On the Need to Rewrite the History of Greek Mathematics

  • Sabetai Unguru
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 240)


The previous string of quotations is (most certainly) not illustrative of the ways in which the history of mathematics has traditionally been written. The authors of the quotations themselves have not always practiced what they occa|sionally preached.4 Indeed, the discipline is exceedingly rich in works written (as it were) as a living illustration of P. W. Bridgman’s exhortation:

...the past has meaning only in terms of the present. The impartial recovery of the past, uncontaminated by the influence of the present, is held up as a professional ideal, and a criterion of technical competence is the degree to which this ideal is reached. This ideal is, I believe, impossible of attainment, and cannot even be formulated without involvement with meaningless verbalisms.5


Geometric Algebra Mathematical Text Symbolic Algebra Greek Mathematic Algebraic Symbolism 
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  1. 1.
    Greek Science Its Meaning For Us (Harmondsworth: Penguin Books, 1953), 311.Google Scholar
  2. 2.
    Plato’s Cosmology (New York: The Liberal Arts Press, 1957), 49.Google Scholar
  3. 3.
    The Mathematical Career of Pierre de Fermat (1601–1665) (Princeton, N.J.: Princeton University Press, 1973), XII–XIII.Google Scholar
  4. 4.
    Ironically, the very works of FARRINGTON and MAHONEY mentioned above are cases in point for the very popular syndrome referred to by HEINE in the following phrase: ‘Sie predigen öffentlich Wasser, Und drinken heimlich Wein’; the difference being, however, that, in this instance, both the ‘preaching’ and the ‘drinking’ take place openly, in the public domain. For an analysis of FARRINGTON’S work see LUDWIG EDELSTEIN, ‘Recent Trends in the Interpretation of Ancient Science’, Roots of Scientific Thought A Cultural Perspective, P.P. WIENER A. NOLAND (eds.) (New York: Basic Books, 1957), 90–121; as to MAHONEY’S book, I will be dealing with it in a future essay review in FRANCIA-Forschungen zur westeuropäischen Geschichte, the journal of the Institut Historique Allemand in Paris.Google Scholar
  5. 5.
    ‘Impertinent Reflections on the History of Science’, Philosophy of Science,17 (1950), 63–73, at 64; it is also there that BRIDGMAN says (among other things): ‘It seems to me that there is a very real danger in a too assiduous devotion to the historical point of view…’ (ibid., 72). Without denying the pregnant philosophical problems stemming from the reconstruction of the past, and accepting the obvious conclusion that ‘the impartial recovery of the past’, etc. is indeed an impossible ideal, it does not follow that abandoning irrevocably this unattainable ideal is tantamount to an abandonment of the historical method. Indeed, to repeat a truism, the fact the historian knows that it is in principle impossible to relive the past and that his reconstruction is inherently deficient and inadequate represents for him the utmost challenge to try and look at the past through sympathetic and understanding eyes and to achieve a reconstruction which does no patent violence to that which is to be reconstructed. That there is something to be reconstructed and understood is taken for granted by any mentally healthy historian worth his salt.Google Scholar
  6. 6.
    Cf., for instance PAUL TANNERY, Mémoires Scientifiques,1, Sciences Exactes Dans L’Antiquité, J.-L. HEIBERG, H.G. ZEUTHEN (eds.) (Toulouse: Edouard Privat and Paris: Gauthier-Villars, 1912), 254–280. Characteristically, the title of this study is “De la solution géométrique des problèmes du second degré avant Euclide”! See also, Mémoires Scientifiques, 3 (1915), 158–187 and 244–250. For ZEUTHEN’S views see his Die Lehre von den Kegelschnitten im Altertum (Hildesheim: Georg Olms, 1966, being a photographic reproduction of the Copenhagen, 1886, edition), 1–38 and Geschichte der Mathematik im Altertum und Mittelalter (Copenhagen: Andr. Fred. Höst, Sön, 1896), 32–64.Google Scholar
  7. P. TANNERY, H.G. ZEUTHEN were not the originators of the concept of ‘geometric algebra’. PIERRE DE LA RAMÉE seems to deserve the doubtful credit for this invention. It was he, apparently, who ‘discerned’ that the algebraic art must underlie some parts of EUCLID’S Elements (Books II and VI) and, perhaps, also Greek analysis. (Cf. MICHAEL S. MAHONEY, ‘Die Anfänge der algebraischen Denkweise im 17. Jahrhundert’, Rete, 1 (1971), 15–31, especially p. 25.)Google Scholar
  8. It is quite interesting (and, as will become clear later, strongly supportive of one argument of this paper) that practically all the founders of modern mathematics (VIÈTE, DESCARTES, and FERMAT) followed RAMUS in his belief that algebra lies at the root of Greek analysis! Remarkably enough, WILLIAM OUGHTRED in the seventeenth century, in his most famous mathematical textbook, Clavis mathematicae, is also of the opinion that algebra can serve as a means of understanding difficult problems in EUCLID, ARCHIMEDES, APOLLONIOS, and DIOPHANTOS (cf. ibid., n. 49, 28). Our nineteenth and twentieth century historians of mathematics can indeed be proud of their lengthy and aristocratic mathematical lineage; in truth they have made OUGHTRED’s ‘insight’ the keystone of their methodological and interpretive approach!Google Scholar
  9. 7.
    Thus spake TANNERY: ‘Je veux parler de tout le livre X d’Euclide et de la théorie des irrationnelles qui s’y trouve renfermée… Ce n’est, rien moins que le détail complet de la solution géométrique de l’équation bicarrée et le commencement de celle de l’équation tricarée, avec l’invention d’une nomenclature destinée à suppléer au défaut de notations’ (op. cit., 1263). In a historical appendix written for his brother’s (JULES) Notions de Mathématiques (Paris: Ch. Delagrave, 1903), PAUL emphatically states: ‘Quoique leurs [i.e., the ancients] procédés d’exposition aient toujours présenté, par rapport aux nôtres, des différences essentielles, leur méthode zététique était au fond beaucoup plus voisine de la nôtre qu’on n’est porté à le croire au premier abord. C’est que, tandis que leur symbolisme algébrique [sic!] se développait péniblement, ils en avaient, dès le quatrième siècle avant notre ère, constitué un pour la géométrie,… Ce langage présentait en même temps tous les avantages de l’emploi des lettres dans l’analyse de Viète [!] au moins pour les puissances 2 et 3. Ils avaient dès lors pu constituer, probablement dès le temps des premiers pythagoriciens, une véritable algèbre géométrique pour les premiers degrés, avec la conscience très nette qu’elle correspondait exactement à des opérations numeriques’ (op. cit., 3, 167, my italics). Then he goes on to say: ‘Quoiqu’ils [i.e., the Greeks] ne se soient pas élevés… au concept général des coordonées, leur façon de considérer les coniques est tout à fait analogue à celle de notre géométrie analytique [!]… L’équation qu’ils établissent [!] revient à la forme générale moderne: y2 =p x ± p/a x2…Google Scholar
  10. Les procédés de transformation des coordonnées chez les anciens sont imparfaits, par suite du défaut de conception générale du problème… Mais ces procédés n’en existent pas moins’ (ibid., 168, my italics). Such examples could be multiplied ad nauseam.Google Scholar
  11. H. G. ZEUTHEN has stated his views repeatedly and in various places. The most cogent and complete statement, however, appears in the two works quoted in the previous note. Thus, in Die Lehre von den Kegelschnitten im Altertum, ZEUTHEN entitles his first chapter ‘Voraussetzungen und Hülfsmittel; Proportionen und geometrische Algebra’ (op. cit., 1). It is there that, in a paragraph remarkable for its non sequiturs, ZEUTHEN says that though the Greeks did not possess the concept of a system of coordinates, they nevertheless used ‘rechtwinklige und schiefwinklige Koordinaten’, and though Algebra was unknown to them, the historian must establish what they used in its stead (op. cit., 2)! He continues by saying that the Greek theory of proportions ‘… enthielt Sätze, welche es ermöglichen die wichtigsten algebraischen Operationen… auszuführen’ (ibid., 4) and ‘Auf diese Weise hat man einen Apparat, mit Hülfe dessen man die Zusammensetzung algebraischen Grössen ausdrücken kann’ (ibid.). Furthermore, after the discovery of incommensurability by PYTHAGORAS or one of his disciples, ‘… wurde der unmittelbaren Anwendung von Zahlen und daran geknüpften Proportionen in der Geometrie, welche Anspruch auf Stringenz sollte erheben dürfen, ein Halt geboten… Indessen konnte es nicht fehlen, dass man praktisch Zahlen und Proportionen auch auf die Geometrie anwandte, wenn auch mit dem Bewusstsein, dass man, um die gewonnenen Resultate anerkannt zu sehen, dieselben hinterher [!] auf einem anderen Wege beweisen müsse’ (ibid., 5).Google Scholar
  12. But, the modern manipulation of proportions is a direct outgrowth of the existence of a symbolic mathematical language in which the symbols themselves are manipulated and operated on. On the other hand, ‘Das Altertum hatte allerdings keine Zeichensprache, aber ein Hülfsmittel zur Veranschaulichung dieser sowie anderer Operationen besass man in der geometrischen Darstellung und Behandlung allgemeiner Grössen und der mit ihnen vorzunehmenden Operationen’ (ibid., 6). And now comes the pregnant statement:Google Scholar
  13. In dieser Weise entwickelte sich eine geometrische Algebra, wie man sie nennen kann, da dieselbe als Algebra teils allgemeine Grössen, irrationale sowohl wie rationale, behandelt, teils andere Mittel als die gewöhnliche Sprache benutzt um ihr Verfahren anschaulich zu machen und dem Gedächtnisse einzuprägen. Diese geometrische Algebra hatte zu Euklids Zeiten eine solche Entwicklung erreicht, dass sie dieselben Aufgaben bewältigen konnte wie unsere Algebra solange diese nicht über die Behandlung von Ausdrücken zweiten Grades hinausgeht, ein Gebiet, welches sie auch,… in ihrer Anwendung auf die Lehre von den Kegelschnitten ausgefüllt hat. Eine solche Anwendung entspricht der Anwendung unserer Algebra in der analytischen Geometrie (ibid., 7).Google Scholar
  14. Having dealt with the ancient theory of proportions, ZEUTHEN passes on to ‘geometric algebra’ proper and establishes that the Greeks had the means to represent the equation ax + ßy + yz += d as follows:Google Scholar
  15. .. auf einer Geraden neben einander Stücke abgetragen würden, die in den Verhältnissen a, ß, y… zu x, y, z… stehen. Der Abstand zwischen dem Anfangspunkt und dem Punkt, den man durch successives Abtragen der Stücke erreicht, wird dann d sein. Auf ähnliche Weise kann man verfahren, wenn andere Vorzeichen in der Gleichung [!] vorkommen. Ebenso wie wir bei der jetzt gebräuchlichen Darstellung im Gedächtnis behalten müssen, was jeder einzelne von unseren Buchstaben bedeutet, ebenso mussten die Alten behalten, was das für Stücke waren, die man abgetragen hatte; dann aber hatten die Alten ebenso wie wir eine Darstellung der GleichungGoogle Scholar
  16. Mit Hülfe einer solchen Darstellung werden Gleichungen ersten Grades auf Wegen gelöst, welche viel mit unserer algebraischen Behandlung gemeinsam haben (ibid., 10, my italics).Google Scholar
  17. This should suffice here. More about TANNERY’S and ZEUTHEN’S views on ‘geometric algebra’ will come to the fore in the balance of this study.Google Scholar
  18. 8.
