Małgorzata Frankowska-Terlecka (Poland)
SOME CONSIDERATIONS
ON THE ROLE OF THE MEDIAEVAL POSTULATES TO BASE SCIENTIFIC COGNITION ON MATHEMATICS*
One day I reached for a book in the hope th a t it m ight drag me aw ay from the problem s of the m ediaeval science. The them e awoke m y in ter est trem endously and I was affected even stronger at once by the way the subject had been treated by the author. But the book did not divert me from my work. On the contrary: the problem s of m y w ork drew nearer, and the longer I read the book, the m ore I felt as if it w ere w rit ten for the specific purpose of helping me to deal w ith my w ork b etter th an heretofore. A ctually, of course, the auth or of the book 1 intended simply to m eet the needs of students and teachers of m athem atics—to aid the form er in the developm ent of th eir own potentialities and to show the la tte r how to develop most effectively th e abilities of th e ir students. The third aim of the author, certainly not th e least in im portance, has been to rouse the curiosity of those who are interested in m ethods lead ing to original inventions and discoveries. I am neither a student nor a teacher of m athem atics but I am deeply interested in the ars inveniendi, not only in the literal sense of this expression as the a rt of m aking scien tific discoveries, b u t also in a m ore general sense — as the a rt of correct reasoning, as the skill of most adroitly approaching all questions, as the ability to resolve problem s—not only m athem atical problems. Even first of all problems th a t are not m athem atical. Because I am not a m athem a tician. And th a t’s why, w hat struck me most forcibly w hen reading G. Polya’s book, was the thought th a t great benefits could derive from it to one interested in historical sciences, to one whose knowledge of m ath
* This article gives a general outline of the problems to be discussed in a mo nograph, bearing the same title, now being prepared by the authoress.
1 The book referred to is B. G. Polya, How to Solve It, Polish translation, Warszawa, 1964.
ematics is lim ited to w hat he learned in the secondary m odern school. Every step in my w ork demands the solution of some problem ; now and again I m ake affirm ations which m ust be dem onstrated. The application of the general modes of procedure appropriate for the solution of m athe m atical problem s and for the dem onstration of m athem atical propositions could facilitate m y w ork considerably and, w h at is more im portant, could ensure for m y conclusions the certitude and tru th . “The certitude w ith out doubth and ...truth w ithout error...”—are these not the words of Roger Bacon? Of course, the m eaning of these words is not th e same as the m eaning of m y statem ent. And the aims are different—w hat I am interested in is the application of the rules of m athem atics to m y own narrow section of research, w hen the m ediaeval philosopher thought of the application of m athem atical rules and achievem ents to practically all departm ents of hum an activity. B ut in both instances the basic thought is the same: it is th e wish to m ake use of m athem atics as of an infallible model of the tru ly scientific thinking, the most perfect model attainable by hum an minds. So it happened th a t a book whose reading had been expected to divert me from m y w ork actually originated the them e of this article. M athem atics as th e model of thinking—the role which it was to play in scientific research according to mediaeval scholars who w ere seeking new methods—these w ere the questions w orthy of a scru pulous investigation. The fact th a t I am not a m athem atician was judged by me to be ra th e r advantageous. For I believe th a t anyone representing the exact sciences would have difficulties in dealing w ith this kind of problems. He w ould n atu ra lly be m ore interested in the concrete m athe m atical knowledge and the creative potentialities of the scientist who is being investigated, and—should he find such a scientist lacking in special value from this point of view—he would probably be inclined to regard as of little w orth the general statem ents of such an author on the significance and role of m athem atics because such statem ents contrib ute nothing of im portance to the history of m athem atical disciplines. For me, on the other hand, such statem ents are of utm ost interest, w hile the m athem atical knowledge and the contribution to the history of m athe m atics of the authors whose tex ts are exam ined are relatively less im por tant. I find th e ground for this attitude of mine in the firm belief th a t a tru ly outstanding m athem atical m ind w ill devote all efforts to m athe matics as such and w ill be fully absorbed in the solution of m athem atical problems, not paying more th an a very lim ited dose of attention to other problems. The interest in other problem s could develop m uch easier w ith philosophers possessing considerable taste for m athem atics b u t who are able to look on it as if from outside and in this w ay to see b etter its value for the scientific cognition in general. This view is supported, I think, by th e m ediaeval texts. Roger Bacon m ay be cited in evidence for this, since he never contributed anything tru ly original to the developm ent of pure
