¹ B. Russell, [Review of Couturat, L., De l'infini mathématique (Paris, 1896)Google Scholar], Mind, 1897, 6, 112–19 (114).Google Scholar

² Letter of 2 December 1904 from Peirce, C. S. to Welby, Lady, in Hardwick, C. S. (ed.), Semiotic and significs. The correspondence between Charles S. Peirce and Victoria Lady Welby, Bloomington, Indiana, 1977, p. 44Google Scholar; but not in MrsGust, H. (ed.), Other dimensions. A selection from the later correspondence of Victoria Lady Welby, London, 1931.Google Scholar

³ Lakatos, I., Proofs and refutations: the logic of mathematical discovery, Cambridge, 1976.CrossRefGoogle Scholar

⁴ Dugas, R., Essai sur l'incompréhension mathématique, Paris, 1940Google Scholar; parts of this rare book are published as ‘L'incompréhension mathématique’, Thalès, 1937–1939, 4, 168–83Google Scholar. Rostand, F., ‘La notion de scrupule dans la psychologie des mathématiques’, Revue générale des sciences, 1952, 59, 325–36Google Scholar; Souci d'exactitude et scrupules des mathématiciens, Paris, 1960Google Scholar; and Sur la clarté des démonstrations mathématiques, Paris, 1962Google Scholar. Among Polyà's writings, I recommend especially Mathematics and plausible reasoning, 2 vols., Oxford, 1954.Google Scholar

⁵ See Bos, H. J. M., ‘Differentials, higher-order differentials and the derivative in the Leibnizian calculus’, Archive for history of exact sciences, 1974, 14, 1–90 (81–4). See also note 13 and text below.CrossRefGoogle Scholar

⁶ See Bunn, R., ‘Quantitative relations between infinite sets’, Annals of science, 1977, 34, 177–91CrossRefGoogle Scholar. Cauchy denied the infinitely large numbers in Sept leçons de physique générale faites à Turin en 1833, in Oeuvres complètes, series 2, xv, p. 422Google Scholar. These lectures were first published in 1868 by Cauchy's follower Moigno, who emphasized Cauchy's point by including his own essay against the actual infinite. Note that 1833 was the year in which Cauchy repeated his 1821 theorem and proof word-for-word in his Résumés analytiques at Turin (loc. cit., x, 55–6). The same misunderstanding of Cauchy's ‘infinite’ occurs in Fisher, G. M., ‘Cauchy and the infinitely small’, Historia mathematica, 1978, 5, 313–31CrossRefGoogle Scholar; and Bloor, D. accepts without criticism Lakatos's history of Cauchy's theorem in his ‘Polyhedra and the abominations of Leviticus’, British journal for the history of science, 1978, 11, 245–72 (262–3).CrossRefGoogle Scholar

⁷ See my The development of the foundations of mathematical analysis from Euler to Riemann, Cambridge, Mass., 1970.Google Scholar I have been asked several times why I have not discussed Lakatos's interpretation of Cauchy's theorem in my own writings on the history of mathematical analysis. The reason is that I learnt of its existence only after seeing it mentioned, after my own work was published, in Cleave, J. R., ‘Cauchy, convergence and continuity’, British journal for the philosophy of science, 1971, 22, 27–37CrossRefGoogle Scholar. Cleave cited the version contained in Lakatos's PhD thesis and now published in the 1976 book form of ‘Proofs and refutations’ (op. cit. (3)), not the version that he has edited for this volume. I had been told by Lakatos's colleagues at the London School of Economics—in error but doubtless in good faith—that the version of ‘Proofs and refutations’ published in 1963 formed the entirety of his thesis. My own interpretation of Cauchy's theorem was contained in my own thesis, which I applied to write under Lakatos's supervision at the School in 1966 (at exactly the time, I now discover, when he wrote the manuscript under review). My application was rejected, and passed over to the School's new Department of Mathematics, where I was supervised by Professor Offord, A. C. and DrRavetz, J. R. (of Leeds University). It was defended in 1969.Google Scholar I had no contact with Lakatos after 1966. In particular, I received no reply to a letter of 26 May 1971, six months after the publication of my work, rejecting the charges of plagiarism which had by then percolated down to me through mutual acquaintances—and continued to do so for more than two years, although at no time were they addressed to me directly.