    B. L. VAN DER WAERDEN, Science Awakening (New York: John Wiley, Sons, Inc., 1963), 265–66. In this instance too, as in so many others (cf. footnote 15, passim), VAN DER WAERDEN mirrors OTTO NEUGEBAUER’S views on the fundamentally algebraic character of APOLLONIUS’ Conics. Thus, NEUGEBAUER thinks that ‘… auch in der scheinbar rein geometrischen Theorie der Kegelschnitte vieles steckt, das uns Aufschlüsse geben kann, über die sozusagen latente algebraische Komponente in der klassischen griechischen Mathematik’ (‘Apollonius-Studien’, 216; full reference in footnote 15). And, speaking of the structure of the Conics, NEUGEBAUER says: ‘Die Behandlung des Evolutenproblems ohne jede Benutzung von Infinitesimal methoden aus rein algebraischen Betrachtungen ist überhaupt ein besonderes Glanzstück des ganzen Werkes. Ebenso ist das ganze Arsenal von Identitäten und zugehörigen Ungleichungen aus Buch VII… rein algebraischer Natur’ (ibid., 218, n. 4, my emphasis).Google Scholar
  19. 9.
    Op. cit., 266. It is clear that not ‘any one can use our algebraic notation’. For somebody to use it, he must have such a notation at his disposal in the first place and he must know to use it, i.e., he must be aware of and conversant with the algebraic way of thinking! The position exemplified by the above quotation is also (though, perhaps, not always presented with the same bluntness) that of PAUL TANNERY and ZEUTHEN. Thus, TANNERY begins his study on the geometrical solution of second degree problems before EUCLID with the following statement: ‘Si nous nous proposons de parler de la solution géométrique des problèmes du second degré avant Euclide, il est clair cependant que ce n’est que dans l’oeuvre de ce dernier que nous pouvons trouver l’exposition de cette solution’ (Mém. Scient., 1, 254) and ZEUTHEN, who, according to his own confession, adopts the point of view of TANNERY (Die Lehre, note 1, 5), says: ‘Um… zu erfahren, wie weit die Bekanntschaft der Alten mit gemischten quadratischen Gleichungen und deren Lösung oder Reduktion auf rein quadratische Gleichungen sich erstreckte, wird es zweckmässig sein zu prüfen, welche Gestalt die quadratische Gleichung in der Sprache der geometrischen Algebra annehmen musste…’ (ibid., 15). ZEUTHEN also categorically proclaims: ‘Wir sehen also, dass die Alten alle Formen der Gleichung zweiten Grades behandelt haben…’ (Gesch. der Math. im Alt. und Mittel., 50). This is also he position of NEUGEBAUER in his ‘Apollonius Studien’ when playing havoc among APOLLONIUS’ geometrical propositions, by transcribing the latter’s rhetorical descriptions into the language of algebraic, manipulative symbolism. There is very little of APOLLONIUS in NEUGEBAUER’S transcriptions as even a glance at HEIBERG’S edition (or VER EECKE’S translation) will show. ‘Bei den vorangehend geschilderten Überlegungen’, says NEUGEBAUER, ‘bin ich nirgends anders von den Apolloniusschen Text abgegangen als durch die äussere Form’ (op. cit., 250). As if this is not precisely the supreme sin a historian of mathematics may perform! (More on the relation between form and content in mathematics, below.) Furthermore, this statement is not even true, since NEUGEBAUER has not respected (among other things) APOLLONIUS’ division into propositions. NEUGEBAUER goes on: ‘Es wäre selbstverständlich auch bei der griechischen Ausdrucksweise der Beweise ohne weiteres möglich gewesen [?] bei analogen Beweisen gleiche Bezeichnungen einzuführen. [This is retrospective, hindsightful history! It is obvious for us that identical notation in analogous proofs is preferable to arbitrary notation, only because for us matters of notation are more than mere name-calling, baptizing! We operate on our notations; for the Greeks this was utterly inconceivable.] Dass die antike Mathematik so gänzlich unempfindlich gegen diese uns so sehr lästige Unsystematik gewesen ist, zeigt, dass man sehr vorsichtig damit sein muss, wenn man behauptet, die Unübersichtlichkeit der Beweise habe ihre Weiterentwicklung schliesslich verhindert. Offenbar [?] überblickte man das Buchstabengewirr einer Konstruktion mit derselben Selbstverständlichkeit vie wir heute komplizierte Formeln’(ibid., 250, n. 28). This is incredible! It presupposes that formulae exist somehow independently of their actual, i.e., written presence, that they are ‘hidden’ within the Greek notational chaos, to be merely disentangled by the penetrating eye of the modern historian of mathematics!Google Scholar
  20. 10.
    VAN DER WAERDEN, ibid.Google Scholar
  21. 11.
    Ibid., italics provided.Google Scholar
  22. 12.
    This qualifier is actually superfluous since this is the only possible content according to the view expounded here.Google Scholar
  23. 13.
    Vorlesungen über Geschichte der antiken mathematischen Wissenschaften, Band I: Vorgriechische Mathematik (Berlin-Heidelberg-New York: Springer-Verlag, 1969); this is an unrevised reprint of a book first published in 1934.Google Scholar
  24. 14.
    Cf. op. cit., 82–147.Google Scholar
  25. 15.
    Ibid., 118–124. VAN DER WAERDEN has essentially adopted in toto NEUGEBAUER’S approach and findings in the latter’s three “Studien zur Geschichte der antiken Algebra’; the first study (I) appeared in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik Google Scholar
  26. 16.
