Abstract
AXIOMS, says Proclus,2 are common to all sciences, though each employs them in its peculiar subject-matter. A little further on3 he cites Aristotle4 as saying that one science is more certain than another, viz. that which emanates from more simple suppositions than that which uses more various principles; and that which tells the why, than that which tells only the simple existence of a thing; and that which is conversant about intelligibles, than that which touches and is employed about sensibles.
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References
Presidential Address delivered by Sir James Cockle, F.R.S., to the London Mathematical Society, on November 8, 1888.
Proclus, "Commentaries on the First Book of Euclid's Elements" (Taylor's Translation, London, 1792), p. 92.
Proclus, op. cit., p. 93.
Taylor (ib. p. 93) supplies the reference to the first Analytics, t. 42.
Proclus, op. cit., p. 79. Hume ("Treatise," vol. i., London, 1739, Book i. Part 3, p. 129, et vid. p. 128) says that geometry falls short of that perfect precision and certainty which are peculiar to arithmetic and algebra.
Proclus, op. cit., p. 93; et vid. pp. 78, 79.
Bacon, "The Proficience and Advancement of Learning" (Oxford, 1633), pp. 49, 50.
Bacon, op. cit., pp. 151, 152; et vid. pp. 119, 120.
Bacon, op. cit., pp. 152, 205, and 231.
Bacon, op. cit., p. 218; et vid. pp. 150, 151.
Bacon, op. cit., pp. 130, 131, 140, and 201; et vid. p. 161.
Bacon, op. cit., pp. 49, 50; et vid. pp. 130, 131, and 140.
Bacon, op. cit., p. 141.
Bacon, op. cit., pp. 130, 140.
Bacon, op. cit., p. 138; conf. pp. 146, 147.
Bacon, op. cit., pp. 150, 151.
Proclus (Taylor's Translation), p. 74.
For this summary of Aristotle's views I am indebted to Mr. Reginald H. Roe, who referred me to Ueberweg' "Hist. of Phil.," p. 164, for a more general statement, and to p. 165 for a list of the best books for its fuller elucidation, adding that in Ritte and Preller's extracts, pp. 288 and 289, will be found all the important passages from Aristotle bearing on the question. As to the views of Boole, see his "Laws of Thought" (London, 1854), pp. 162 et seq.; see also p. 419. Boole treats of space at pp. 163, 175, and 418; and at p. 175 he quotes Aristotle's statements respecting the existence of space in three dimensions.
Newton, "Fluxions," pp. 26 and 38 of the small edition (London, 1737). This is a genuine work of Newton's. As to its bibliography, see Notes and Queries, 2nd S., vol. x. pp. 163, 232, 233; 3rd S., vol. xi. pp. 514, 515; 4th S., vol ii. p. 316; 5th S., vol. iv. p. 401; 6th S., vol. iv. pp. 129, 130; vol. v. pp. 263, 264, 304, 305, and 426. This octavo edition is very scarce. Indeed, I only know of two copies, viz. my own copy and one in the library of the Royal Astronomical Society.
De Morgan, "On the Foundation of Algebra," Cambridge Transactions, vol. vii. pp. 173–87; see pp. 175, 176. The remarks of Prof. Cayley on Whewell, at p. 18 of his Southport Address, are applicable to Rowan Hamilton .
Newman, "Lectures on Logic, or on the Science of Evidence," &c. (Oxford, 1838), p. 15.
Newman, op. cit., pp. 32–34.
Boole, "An Investigation of the Laws of Thought" (London, 1854), pp. 162 et seq.
See the account of the "Novum Organon" in the "Library of Useful-Knowledge," p. 10.
Bacon, "Advancement of Learning" (cited supra), p. 193.
Bacon . op. cit., p. 132.
T. S. Davies, Geometrical Notes, Mechanics' Magazine, vol. liii. (1850), pp. 150, 169, 262, 291, 442. Davies points out "the connection between parallels and similar triangles." He thinks that Aristotle's Secession from the school of Plato arose from his enforcement of his own logical doctrines. Davies rejects the notion of a geometry build upon definitions along without the assistance of axioms.
Potts (Robert), "Euclid's Elements of Geometry," &c. (Cambridge and London, 1845); Notes to Book i., p. 41.
Thomas Reid . "An Inquiry into the Human Mind on the Principle of Common Sense" (1764). My pagings refer to the Calcutta Reprint of 1869. Chapter vi. treats (pp. 94–277) of seeing; its Section vii. (pp. 120–24), of Visible Figure and Extension; and its Section ix. (pp. 132–45), of the Geometry of Visibles. In Section viii (pp. 125–35), we have Some Queries concerning Visible Figure answered.