⁸ For these and other criticisms of Kuhn see the articles by Masterman, M. and Popper, K. R. in Lakatos, I. and Musgrave, A. (eds.), Criticism and the growth of knowledge, Cambridge, 1970, pp. 59–89 and 51–8 respectivelyCrossRefGoogle Scholar; and Stegmüller, W., Structure and dynamics of theories, Berlin, 1976, pp 170–80CrossRefGoogle Scholar. Kuhn responded to some criticisms in the 2nd edn. of his The structure of scientific revolutions, Chicago, 1970Google Scholar; and further in ‘Second thoughts on paradigms’ (1974)Google Scholar in The essential tension, Chicago, 1977, pp. 293–319Google Scholar. Kuhn's defence of his objectivity in chapter XIII of this book seems anaemic to me, since it is concerned only with the criteria under which scientists take decisions. Thus—once again—he talks mediodology rather than epistemology. For a brief but enlightening survey of the history of methodology in science, see Whitrow, G. J., ‘An analysis of the evolution of scientific method’, L'âge de la science, 1970, 3, 255–80.Google Scholar

⁹ Until the mid-eighteenth century Newtonian mechanics faced substantial rivalry from vortex theories (see Aiton, E. J., The vortex theory of planetary motion, London & New York, 1972)Google Scholar. Although Lakatos claims various examples of the power of his historiography in the history of astronomy, he gives this one only a slight mention (1, 217). Another area of rivalry in which Newtonianism largely won by the mid-eighteenth century was Newton's conception of matter as opposed to Leibniz's.

¹⁰ Popper, K. R., ‘Replies to my critics’, in Schilpp, P. A. (ed.), The philosophy of Karl Popper, La Salle, Illinois, 1974, pp. 961–1117 (1007).Google Scholar

¹¹ Kuhn, T. S., ‘Mathematical versus experimental traditions in the development of physical science’ (1976)Google Scholar, in Essential tension, op. cit. (8), pp. 31–66Google Scholar. In footnote 10 of his ‘Second thoughts on paradigms’ (ibid.) Kuhn uses the unfortunate expression ‘law sketches’ to describe Hamiltonian and Lagrangian principles in mechanics, since the particular function needs specification for a given problem. The phrase ‘law schemata’ more precisely conveys the idea, and also avoids the sense of incompleteness that ‘sketches’ conveys.

¹² Fourier himself exemplifies the hope of merging mathematics and experimentation in the content of heat diffusion, although the idea was not original with him (see, for example, his comments quoted in my and Ravetz's, J. R.Joseph Fourier 1768–1830, Cambridge, Mass., 1972, p. 35Google Scholar). His experimental work shows the disappointing results of comparison (ibid., chapter XX). In addition, he is a fine example of a de-simplifier of a theory (ibid., chapters IV–XIX). A more detailed summary of early nineteenth-century developments is given in my ‘Mathematical physics in France, 1800–1840: knowledge, activity and historiography’, Historia mathematica (forthcoming).

¹³ Robinson himself held the same mistaken view of the bearing on history of his new theory. On 15 November 1965, while his book containing his new theory of infinitesimals was in proof, he gave a lecture at Popper's seminar at the London School of Economics on the history of the foundations of the calculus and its ‘correction’ by his theory. The book was published as Non-standard analysis, Amsterdam, 1966.Google Scholar

¹⁴ Wood, P., review of Lakatos's Philosophical papers, Annals of science, 1979, 36, 289–98Google Scholar. To be fair to Lakatos, according to his editors he regarded this essay ‘as in need of substantial revision’ (i, 193)Google Scholar; but the editors published it just the same, and also several other essays that he felt to be unready. They also omitted from i, 8 an important question mark from the title of a section.

¹⁵ See my ‘Notes on some methodological problems in the history of science’, Rete, 1971–1972, 1, 1–14.Google Scholar

¹⁶ For an example—Newton's conceptions in astronomy—contrast Popper's ‘The aim of science’ (1957)Google Scholar, in Objective knowledge, Oxford, 1972, Chapter VGoogle Scholar, with Whiteside, D. T., ‘Before the Principia: the maturing of Newton's thoughts on dynamical astronomy, 1664–1684’, Journal for the history of astronomy, 1970, 1, 5–19.CrossRefGoogle Scholar

¹⁷ The place in science of conceptual questions of this kind is welcomely stressed in Laudan, L., Progress and its problems, Berkeley, Los Angeles, & London, 1977Google Scholar. See also his criticisms of Lakatos, 's methodology, especially on pp. 76–88, 168–70, 232–4.Google Scholar

¹⁸ For Popper's views on metaphysical research programmes, see his Unended quest, London, 1976, sections 33 and 37Google Scholar. This book is a reprint of Popper's autobiographical introduction to Schilpp, op. cit. (10).