    Abteilung B: Studien, 1–27; the second (II), carrying the additional title ‘Apollonius- Studien’, came out in the same volume, same section (Studien) of the same journal, pp. 215–254; finally, the third (III), entitled ‘Zur geometrischen Algebra’, saw the light of the day in volume 3 (1936) of the same journal, same section, pp. 245–259. NEUGEBAUER summarized his well- known views on Greek ‘geometric algebra’ in his The Exact Sciences in Antiquity (Princeton, N.J.: Princeton University Press, 1952); I have used the second edition of this work (New York: Dover Publications, Inc., 1969). In The Exact Sciences, NEUGEBAUER confesses that there is no documentary evidence for what he calls ‘Oriental influence’ on theoretical Greek mathematics (p. 147). His ‘working hypothesis’, however, is: ‘the theory of irrational quantities and the related theory of integration [?] are of purely Greek origin, but the contents of the “geometrical algebra” utilize results known in Mesopotamia’ (ibid.). The only evidence for this mathematically beautiful ‘working hypothesis’ that NEUGEBAUER is able to produce is the fact that both the Babylonian numerical-arithmetical material and some Greek geometrical propositions lend themselves rather easily to an algebraic rendering which, when performed, shows them to be identical. There is no question, indeed, about their identity for NEUGEBAUER (and any modern mathematician) who has at his disposal the algebraic language and the rules of translations into it. The real question is: Did the ancients (Babylonians and Greeks) know the algebraic language and the rules of translating it into either number manipulations or geometric propositions? Pointing out that the problem of application of areas, which he calls ‘the central problem of the geometrical algebra’ (p. 149) is ‘rather difficult to motivate’ (ibid.) in any other way than by translating it into the language of equations (the same procedure as that followed in the transcription and solution of so-called Babylonian ‘problems of second degree’), NEUGEBAUER is really telling us something about his own motivations, idiosyncrasy, and background rather than anything significant about the ancients. Why is the problem of the application of areas a ‘strange geometrical problem’ (ibid., 150)? What is strange about it? Was it strange for the Greeks? Why does it need any motivation? Why is the Babylonian method of solution by reduction to the ‘normal form’ not in need of any motivation? NEUGEBAUER acknowledges that attempts to explain the problem of application of areas independently of algebraic translations have been made, but he claims that the algebraic explanation is ‘by far the most simple and direct explanation’ (ibid.). Fully aware that simplicity does not amount to historical proof, NEUGEBAUER rests his case on the plausibility of his algebraic interpretation and on the historical likelihood of contacts between the Babylonian and the Greek civilizations in Hellenistic times (ibid., 150–151). NEUGEBAUER has dealt at great length with the historical problem of the alleged relations between Babylonian and Greek mathematics in the ‘Schlussbemerkungen’ to his third study on ancient algebra (q.v.). It is there that after stating that in the realm of elementary geometry as well as in the realms of the theory of proportions and the theory of equations (!), Babylonian mathematics contains the entire substantive material on which Greek mathematics continued to erect its structures, NEUGEBAUER points out that, in spite of the total lack of explicit citations of sources, he is convinced of the indubitable influence of Babylonian on Greek mathematics. His conviction is based on the following three factors: 1) The specific evidence of the relation between the two (by which he means their identity when submitted to the same algebraic treatment); 2) The historical fact of a widely spread Hellenistic culture reaching the ‘Orient’; and, finally, 3) The numerous Greek citations referring to Greeks having studied in the ‘Orient’ (loc. cit., 258). According to NEUGEBAUER, the period during which contacts between Greek and Babylonian mathematics took place should be taken as the period from PLATO to HIPPARCHUS. A notable result of these contacts is the Greek geometric algebra, which was later applied to conic sections, achieving there its most remarkable results (ibid.) A few questions naturally arise. If the Greeks were so smart to take over ‘Babylonian algebra’ and geometrize it, why did they adopt the Babylonian dainties rather selectively? Specifically, why did they not adopt a positional number system from the Babylonians rather than clinging to a dreadful one? Why did they fail to see the great ‘advantages’ of the Babylonian approach to astronomy, sticking exclusively to geometrical models rather than to arithmetical sequences? Why did they not deal with the ‘irrational’ like the Babylonians? (There would not have been then any ‘crisis of the irrational’!) The above is by no means an exhaustive list of troublesome questions stemming from NEUGEBAUER’S hypothesis.Google Scholar
  27. 16.
    This is precisely ABEL REY’S point of view in La Science dans l’Antiquité, 3 (La Maturité de la Pensée Scientifique en Grèce), (Paris: Albin Michel, 1939), note 1, 391. In that note REY seems to imply that there was in existence an ‘algebra numerosa’ and that the ancient Greeks had, therefore, a real choice between this algebra and the geometrical symbolism of ‘geometrical algebra’ and that, furthermore, they preferred (wisely) the latter! But if this is the case, where are the traces of this ‘algebra numerosa’ during the first six centuries B.C.? There are no such traces, and this is for a very simple reason, mentioned by A. REY on a previous page of his book: ‘Le mathématicien grec est un geomètre. II n’arrivera à l’algèbre numérique, et bien imparfaitement encore, qu’à l’extrême fin de la période gréco-romaine, au IVe siècle après J.C.’ (ibid., 349). Moreover, this point of view of A. REY clearly contradicts what its author said elsewhere in this work (and in other works) about the inherent fundamental differences between geometry and algebra, the latter requiring a new way of thinking, etc. (More about this, below.)Google Scholar
  28. 17.
    Cf. what A. REY has to say concerning the characteristics of the geometrical method: ‘La construction géométrique… nécessite… une intuition plus singulière que les formules de l’algèbre… D’abord elle a besoin d’une intuition concrète. L’esprit comme dira Descartes, y est asservi aux lignes, aux angles, et aux figures, aux agencements complexes de leurs traces et, comme les Grecs le professaient, à la règle et au compas. Il y a là effort pénible prétendra encore Descartes, nous ajouterons limitatif, d’imagination: limitatif parce que l’image est quelque chose de limité et de singulier en face de l’acte conceptuel, de la relation saisie toute nue.Google Scholar
  29. Ensuite,… [1]a construction géométrique est une synthèse où chaque pas prépare le suivant, où les inventions se lient et se commandent. Mais dans l’invention elle-même, chaque construction nécessite encore un tour de main, un biais, une intuition, une finesse particulière…Google Scholar
  30. Le symbolisme géométrique reste toujours en deça du symbolisme algébrique. Il faut, pour atteindre les articulations de pensée dans l’algèbre s’affranchir de la nécessité de construire, de même que la «construction» permettait de s’affranchir de la necéssité de compter et de calculer, et du spécifisme qui es [sic] affectait’ (ABEL REY, Les Mathématiques en Grèce au Milieu du Ve Siècle (Paris: Hermann, Cie, 1935), 55–56).Google Scholar
  31. 18.
    MICHAEL S. MAHONEY, ‘Die Anfänge der algebraischen Denkweise im 17. Jahrhundert’, Rete, 1 (1971), 15–31.Google Scholar
  32. 19.
    Op. cit., 16–17. There is a slight variation of this characterization, appearing in an enlightening essay review of the reprint of NEUGEBAUER’S Vorgriechische Mathematik: MICHAEL S. MAHONEY, ‘Babylonian Algebra: Form Vs. Content’, Studies in History and Philosophy of Science,1(1971), 369–380; see particularly p. 372.Google Scholar
  33. 20.
    ABEL REY, in his highly illuminating analysis of the features of algebraic thinking (which makes some of his other hackneyed and erroneous conclusions stand out like the proverbial sore thumb), is in substantial agreement with the shorter characterization of MAHONEY. To illustrate: ‘La condition sine qua non d’une algèbre sera… un système de symboles et de règles mécaniques pour agencer ces symboles. C’est une spécieuse universelle, et c’est par ce mot que Viète l’a distinguée du calcul numérique. Son signe éminent c’est l’évasion hors de tout concret dans le pur abstrait. Il faut donc y faire abstraction des nombres et du calcul numérique, opérer sur des termes qui en soient des substituts universels, à l’aide d’un symbolisme opératoire. Les inconnues ont par là-même la même nature, et jouent le même rôle dans l’opératoire que les quantités connues’ (Les Math. en Grèce, 38). ‘… les signes opératoires… se substituent en algèbre aux articulations du raisonnement’ (ibid.). [En algèbre] On n’opère plus sur des nombres, sur des quantités, des valeurs des termes. On opère sur des relations. Les termes ici sont déjà des relations, car ils sont imbriqués les uns avec les autres, et pour employer le mot dans un sens très général mais qui prélude à son sens technique moderne, ils sont fonction les uns des autres’ (ibid., 40). ‘Le besoin du symbole et sa création montrent que la pensée ne peut plus, pour l’objectif qu’elle vise et qu’elle trouve, utiliser une représentation concrète et particulière. Le saut, le voilà…’ (ibid., 45). ‘L’algèbre seule peut permettre de transcender l’espace de la perception’ (ibid., 56).Google Scholar
  34. 21.
    See M. MAHONEY, ‘Babylonian Algebra: Form Vs. Content’ and ‘Die Anfänge der algebraischen Denkweise im 17. Jahrhundert’; ABEL REY, Les Math. en Grèce, 30, 32, 34, 36–37, 41, 44, passim; also LÉON RODET, Sur les Notations Numériques et Algébriques antérieurement au XVIe Siècle (Paris: Ernest Leroux, 1881), passim, especially 69–70; JACOB KLEIN, Greek Mathematical Thought and the Origin of Algebra (Cambridge, Mass.: M.I.T. Press, 1968), passim; Á. SZABó, Anfänge der griechischen Mathematik (Müchen-Wien: R. Oldenburg, 1969) passim, but especially 28, 34, 35–36, and primarily the ‘Anhang’ appearing on 455–488; PAUL-HENRI MICHEL, De Pythagore à Euclide: Contributions à l’histoire des mathématiques Préeuclidiens (Paris: Les Belles Lettres, 1950), 639–646; G. A. MILLER, ‘Weak Points in Greek Mathematics’, Scientia,39 (1926), 317–322.Google Scholar
  35. Some of the works cited here establish the point solidly and unambiguously (SZABÓ and to a lesser extent KLEIN); some of them are, at best, ambiguous (MAHONEY and REY) succeeding in determining at one and the same time the ontological incommensurability of the geometric and the algebraic way of thinking and, yet, accepting (openly or implicitly), without realizing the contradiction involved, the historical legitimacy of the concept ‘geometric algebra’; finally, some, though presenting a less clearcut point of view (RODET), or an unacceptable, ahistorical point of view (MICHEL and, especially, MILLER), enable the astute eye of the historically minded reader to reach easily a conclusion opposite to that presented by the author.Google Scholar
  36. 22.
    Cf. A. REY, Les Math. en Grèce, 34, 41.Google Scholar
  37. 23.
