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Perspectives victoriennes

Experience and Universal Judgements: William Kingdon Clifford and Non-Euclidean Geometries

Expérience et jugements universels : William Kingdon Clifford et les géométries non-euclidiennes
Christophe Duvey
p. 490-502

Résumé

La pensée de W. K. Clifford se distingue par un questionnement radical de la valeur de la connaissance humaine. Fondée sur une doctrine exclusivement empiriste, elle n’adhère cependant pas pleinement aux conceptions de l’expérience alors défendues par des philosophes comme J. S. Mill ou H. Spencer. Récusant l’apriorisme kantien ainsi que toute proposition universelle et nécessaire, Clifford défend un relativisme, dont l’origine est fortement marquée par son étude des mathématiques, et notamment par son enthousiasme pour les géométries non-euclidiennes alors en plein essor. C’est précisément cet intérêt qui fait de ce mathématicien philosophe l’un des penseurs les plus originaux de la période victorienne.

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Texte intégral

  • 1  James R. Newman, “William Kingdon Clifford,” Scientific American, (188), February, 1953, 78.
  • 2  It should be noted that the valuable remarks made by Bernard Lightman in his book on The Origins o (...)
  • 3  There is a ruthless caricature of Clifford in W. H. Mallock’s The New Republic, or Culture, Faith, (...)
  • 4  He wrote that “I can find no evidence that seriously militates against the rule that the priest is (...)
  • 5  “The dim and shadowy outlines of the superhuman deity fade slowly away from before us; and as the (...)
  • 6  Quoted in B. Lightman,  op. cit., 93 and 95.
  • 7  Matthew Arnold, God and the Bible, New York: Macmillan and Co., 1883, xiv.
  • 8  A. O. J. Cockshut, The Unbelievers, English Agnostic Thought, London: Collins, 1964, 67-8, 122-3. (...)

1In an article published in the magazine Scientific American in 1953, James R. Newman observed that “[t]he neglect of Clifford is difficult to explain.”1 Since then, and apart from scattered references here and there,2 few, if any, studies have been devoted to his thought per se. Clifford’s reputation is more often linked to his provocative writings on ethics and religion, or to his impressive scientific achievement in the field of mathematics, than to anything else.3 The vigour of his anticlericalism4 and, at times, his almost Nietzschean vein,5 certainly strike the modern reader as much as they did his Victorian contemporaries. If William James spoke of him as “that delicious enfant terrible,” and T. H. Huxley as “the finest scientific mind born in England for fifty years,”6 Matthew Arnold deemed Clifford’s historical judgement on religion to be “[…] merely the crackling fireworks of youthful paradox,”7 while others go so far as to take him as “a dogmatic atheist.”8 Yet an examination of his works should not be restricted to these more conspicuous aspects, even though they must obviously not be underestimated. Besides, one may wonder if such an emphasis put on religion and science in his case does not tend, as is often the case, to hide the real significance of the controversies of the period, which should be more appropriately analysed from a philosophical viewpoint rather than interpreted as a so-called clash of “religious” and “scientific” conceptions of the world.

2Confronted with Clifford’s wide range of interests as regards intellectual matters, it is obviously impossible to offer a comprehensive view of his published material in a single article. Therefore, rather than aiming at a general survey of the whole of his thinking, the following pages will be restricted to a study of Clifford’s ideas on the nature of universal judgements in their relationship with experience. For, even though Clifford’s epistemology can be broadly defined as naturalistic, it certainly ranks as one of the most original doctrines of the Victorian era, especially as regards his enthusiasm for non-Euclidean geometries, which were then being devised by the most prominent mathematicians of the period. The consequences he drew from them in favour of a radical empiricism should thus be closely examined.

Clifford on experience

  • 9  W. K. Clifford, “The Philosophy of the Pure Sciences,” LE, vol. 1, 308. Clifford explains that “[i (...)
  • 10  “Whereas, therefore, I think I see a solid scene, having depth, and relief, and distance in it, re (...)
  • 11 Ibid, 304.