m athem atics bu t reached m any a valuable and independent m ethodolo gical conclusion based on m athem atical concepts, sum m ed up in his asser tion th a t m athem atics w as “the door and key of th e sciences and things of this w orld”. 2
The history of m athem atics generally speaking does not pay much attention to th e Middle Ages taking it for granted that, after the achieve m ents of the antiquity, they are a period in which m athem atical thought had been restricted m ainly to m aking compilations. Of course, even in the Middle Ages it is possible to find nam es of persons distinguished in the history of m athem atics, such as Leonardo Fibonacci, Nicole Oresme, or Regiomontanus—whose studies clearly show a degree of ripe scholarship and independence—b u t these exceptional m en ra th e r sharply contrast w ith th eir contem poraries who, as a rule, did not go beyond purely prac tical objectives, particu larly those connected w ith the growing com m er cial interests. It is not suprising, therefore, th a t in practically all Euro pean m anuals on the history of m athem atics the period of European mediaeval history is altogether o m itte d .3 Thus, it could appear th a t the study of th e m ediaeval m athem atical conceptions m ight now serve no other purpose than—commendable, of course—satisfaction of the curiosity concerning the roads and devious path s traversed by the hum an m ind before it reached th e present heights of abstraction. In spite of all that, and con trary to opinions held by the historians of m athem atics, it is ju st in the Middle Ages w here we can pick even now quite a few most interesting questions, particu larly actual today in view of the cur ren t postulates of the integration of science and penetration of m athem a tics into an ever growing num ber of fields of hum an activity. And I do believe it to be w orth undertaking p roper research w ork th a t one m ight show m athem atics in the m ediaeval tex ts as th e foundation of all scientific studies, as a model of scientific thinking, as a specific exponent of the existing reality—and to point out the postulates advanced at th a t tim e of applying m athem atics to practical life. N aturally, this paper makes no pretence to be an attem pt to present fully th e questions m entioned above. N either shall the problem find a totally com prehensive exposition in the monograph I am now w orking on. I shall endeavour to give here but a sketchy outline of some selected questions w hich I propose to discuss at length in the intended publication.
It was usual to regard it as self-evident th a t more or less u n til the tim e of Francis Bacon the concept of the un ity of sciences had been de
2 Opus maius, ed. J. H. Bridges, Oxford, 1900, I, 4, p. 97.
3 As a valuable exception one should mention probably the only book in the literature of this subject which treats synthetically the history of mathematics in the Middle Ages, written by an eminent Soviet historian of mathematics, A. P. Juszkiewicz: The History of Mathematics in the Middle Ages, Polish translation, Warszawa, 1969. However, the methodological function of mathematics is discussed here rather inadequately.
term ined in its essence by religion and reduced to the idea of uniting all sciences in the service of God. Such an opinion does not seem to be based on the consideration of all aspects of the idea of scientific universalism in the Middle Ages. This idea derives indeed from th e thought th a t the glory of God and the eternal salvation are the final goals of all human activity, the scientific activity not excepted—because the theological universalism predom inant at th at tim e postulated it. But m any thinkers of the Middle Ages—especially the representatives of the new orienta
tion tow ard the study of natu re—not questioning the above general goals, w ere vividly interested in “p articu lar” aims and concrete objectives of the scientific disciplines which could be tu rn ed to the service of society at large and of individuals. Such interests w ere the product of the new socio-economic and cu ltu ral conditions which took shape w hen the me diaeval mode of life was at its highest, th a t is, in the tw elfth and th ir teenth centuries. A growing dem and for an effective advancem ent of knowledge was the foundation on w hich new scientific m ethod was to appear. This m ethod—one for all sciences—w as to g u aranty to all disciplines the absolute certitude and tru th of their conclusions and to facilitate fu rth e r studies. The postulate of a single scientific method for all branches of scientific cognition becomes, in a sense, a second cause of the idea of the u n ity of sciences, this time based on the methodological universalism. Being theoretically tied as regards its origin and the final goal w ith the theological universalism, the methodological universalism is autonomous and independent from the form er for all practical purpos es. The m ethod which should unite all disciplines was to be founded on experim ent on th e one hand and, on the other hand, on m athem atics, the most perfect among the sciences because it commands the form al demon stration which, being made use of in other sciences, will ensure for them an accurate and precise mode of reasoning.