    Ibid., 43, 45, 91–92. Against this view, for NEUGEBAUER, it seems, mathematics has always historically been algebra in various disguises and shapes. Thus, the first stage in the development of mathematics (algebra) was represented by the Babylonian sexagesimal place-value system and the operations with numbers made possible by the existence of such a system (‘Zur geometrischen Algebra’, 247). The second stage was represented by ‘Babylonian algebra’ proper, in which problems are reduced to quadratics of the normal form; the third stage is illustrated by the translation of algebraic techniques into the language of geometry – Greek mathematics or geometrical algebra (ibid.), and, finally, the fourth stage in the development of mathematics (algebra) is ‘… [die] Periode der neuerlichen Rückübersetzung der geometrischen Algebra in eine “algebraische” Algebra’ (ibid., 249). Besides, for NEUGEBAUER, geometry has always had secondary, derivative character: ‘Die grossen Fortschritte der Geometrie sind in allen Phasen [!] immer unlösbar mit der Entwicklung anderer Disziplinen verknüpft (analytische Geometrie und elementare Algebra, Differentialgeometrie und Analysis, Topologie und Riemannsche Flächen + abstrakte Algebra), so dass das Geometrische an sich immer erst nachträglich [!] wieder aus dieser Verknüpfung gelöst werden musste [?]. Für die Frühgeschichte der Mathematik ist eine,,reine“(,,synthetische“) Geometrie viel zu schwierig [Why should this be so? Is there any substantiation for this unqualified claim, except algebraic hindsight?]. Das primäre Hilfsmittel ist hier die Verknüpfung mit dem Bereich der (rationalen) Zahlen und ein wesentlicher Fortschritt der Geometrie ist immer erst möglich, wenn die ungeometrischen Hilfsmittel weit genug entwickelt sind’ (op. cit., 246). It is true that NEUGEBAUER knows very well the importance of symbolism for the development of mathematics (cf. op. cit., 246–247); (after all this should not come as a surprise from the part of somebody who, for all practical purposes, identifies mathematics with algebra!) But he draws from this awareness what seems to me to be unwarranted conclusions. Even if he is right about the precedence of computational techniques in pre-Greek civilizations over geometrical considerations, it does not follow that algebra preceded geometry in the Hellenic civilization. Logistic is most certainly not algebra, and quoting ARCHYTAS (p. 245) to the effect that logistic takes precedence over the arts, including geometry (DIELS-KRANZ, Die Fragmente der Vorsokratiker, 5th ed., 47B4) does not prove that algebra preceded geometry in Greek mathematics!Google Scholar
  38. 24.
    Indeed the nineteenth century originators of the concept of ‘geometric algebra’, ZEUTHEN and TANNERY, wrote before NEUGEBAUER (the ‘discoverer’ of Babylonian algebra) and VAN DER WAERDEN (the articulate spokesman for the view that ‘Babylonian algebra’ became Greek ‘geometric algebra’) and yet, did not hesitate to speak freely of Greek ‘geometrical algebra’ when they encountered in the Elements propositions which seemed to them out of place, unwieldy, incongruous!Google Scholar
  39. 25.
    Sci. Awak., 172, my italics. Once more, VAN DER WAERDEN follows in the footsteps of O. NEUGEBAUER. It was NEUGEBAUER who, in his ‘Apollonius Studien (Studien zur Geschichte der antiken Algebra II.)’, already stated that, though there are recognizable structures and algorithms all over the Conics, which the trained eye of the mathematician can disentangle, these structures, algorithms, and methods of proof have been subsequently completely hidden, camouflaged (‘… nachträglich völlig erdeckt’…; see loc. cit. (footnote 15), 253). Furthermore, speaking of his own analytical transcriptions and manipulations of APOLLONIUS’ geometrical rhetoric, NEUGEBAUER says: ‘Diese höchst einfache Schlussweise gibt den Schlüssel zu sämtlichen hier zusammengestellten Konstruktionen. Bei Apollonius ist nur alles mit grosser sorgfalt auf den Kopf gestellt und verschleiert’ (ibid., 251).Google Scholar
  40. 26.
    NEUGEBAUER’S unmitigated enthusiasm for geometric algebra (for which he erroneously takes ZEUTHEN as the originator) is typical: ‘Zeuthen verdankt man die für das Verständnis der ganzen griechischen Mathematik grundlegende Einsicht, dass es sich insbes. in den Büchern II und VI von Euklids Elementen um eine geometrische Ausdrucksweise eigentlich algebraischer Probleme handelt. Insbesondere hat er an vielen Stellen darauf hingewiesen, dass in den,,Flächenanlegungs“-Aufgaben von Buch VI und zugehöriger Sätze der Data die vollständige Diskussion der Gleichungen zweiten Grades steckt. Er hat dann weiter gezeigt, wie diese,,geometrische Algebra“die Basis für die,,analytische Geometrie“der Kegelschnitte des Apollonius bildet, deren Bezeichnungen,,Ellipse“,,,Hyperbel“,,,Parabel“noch heute auf die Fundamentalfälle der,,Flächenanlegung“zurückwiesen’ (‘Zur geometrischen Algebra’, Q. U.S., 3 B (1936), 249). There are hardly any unambiguous, clear-cut exceptions to the rule. Even those who, for one reason or another, began doubting the inherited interpretation (and this ‘doubting’ got under way only in recent times) did not, as a rule, abandon the concept of ‘geometric algebra’. A case in point is represented by MICHAEL MAHONEY. (Cf., for instance, his otherwise enlightening article, ‘Another Look at Greek Geometrical Analysis’, Archive for History ofExact Google Scholar
  41. Sciences, 5 (1968), 318–48, where he says: ‘For example, Proposition VI, 28 [of the Elements], which is part of the “geometric algebra” of the Greeks…’ (328) or ‘The earliest techniques of analysis evolved from the researches of the… Pythagoreans, and are brought together in the major contribution of this mathematico-philosophical school to Greek mathematics: geometrical algebra. Geometrical algebra was one of the basic tools of the mathematical analyst. In the Data… Euclid gave prominent place to the doctrine of the application of areas, which is the essence of Greek geometrical algebra’ (ibid., 330–31); even in his cogent and powerful criticism of NEUGEBAUER’S ahistorical procedures (‘Babylonian Algebra: Form vs. Content’), MAHONEY somehow considers the ahistorical concept of ‘geometrical algebra’ as legitimate, since he says that ‘Greek geometrical algebra’ could construct ‘… a quadratic system of equations in two unknowns from the values of those unknowns…’(376), but could not construct ‘… a single quadratic equation from its two roots…’ (ibid).). Another, earlier instance of the same syndrome is illustrated by ABEL REY’S writings quoted above. (Incidentally, ‘Chapitre IX’ of ‘Livre III’ of La Maturitéde la Pensée Scientifique en Grèce reproduces verbatim, in toto, ‘Chapitre IV’ (‘Arithmétique et Système Métrique Algèbre, Géométrie et Algèbre Géométrique’) of Les Mathématiques en Grèce au Milieu du Ve Siècle, without any hint whatever to the reader!) To my knowledge, it is only ÁRPÁD SZABÓ, who, in the introduction and (primarily) in an appendix appearing in op. cit., 455–88 (about which more will be said below), unequivocally and forcefully calls attention to what is wrong with the concept of ‘geometric algebra’ and asks for its abandonment. I had arrived at my ideas concerning the historical unsoundness of the notion of ‘geometrical algebra’ independently, while, as a graduate student, I immersed myself in reading Greek mathematical texts and the modern commentaries on them. I reached my final conviction about the necessity to discard and repudiate ‘geometric algebra’ as an explanatory device in the study of the history of Greek mathematics and about the need, growing out of this rejection, to rewrite that history on a sound basis, while teaching a graduate seminar on EUCLID’S Elements at the University of Oklahoma in the fall of 1972. I gave a talk on this topic at the Hebrew University of Jerusalem in the late fall of the same year, which got (so far as I can judge) a mixed reception: historians and the (very few) historically-minded mathematicians present seemed to like its conclusions, while the mathematicians (to put it mildly) remained unconvinced.Google Scholar
  42. 27.
    See CARL B. BOYER, A History of Mathematics (New York-London-Sydney: John Wiley, Sons, Inc., 1968), 85–87, 114–15, 121–131, passim and HOWARD EVES, An Introduction to the History of Mathematics (New York: Holt, Rinehart and Winston, 1964, rev. ed.), 64–69, passim.Google Scholar
  43. 28.
    See J.F. SCOTT, A History of Mathematics (London: Taylor, Francis Ltd., 1960), 23, passim; DIRK J. STRUIK, A Concise History of Mathematics (New York: Dover Publications, Inc., 1948, 2nd rev. ed.), 58–60, passim; FLORIAN CAJORI, A History of Mathematics (New York: The MacMillan Co., 1919), 32–33, 39; DAVID EUGENE SMITH, History of Mathematics, 2 vols. (New York: Dover Publications, Inc., 1958), 1, 106 and 2, 290; EDNA E. KRAMER, The Nature and Growth of Modern Mathematics, 2 vols. (Greenwich Connecticut: Fawcett Publ. Inc., 1974), 1, 108, 137–40, 146.Google Scholar
  44. 29.
    The Works of Archimedes (Cambridge: At the University Press, 1897), Apollonius of Perga Treatise On Conic Sections (Cambridge: W Heffer, Sons Ltd., 1961), The Thirteen Books of Euclid’s Elements, 3 vols. (Cambridge: At the University Press, 1908); HEATH’S edition of EUCLID will be referred to in the future as EUCLID, Elements.Google Scholar
  45. 30.
    Cf., in this context, SZABÓ’S remark: ‘Es wurde also eben betont, dass man auf die Übersetzungen der Quellen – vom Gesichtspunkt der Mathematikgeschichte aus – sich häufig nicht verlassen kann, auch dann nicht, wenn die fraglichen Übersetzungen manchmal philologisch so gut wie tadellos sind’ (op. cit., 16).Google Scholar
  46. 31.
    This is also ZEUTHEN’S view. Cf., for instance, Gesch. der Math. im Alt. und Mittel., 42.Google Scholar
  47. 32.