3Clifford contends that a sensation cannot be taken as representing an elementary entity, but that it is really composed of two distinct elements: “In every sensation,” he writes, “there is besides the actual message, something that we imagine and add to the message.”9 Whenever there is a perception, one must distinguish what is immediately experienced – “the actual message” – and a supplement which cannot be said to belong to it. Among the examples given by Clifford, there is the perception of a theatre. The contents of the message are mainly obtained through the sense of sight, and physiological studies show that the image captured by both retinas does not exactly correspond to the perception of the theatre. For, while the retinal images are two-dimensional and, as such, do not give a representation of solidity, the perception of the theatre consists of three dimensions, the theatre itself looking solid.10 According to Clifford, we are therefore compelled to conclude that from the original message, we “must have imagined the rest.”11 We must now concentrate on the properties and source of this supplement.

4The first peculiarity to be noted by Clifford is that men add to their experience in keeping with definite principles or “rules”, which can be grouped in three categories according to their characteristics:

  • 12 Ibid., 315.

The rules about Space and Motion constitute the pure science of Geometry and Kinematic.
The rules about Things and Uniformity have been said to belong to a pure science of Nature.
The rules about Numbers and Classes constitute the pure sciences of Arithmetic and Formal Logic.12

  • 13 Ibid., 316.
  • 14 Ibidem.

5The second peculiarity of the complement to the message pertains to its accuracy : “[…] in general this filling-in of experience is right: and, […] so far as these rules are concerned, it is not only right in general, but always right.”13 This means that when particular perceptions are predicted from distinct sensations, we observe that it is these adequate rules which enable us to do this correctly. Clifford illustrates his point with an example: when thinking of a geometrical figure such as a parallelogram, its opposite sides must be conceived of as equal in length. Accordingly, if such a figure is drawn on a sheet of paper and folded so as to put one of its sides on the opposite one, no difference will be found, and the conclusion that in a parallelogram the opposite sides are equal will be confirmed. In other words, “[t]he rule by which I supplement my perception is also a true statement about objects; it is capable of a certain kind of verification, and it always stands this test.”14

  • 15  “With Euclid’s system elevated to an immobile, transcendental heaven, it was thought, not merely t (...)

6If the rules of space which make up the laws of geometry are more closely examined, it can be said that they have long been believed to be “theoretically” true, in Clifford’s meaning of the word, i.e., absolutely and universally exact. Under the influence of Euclid, geometry was generally deemed to be a pure and applied science, describing the nature of space with theoretical correctness.15 In his introduction to Clifford’s The Common Sense of the Exact Sciences, J. R. Newman argues that this state of affairs could be deplored on two counts:

  • 16 Ibid., xxxiii.

First, as a description of actual space Euclidean geometry could not pretend to universality: its postulates and theorems having been tested only in a most limited range might not be valid outside that range – in the domain of the very small or of the very large. Second, by turning Euclidean postulates into commandments, the freedom of mathematical inquiry was more effectively throttled than by ecclesiastical ban. For there appeared to be no possibility of constructing new geometries based upon non-Euclidean postulates, such postulates being, clearly, “contrary to nature.” Any discussion of space not known to the senses was thus forbidden.16

7It is not surprising, then, that philosophers came to wonder about the origin of what was then supposed to be theoretical knowledge. Among the existing schools trying to give an acceptable account of it, two are especially selected by Clifford: that maintaining the empirical nature of all knowledge (and more particularly J. S. Mill and H. Spencer), and Kant’s Critical philosophy.

Clifford’s criticisms of Mill, Spencer and Kant

8The main thesis the various empiricist doctrines share relates to the source of the supplement added to the message immediately experienced to form a perception. Following the fathers of empiricism, those who held such doctrines maintain that it arises solely from experience coupled with a process of association and reinforced by habit. Clifford writes:

  • 17  W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 319.

According to these explanations of Locke and Hume […] all the knowledge we have that the rules are right, or may be objectively verified, is really derived from experience; only it is just experience, which we have had so often and got so accustomed to that it is now really a part of ourselves.17

9Mill’s contention in this matter consists in asserting that the distinctive attributes of geometrical figures, for example, are the result of experience. He declares that

  • 18  J. S. Mill, A System of Logic, Ratiocinative and Inductive, London: Parker, Son and Bourn, 5th ed. (...)