¹⁹ Popper, K. R., The logic of scientific discovery, London, 1959Google Scholar; the quotations are taken successively from pp. 42, 83 and 273, and all occur in the original 1934 German text of which this book is Popper's own translation.

²⁰ For a survey of views on the Duhem-Quine thesis, see Harding, S. G. (ed.), Can theories be refuted?, Dordrecht, 1976.CrossRefGoogle Scholar In this connexion I may mention that Lakatos's claim that ‘theories with a ceteris paribus clause, have no empirical basis’ (i, 18) must be wrong; the exploration of the referents of such a clause for sources of hitherto overlooked but possibly relevant auxiliary hypodieses is as empirical an activity as one could imagine.

²¹ See, for example, Popper, , op. cit. (19), pp. 53–4, 108–11.Google Scholar

²² Popper, , op. cit. (10), pp. 1004–13Google Scholar. On p. 1009 he says that the falsification of a theory is a ‘matter of logic’ while its rejection ‘is a question of methodology’. However, on the next page he says that his theory of falsifiability is ‘methodological or philosophical’. This puts it in the same category as the rejection of a theory, whereas a sharp distinction between falsification and rejection has just been made. I take Popper's use of ‘methodological’ on p. 1010 to be a slip of the pen.

²³ For a survey of past and especially current theories and definitions of truth, see Haack, S., Philosophy of logics, Cambridge, 1978, chapter VIICrossRefGoogle Scholar. I shall not distinguish here between a sentence and a proposition.

²⁴ On Tarski's definition of truth and Popper's use of it, see Haack, S., ‘“Is it true what they say about Tarski?”’, Philosophy, 1976, 51, 323–36.CrossRefGoogle Scholar

²⁵ My schema bears some relation to the table of ontological distinctions proposed in Bhaskar, R.'s difficult but important book A realist theory of science, 2nd edn., Hassocks, Sussex, 1978, pp. 56–62 (see also pp. 13–15, 46–8).Google Scholar For a bewailing of scientists' lack of interest in ontology, see Thom, R., ‘La science malgré tout…’, in Encyclopaedia universalis, vol. XVII, Organum, Paris, 1968, pp. 5–10Google Scholar; but a more optimistic view, especially in the context of cosmology, is advocated in Ruyer, R., La gnose de Princeton. Des savants à la recherche d'une religion, Paris, 1974.Google Scholar

²⁶ On Popper's World 3, see his op. cit. (16), especially chapters III and IV; and his chapters in Popper, and Eccles, J. C., The self and its brain, New York, 1977CrossRefGoogle Scholar. See also the review of the latter volume by Place, U. in Annals of science, 1979, 36, 403–8.Google Scholar

²⁷ See Ruse, M., ‘Karl Popper's philosophy of biology’, Philosophy of science, 1977, 44, 638–61.CrossRefGoogle Scholar

²⁸ For Lakatos's acceptance of Popper's World 3, see especially i, 110, 119; and ii, 108–9, 226–35. The curious nature of World 3, and also the difficulties attending Lakatos's mediodology, are noted in Skolimowski, H., ‘Evolutionary epistemology’, in Cohen, R. S. et al. (eds.), PSA 1974. Proceedings of the 1974 Biennial Meeting of the Philosophy of Science Association, Dordrecht, 1976, pp. 191–213.Google Scholar

²⁹ In his Peirce's philosophy of science, Notre Dame, 1978, pp. 27–30Google Scholar, N. Rescher rejects the analogy with geographical exploration as ‘fundamentally mistaken’ because the analogy assumes that ‘science progresses by cumulative accretion’ and ‘the magnitude of these additions is steadily decreasing’ (p. 29). My use of the analogy makes no use of these assumptions, which I do not accept, so do not find Rescher's grounds for rejecting it to be conclusive. See also note 36 below.