    EUCLID,Elements, 1, 372.Google Scholar
  48. 33.
    Ibid. Cf. also ZEUTHEN, Die Lehre von den Kegel., 12.Google Scholar
  49. 34.
    Why is such a procedure ‘algebraical’?Google Scholar
  50. 35.
    Elements, 1, 373.Google Scholar
  51. 36.
    Ibid., 374.Google Scholar
  52. 37.
    Ibid. Cf. also ZEUTHEN, Die Lehre, 14 and TANNERY, Mém. Scient. 1, 256–57.Google Scholar
  53. 38.
    Science Awakening (which was originally published in Dutch as Ontwakende Wetenschap (Groningen, 1950) appeared first in English translation at Groningen in 1954; since then the scholarly world was supplied with a paperback edition (New York: John Wiley, Sons, 1963), used in this study, in which the beautiful illustrations of the hard cover edition are marred by imperfect typographical reproductive processes, and, very recently (what a dream for a publishing house!), with a new hardcover ‘Third Edition’ in English (Groningen: Noordhoff, n.d.).Google Scholar
  54. 39.
    Sci. Awaken., 118, my italics. Cf. also, ZEUTHEN, Die Lehre, 12–13. Interestingly, G.H.F. NESSELMANN in his Die Algebra der Griechen (Berlin, 1842) – a photographic reprint (Frankfurt: Minerva, 1969) is available – considers Book II as arithmetical (not algebraic) in character: “Jedenfalls… müsen wir… das zweite Buch… zu den arithmetischen zählen, da von seinen vierzehn Sätzen die ersten zehn gleichfalls nur geometrisch ausgesprochene und bewiesene, aber ihrem Wesen nach [?] lauter arithmetische Wahrheiten enthalten’ (op. cit., 154). NESSELMANN (about whose book L. RODET remarked that a better title would be ‘le calcul chez les Grecs’ (op. cit., 57)) then goes on to transcribe the first ten propositions in algebraic symbolism! Incidentally, we do possess an arithmetical translation of these ten propositions dating from the 14th century by a Byzantine monk, BARLAAM, entitled iρι#μητικy iπ6δeιξ ι’ τω¯ν γραμμικω¯ Rν τω~ δeNρω τω¯ν στοιχegων iποδeιχ#Nντων and another arithmetical translation by CONRAD DASYPODIUS published with the original Book II of EUCLID in 1564. The proofs in these arithmetical translations are patterned after those appearing in the so-called ‘arithmetical Books’ of the Elements (VII–IX). For an example of BARLAAM’S translation and proofs, see NESSELMANN, op. cit., 155, where the proposition dealt with is II. 4.Google Scholar
  55. 40.
    Op. Cit., 119, my italics. Again, NEUGEBAUER espoused similar views long before VAN DER WAERDEN. Thus, describing the contents of the Conics, NEUGEBAUER said: ‘Im ersten Buch werden die Grundgleichungen [!] der Kurven und ihrer Tangenten entwickelt…’ (‘Apollonius Studien’, 218, italics added). Referring in a more detailed fashion to the contents of Book I, NEUGEBAUER again spoke of, ‘Gewinnung der Grundgleichung a) zunächst in unmittelbar geometrischen Form, b) Umformung in eine solche Gestalt, wie sie für die Anwendung bequemer ist… Schliesslich wird gezeigt: ist ein Kegelschnitt durch seine Gleichung gegeben, so gibt es auch… einen Kegel… auf dem er liegt. Zusammen mit der ursprünglichen Gewinnung der Gleichung aus dem räumlichen Schnitt ist damit die volle Äquivalenz von räumlicher und analytischer Darstellung bewiesen’ (ibid., 219, my emphasis).Google Scholar
  56. 41.
    Ibid. Needless to say, the originator of this view is OTTO NEUGEBAUER. Thus, in his ‘Zur geometrischen Algebra’, speaking of the title he chose for this study, NEUGEBAUER confesses that, although it may be too narrow for his purposes, it was selected, ‘… um anzudeuten welchen Punkt ich für den eigentlichen Schlussstein für das Verständnis des Verhältnisses der griechischen Mathematik zur babylonischen halte’ (op. cit., 246). Having accepted ZEUTHEN’S views on the nature of ‘geometric algebra’ in toto, NEUGEBAUER goes on to ask what was the historical origin of the problem of application of areas, a question left unanswered by ZEUTHEN. NEUGEBAUER’S answer runs as follows: ‘Die Antwort… liegt einerseits in der aus der Entdeckung der irrationalen Grössen folgenden Forderung der Griechen der Mathematik ihre Allgemeingültigkeit zu sichern durch Übergang vom Bereich der rationalen Zahlen zum Bereich der allgemeinen Grössenverhältnisse, andererseits in der daraus resultierenden Notwendigkeit [Is this logical necessity or historical necessity? Clearly, the former! And so, NEUGEBAUER has sinned once more against history, by substituting logical for historical criteria in his analysis.], auch die Ergebnisse der vorgriechischen “algebraischen” Algebra in eine “geometrischie” Algebra zu übersetzen’ (ibid., 250). Is there any historical proof for the above italicized statement? As NEUGEBAUER would ask: ‘Ist diese mächtige Behauptung textlich belegt?’ No! What, then, is NEUGEBAUER’S basis for making such a statement? He tells us in what immediately follows the above passage: ‘Hat man das Problem einmal in dieser Weise formuliert, so ist alles Weitere vollständig trivial und liefert den glatten Anschluss der babylonischen Algebra an die Formulierungen bei Euklid (ibid.) In other words, NEUGEBAUER begins with what one would normally expect the historian to conclude (namely, that the Greeks knew the Babylonian stuff and ‘translated’ it into geometrical language), and from this historically totally unfounded assumption, by transcribing both the Greek geometrical propositions and the Babylonian numerical manipulations into algebraic symbolism, ‘manages to show’ that they are both the same and ‘therefore’ the Greeks copied the Babylonians. The vicious circle of his reasoning is obvious! Having thus shown the complete mathematical equivalence between the Babylonian ‘normal form’ and the simplest case of application of areas, NEUGEBAUER then exclaims in pleasant amazement: ‘Das ist aber genau die einfachste Formulierung der Flächenanlegungsaufgabe des,,elliptischen“Falles, wie sie bei Euklid VI, 28 steht… Euklid VI, 29 steht dann die Ubersetzung der Normalform (2), d.h. der,,hyperbolische“Fall’ (ibid.). What does this prove? To my mind, nothing else than the fact that if one performs the historically impermissible translation of the Babylonian and Greek mathematical stuff into algebraic symbolism, one can see that they are the same. It certainly does not prove that the Greeks knew the Babylonian stuff! But all this is not enough, since NEUGEBAUER goes on: ‘Damit ist gezeigt, dass die ganze Flächenanlegung nichts anderes ist, als die mathematisch evidente geometrische Formulierung der babylonischen Normalformen quadratischer Aufgaben’ (ibid., 251). Aber ist was ist mathematisch evident auch historisch evident? Das scheint mir nicht! Neugebauer continues in the same vein by showing ‘… dass. auch die griechische Lösungsmethode nicht anderes ist als die wörtliche Übersetzung der babylonischen Formel… xl = b2; 2 1b2 22 — c2 p, (ibid.). Having done this, he remarks: ‘Die yr einzig neue Überlegung ist hier die Bemerkung über die Grösse der Gnomon figur, also etwas, was so nahe liegt, dass es gewiss keiner besonderen Motivierung bedarf, wenn man den Ausgangspunkt so wählt, wie es /hier geschehen ist, nämlich in der Aufgabe, die algebraische Formel (2) Google Scholar
  57. 45.
  58. 46.
    The reader may like being reminded that, strictly speaking, there are no Babylonian equations, normalized or ‘abnormalized’. There are only tablets containing numbers and operations executed on these (specific) numbers, which the modern mathematician can translate, if he so wishes, into equations. On this whole issue, I urge the reader to peruse MAHONEY’S ‘Babylonian Algebra: Form Vs. Content’. It is there that MAHONEY says: ‘All that Babylonian texts contain is series of arithmetical operations that lead to (usually) correct results. The rest is interpretation by the historian. The Babylonians state the problem and compute the solution; the derivation of that solution is the work of the historian, and one may question whether the derivation tells us more about the historian’s mathematics than about Babylonian mathematics’ (op. cit., 375). Indeed if NEUGEBAUER was able at all to speak of a ‘Babylonian Algebra’ and to tie it in with Greek geometry, this was due to his rather radical methodological innovation. As he tells us in his ‘Studien zur Geschichte der antiken Algebra I’ (full reference in footnote 15): ‘Dabei verstehe ich unter,,antiker Algebra“einen wesentlich weiteren Problemkreis als dies üblicherweise der Fall ist. Einerseits fasse ich das Wort,,Algebra“sachlich möglichst weit, d.h. ich ziehe auch stark geometrisch betonte Probleme mit in Betracht, wenn sie mir nur auf dem Wege zu einen letztlich,,algebraischzu nennenden formalen Operieren mit Grössen zu liegen scheinen. Andererseits gehe ich zeitlich weit über das übliche Mass hinaus,…’ (op. cit., 1, italics added).Google Scholar
  59. 47.
    Indeed it is far from chancy! It is due to the conscious premeditated talents of the modern mathematician, turned historian, who has managed to translate (‘traduttore traditore’) both Babylonian numerical manipulations and Greek geometrical propositions into algebraic language.Google Scholar
  60. 48.
