimedes, 31. 
 
 As if, 79. 
 
 Astral geometry, 15. 
 
 Atomic fiction, 81. 
 
 Ausdehnungslehre, Grassmann's, 28, 
 29n., 30. 
 
 "Axiom," Kuclid avoided, i, 127; 
 Hiibert's use of, 128. 
 
 Axioms, iff; not Common Notions, 4. 
 
 Ball, Sir Robert, on the nature of 
 
 space, i23f. 
 Bernoulli, 9. 
 
 Ficsscl, Letter of Gauss to, t2ff. 
 Hillingslcy, Sir H., 82. 
 Bolyai, Janos, 22ff, 98; translated, 27. 
 Boundary concepts, Utility of, 74. 
 Iloundarics, 78, 129; produced by 
 
 halving. Even, 85, 86. 
 lirid^cs of Konigsberg, I02f. 
 IJiioch, Wilhclin, 115. 
 
 Carus, Paul, Fundamental Problems, 
 39n ; Kant's Prolegomena, 39n, 1 22n ; 
 Primer of Philosophy, 4on. 
 
 Causation, a priori, 53 ; and trans- 
 formation, 54; Kant on, 40. 
 
 Cayley, 25. 
 
 Chessboard, Problem of, 10 1. 
 
 Circle, Squaring of the, 104: the sim- 
 plest curve, 75. 
 
 Classification, 79. 
 
 Clifltord, 16, 32, 60; Plane constructed 
 by, 69. 
 
 Common notions, 2, 4, 128. 
 
 Comte, 38. 
 
 Concreteness, Purely formal, absence 
 of, 60. 
 
 Continuum, 78flF. 
 
 Curved space, 106; Ilelmholtz on, 113. 
 
 Definitions of Euclid, 1, 128. 
 Delboeuf, B. J., 27. 
 De Morgan, .'\ugustus, 10. 
 Determinism in mathematics, 104. 
 Dimension, Definition of, 85. 
 Dimensions, Space of four, golf. 
 Directions of space, Infinite, 117. 
 Discrete units, 78ff. 
 Dual number, 89. 
 
 Edward's Dream. 115. 
 
 Kgg-shaped body, ii3f. 
 
 Elliptic geometry, 25. 
 
 I'mpiricism, Transcendentalism and. 
 38ff. 
 
 Engel, Fricdrich, 26. 
 
 Euclid, 1-4, 3 1 IT; avoided "axiom," i. 
 127; Expositionsof, rearranged, 128; 
 Ilalstcd on, 31 f. 
 
 Euclidean geometry, classical, 31, 121. 
 
 Even boundaries, 122; as standards of 
 tneasureniciit, 69(T, 85. 86; ptnduccd 
 by h.ilving, 85, 86. 
 
 Experience, Physiological space orig- 
 inates through, 65. 
 
 Faust, 133. 
 
 Fictitious spaces, logff.
 
 140 
 
 thp: foundations of mathematics. 
 
 Flatland, 115. 
 
 Form, 60, 133; and reason, 48. 
 
 Four-dimensional space and tridimen- 
 sional beings, 93. 
 
 Four dimensions, 109; Space of, goff. 
 
 Fourth dimension, 25; illustrated by 
 mirrors, 93 ff. 
 
 Gauss, I iff; his letter to Bessel, laff; 
 
 his letter to Taurinus, 6f, i3f. 
 Geometrical construction, Definiteness 
 
 of, 99 ff. 
 Geometry, a priori, iigff; Astral, 15; 
 
 Elliptic, 25; Question in, 72, 73, 121. 
 Geometries, (7 priori constructions, 127. 
 God, Conception of, 136. 
 Grassmann, 27ff, 127, i3on. 
 
 Ilalsted, Gegrge Bruce, 4n, 2on, 23, 
 26, 27, 28n, 10 1 ; on Euclid, 3 if. 
 