The most decided champion of the m athem atical m ethod (jointly w ith the experim ental method) in the Middle Ages had been undoubted ly Roger Bacon who, in his works, now and again expressed his deep belief in the w eight and significance of m athem atics as a model of scientific thinking, and in its im m easurable practical u tility .4 He was not alone in these considerations. One should look for tex ts expressing thoughts identical with, or sim ilar to those of Bacon—first of all among the representatives of the Oxford school, taking Robert Grosseteste as
4 The following works of Roger Bacon contain most of the material on this subject: Opus maius (ed. J. H. Bridges, vol. I-III, Oxford, 1900); Opus minus and
Opus tertium (ed. J. S. Brewer, in the Rerum Britannicarum Medii A evi Scriptores,
vol. 15, London, 1859); Communia Naturalia (ed. R. Steele, Opera hactenus inedita
Rogeri Baconi, fasc. Ill, Liber primus communium naturalium Fratris Rogeri, Oxford,
1911) and Communia mathematica (ed. R. Steele, Opera hactenus inedita Rogeri
Baconi, fasc. XiVI; Communia mathematica Fratris Rogeri, part I and II, Oxford,
the principal exponent of this type of opinions and—going back— in the school of C hartres which turned to P lato and the Pythagoreans. Undoubtedly, the P latonist and P ythagorean schools of thinking w ere the source of all concepts connected w ith the im portance and methodo logical functions of m athem atics. Such is the opinion of the au th o r of the fundam ental work on the m athem atical method, L .B runschvicg,5 who asserts th a t the Pythagoreans were the first to appreciate th e real significance of m athem atical cognition.
As a parenthetic rem ark, it is w orth m entioning th a t the two basic w o rk s 6 which, judged by th eir titles, should have contained inform ation of considerable value to one interested in th e m athem atical method in the Middle Ages — bring disappointm ent w hen th eir content is read. For in L.Brunschvicg the mediaeval period of history, and in M .Grab- m ann the m athem atics — are not to be found. L.Brunschvicg omits the period of the Middle Ages because he believes th a t after the achievem ents of th e Pythagoreans it was really only Descartes w ith his idea of the m athesis universalis who opens the road for th e modern trium ph of mathematics. M .Grabmann, on the other hand, m entions the m athem a tical method only on a couple of pages of the first volume, w hen he analyses the theological w ritings of Boethius, 7 ap art from th a t dedicating his efforts prim arily to a deep study of the dialectical method. This is no proof, in m y opinion, of the lack in the Middle Ages of m aterial for the history of the m athem atical method—though, of course, it provides additional evidence in support of the old tru th th a t peculiar interests and personal convictions of authors affect th e scope of, and the treatm en t of the subject in th eir books.
To retu rn to the Pythagoreans—and exceptionally valuable te x t in connection w ith th e discussed problem is th e brief exposition of the Pythagorean conceptions concerning the universal significance of the num ber, included in the w ork of Sextus Em piricus Against the Logi cians. 8 It may be w orthw hile to analyse th e respective fragm ent of this work.
Considering the criteria of tru th Sextus Em piricus affirm s th a t ac cording to the laws of nature things of a given n atu re are perceived by senses of a sim ilar nature. Thus, the sense of vision has th e same n ature as light and because of this fact our eyes are able to perceive light and w hatever is connected w ith it. On th e same principle our
5 L. Brunschvicg, Les étapes de la philosophie mathématique, Paris, 1947. 6 In addition to the above-mentioned work of L. Brunschwicg I refer here to M. Grabmann, Die Geschichte der scholastischen Methode, vol. I—II, Berlin, 1957. This work covers the period, chronologically, up to the beginnings of the thirteenth century.