  61. 49.
    M. MAHONEY, ‘Babyl. Alg.”, 370.Google Scholar
  62. 50.
    Recently, more attention is being paid to mathematics as a reflection of culture. Cf. in this context DAVID BLOOR, ‘Wittgenstein and Mannheim on the Sociology of Mathematics’, Studies in History and Philosophy of Science, 4 (1973), 173–91. It is there that one finds the following interesting remark: ‘As evidence for the idea that mathematical notions are cultural products, consider the historical case of the concept zero. Our present concept is not the one that all cultures have used. The Babylonians, for example, used a place-value notation but had a different, though related, concept. Their nearest equivalent to zero operated in the way that ours does when we use it to distinguish, say, 204 from 24. They had nothing correspond – [sic!] to our use when we distinguish, say 240 from 24. As Neugerbauer [sic!] puts it, ‘context alone decides the absolute value in Babylonian mathematics’… If the Babylonians used a zero which left some aspects of a calculation context dependent, then,thus far, their concept of zero differs from ours’ (ibid., 186, italics provided).Google Scholar
  63. 51.
    As an illustration of the kind of mathematician I have in mind, I shall refer the reader to G.A. MILLER (see reference in note 21, above). His article on ‘Weak Points in Greek Mathematics’ is a genuine chef-d’oeuvre and should be read in its entirety. Space limitations, however, permit me to quote here only sparingly. Thus, after deploring the ‘… undue emphasis on the geometric view…’ (317) of the ancient Greeks, MILLER declares emphatically that ‘the lack of emphasis on the formal algebraic side of mathematics doubtless constituted the greatest inherent weakness of Greek mathematics’ (op. cit., 317–18, my italics). He illustrates this ‘drawback’ with ‘the Greek attitude towards the solution of the quadratic equations. Not only did they solve certain quadratic equations geometrically, but… it appears clear that they had three general formulas [! ?] for the algebraic solutions of such equations… they failed to see the general significance of the formulas and hence… did not succeed in obtaining a general solution of the quadratic equation in the modern sense. It seems therefore unfortunate that many writers claim that they solved the quadratic equation’ (ibid., 318). ‘By overlooking the fact that the algebraic equation frequently gives us much more than what we explicitely put into it, the Greeks made a blunder and failed to put into their work one of the most fruitful ideas of later mathematics’ (ibid., my italics).Google Scholar
  64. ‘The awe inspired by the immortal Elements… is partly offset by the short-sightedness exhibited by the Greeks when they failed to extend the number concept so as to include the negative and the ordinary complex numbers [!]. In fact, the earlier Greek writers did not include the irrational numbers in their concept of numbers [Imagine, such nasty behaviour!]’ (ibid., 318–19).Google Scholar
  65. ‘It was well that the Greeks developed the theory of conic sections without awaiting the discovery of the usefulness of this theory in the study of our solar system… (ibid., 320, my italics).Google Scholar
  66. Finally: ‘The painstaking care which the modern scientist employs in making accurate measurements was foreign to the Greek mind. They devoted their attention to the shorter and easier routes leading to scientific truths’ (ibid., 320–21).Google Scholar
  67. I have burdened the reader with this string of quotations for two main reasons: 1. Most of the views expressed by MILLER relate to issues discussed in this study and 2. These views, though representing a much lower degree of sophistication than those embodied in the term ‘geometric algebra’, stem ultimately from the same condemnable approach to the history of mathematics. If they, rightly, seem offensive and simple-minded, let the reader keep in mind that they, at least, condemn Greek mathematics for not being algebraic rather than (which I think is potentially much more dangerous) making it algebraic and then discussing it as such.Google Scholar
  68. 52.
    Cf. TANNERY, Mém. Scient., 1, 264–67; ZEUTHEN, Gesch. d. Math. im. Alt. u. Mittel., 56, 158–161; ZEUTHEN, Die Lehre, 24–26; HEATH’S edition of EUCLID, Elements, 3, passim; VAN DER WAERDEN, Sc. Awaken., 168–172; BOYER, A Hist. of Math., 128–29; etc., etc.Google Scholar
  69. 53.
    Op. cit., 170.Google Scholar
  70. 54.
    Elements, 3, 5.Google Scholar
  71. 55.
    Ibid., 4–5; these expressions are taken over approvingly by HEATH from ZEUTHEN’S Gesch. d. Math. im. Alt. u. Mittel., 56.Google Scholar
  72. 56.
    Ibid., 5, my italics.Google Scholar
  73. 57.
    Ibid., 7. By the way, Sir THOMAS avows somewhat belatedly, in a confessional slip, while discussing the character of the first five propositions of Book XIII of the Elements, that, ‘… the method of [proof of] the propositions is that of Book II., being strictly geometrical and not algebraical…’ (ibid., 441).Google Scholar
  74. 58.
    EUCLID,Elements, 1, 382.Google Scholar
  75. 59.
    Ibid., 382–83.Google Scholar
  76. 60.
    Cf. ‘Babyl. Algebra’, 372; see also ‘Die Anfänge der algebr. Denkweise’, 17–18, passim.Google Scholar
  77. 61.
    Elements, 383.Google Scholar
  78. 62.
    Ibid.; cf. also ZEUTHEN, Die Lehre, 19 and TANNERY, Mém. Scient., 1, 257–59. Also, ZEUTHEN, Geschichte d. Math. im. Alt. u. Mittel., 47–48 and 52.Google Scholar
  79. 63.
    De Pythagore à Euclide, 639, my italics.Google Scholar
  80. 64.
    Op. cit., 640. It is clear that MICHEL has a weird (if unoriginal) view of both algebra and the historical method. Thus he says: ‘Nous ne considérons pas l’algèbre comme nécessairement liée à un certain système de symboles, mais, à la suite de M. Thureau-Dangin, comme “une application de la méthode analytique [des Grecs] à la résolution des problèmes numériques”… Pour qu’il y ait algèbre (non pas algèbre “lettrée” mais algèbre “parlée”) il faut mais il suffit qu’une quantité inconnue soit posée d’emblée comme connue. Dès que le mathématicien adopte cette méthode, son discours est susceptible d’être traduit en équations, ce que nous faisons couramment pour la commodité du lecteur’ (ibid., 641–42, my italics).Google Scholar
  81. 65.
    Ibid., 643.Google Scholar
  82. 66.
    Ibid., 643–44.Google Scholar
  83. 67.
    This is what II.5 is: a theorem, a geometrical theorem and not a quadratic equation, or a problem leading to a quadratic equation!Google Scholar
  84. 68.
    Op. cit., 644.Google Scholar
  85. 69.
    Op. cit., 645, my italics.Google Scholar
  86. 70.
  87. 71.
    Ibid., 646, my italics.Google Scholar
  88. 72.
    Cf. the following: ‘… chaque type de problème arithmétique nécessite une invention de l’esprit particulière à ce problème, adaptée à sa solution, et que ne peut pas servir à d’autres types dérectement [sic]; car, indirectement, toute opération contribue bien à former l’esprit arithmétique et à faciliter les inventions nouvelles’ (Les Math. en Grèce, 55–56).Google Scholar
  89. 73.
    See MICHEL, op. cit., 646; A. REY, La Science dans l’Antiquité, 3, 388–91; G.A. MILLER, op. cit., passim.Google Scholar
  90. 74.
    See A. REY, Les Math. en Grèce, 32, 45 (note 1), passim; MAHONEY, ‘Die Anfänge der algebr. Denkweise’, 18, 23, passim.Google Scholar
  91. 75.
    That there was such a ‘scandal’ in the mathematical world is, at best, doubtful. Cf., in this respect, Á. SZABÓ, op. cit., 115.Google Scholar
  92. 77.
  93. 78.
    Ibid., 121.Google Scholar
  94. 79.
    I wonder if this really solves the ‘problem’! Let me state emphatically that there is a problem only if one transcribes II.5 and II.6 into modern symbolism. It is only due to this totally unacceptable procedure that VAN DER WAERDEN could make his initial claim that II.5 and II.6 are nothing but the same algebraic formula! If one stays within the EUCLIDEAN realm (and this is the only admissible procedure), i.e., if one does not transcend the boundaries of geometry, then, most clearly, II.5 and II.6 are not the same proposition. Specifically, in the language of application of areas, II.5 asks to apply a rectangle to a given line such that it will be equal to a given square and fall short by a square figure, while II.6 asks for the application of a rectangle to a given line such that it will be equal to a given square and exceed by a square figure! (Cf. EUCLID, Elements, 1, 385–86). These can be shown to be the same only by somebody who has the benefit of formulaic expression.Google Scholar
  95. 80.
    Á. SZABÓ, Anfänge der griechischen Mathematik, 458–59.Google Scholar
  96. 81.
    Ibid., 457.Google Scholar
  97. 82.
    Ibid., 458. SZABó aims a scathing criticism at the TANNERY-ZEUTHEN thesis: ‘H.G. ZEUTHEN, dessen “Verdienste” um die Entdeckung der sog. “geometrischen Algebra der Griechen” durch O. NEUGEBAUER… so übertrieben hervorgehoben wurden, hat in Wirklichkeit in seinen beiden Werken (Die Lehre von den Kegelschnitten im Altertum,… und Geschichte der Mathematik im Altertum und Mittelalter,…) – was die “geometrische Algebra” betrifft – nur den irreführenden Vergleich von P. TANNERY weitergebaut. (Man hätte sich nämlich erst einmal fragen müssen, inwiefern überhaupt erlaubt ist, im Zusammenhang mit EUKLIDS geometrischen Konstruktionen über Lösungen von algebraischen Gleichungen zu reden!)’ (ibid., note 6, 457). For additional elements of this criticism, cf. op. cit., 35–36, 474, 488.Google Scholar
  98. 83.