 […] we should not be authorized to substitute observation of the image in our mind, for observation of the reality, if we had not learnt by long-continued experience that the properties of the reality are faithfully represented in the image.18

10The mental image which is formed of a geometrical figure and its attributes can be said to correspond truly to, and replace accurately, an observed reality, only because men have slowly learned that experience certifies the legitimacy of the congruence,

  • 19  J. S. Mill, Ibid.

[…] just as we should be scientifically warranted in describing an animal which we had never seen, from a picture made of it with a daguerreotype; but not until we had learnt by ample experience, that observation of such a picture is precisely equivalent to observation of the original.19

11Yet Clifford thinks that Mill’s explanation is acceptable only if restricted to an observed order of facts, but that it becomes pointless as soon as anything transcending experience is to be supposed:

  • 20  W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 330 (original emphasis).

I must know both of two things to know that one faithfully represents the other. Experience then tells me that my mental images of geometrical figures are faithful representations of those realities which are within the bounds of experience. But what is to tell me that they are faithful representations of realities that are beyond the bounds of experience? Surely no experience can tell me that.20

  • 21  “It seems to me, then, that in admitting the universality of certain statements Mr. Mill knows som (...)

12Clifford believes that Mill is not allowed to recognise, as he seems to do, the universal necessity of any proposition (defined as the inconceivability of their negation), because the criteria he defends are confined to a limited experience.21 It should be noticed that, in Clifford’s opinion, Mill’s supposed inconsistency does not lie so much in his assertion that knowledge is wholly dependent on experience as in his belief in the universal truth of certain propositions.

  • 22  “The universal law that, other things equal, the cohesion of psychical states is proportionate to (...)

13Clifford’s criticism of Herbert Spencer rests on similar grounds. Spencer also held that all knowledge was experience-based. He added, however, that this experience should not be limited to individuals, as Mill and others supposed, but should include that of all the ancestors who had gone before them. By a process of hereditary descent, they pass on, as it were, their various psychical characteristics to their descendants, who in their turn transmit them slightly modified to their offspring (through natural selection). Thanks to this explanation, which combined both a theory of association and heredity, Spencer hoped to have reconciled the doctrine of experience, upheld by philosophers like Mill or Bain, with the Kantian theory of the “forms of thought.”22 These “forms of thought” become a product of an evolutionary history, i.e., they can be empirically accounted for, having steadily been ingrained in the structure of the brain of modern man by a long chain of experiences:

  • 23  H. Spencer, op. cit., vol. 1, 470-1 ; also quoted in W. K. Clifford, op. cit., 334.

But these pre-determined internal relations, though independent of the experiences of the individual, are not independent of experiences in general; they have been determined by the experiences of preceding organisms. The corollary here drawn from the general argument is, that the human brain is an organized register of infinitely-numerous experiences received during the evolution of life, or rather, during the evolution of that series of organisms through which the human organism has been reached. The effects of the most uniform and frequent of these experiences have been successively bequeathed, principal and interest; and have slowly amounted to that high intelligence which lies latent in the brain of the infant – which the infant in after life exercises and perhaps strengthens or further complicates – and which, with minute additions, it bequeaths to future generations.23

14Clifford criticises Spencer’s theory for the same reason he did Mill’s. It seemed to him that Spencer’s analysis left Kant’s interpretation just where it had been before, since he could certainly not profess to know more than what there was in experience, that is to say, the explanation did not give a satisfactory account of a certain kind of knowledge which claimed to be absolutely and universally (i.e., “theoretically”, and not only “practically”) true:

  • 24  W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 335.

How am I to know that the angles of a triangle are exactly equal to two right angles under all possible circumstances; not only in those regions of space where the solar system has been, but everywhere else? The accumulated experience of all my ancestors for a hundred and fifty million years is no more competent to tell me that than my own experience of the last five minutes.24

15Yet again, it should be observed that Clifford’s commentary does not controvert the assertion that knowledge is grounded on experience only, but the argument that the existence of universal statements can be sustained from such a hypothesis.