7 Ibid., ch. Ill: Die scholastischen Methode in den Opuscula sacra des Boethius, pp. 163-77.
8 Sextus Empiricus, Against the Logicians, I, 93-109, Polish translation, War szawa, 1970.
sense of hearing catches sounds. And in th e same m anner the mind com prehends the essence of reality, because th e m ind is of the same n atu re as the essence of the universal beings. The Pythagoreans proclaim th a t th e num ber is exactly this essence of being, since in th e Universe reigns the perfect harm ony of num erical proportions. The most perfect num ber is 10 because it is the sum of 1, 2, 3 and 4, w hile these in their tu rn reflect anything th at exists. For one corresponds to a point, two— to a line, th ree—to a surface and four—to a solid. According to the Pythagoreans—says Sextus Em piricus—it is impossible to comprehend anything in existence w ithout the notion of the num ber. All corporeal things have dimensions, composite things comprise qu an tity—and these are comprehensible only m athem atically. And even incorporeal things are expressed in num erical categories—as, for example, the time which we divide into years, months, days and hours. All m eans by which we facilitate our daily life and w hich m ake our technical skills more effec tive, such as m easures, weights, the m onetary system w ith the rules of doing commerce, w ith loans, bills of exchange and the like—are reducible to num erical relations. Likewise all hum an skills, such as the plastic arts or architecture, are based on certain proportions and these are founded on num bers. Thus it is clear th a t actually th e real essence of everything in existence is the num ber—cognizable by the hum an m ind thanks to the affinity of its n atu re to the n atu re of the number. So m uch Sextus Empiricus. This is a capital text, actually good enough to serve, as it is, for a conspectus of the main problem of a paper on the role and significance of m athem atics. The one, m athem atical nature of the existing universe gives ground for postulating one only method of cognition: the m athem atical method. The universe is perfect and perfection is beautiful—m athem atics, therefore, is also the exponent of beauty (Plato w ill speak later on the beauty of a straight lin e s). M athem atics is also th e base of our daily life, of all hum an arts and skills. Now, all these a re the ideas we can find again in the Middle Ages, enriched by th e trends and thoughts of m any centuries and, which is most im portant, Christianized.
Neoplatonist, A ugustinian and A rab philosophic and scientific thought evolved various aspects of the Pythagorean philosophy of the num ber and jointly had prepared the ground on which, w hen scholastic learning attained its summit, blossomed new postulates for bringing m athem atics into the m ethods of scientific cognition. For Augustine, the C hristian philosopher of perhaps th e greatest consequence in the Middle Ages, the m athem atical tru th s w ere the firm foundation of all knowledge. A ugustine expressed this opinion repeatedly in his works, but prob ably m ost convincigly and vividly in the following passage “A. So
now you know for certain th a t to form a figure of straight lines at least th ree lines are necessary. W ould you retra ct from your opinion if a proof w ere found contradicting it? E. Should anyone dem onstrate to me evidently the falsity of this opinion, I shall lose all hope for the possi bility of acquiring any knowledge.” 10 The above sentences have for us an unexpectedly modern tinge, quite free from the P ythagorean— P latonist mysticism. Yet it is known th a t m ediaeval thought throve predom inantly on A ugustine’s mysticism, on his theological conceptions, his w ithdraw al from nature. The A ugustinian theory of illum ination weighed heavily for centuries on C hristian philosophy. It is interesting to observe, nevertheless, how the A ugustinian system, thoroughly ide alistic, perm eated by religious elations, turning its back on n atu re— and undisputably conservative in character w hen scholastic philosophy reached its sum m it—generated premisses for new, original conclusions concerning a scientific method. Obviously, an im portant role in this process played th e works, hitherto unknow n to mediaeval scholars, w hich contained practically the entire heritage from the an tiquity and from the Arabs. And it was exactly the A ugustinian theory of illum ination in conjunction w ith th e Arabic m etaphysics of light th a t blossomed out, in the Oxford school of the th irteen th century, into the theory of the m ultiplication of species—the theory w hich contributed much to the high appraisal of the u tility of m athem atics for the study of reality. According to this theory th e light is th e substance of reality and is therefore the basis for explaining the world. And the light can be know n only m athem atically—the laws of the diffusion of light are defined by geometry. For this reason the only right m ethod of scientific cogni tion is the m athem atical method, which is certain and infallible. No doubt, it is one of the m any paradoxes in the history of hum an thought th a t the germs of new ideas often come into existence in surroundings satu rated w ith old ideas and seemingly incapable of creatively affecting the hum an mind—w hile new systems are frequently born w ith a hidden stigm a ham pering the free developm ent of an independent searching thought.