    Ibid., 36.Google Scholar
  99. 84.
    Ibid., 465, passim.Google Scholar
  100. 85.
    Ibid., 487.Google Scholar
  101. 86.
  102. 87.
    EUCLID,Elements, 2, 260.Google Scholar
  103. 88.
    Ibid., 260–62.Google Scholar
  104. 89.
    Op. cit., 121–22; cf. also ZEUTHEN, Die Lehre, 19–20, 29–31 and Gesch. d. Math. im Alt. u. Mittel. 47–48.Google Scholar
  105. 90.
    Elements, 2, 263.Google Scholar
  106. 96.
    This follows from a scholium to X.9 and from PAPPUS’ commentary on Book X, preserved in Arabic; cf., however, concerning the veracity of these sources, Á. SZABÓ, op. cit., 100–111.Google Scholar
  107. 97.
    Elements, 3, 28.Google Scholar
  108. 98.
    Incidentally, EUCLID takes this for granted, i.e., without further ado, he assumes that ratios the duplicates of which are equal are themselves also equal; the converse of this assumption was employed in the preceding stage of the proof.Google Scholar
  109. 99.
    Ibid., 28–30. There is a porism (and a lemma) after this proposition; they do not interest us here.Google Scholar
  110. 100.
    This is how HEATH transcribed EUCLID’S enunciation: ‘If a, b be straight lines, and a: b =m: n, where m, n are numbers, then a2: b2 = m2: n2 and conversely’ (ibid., 30).Google Scholar
  111. 101.
    Ibid., 31.Google Scholar
  112. 102.
  113. 103.
    Ibid., 50.Google Scholar
  114. 104.
    Ibid., 50–51.Google Scholar
  115. 105.
    Ibid., 115.Google Scholar
  116. 106.
    Ibid., 115–16.Google Scholar
  117. 107.
    Cf. ibid., 56–57, especially the beginning of 57.Google Scholar
  118. 108.
    Ibid., 116.Google Scholar
  119. 109.
    Ibid., 194. To enable the reader to grasp the meaning of the proposition and, at the same time, to give him an inkling of its complexity, I shall define the crucial concepts appearing in it: An apotome is an irrational straight line obtained by subtracting from a rational straight line another rational straight line, the two rational straight lines being commensurable in square only.Google Scholar
  120. The Annex is the straight line which, when added to a compound irrational straight line obtained by subtraction (like an apotome) makes up the greater term, i.e. the annex is the negative term in an apotome.Google Scholar
  121. Asecond apotome is an apotome having the following characteristics: Given a rational straight line and an apotome, if the square on the whole be greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the annex be commensurable in length with the rational straight line set out, the apotome is called a second apotome.Google Scholar
  122. A medial straight line is a mean proportional between two rational straight lines commensurable in square only.Google Scholar
  123. A first apotome of a medial straight line is an irrational line obtained by subtracting from a medial straight line another medial straight line commensurable with the former in square only and which contains with it a rational rectangle.Google Scholar
  124. 120.
  125. 121.
  126. 122.
    On ‘genuine’ Greek analysis see R. ROBINSON, ‘Analysis in Greek Geometry’, Mind, N.S., 45 (1936), 464–73 and M. MAHONEY, ‘Another Look at Greek Geometrical Analysis’ (full reference in note 26, above). MAHONEY’S article is, on the whole (as are his other studies), perceptive, penetrating and insightful. Concerning his attitude toward ‘geometrical algebra’, however, these qualities are lacking. Thus, he says: ‘As they are given in the Data, however, the theorems pertaining to geometrical algebra are cumbersome [?], involving as they do the intricate construction of plane figures. Working mathematicians used a simpler form of geometrical algebra, an algebra of line lengths… Although it is an example of theoretical, rather than problematical, analysis, the analysis of Euclid XIII, 1… illustrates the use of the simplified algebra of line lengths…’ (op. cit., 331, my italics).Google Scholar
  127. Where is there any concrete, specific proof for the use of ‘geometric algebra’ in pre-EUCLIDEAN, or even EUCLIDEAN, times? There is none! The reference to the scholiast’s interpolation to XIII.1 misses the point, I think, since the exact date of the interpolation is unknown; (and others) is that it is spurious, or, as HEATH put it, ‘… altogether alien from the plan and manner of theElements’ (Elements, 3, 442). It was interpolated perhaps as late as 500 years after the writing of the Elements (HEATH says that all the interpolations to XIII.1–5 ‘… took place before Theon’s time…’ (ibid.)–i.e., fourth century A.D.), and the method of proof it displays is totally foreign to classical Greek mathematics. There is not one shred of reliable historical evidence to support the speculations of BREITSCHNEIDER, HEIBERG, etc. that this method represents ‘… a relic of analytical investigations by Theaetetus or Eudoxus…’ (ibid.); indeed the whole history of Greek mathematics seems to exclude such an inference. But even if one believes HEIBERG’S later dating (in ‘Paralipomena zu Euklid’, Hermes, 38 (1903), 46–74, 161–201, 321–356), namely that the author of these interpolations is HERON OF ALEXANDRIA, this would still make these additions some 400 years younger than EUCLID and would place them comfortably (to the exclusion of PAPPUS and DIOPHANTUS) after the decline of classical Greek mathematics.Google Scholar
  128. MAHONEY also speaks of the ‘… increased use of an informal, but subtle and penetrating, algebra of line lengths’… (op. cit., 337, my italics) in the works of the post-EUCLIDEAN mathematicians of the 3rd century, APOLLONIUS and ARCHIMEDES. Then he goes on saying, ‘ARCHIMEDES provides an example of these analyses’ (ibid.). His reference to proposition II.1 of On the Sphere and the Cylinder, however, does not warrant any allusion to ARCHIMEDES’ proof as an ‘algebra of line lengths.’ (Cf. J.L. HEIBERG, ed., Archimedes Opera Omnia, 2nd ed. (Leipzig: Teubner, 1910) 1, 170–74.) The proof is still essentially geometrical in the best tradition of EUCLID’S Elements! MAHONEY, then, proceeds to give a detailed example of ‘… Greek geometrical analysis in action, one which proceeds by an algebra-like manipulation of line lengths…’(ibid., my italics). His chosen example is proposition II.4 of ARCHIMEDES’ On the Sphere and the Cylinder. I must say, however, that I am unconvinced. Again, what ARCHIMEDES is doing in II.4 of On the Sphere and the Cylinder is very much like what EUCLID is doing in the Elements (though there are obviously differences, some of which do point toward a freer manipulation of lines; interestingly, however, in both examples given by MAHONEY the line lengths are closely associated with two or three-dimensional figures!); I cannot see how somebody whose mind was not ‘corrupted’ by algebraic reasoning and manipulations can describe ARCHIMEDES’ proof as ‘algebra-like manipulation of line lengths’, though, to be sure, this name is less offensive than ‘geometrical algebra’.Google Scholar
  129. One more remark. Speaking of EUCLID’S Porisms, MAHONEY says that it represents ‘… the best example of the sort of treatise included in the Treasury of Analysis. It also illustrates well TANNERY’S remark that the Greeks lacked not so much the methods as the language to express them’ (ibid., 343–44). There is another laudatory reference to TANNERY’S saying at the end of the article. According to MAHONEY, the Porisms indicates ‘… why the lack of a suitable mode of exposition – such as symbolic algebra – prevented the Greeks from pursuing geometrical analysis further and from being able to express clearly what they had accomplished [!]. In the realm of geometrical analysis in particular, TANNERY’S remark holds true; the Greeks did not so much lack methods of mathematics as means to express them’ (ibid., 348). Finally the motto itself of MAHONEY’S article is, once more, TANNERY’S original saying: ‘Ce qui manque aux mathématiciens grecques [sic] ce sont moins les méthodes… que des formules propres à l’exposition des méthodes’ (ibid., 318). MAHONEY is not alone in praising this famous ‘fliegende Wort’ of TANNERY; so do ZEUTHEN, HEATH, etc. And yet, is not this famous saying an unwitting confession that ‘geometric algebra’ is a pernicious and historically stillborn concept to use? Furthermore, is it not absurd to talk of the methods when the means to express them, i.e., to use them, are not available? How could one use a method which is de facto inexpressible, i.e., unthinkable? Within the given limits of coherence of a mathematical culture, the methods available to that culture are exactly those by means of which the culture reached and expressed its mathematical achievements. The methods are contained in the tangible products of that mathematical culture. In the absence of treatises on the methodology of mathematics, the methods are those embodied in and displayed by the actual mathematical works available to the historian. The question is really very simple: To what extent does one possess the method if he lacks the means to put it to use? And the answer seems to me obvious. ‘Wovon man nicht sprechen kann darüber muss man schweigen’ has not only hortatory and prescriptive consequences; it is also, mutatis mutandis, a correct description of the historical state of affairs in intellectual history: ‘Wovon man nicht sprechen kann darüber schweigt man’. If a culture (any culture!) cannot speak it does not speak. It remains silent. It certainly does not hide its impotence. Part of being ignorant of something is being ignorant of your ignorance. If you know that you are ignorant, your ignorance stricto sensu has disappeared. And the Greeks, clearly, did not know that they did not know algebra. So they did not hide their ignorance behind a geometrical screen. There is nothing lurking in hiding behind Greek geometry!Google Scholar
  130. 123.