  • 25  “But if a man felt absolutely sure that two straight lines perpendicular to the same line would ne (...)
  • 26 Ibid., 325-6.
  • 27  “Now the doctrine of evolution itself forbids me to admit any transcendental source of knowledge; (...)

16This is why, Clifford remarks, Kant judged that such knowledge could not have experience as its source, and the reasoning of the transcendental philosopher seemed quite unanswerable at the time he formulated it. In his opinion, the cause is to be found in the firmness of the belief in the universal and necessary truth of the axioms of mathematics and geometry up to then. If such was really their nature, mere experience could not provide an adequate explanation of the origin of the conviction. The absolute quality of the knowledge exceeded anything empirical circumstances could attain to, hence Kant’s opinion that it preceded experience, that it was a priori.25 Nevertheless, Clifford thinks that, even if the issue raised by the Critical philosophy was crucial, notably as regards the distinction between the thing-in-itself and the subjective supplement added to every perception, the way Kant tried to solve it was unsatisfactory, because he took the existence of necessary and universal truths for granted, showing they were necessarily part of the mind of the subject. But, as Clifford points out, “[i]t requires, therefore, some infallible way of judging what characters are necessary and universal,” and it is precisely that which seems doubtful to him, because “judgments of this kind may very possibly be mistaken.”26 A simple alternative presents itself then: if the existence of universal and necessary truths is maintained, we are compelled to follow Kant and uphold that experience is not enough to warrant them, and consequently, that a part of man’s knowledge must be derived from another source, i.e., be a priori; or it may be argued that we are misguided when we think that universal truths are known at all. Since Clifford resolutely supports the empirical character of the hypothesis of evolution, he is led to rule out Kant’s transcendental position and to assert that a conviction about the universal quality of any proposition is baseless.27

  • 28  “Clifford made Kant’s transcendental deduction of space and time stand or fall with Euclidean geom (...)

17In the part devoted to the “Transcendental Aesthetic” in his Critique of Pure Reason, Kant made use of geometrical thought to hold that space was a necessary form of intuition. It is understandable that Clifford should have opposed the Kantian philosophy on this, because, as a mathematician, his study of non-Euclidean geometries brought him to the conclusion that Euclidean postulates did not have a universal validity.28

Clifford and the non-Euclidean geometries

  • 29  J. R. Newman, in W. K. Clifford, The Common Sense of the Exact Sciences, xxiii.
  • 30  Riemann’s work was posthumously published in 1867 and translated by Clifford in 1873. Cf. W. K. Cl (...)

18Clifford was an “early and ardent disciple of the new doctrines,”29 and was one of the first in England to align himself with the great nineteenth-century mathematicians Nikolai I. Lobatchevsky (1793-1856) and Georg F. B. Riemann (1826-1866). Lobatchevsky began in the 1820s to question Euclid’s fifth postulate (on parallels), and he built a non-Euclidean geometry, assuming that more than one parallel passed through a point lying outside a straight line, then forming a geometry in which the sum of the angles between the sides of a triangle was inferior to two right angles. In 1854, Riemann finalised his celebrated work, later translated into English by Clifford, “On the Hypotheses which Lie at the Bases of Geometry,” in which he aimed at “constructing the notion of a multiply extended magnitude out of general notions of magnitude,” and showing that “a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude.”30 He goes on:

  • 31  G. F. B. Riemann, translated by W. K. Clifford, in Mathematical Papers, op. cit., 56; my italics.

But hence flows a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but that the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience.31

  • 32 Ibidem. Clifford fully endorsed Riemann’s views, as the following passage shows : “And the conclusi (...)

19It follows that Riemann was now in a position to look for “the simplest matters of fact from which the measure-relations of space may be determined,” but the problem was complicated by the fact that there was more than one system which could adequately establish “the measure-relations of space,” the most famous being, of course, Euclid’s. It remains that these “matters of fact” cannot be said to be absolutely necessary and are only founded on experience; they are mere “hypotheses,” and even if Riemann admits that their probability is “very great” “within the limits of observation,” their worth becomes quite uncertain on the side of the “infinitely great” as well as on that of the “infinitely small.”32 Riemann created a different non-Euclidean geometry from Lobatchevsky’s. He developed an elliptical geometry, postulating that lines always meet, therefore rendering parallels impossible, and in which the sum of the angles between the sides of a triangle is superior to two right angles.