Montucla, "Histoire" (2de édition, An. vii.), p. 21.
See Cayley, Southport Address, pp. 11, 12.
See Bacon, "Advancement of Learning," p. 108.
Kant, "Prolegomena," p. 52.
William Kingdon Cliff rd, "Mathematical Papers" (London, 1882). See pp. xl and xlii. of the Introduction, by H. J. S. Smith .
De Morgan, "On the General Principles of which the Composition of Aggregation of Forces is a Consequence" (Camb. Trans., vol. x., part 2, pp. 294, 295, footnote).
I should have been glad to have given Locke's and Kant's descriptions of space and time, and to have compared them with Newton's. But I cannot omit to refer to a Smith's Prize paper, by Mr. Robert Franklin Muirhead, printed in the Philosophical Magazine for June 1887, S. 5, vol. xxiii. pp. 473–89.
Bacon, "Advancement of Learning," p. 47.
Clifford, "The Universal Statements of Arithmetic" Nineteenth Century (1879, vol v. pp. 513–22; vide p. 22).
A paper in Mind, by Prof. von Helmoholtz, elicited a criticism from Prof. Land, which produced a reply; and with a brief not appended to a peper on another subject, by Prof. Land, the discussion closed. See Mind, vol. i. pp. 301–21; vol ii. pp. 38–46; vol. iii. pp. 212–25 and 551–55; and vol. iv. pp. 591–96.
De Morgan, "On Infinity," &c. (Camb. Trans., vol. xi. Part 1, 1865, pp. 145–89; vid. pp. 173, 176, 180, 147). In connection with this paper, the comments of Mr. W. S. B. Woolhouse in the Educational Times (Reprint, vol. vi. pp. 49–52) should be considered. And in connection with a paragraph at pp. 161, 162, of De Morgan's paper, the leading paragraph of p. 424 of a previous paper of his, "On the Theory of Errors of Observation" (C. T. vol. x. Pt. 2, 1862), should be read. In the last-mentioned passage he distinguishes between the zero and the indivisible of probability. Hamilton, of Edinburgh, following earlier authorities, expressly restricts the application of logic to finite things. But it does not therefore follow that logicians in general turn of deaf ear to all reasoning upon infinities and infinitesimals, and that they reject results stamped with authority and universally accepted.
This sufficiently appears from a statement at p. 15 of his paper, "On the Root," &c. (Camb. Trans., vol. xi. Pt. 2).
Bacon, "Advancement of Learning," p. 143; conf. pp. 130, 140. See also pp. 192, 209.
My pagings refer to the 2nd ed. of the "Mécanique Celeste," vol. i. (Paris, 1829).
Boole, C. and D. M. J., vol. vi, p. 286; "L. of T.," pp 321–26; Phil. Mag., S. 4, vol. vii. pp. 29–32; vol xxiii., pp. 361–63; Wilbraham, Phil. Mag., S. 4. vol. vii. pp. 465–76; Cayley, Phil. Mag., S. 4, vol vi. p. 259; S. 4, vol. xxiii. pp. 352–65, and 470. A short letter by Boole (Phil. Mag., S. 4, vol xxiv. (1862), p. 80, concludes the discussion.
Glanvil (Joseph), "Scepsis Scientifica," &c. (Lond. 1665, 4to); Lond., 1885, 8vo. On Causation, I have only mentioned comparatively recent authors. But, going further back, we find Thales (with his elemental analysis), Xenophanes (with his one cosmic substance), and Pythagoras (with his arithmetical and geometrical combinations), all recognizing invariable sequences in nature; and Socrates admitted a class of phenomena wherein the connection of antecedent and consequent was invariable and ascertainable by human study (Grote, "History of Greece," vol. i., 1846, pp. 495–98). Socrates applied similar scientific reasonings to moral and social phenomena (ib., p. 504).
David Hume, "A Treatise of Human Nature," &c. (Lond., vols. i. ii., 1740: his name does not appear on the title-pages). "Philosophical Essarys concerning Human Understnading" (2nd ed., Lond., 1750.) "An Inquiry concerning Human Understanding" (Lond., 1861) marks the issue to which I refer.
Thomas Brown, "Inquiry into the Relation of Cause and Effect" (3rd ed., Edinburgh, 1818). Draper does not admit the construction put upon Algazel's words by Whewell ("Hist. Ind. Sc.," Lond, 1837, i. p. 251). A facsimile reprint of Glanvil has been published within the last few years. Buckle pronounced Brown's to be one of the best books ever written.
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On the Confluences and Bifurcations of Certain Theories 1 . Nature 39, 521–523 (1889). https://doi.org/10.1038/039521a0
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DOI: https://doi.org/10.1038/039521a0