    Elements, ibid., 248–50.Google Scholar
  131. 124.
    In its sense of ‘corollary’.Google Scholar
  132. 125.
    Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik (Abteilung B: Studien), 3 (fasc. 1, 1934), 18–105 and 3 (fasc. 2, 1936), 122–235. Ironically, KLEIN’S insightful articles were followed in each case by NEUGEBAUER’S unbridled transcriptions of ancient mathematical texts into algebraic language; thus, in fasc. 1, NEUGEBAUER published ‘Serientexte in der babylonischen Mathematik’ (ibid., 106–114), while in fasc. 2 KLEIN’S article was succeeded by ‘Zur geometrischen Algebra’! To make things even more piquant, the article immediately following the last part of KLEIN’S study was ‘Eudoxos Studien III. Spuren eines Stetigkeitaxioms in der Art des Dedekind’schen zur Zeit des Eudoxos’ (op. cit., 236–244), by OSKAR BECKER!Google Scholar
  133. 126.
    Greek Mathematical Thought and the Origin of Algebra (Cambridge, Mass.: 1968, M.I.T. Press). The book includes an ‘Appendix’, containing VIETA’S In artem analyticem [sic] isagoge (Tours, 1591), translated into English by the Reverend J. WINFREE SMITH. (Parenthetically, let me urge those readers who have a choice and wish to read KLEIN’S highly interesting study to refer back to the original German articles: somehow the pomposity, stuffiness, and turgidity of the author’s Jene Vermutungen die die “geometrische Algebra der Pythagoreer” als Übernahme bzw. griechische Weiterentwicklung von ursprünglich babylonischen Gedankengängen auffassen wollten, waren voreilig. Der Zusammenhang dieser Art Kentnisse mit der “babylonischen Wissenschaft” ist in Wirklichkeit nirgends erwiesen. Im Gegenteil! Man hat eher den Eindruck, dass die hier behandelte ‘Flächengeometrie der Pythagoreer’ eine rein griechische Errungenschaft war. 133 style are better accomodated by the Teutonic cadences than by the more friendly sounds of the perfidious Albion…)Google Scholar
  134. 127.
    Les Math. en Grèce, 30.Google Scholar
  135. 128.
    REY, op. cit., 32, 44, 45, 48, 91; MAHONEY, ‘Die Anfänge der algebr. Denkweise’, passim.Google Scholar
  136. 129.
    This makes their acceptance of the legitimacy of the term ‘geometric algebra’ the more difficult to understand. Instances of this acceptance in REY can be found in op. cit., 33, 46, 49–51, 52 (‘Le théorème de Pythagore lui-même est une résolution intuitive de l’equation x2 +y2 = z2), 56–57; also, La Science dans l’Antiquité, 352; for examples of MAHONEY’S acceptance see note 26 above.Google Scholar
  137. 130.
    MAHONEY, ‘Babyl. Algebra’, passim; Rey, op. cit., 34, 36–37, 41, 91–92. The last reference in REY is a scathing attack against NEUGEBAUER’S interpretation of Babylonian mathematics. It is there that REY says: ‘Du reste la preuve convaincante c’est que si, bien avant l’ère chrétienne et surtout avant Diophante, on avait eu l’idée algébrique des équations et, peu on prou, la pensée algébrique, toute la face de la mathématique en eût été changée’ (ibid., 91).Google Scholar
  138. 131.
    Most of them appear in an appendix to the book.Google Scholar
  139. 132.
    Op. cit., 28, 36, passim.Google Scholar
  140. 133.
    Op. cit., 488.Google Scholar
  141. 134.
    Ibid., note 21, 487. It seems to me, therefore, a concession to the prevailing mode of writing the history of mathematics, which SZABó so eloquently denounced, when he himself starts, somewhat indiscriminately, using algebraic notation in his geometrical discussions (cf. op. cit., 483).Google Scholar
  142. 135.
    Op. cit., 472–73.Google Scholar
  143. 136.
    La science dans l’antiquité, 390.Google Scholar
  144. 137.
    Op. cit., 118.Google Scholar
  145. 138.
    Modern algebra (the only true algebra) is a creation of the sixteenth and seventeenth centuries. Its great protagonists are VIÈTE, DESCARTES, and FERMAT. It marks the passage from an old way of thinking in mathematics (the geometrical way, the mos geometricus) to a new way (the symbolic way, the mos per symbola). Its historical development is rightly connected with the reintroduction into the West of the great works of classical Greek mathematics which, however, contained the old way of thinking, to be discarded by modern mathematics. With VIÈTE algebra becomes the very language of mathematics; in DIOPHANTUS’ Arithmetica, on the other hand, we possess merely a refined auxiliary tool for the solution of arithmetical problems (cf. M. MAHONEY, ‘Die Anfänge der algebr. Denkweise’, passim). In the seventeenth century, algebra was called ars analytica, a pregnant name indeed. It shows the difference between the Greek approach and that of the seventeenth century. For the Greeks, mathematics was not an art, a manipulative technique (techne) but a science (episteme, scientia). Furthermore, for the Greeks analysis was merely a means of discovery, a heuristic tool. Mathematics, episteme, was limited to synthesis. In the seventeenth century, on the other hand, one is faced with algebraical analysis without any synthesis. This new approach meant (among other things) a certain loosening of the Greek strictures of rigor and a new mathematical style. MAHONEY identifies the necessarily external factors which led to this development as PETRUS RAMUS’ pedagogical endeavours and the search for a universal symbolism (characteristica universalis) starting with RAMON LULL in the thirteenth century. These two factors were united in RAMUS, who contributed to a separation of the universal symbolism from its ties with magic via the ars memoriae. According to MAHONEY, RAMUS seems to have been the first to demand more respectability and status for the algebraic art, practiced, as a rule, outside the walls of the university establishment (cf. ibid., 25).Google Scholar
  146. 139.
    From the dust jacket of KLEIN’S book.Google Scholar
  147. 140.
    Elements, 2, 353, my italics.Google Scholar
  148. 141.
    Op. cit., 301–303.Google Scholar
  149. 142.
    KLEIN, I think rightly, sees DIOPHANTUS’ Arithmetica as an exercise in theoretical logistic (cf. op. cit., 127–149, passim).Google Scholar
  150. 143.
    LÉON RODET in op. cit. (see note 21 above for full reference) demolishes NESSELMANN’S taxonomy. It is there that RODET says: ‘… il faut reconnaître que cette distinction des trois étapes successives du langage algébrique a quelque chose de séduisant. Il n’y a qu’un malheur: c’est qu’elle est bâtie uniquement sur un échafaudage d’inexactitudes…’ (ibid., 56). RODET points out that even admitting the truthfulness of NESSELMANN’S classification, it is wrong to call it historical! The three stages do not correspond to three historically successive stages even on NESSELMANN’S own account, since the lowest rank of this classification is occupied by the Arabs and by Italian mathematicians writing between the Crusades and the sixteenth century, while DIOPHANTUS (3rd century A.D.) corresponds to the middle stage and the Hindus, reported masters of the Arabs, are occupying the highest rank, i.e., the same spot as modern symbolic algebra! RODET destroys especially this characterization of Hindu mathematics and reveals its absolute historical falsehood due to NESSELMANN’S ignorance of ‘… les notations algébriques des Indiens’ (ibid., 57). Speaking of Hindu ‘algebraic notation’, LÉON RODET says: ‘Il lui manque, pour être mise en parallèle avec la nôtre, deux choses essentielles: des signes spéciaux pour les deux opérations directes de l’addition et de la multiplication, et le moyen de représenter autrement que par des nombres particuliers les paramètres qui entrent, simultanément aux variables proprement dites, dans nos expressions algébriques. Enfin, comme chez Diophante, les symboles qu’elle emploie ne sont que les initiales des noms des quantités qu’elle veut représenter. L’algèbre Indienne mérite tout autant que celle des Grecs et des Européens entre le XIIe et le XVIIe siécles, le nom d’ Algébre syncopée…’ (ibid., 60).Google Scholar
  151. 144.
    Op. cit., 69–70.Google Scholar
  152. 145.
    Cf., in this connection, MICHEL’s statement: ‘D’une façon générale, le vocabulaire de Diophante reste imprégné de géométrie, comme en témoignent ces énoncés de problémes’ (op. cit., 641); also, NESSELMANN: ‘So finden wir wirklich selbst bei Diophant Beispiele von gänzlicher Vernachlässigung des Gebrauches der Abbreviaturen…, die also ganz der rhetorischen Stufe angehören’ (op. cit., note 15, 304).Google Scholar
  153. 146.
  154. 147.
    KLEIN, op. cit., 123.Google Scholar
  155. 148.
    Op. cit., 123–24.Google Scholar
  156. 149.
    See text to note 145 above; cf. also, KLEIN, op. cit., 146–47.Google Scholar
  157. 150.
    EUCLIDElements, 1, 37. I have striven in this paper to demolish the validity of the concept of ‘geometric algebra’ as a useful historiographic term. In this, if POPPER is right, I must have achieved the highest level of understanding of the true underpinnings of that concept… According to Sir KARL, there are three levels of understanding: 1. The lowest represented by the pleasant feeling of having grasped the argument 2. The medium level, represented by the ability to repeat the argument 3. The highest level, represented by the ability to refute the argument (Cf. IMRE LAKATOS, ‘Proofs and Refutations (II)’, British Journal for the Philosophy of Science, 14 (1963–64), 120–139, at 131.)Google Scholar

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  • Sabetai Unguru

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