  • 33  B. Lightman, op. cit., 162.

20This questioning of the foundations of geometry transformed it either into a mere “formal exercise in logic,” if it was taken to be “a pure science of ideal space,”33 or an empirical science liable to verification, when referring to real spatial relations. In that case, it could not claim to be universally true but only imperfectly or “practically” accurate.

  • 34  “What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobatchewsky [sic] to Euclid (...)

21Such revolutionary thinking in the domain of geometry deeply impressed Clifford, who goes so far as to compare Lobatchevsky to Copernicus.34 He contends that the influence of both thinkers’ work and that of their successors gave rise to a renewed idea of the universe. While, according to him, Copernicus thoroughly modified the image man had formed of the universe in which he lived, showing that he did not possess comprehensive knowledge about it in all its eternity and immensity, his science only extending to “Here and Now”, i.e., that it was limited to the solar system, Lobatchevsky achieved a similar transformation when he showed that the legitimacy of the postulates of space in the Euclidean system did not reach infinite space, as geometers had once thought:

  • 35 Ibid., 358-9. Clifford even writes: “In both of these the knowledge of Immensity and Eternity is re (...)

The geometer of to-day knows nothing about the nature of actually existing space at an infinite distance; he knows nothing about the properties of this present space in a past or a future eternity. He knows, indeed, that the laws assumed by Euclid are true with an accuracy that no direct experiment can approach [i.e., are practically true], not only in this space where we are, but in places at a distance from us that no astronomer has conceived; but he knows this as of Here and Now; beyond his range is a There and Then of which he knows nothing at present, but may ultimately come to know more.35

  • 36  Cf. J. R. Newman, in W. K. Clifford, The Common Sense of the Exact Sciences, op. cit., xxx. On the (...)
  • 37  On the difference between Clifford’s and Einstein’s ideas on the Kantian notion of space, cf. B. L (...)

22Yet, it was mainly Riemann’s work which stimulated Clifford to carry on his geometrical studies.36 The German mathematician’s postulate, which would later be used in the theory of general relativity, helped Clifford to develop ingenious views on matter and space, even anticipating, it has been argued, some of Einstein’s ideas on space curvature in time.37

  • 38  W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 384.
  • 39  W. K. Clifford, “On the Aims and Instruments of Scientific Thought,” LE, vol. 1, 157, my italics.
  • 40 Ibid.
  • 41  B. Lightman, op. cit., 163.

23But what is interesting to notice once more in this particular context is Clifford’s emphasis on the mere “practical” and relative character of scientific laws. He says that Riemann’s efforts in examining the assumptions of geometry were “sufficient to deprive them for the geometer of their exactness and necessity.”38 For Clifford, any scientific law or axiom cannot claim to be universally true in the theoretical sense, for “[…] a law would be theoretically universal if it were true of all cases whatever; and this is what we do not know of any law at all.”39 And the reason Clifford adduces to uphold such limited universality is that “it pays us to assume it.”40 As Bernard Lightman rightly remarks: “Utility is the criterion of verification.”41

Conclusion

  • 42  Obituary notice, in the Athenaeum, quoted in W. K. Clifford, The Common Sense of the Exact Science (...)
  • 43  “His mathematical work was prophetic, and its merit is still untouched after three quarters of a c (...)
  • 44  In W. K. Clifford, The Common Sense of the Exact Sciences, p. ix. As a young man, Bertrand Russell (...)

24The reception of Clifford’s work in mathematics and in the philosophy of science was usually highly praised by his contemporaries and viewed as a lasting contribution by modern writers. Clifford was not uncommonly acclaimed as “[…] the most remarkable mathematician of his generation,”42 and some have stressed the “prophetic” character of parts of his research.43 Bertrand Russell adds that “Clifford was much more than a mathematician: he was a philosopher of considerable merit in what concerned the foundations of mathematical knowledge.”44

25Having argued in favour of the empirical nature of all human knowledge, and having asserted its “practical” character, Clifford even went as far as to challenge the traditional conception of science inherited from Newton, which took the deterministic quality of scientific inferences for granted. In this, he made use of probabilistic theories, such as the atomic theory (developed thanks to J. C. Maxwell’s kinetic theory of gases) and the Darwinian theory of evolution, which he interpreted as resting on statistical calculations.