The only way to get the proper picture of m athem atics in the Middle Ages as a discipline indispensable for a tru ly scientific cognition of the w orld is, naturally, one w hich leads through an accurate analysis of the mediaeval texts. The choice of th e tex ts from w hich to s ta rt the analysis is not an indifferent m atter, because th e adoption of a p a rti cular point of d eparture often affects the direction in w hich th e w ork is going to evolve; it m ight change the placing of emphasis and influ ence the perspective in w hich the questions are viewed. In our case, I think, we should sta rt by getting acquainted w ith the m ediaeval
10 Augustine, De quantitate animae, VIII, 13, Polish translation, Warszawa, 1953. 9 — O r g a n o n 8/71
classifications of sciences. The task of classifying the sciences is usually undertaken w hen it becomes necessary to order anew the accumulated knowledge — w h eth er on account of the increased tem po of the accu m ulation, or under the influence of new intellectual trends for which old moulds no longer suffice. This is exactly w hat had happened in the tw elfth and th irteen th centuries. The classifications then made were, by and large, compilations reflecting more or less accurately the average contem porary scientific thought, but also, as they took into considera tion new sources, foretelling (though not always in a direct way) novel trends, often such as w ere actually to become prevalent m any years later. A careful scrutiny of th e place occupied in these classifications by m athem atics m ay greatly facilitate fu rth e r research work on the significance in the Middle Ages of the postulates for increasing the range of uses of m athem atical sciences. The position in w hich m athem a tics was placed among other sciences by the authors of the various classifications w ould reflect more or less the average scientific opinion of th e period. It w ill therefore supply a specific scale by which to appraise the attitu d e to m athem atics of various authors and in texts other th an those concerned w ith the classification. The average scientif ic opinion w ill become a zero grade on our scale, something like the zero on the Celsius’ therm om eter. “Above” will be placed original, new thoughts, “below”—statem ents which are b u t an echo of the old theories and convictions.
S everal classifications of th e tw elfth and th irteen th centuries should be reviewed as examples. Even a brief and, natu rally, ra th e r super ficial analysis of selected tex ts should give us an interesting picture of the changes w hich the position of m athem atics among other mediaeval sciences had been subjected to.
Among the classifications of the tw elfth century two are especially notew orthy: those of Hugh of St. Victor and of Dominicus Gundissalinus. Hugh of St. Victor, using exclusively L atin sources, divides the whole knowledge into theory, practice, mechanics and logic—and the theory he subdivides into theology, m athem atics and p h y sics.11 This is a typical A ristotelian division, in w hich theoretical sciences are arranged in a sequence according to th e increasing degree of abstraction: from physics as the least abstract science, to the most abstract among sci ences—metaphysics. M athem atics which, according to H ugh of St. Victor, treats diverse kinds of q uantity is divided into four disciplines in con form ity w ith the scheme of the quadrivium: arithm etic, music, geom etry and astronom y. “M agnitudinis vero alia sunt mobilia, u t sphaera mundi, alia immobilia u t terra. M agnitudinem ergo quae per se est, arithm etica
speculatur, illam autem quae ad aliquid est: musica. Imm obilis m agni- tudinis geom etria pollicetur notitiam . Mobilis vero scientiam astrono- micae disciplinae peritia sibi vindicat. M athem atica igitur d iv iditur in arithm eticam , musicam, geom etriam, astronom iam .” 12 W ith evident gusto Hugh of St. Victor explains on each occassion th e origin of th e nam e of th e particu lar science (mostly indicating th a t the nam e derives from Greek) and he gives also the fu rth e r divisions of each of th e four m athem atical disciplines.13 B ut th e most im portant among his affirm a tions is this: th a t m athem atics, jo in tly w ith logic, should be learned before one begins to study physical sciences, because m athem atics and logic are in stru m ental to th e cognition of reality: “Quia enim logica et m athem atica priores sunt ordine discendi quam physica et ad earn quodammodo in s tru m e n t vice funguntur, quibus unum quem que prim um inform ari oportet antequam physicae speculationi operam det: necesse fuit, u t non in actibus rerum , ubi fallax experim entum est, sed in sola ratione, ubi inconcussa veritas m anet, suam considerationem ponerent, deinde ipsa ratione praevia ad experientiam reru m descenderent.” 14
The w ork of Dominicus Gundissalinus De divisione philosophiae comprises, in a general outline, sim ilar ideas to those included in the w ork of Hugh of St. Victor. However, Dominicus G undissalinus makes use of Arabic sources as w ell as of L atin ones. His w ork is based first of all on two w orks of A lfarabi: the De ortu scientiarum and De scien- tiis. This is very pronouncedly noticeable from the w ay he distinguishes th e particular m athem atical disciplines. Dominicus G undissalinus divid es th e entire hum an knowledge into theoretical and practical branches. Logic, gram m ar and rhetoric—coming first in the order of knowing— are, in his opinion, subservient in ch aracter as introductory disciplines to the study of philosophy. The theoretical philosophy includes physics, m athem atics and th eology.15 M athem atics is, according to Dominicus Gundissalinus, an abstract science whose subject—m atter is qu an tity abstracted from corporeal bodies. Though in reality there are no lines, surfaces, circles, triangles etc. existing separately from bodies—m ath e matics, nevertheless, exam ines them as beings abstracted from m atter and only as such, assuming conventionally th eir objective existence. Arithm etic, music, geom etry and astronom y are strictly p arts of m athe matics, w hile one counts also to m athem atical sciences optics, the sci ence of weights and the science of inventions. M athem atical sciences should be studied after physical sciences: here, the order of cognition agrees