³⁰ See i, 141, quoting Popper, , op. cit. (19), p. 53Google Scholar. For Popper's criticisms of Lakatos see his op. cit. (22). For strong criticisms of Popper's eschewance of cognitive aspects of scientific theorizing see Rescher, ibid., especially pp. 51–7.

³¹ See the articles by Tichý, P., Harris, J. R., and Miller, D. in British journal of the philosophy of science, 1974, 25, 155–88CrossRefGoogle Scholar. A useful survey of the problem is given by Andersson, G., ‘The problem of verisimilitude’, in Radnitzky, G. and Andersson, G. (eds.), Progress and rationality in science, Dordrecht, 1978, pp. 291–310.CrossRefGoogle Scholar

³² Popper, , Knowledge, op. cit. (16), p. 55.Google Scholar

³³ N. Koertge offers an interesting defence of probabilistic considerations against both Popper and Lakatos, and an appealing contrast between the two, in ‘Towards a new theory of scientific enquiry’, in Radnitzky, and Andersson, , op. cit. (31), pp. 253–78.Google Scholar

³⁴ See Popper, K. R., Conjectures and refutations, London, 1963, especially pp. 1–3Google Scholar for the synonymy of ‘pseudo-science’ and ‘metaphysics’ and the status of astrology. For Lakatos, 's agreement with Popper on astrology see for example, ii, 227.Google Scholar

³⁵ Kuhn makes a similar defence of astrology in his ‘Logic of discovery or psychology of research?’ (1974)Google Scholar, in his Essential tension, op. cit. (8), p. 274.Google Scholar A clear account of the basic rules of astrology is given in Carter, C. E. O., The principles of astrology, 5th edn., London, 1963Google Scholar. Information on Hindu astrology is contained in, for example, Oken, A., Astrology: evolution and revolution, New York, 1976Google Scholar. For Ebertin's approach, see his Applied cosmobiology, Aalen, 1972Google Scholar. Reports on the current state of research into the birth chart, the testing of astrological theories, and the gullibility of the public, form sections of Dean, G. and Mather, A., (comps.), Recent advances in natal astrology. A critical review 1900–1976, London, 1977.Google Scholar

³⁶ Bertrand Russell once even drew on the geographical analogy which I used in sub-section 7.4 with regard to mathematical objects: ‘Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians’ (The principles of mathematics, 2nd edn., London, 1937, p. 451Google Scholar). Recent discussions of the problem include: Smith, B., ‘The ontogenesis of mathematical objects’, Journal of the British Society for Phenomenology, 1975, 6, 91–101CrossRefGoogle Scholar; Kitcher, P., ‘The plight of the Platonist’, Noûs (forthcoming)Google Scholar; and Thom, R., ‘Les mathématiques et l'intelligible’, Dialectica, 1975. 29, 71–80CrossRefGoogle Scholar. Lakatos, 's ‘A renaissance of empiricism in the recent philosophy of mathematics?’ of 1967 (ii, chapter II) is more concerned more with proof-methods in mathematics than with the discovery of objects.Google Scholar

³⁷ This question is raised for moral discovery in the best short introduction to Popper's philosophy that I know: Sharratt, M., ‘Introducing Karl Popper’, The clergy review, 1974, 59, 390–400, 467–80, 543–55Google Scholar. It is discussed with insight in Pirsig, R.'s deservedly popular Zen and the art of motorcycle maintenance, New York, 1974Google Scholar. It ought to be prominent also in Ward, B. N., What's wrong with economics?, London, 1972CrossRefGoogle Scholar, for Ward compares science with economics. But in taking Kuhn as his main guide to science, Ward unfortunately highlights puzzle-solving as the principal scientific activity, and thus largely side-steps epistemological questions—such as the possibility of a logic of testing economic theories.

³⁸ On the logical difficulties concerning self-referential criticism, see Watkins, J. W. N., ‘CCR: a refutation’, Philosophy, 1971, 46, 56–61CrossRefGoogle Scholar. ‘CCR’ stands for ‘comprehensive critical rationalism’.

³⁹ For example, while impressed by Laudan's advocacy of a historiography based on research traditions (op. cit. (17)), I am sceptical of his claims for its characterization of rationality in science.