26Clifford’s epistemology, and particularly his interest in non-Euclidean geometries and his questioning of some fundamental principles of Newtonianism make him one of the prominent thinkers of the Victorian era, even though there is another side of Clifford’s thinking (his idealist monism) which does not loom as large as the rest of his ideas, with his doctrine of “mind-stuff” being either misunderstood or discarded.

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Bibliographie

Arnold Matthew, God and the Bible, New York: Macmillan and Co., 1883.

BENN Alfred W., The History of English Rationalism in the Nineteenth Century, London, New York and Bombay: Longmans, Green, and Co. 1906, vol. 2.

CLIFFORD William K., Lectures and Essays, 2 vol. edited by L. Stephen and Sir F. Pollock, London: Macmillan, [1879] 1901.

------, Mathematical Papers, edited by R. Tucker, [1882], reprinted in 1968, New York: Chelsea Publishing Company.

------, Seeing and Thinking, London: Macmillan, 1879.

------,The Common Sense of the Exact Sciences, originally edited by K. Pearson and newly edited by J. R. Newman, New York: Dover Publications, [1946] 1955.

------, “Problems of Life and Mind,” The Academy, February 7, 1874, 148-150.

COCKSHUT A. O. J., The Unbelievers, English Agnostic Thought, 1840-1890, London: Collins, 1964.

LIGHTMAN Bernard, The Origins of Agnosticism, Victorian Unbelief and the Limits of Knowledge, Baltimore and London: The Johns Hopkins University Press, 1987.

MILL John S., A System of Logic, Ratiocinative and Inductive, London: Parker, Son, and Bourn, 5th ed., 1862, vol. 1.

NEWMAN James R., “William Kingdon Clifford,” Scientific American, (188), February, 1953, 78-83.

SMITH D. E., “Clifford’s Genius Shown as a Boy,” American Mathematical Monthly, (29), 1922, 157-158.

SPENCER Herbert, The Principles of Psychology, London: Williams and Norgate, 2nd ed., vol. 1, 1870.

STEPHEN Leslie, “Clifford, William Kingdon”, Dictionary of National Biography, edited by L. Stephen, London: Smith, Elder and Co., vol. XI, 1887, 82-85.

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Notes

1  James R. Newman, “William Kingdon Clifford,” Scientific American, (188), February, 1953, 78.

2  It should be noted that the valuable remarks made by Bernard Lightman in his book on The Origins of Agnosticism (Baltimore and London: The Johns Hopkins University Press, 1987, x-249 p.) are an important contribution to a better understanding of Clifford.

3  There is a ruthless caricature of Clifford in W. H. Mallock’s The New Republic, or Culture, Faith, and Philosophy in an English Country House (1877), in which Clifford is painted as always disconcerting his fellow agnostics by expressing shocking ideas, notably in the domains of morals and religion. Cf. B. Lightman, op. cit., 161.

4  He wrote that “I can find no evidence that seriously militates against the rule that the priest is at all times and in all places the enemy of all men – Sacerdos semper, ubique, et omnibus inimicus.” W. K. Clifford, “The Ethics of Religion,” in Lectures and Essays (abbr. LE in these notes), London: Macmillan, 1901, vol. 2, 237-8. See also W. K. Clifford, “The Influence upon Morality of a Decline in Religious Belief,” LE, vol. 2, 255-6.

5  “The dim and shadowy outlines of the superhuman deity fade slowly away from before us; and as the mist of his presence floats aside, we perceive with greater and greater clearness the shape of a yet grander and nobler figure – of Him who made all Gods and shall unmake them. From the dim dawn of history, and from the inmost depth of every soul, the face of our father Man looks out upon us with the fire of eternal youth in his eyes, and says, ‘Before Jehovah was, I am !’ ” (W. K. Clifford, “The Ethics of Religion,” LE, vol. 2, 245). Cf. B. Lightman, op. cit., 136.