12 Ibid., 755. w Ibid., 755-7. 14 Ibid., 758-9.
15 A general characteristic of the entire division of knowledge Dominicus Gun dissalinus gives in the prologue to his work De divisione philosophiae, ed. L. Baur,
Beiträge zur Geschichte der Philosophie des Mittelalters, IV, 2-3, Münster 1903,
w ith the doctrinal order, because our senses perceive first the form together w ith the m atter and only la ter our intellect can attain the perception of the form abstracted from th e m a tte r .16 W ithin each of the m athem atical disciplines Dominicus Gundissalinus distinguishes its theoreticaol p a rt which considers the basic principles of this discipline, and the practical p art which teaches how these principles could be applied to practical ends. F or instance, th e science of w eights (scientia de ponderibus) trea ts the theoretical principles of heaviness and w eight b u t deals also w ith the instrum ents w hich serve to raise heavy things and transp ort them from place to p la c e .17 The science of inventions (scientia de ingeniis)—w hich closes the list of m athem atical sciences —is w orthy of attention “Scientiae ergo ingeniorum docent modos excogitandi et adinveniendi...” 18 The aim of this discipline is to utilize the theoretical principles of all m athem atical sciences for diverse useful purposes, such as the construction of m easuring implements, musical and optical instrum ents, tools for bricklaying, carpentry and other “mechanical a rts ”.
The characteristic feature of the classification of sciences expounded by Dominicus Gundissalinus is his insistence on the great im portance of the utility of each of the discussed disciplines. The same attitude, possibly even toned up, prevails also in th e classification deriving from the first half of the th irteen th century, w ritten by Michael S c o t.19 His division of philosophy is a stereotype: the theoretical and practical branches, the theory comprising physics, m athem atics and theology. The sub ject-m atter of m athem atics is quantity, continuous and discon tinuous, and—in accordance w ith the kind of q u an tity considered— m athem atics is divided into arithm etic, music, geom etry and astronom y. The m athem atical science should be studied afte r physics because, in the opinion of Michael Scot, th e sensory perception proper to physical sciences which consider the form in corporeal bodies is prior in the order of know ing to rational cognition on w hich m athem atics is based. 20 Evidently, in these speculations Michael Scot does not differ from his predecessors. B ut his argum entation concerning the practical branches of knowledge is notew orthy. He divides the practical philosophy into
16 Ibid., pp. 28-35. The author discusses mathematics at first in a general way,
presenting—according to his wont observable in the whole work—its subject-mat ter, its genus, division, aim, its particular manner of reasoning and the like ques tions, and then proceeds to analyse thoroughly, one after another, each part of mathematics (ibid., pp. 90-124). Almost one third of the whole work of Dominicus Gundissalinus is given up to mathematics.