 Hclmholtz, 26, 83; on curved space, 
 113; on two-dimensional beings, 
 II of. 
 
 Hubert's use of "axiom," 128. 
 
 Homaloidal, i8, 74. 
 
 Homogeneity of space, 66ff. 
 
 Hypatia, 31. 
 
 "Ideal" and "subjective," Kant's iden- 
 tification of, 44f; not synonyms, 64. 
 
 Infinite directions of space, 117; di- 
 vision of line, 117; not mysterious, 
 118; Space is, 116, 126; Time is, 
 116. 
 
 Infinitude, ii6ff. 
 
 Kant, 35, 40, 61, 84; and the a priori, 
 36, 38; his identification of "ideal" 
 and "subjective," 44f; his term 
 Anschaunng, 32, 97; his use of 
 "transcendental," 41. 
 
 Kant's Prolegomena, 39n. 
 
 Keyser, Cassius Jackson, yj. 
 
 Kineniatoscope, 80. 
 
 Klein, Felix, 25. 
 
 Konigsberg, Seven bridges of, i02f. 
 
 Lagrange, lof. 
 
 Lambert, Johann Heinrich, gf. 
 
 Laws of nature, 132. 
 
 Legendre, 11. 
 
 Line created by construction, 83; in- 
 dependent of position, 62; Infinite 
 division of, 117: Shortest, 84; 
 Straightest, 75, 127. 
 
 Littre, 38. 
 
 Lobatchevsky, 10, 2off, 75, 98; trans- 
 lated, 27. 
 Lobatchevsky's Tlteory of Parallels, 
 
 lOI. 
 
 Logic is static, 53. 
 
 Mach, Ernst, 27, 65. 
 
 Mathematical space, 63ff, 67, 109; a 
 
 priori, 65; Apriority of, 121, 129. 
 Mathematics, Analogy of, to religion, 
 
 134; Determinism in, 104; Reality 
 
 of, 77; Teaching of, i27ff. 
 Measvuement, Even boundaries as 
 
 standards of, 69ff, 85, 86; of star 
 
 parallaxes, 125; Standards for, 74. 
 Mental activity. First rule of, 79. 
 Metageometry, sff; History of, 26; 
 
 Mathematics and, 82ff. 
 Mill, John Stuart, 38. 
 Mind develops through uniformities, 
 
 52; Origin of, 51. 
 Mirrors, Fourth dimension illustrated 
 
 by, 93ff- 
 Monist, 4n, 27, 125. 
 
 Names, Same, for parts of figures, 130. 
 
 Nasir Eddin, 7. 
 
 Nature, 16; a continuum, 78; Laws 
 
 of, 132. 
 Newcomb, Simon, 25. 
 
 Open Court, 2on, iisn, i3on. 
 Order in life and arithmetic, 80. 
 
 Pangeometry, 22. 
 
 Pajipus, 31. 
 
 Parallel lines in spherical space, 84; 
 theorem, 4n, 25f, 98, 129. 
 
 Parallels, Axiom of, 3. 
 
 Path of highest intensity a straight 
 line, 58. 
 
 Peirce, Charles S., on the nature of 
 space, i23f. 
 
 Physiological space, 63ff, 67; origi- 
 nates through experience, 65. 
 
 Plane, a zero of curvature, 82; con- 
 structed by Clifford, 69; created by 
 construction, 83; Nature of, yz; 
 Significance of, 129. 
 
 Plato, 135. 
 
 Plutarch, 135. 
 
 Poincare, H., 27. 
 
 Point congruent with itself, 71.
 
 INDEX. 
 
 141 
 
 Population of Great Britain deter- 
 mined, iigi. 
 
 Position, 62. 
 
 Postulates, 2, 128. 
 
 Potentiality, 63. 
 
 Proclus, 4, 31. 
 
 Pseudo-spheres, 83. 
 
 Pure form, 63; space. Uniqueness of, 
 61 ff. 
 