6  Quoted in B. Lightman,  op. cit., 93 and 95.

7  Matthew Arnold, God and the Bible, New York: Macmillan and Co., 1883, xiv.

8  A. O. J. Cockshut, The Unbelievers, English Agnostic Thought, London: Collins, 1964, 67-8, 122-3. See also p. 123: “Clifford is the extreme point of Victorian atheism.” Bernard Lightman disapproves of these expressions (B. Lightman, op. cit., 195, n. 63). Alfred W. Benn also seems to consider Clifford an atheist (Alfred William Benn, The History of English Rationalism in the Nineteenth Century, London, New York and Bombay: Longmans, Green and Co., 1906, vol. 2, 374 and 388).

9  W. K. Clifford, “The Philosophy of the Pure Sciences,” LE, vol. 1, 308. Clifford explains that “[i]t is convenient to use the word sensation as meaning the whole phenomenon, not only the immediate experience, but also the supplement” (Ibid., 314).

10  “Whereas, therefore, I think I see a solid scene, having depth, and relief, and distance in it, reflection tells me that I see nothing of the kind ; but only (at the most) two distinct surfaces, having no depth and no relief, and only a kind of distance which is quite different from that of the solid figures before me.” (Ibid , 303).

11 Ibid, 304.

12 Ibid., 315.

13 Ibid., 316.

14 Ibidem.

15  “With Euclid’s system elevated to an immobile, transcendental heaven, it was thought, not merely that spatial relations with a limited range are thus and so, as Euclid said, but that they must be thus and so, everywhere and for ever.” J. R. Newman, in W. K. Clifford, The Common Sense of the Exact Sciences, xxxii.

16 Ibid., xxxiii.

17  W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 319.

18  J. S. Mill, A System of Logic, Ratiocinative and Inductive, London: Parker, Son and Bourn, 5th ed., 1862, vol. 1, 265. Also quoted in W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 329.

19  J. S. Mill, Ibid.

20  W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 330 (original emphasis).

21  “It seems to me, then, that in admitting the universality of certain statements Mr. Mill knows something which on his own principles he has no right to know.” Ibid., 331.

22  “The universal law that, other things equal, the cohesion of psychical states is proportionate to the frequency with which they have followed one another in experience, supplies an explanation of the so-called “forms of thought,” as soon as it is supplemented by the law that habitual psychical successions entail some hereditary tendency to such successions, which, under persistent conditions, will become cumulative in generation after generation.” H. Spencer, The Principles of Psychology, London: Williams and Norgate, 2nd ed., vol. 1, 1870, 466; also quoted in W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 332-3.

23  H. Spencer, op. cit., vol. 1, 470-1 ; also quoted in W. K. Clifford, op. cit., 334.

24  W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 335.

25  “But if a man felt absolutely sure that two straight lines perpendicular to the same line would never meet, however far produced, he could not maintain against Kant that all knowledge is derived from experience. He was obliged to admit the existence of knowledge a priori, that is, knowledge lying ready in the mind from the first, antecedent to all experience.” Ibid., 320-1.

26 Ibid., 325-6.

27  “Now the doctrine of evolution itself forbids me to admit any transcendental source of knowledge; so that I am driven to conclude in regard to every apparently universal statement, either that it is not really universal, but a particular statement about my nervous system, about my apparatus of thought; or that I do not know that it is true.” Ibid., 335-6.

28  “Clifford made Kant’s transcendental deduction of space and time stand or fall with Euclidean geometry […] and it was Clifford’s adherence to non-Euclidean geometry which led him to dismiss Kant hastily.” B. Lightman, op. cit., 163; cf. F. Pollock, in W. K. Clifford, LE, 18.

29  J. R. Newman, in W. K. Clifford, The Common Sense of the Exact Sciences, xxiii.

30  Riemann’s work was posthumously published in 1867 and translated by Clifford in 1873. Cf. W. K. Clifford’s Mathematical Papers, edited by R. Tucker, [1882], reprinted in 1968, New York: Chelsea Publishing Company, 55-69 (quotation from pp. 55-6; my italics).