« Ibid., pp. 121-2.
i® Ibid., p. 122. The whole science of inventions is expounded on pp. 122-4. is The work of Michael Scot on the classification of sciences has been preserved only in fragments included in the Speculum doctrinale of Vincent of Beauvais. These fragments were edited by L. Baur as an appendix to the De divisione philo-
sophiae of Dominicus Gundissalinus, Beitrdge..., IV, 2-3, pp. 398-400.
th ree p arts corresponding to the three p arts of the theoretical philo sophy. M athem atics corresponds to the second p art of the practical philosophy quae adinventa est ad sim ilitudinem doctrinalium , u t negotiatio, Carpentaria, fabrilis, cem entaria, textoria, sutoria, et aliae huiusm odi m ultae quae spectant ad m echanicam et sunt quasi practica illius.” 21 The skills m entioned tu rn to their benefit the theoretical know l edge of the various m athem atical disciplines, in this w ay im proving their artefacts and making them easier to produce.
L. B aur considers the th irteen th century classification of Robert K ilw ardby, contained in his w ork De ortu et divisione philosophiae, to be the best among the mediaeval classifications.22 However, taking into account only the division of m athem atics and its place in th e whole classification—th ere is nothing p articularly in K ilw ardby, w hen we compare his ideas w ith th e classifications discussed heretofore. We find w ith him the same considerations on the su b ject-m atter of m athe matics: the diverse kinds of quantity; the same position of m athem atics in the list of theoretical sciences, between physics and theology, as befits the scheme of the increasing degree of abstraction. Robert K ilw ard by utilized in a large way the Latin and Arabic sources, the form of his classification is finished, bu t it does not seem to contain any tru ly original thoughts.
On the contrary, a really original division of science can be found, I am convinced, in the works of Roger Bacon. His division is not of the type discussed above, because Bacon never w rote a separate work on the classification of sciences, like his predecessors. W hat he did was to adopt a certain order of all disciplines w hen he planned his ency clopaedia on sciences. B ut he is w orth m entioning for the sake of com parison. Bacon divides all sciences into four principal parts. P a rt one comprises gram m ar and logic—the disciplines in strum ental in charac ter, subservient to the other sciences. They give the knowledge of the rules of thinking and of correctly expressing thoughts. The second p art consists of m athem atical disciplines. The th ird p art contains the phys ical sciences, and the fourth—metaphysics and m oral science. 23 Bacon places m athem atics before the physical sciences because, he m aintains, one should begin all studies by learning m athem atics. M athematics, in his opinion, which considers all kinds of quantity, is the most simple science, containing tru th s inborn to men. It is n atu ra l for hum an cogni tion to proceed gradually from easier subjects to more difficult ones. Bacon retains the division into four p arts also for mathem atics, which
Ibid., p. 399.
22 L. Baur gives the content« of, and discusses the work of R. Kilwardby in the Beiträge..., IV, 2-3, pp. 369-75.
23 For an extensive analysis of the classification of sciences in Roger Bacon see M. Frankowska, “Scientia” w ujqciu Rogera Bacona (“Scientia” as Interpreted by Roger Bacon), Wroclaw, 1969.
he divides into arithm etic, geometry, astronom y and music. However, following A lfarabi’s division of sciences, he sees two principal parts in m athem atics: a general part, treating the elements and basic princi ples of m athem atics as a whole—this p art is a species of introduction to the second part, which is divided into the four particu lar disciplines m entioned above. Each of the p articu lar disciplines is, in its turn, di vided, in two indissoluble parts: the theoretical and the practical part. The theoretical disciplines are indissolubly united w ith th eir practical counterparts “quoniam vero speculativa com pletur per suam practicam, et evidentius per earn apparet, et e converso, ideo conj ungam quam libet practicam cum sua speculativa correspondente.” 24 Thus, the theoretical arithm etic considers diverse kinds and properties of num bers, while the practical arithm etic transfers these considerations to the concrete problem s connected w ith commerce, bank accounts, modes of lending money on interest, etc., and even teaches various games based on arith m etical prin ciples.25 Roger Bacon devoted considerable space in his w orks to the discussion of various m athem atical disciplines. He attached great im portance to m athem atics as the means of scientific cognition and asserted th a t dem onstrations and the m anner of reasoning proper to m athem atics, w hen applied to other sciences make possible a more efficient conduct of research w ork and g uaranty th e certitude and tru th of their conclusions. The utility of mathem atics, according to Bacon, is even more many-sided, since its range includes not only scien tific cognition b ut also the practical activities of men, undertaken for the benefit of the state and the Church.