 Purely a priori, The, 55: formal, ab- 
 sence of concreteness, 60. 
 
 Question in geometry, -2, 73, 121. 
 
 Ray a final boundary, 58. 
 
 Reason, Form and, 48; Nature of. 76. 
 
 Rectangular pentagon, 98. 
 
 Religion, Analogy of mathematics to, 
 
 134- 
 
 Riemann, isff, 37, 62, 96, 104, io6fT, 
 123. 
 
 Right angle created by construction, 
 83; Nature of, 73; Significance of, 
 129. 
 
 Russell, Bertrand A. W., 26; on non- 
 Euclidean geometry, 119. 
 
 Saccheri, Girolamo, 8f. 
 
 Schlegel, Victor, 30. 
 
 Schoute, P. H., 3on. 
 
 Schumaker, Letter of Gauss to, 11. 
 
 Schweikart, 15. 
 
 Sense-experience and space, i22ff. 
 
 Shortest line, 84. 
 
 Space, a manifold, 108; a spread of 
 motion, s6fT; Apriority of mathe- 
 matical, 121, 129; curved, 126; 
 founded on "anyness," 60; Helm- 
 holtz on curved, 113; Homogeneity 
 of, 66ff; homaloidal, 74; Infinite di- 
 rections of, ir7; Interference of, 
 loiff; is infinite, 116, 126; Mathe- 
 matical, 63ff, 67, 109; Mathematical 
 and actual, 62; of four dimensions, 
 9off; On the nature of, i23f; Phys- 
 iological, 63fT, 67; Sense-experience 
 and, i22fT; the juxtaposition of 
 things, 67, 87; the possibility of 
 motion, 59; the potentiality of meas- 
 uring, 61; Uniqueness of pure, 61 ff. 
 
 Space-conception, how far a pr'.oriT 
 59f; product of pure activity, 53. 
 
 Space-measurement, Apriority of, logff; 
 Various systems of, i04ff. 
 
 Spaces, Fictitious, logff. 
 
 Squaring of the circle, 104. 
 
 StSckcl, Paul, 26. 
 
 Standards of measurement. 74; Even 
 boundaries as, 69!!, 85, 86. 
 
 Star parallaxes. Measurements of, 125. 
 
 Straight line, 69. 71, 112, 122; a path 
 of highest intensity, 59; created by 
 construction, 83; does not exist, 72; 
 indispensable, 72ff; Nature of, 73; 
 One kind of. 75; Significance of, 
 129; jiossible, 74. 
 
 Straightest line, 75, 127. 
 
 Subjective and ideal, Kant's identifi- 
 cation of, 44f; not synonyms, 64. 
 
 Su])erreal. The, 76IT. 
 
 Taurinus, Letter of Gauss to. 6f, i3f. 
 
 Teaching of mathematics. I27ff. 
 
 Teiitanien, 23. 
 
 Theon, 31, 32. 
 
 Theory of Parallels, Lobatchevsky's, 
 lot. 
 
 Thought- forms, systems of reference, 
 61. 
 
 Three, The number. 88f. 
 
 Ttme is infinite. 116. 
 
 "Transcendental," Kant's use of, 41. 
 
 Transcendentalism and Empiricism, 35. 
 38ff. 
 
 Transformation, Causation and, 54. 
 
 Tridimensional beings. Four-dimen- 
 sional space and, 93; space. Two- 
 dimensional beings and, 91. 
 
 Tridimensionality, 84 ff. 
 
 Trinity, Doctrine of the, 89. 
 
 Two-dimensional beings and tridimen- 
 sional space, 91. 
 
 Uniformities, 132; Mind develops 
 
 through, 52. 
 I'nils, Discrete. 78tT: Positing of. 80, 
 
 Wallis, John, 7f. 
 \\ hy ? 100. 
 
 Zambcrti, 32. 
 
 Ziwet, Professor, l3Sn.
 
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