31  G. F. B. Riemann, translated by W. K. Clifford, in Mathematical Papers, op. cit., 56; my italics.

32 Ibidem. Clifford fully endorsed Riemann’s views, as the following passage shows : “And the conclusion to which these investigations lead is that, although the assumptions which were very properly made by the ancient geometers are practically exact – that is to say, more exact than experiment can be – for such finite things as we have to deal with, and such portions of space as we can reach; yet the truth of them for very much larger things, or very much smaller things, or parts of space which are at present beyond our reach, is a matter to be decided by experiment, when its powers are considerably increased.” (W. K. Clifford, “On the Aims and Instruments of Scientific Thought,” LE, vol. 1, 155).

33  B. Lightman, op. cit., 162.

34  “What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobatchewsky [sic] to Euclid.” (W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 356).

35 Ibid., 358-9. Clifford even writes: “In both of these the knowledge of Immensity and Eternity is replaced by knowledge of Here and Now. And in virtue of these two revolutions [Copernicus’s and Lobatchevsky’s] the idea of the Universe, the Macrocosm, the All, as subject of human knowledge, and therefore of human interest, has fallen to pieces.” (W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 359; cf. W. K. Clifford, “Cosmic Emotion,” LE, vol. 2, 277-8).

36  Cf. J. R. Newman, in W. K. Clifford, The Common Sense of the Exact Sciences, op. cit., xxx. On the contrary, it is interesting to note that a physicist like James Clerk Maxwell was no advocate of the German mathematician’s work: “The Riemannische Idee is not mine,” he wrote in 1874. (Cf. P. M. Harman, The Natural Philosophy of James Clerk Maxwell, Cambridge: Cambridge University Press, 1998, 196).

37  On the difference between Clifford’s and Einstein’s ideas on the Kantian notion of space, cf. B. Lightman, op. cit., 163.

38  W. K. Clifford, “The Philosophy of the Pure Sciences,” op. cit., 384.

39  W. K. Clifford, “On the Aims and Instruments of Scientific Thought,” LE, vol. 1, 157, my italics.

40 Ibid.

41  B. Lightman, op. cit., 163.

42  Obituary notice, in the Athenaeum, quoted in W. K. Clifford, The Common Sense of the Exact Sciences, lviii, n. 58. “[…] one of the most promising of the British mathematicians of his day.” ([D. E. Smith], “Clifford’s Genius Shown as a Boy,” American Mathematical Monthly, (29), 1922, 157). Leslie Stephen considered Clifford “a real man of genius,” and John Tyndall spoke of him as the “gifted Clifford” (cf. B. Lightman, op. cit., 95).

43  “His mathematical work was prophetic, and its merit is still untouched after three quarters of a century of immense progress in mathematics;” (J. R. Newman, “William Kingdon Clifford,” Scientific American, (188), February 1953, 78).

44  In W. K. Clifford, The Common Sense of the Exact Sciences, p. ix. As a young man, Bertrand Russell enjoyed reading this book of Clifford’s (cf. My philosophical development, French translation, Histoire de mes idées philosophiques, Paris: Gallimard, [1959] 2001, 43).

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Christophe Duvey, « Experience and Universal Judgements: William Kingdon Clifford and Non-Euclidean Geometries », Revue LISA/LISA e-journal, Vol. VII – n°3 | 2009, 490-502.

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Christophe Duvey, « Experience and Universal Judgements: William Kingdon Clifford and Non-Euclidean Geometries », Revue LISA/LISA e-journal [En ligne], Vol. VII – n°3 | 2009, mis en ligne le 19 mai 2009, consulté le 28 septembre 2020. URL : http://journals.openedition.org/lisa/139 ; DOI : https://doi.org/10.4000/lisa.139

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Christophe Duvey

Dr. , (Montpellier, France)
Christophe Duvey est Maître de Conférences à l’université Paul Valéry – Montpellier III. Sa thèse de doctorat, intitulée « L’homme et la nature dans la pensée du naturaliste et philosophe St. George Jackson Mivart (1827-1900). Contribution à l’histoire des idées et des controverses philosophiques à l’époque victorienne », a été rédigée sous la direction de René Gallet. Ses travaux portent sur l’histoire des idées au XIXe siècle, notamment sur les différents courants de la pensée philosophique de cette période.

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