Roger Bacon evidently makes extensive use of both the Latin and Arabic sources. Like Dominicus Gundissalinus, he relies mostly on the classification of Alfarabi. The evidence of his having draw n from the above sources is clearly visible, b u t Bacon certainly contributes much of his own original thought to the considerations he expounds. Bacon’s presentation of the distinction between the theoretical and practical p arts of p articu lar disciplines is consistent and decided beyond compare w ith any earlier sim ilar effort. The stress he puts on the application of m athem atical theories in practice and the firm assertion th a t theoret ical considerations for their own sake m ake no sense—though both ideas derive from affirm ations of his predecessors—w ere for the first time advanced so clearly and so forcibly. The grasp of the wide range of the u tility of m athem atics is also found for the first tim e expressed in such a large w ay in the opinions of Roger Bacon. This, of course, is the result of Bacon’s special interest in the methodological function of m athem atics w ith regard to other sciences.
24 Roger Bacon, Communia mathematica, p. 39. w Ibid., p. 47.
It is interesting to note how even a superficial review of several selected classifications of sciences—such as the one made here—helps to bring out the course of the changes in the position of m athem atics among other disciplines. For Hugh of St. Victor, who relied exclusive ly on the Latin sources, like for A ristotle, m athem atics is a science on a higher level of abstraction th a n physics; it treats various kinds of quantity, abstracted from corporeal bodies. Hugh of St. Victor, how
ever, is aw are of the methodological function of m athem atics and places it—in the order of cognition—before physics. Though the methodological aspect disappears from la ter classifications, w hich drew freely from the Arabic sources, yet the practical u tility of m athem atical disciplines commes to the fore. The science of invention of Dominicus G undissalinus and the p arts of the practical philosophy in Michael Scot testify to this fact. In Roger Bacon both aspects appear together. The practical application of the achievem ents of m athem atical disciplines and the methodological role of m athem atics in relation to other sciences are presented by him on the same plane, and he discusses both problem s in a considerably more m ature w ay than his predecessors ever did. For instance, the practical p arts are set w ith th eir theoretical counterparts more logically—w ithout linking theoretical m athem atical knowledge in any w ay w ith bootm aking or weaving, as we saw it done in Michael Scot.
The analysis of the m ediaeval classifications of science could thus undoubtedly make a good starting point for a fu rth e r scrupulous ana lysis of the texts, for the purpose of preparing a synthetic picture of the role played by the postulates p u t forw ard in the Middle Ages for assigning to m athem atics the universal methodological function in the scientific cognition. N aturally, in the scientific cognition first of all. I t seems, however, th a t the picture would not be complete should it fail to include at least an outline of the im portance of m athem atics in the everyday life of this period of history, at least an idea of how it was reflected in the arts, literature, music. One should rem em ber th a t m athem atics in the Middle Ages was not only a science indispensable for acquiring knowledge, not only a discipline of enormous practical utility, b u t also a science reflecting the ideal of beauty. M athem atics is a beautiful science indeed, and it is w orth showing th a t in the Middle Ages people had been aw are of this fact. To avoid being accused of idle talk, I propose to finish this article by quoting a tw elfth century tale of a m arvellous robe made by fo ur fairies:
L’uevre del drap et le portret. Quatre fées l ’avoient fet
Par grant san et par grant mestrie. L’une i portrest Geometrie, Si come ele esgarde et mesure,
Con li ciaus et la terre dure, Si que de rien nule n i faut, Et puis le bas et puis le haut, Et puis le lé et puis le lonc... Et la seconde mist sa paine An Aritmetique portreire, Si se pena mout de bien feire, So corne ele nonbre par sans Les jors et les ores del tans, Et l’eve de mer, gote a gote, Et puis après l’arainne tote Et les estoiles tire a tire,... La tierce oevre fu de Musique, A cui toz li deduiz s’acorde, Chant et deschant, et son de corde, De harpe, de rote et viële.
Ceste oevre fu et buene et bele; Car devant li gisoient tuit Li estrumant et li déduit. La quarte qui après ovra, A mout buene oevre recovra; Car la mellor des arz i mist. D’Astronomie s’antremist Cèle, qui fet tant mervoille, Qui as estoiles se consoille Et a la lune et au soloil;...26
26 Chrétien de Troyes: Erec et Enide; La robe des quatre fées, in: Antologia
poezji francuskiej (An Anthology of French Poetry), ed. by J. Lisowski, vol. I,
