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Another Mathematician's Apology: Theological Reflections Upon the Role of Proof in Mathematics
Published online by Cambridge University Press: 30 January 2009
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Not that I have already obtained this or have already been made perfect (teteleiomai); but I press on to make it my own, because Christ Jesus has made me his own. Brethren, I do not consider that I have made it my own; but one thing I do, forgetting what lies behind and straining forward to what lies ahead, I press on toward the goal for the prize of the upward call of God in Christ Jesus. Let those of us who are mature (teleioi) be thus minded; and if in anything you are otherwise minded, God will reveal that also to you.
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- Copyright © Scottish Journal of Theology Ltd 1993
References
1 On this point, cf. Torrance, T. F., Reality and Scientific Theology (Edinburgh: Scottish Academic Press, 1985), pp. 22f.Google Scholar
2 Particularly important for the general theme of this essay is the work of Michael Polanyi. On the increasing importance of Polanyi in theology, see Gunton, C., Enlightenment and Alienation: An Essay towards a Trinitarian Theology (Basingstoke: Marshall Morgan and Scott, 1985)Google Scholar; Torrance, , Transformation and Convergence in the Frame of Knowledge: Explorations in the Interrelations of Scientific and Theological Enterprise (Grand Rapids: Eerdmanns, 1984), esp. chs. 2–3, pp. 61–173Google Scholar; and esp., Torrance, , ed., Belief in Science and in Christian Life: The Relevance of Michael Polanyi's Thought for Christian Faith and Life (Edinburgh: The Handsel Press, 1980)Google Scholar. For Polanyi's own discussion of mathematics, see Personal Knowledge: Towards a Post-Critical Philosophy (Cor. ed.; Chicago: The University of Chicago Press, 1962), pp. 117–131, 184–93Google Scholar. For a helpful comparison of mathematics and theology, though from a slightly different perspective than mind, see Puddefoot, J. C., Logic and Affirmation: Perspectives in Mathematics and Theology (Edinburgh: Scottish Academic Press, 1987).Google Scholar
3 Cf. Hardy, G. H., A Mathematician's Apology (Cambridge: Cambridge University Press, 1941).Google Scholar
4 Baillie, J., Our Knowledge of God (Oxford: Oxford University Press, 1949), pp. 134–143Google Scholar. References to this work will be given parenthetically.
5 Cf. Barth, K., Anselm: Fides Quarenes Intellectum, Robertson, I. W., trans. (London: SCM Press, 1960), esp. pp. 59fGoogle Scholar; and Voegelin, E., ‘The Beginning and the Beyond’, in The Collected Works of Eric Voegelin, vol. XXVIII: What Is History? and Other Late Unpublished Writings. Hollweck, T. A. and Caringella, P., eds. (Baton Rouge: Louisiana State University Press, 1990), pp. 192f.Google Scholar
6 R. G. Collingwood, Essay on Philosophic Method, quoted in Baillie, op. cit., p. 140.
7 For a more recent expression of a similarly sharp disciplinary distinction, cf. Macquarrie, J., Principles of Christian Theology (Rev. ed.; London: SCM Press, 1977), esp. pp. 56–57.Google Scholar
8 Cf. Voegelin, , in Cahn, E. and Going, C., eds., The Question as Commitment: A Symposium (Montreal: Thomas More Institute, 1979), p. 126Google Scholar: ‘It is one of the banes of our academic life that people accept the results of science, and their applications, as ready-made pieces of information and remain unaware of the processes of questioning and research in the background.’
9 Cf. Kline, M., Mathematics: The Loss of Certainty (New York: Oxford University Press, 1980).Google Scholar
10 Cf., e.g., Kūng, H., Does Cod Exist?: An Answer for Today, Quinn, E., trans. (Garden City: Doubleday & Co., 1980), pp. 31f.Google Scholar
11 A typical example is the largely negative reception of Whitehead and Russell's Principia Mathemalica by working mathematicians. Cf. Lowe, V., Alfred North Whitehead: The Man and His Work, vol. I: 1861–1910 (Baltimore: The Johns Hopkins University Press, 1985), p. 290.Google Scholar
12 On this point, cf. Puddefoot, Logic and Affirmation, p. xiii: ‘While I was an undergraduate at Oxford I found myself in conversation with a fellow student reading the joint honours degree in mathematics and philosophy. He was giving up philosophy because he had found that it interfered with his mathematics: by thinking too much about what he was doing he had found that he could no longer do mathematics.…’
15 My formulation of this issue will seem very superficial to a philosopher or historian of mathematics. For a further discussion of constructive mathematics, see Cassirer, E., The Philosophy of Symbolic Forms, Manheim, R., trans., vol. III: The Phenomenology of Knowledge (New Haven: Yale University Press, 1957), pp. 357–378Google Scholar; and Kline, op. cit., pp. 238f.
14 Cf. Berberian, S.K., Lecturers in Functional Analysis and Operator Theory (New York: Springer-Verlag, 1974)CrossRefGoogle Scholar, p. v: ‘An “analyst” is a mathematician who is seen habitually in the company of the real or complex numbers; a “functional analyst” is an analyst who is not squeamish about using Zorn's Lemma, definitely relishes the use of topology, and does not stand in the way of the internal algebraic impulsesof the subject’ (emphasis added).
15 Cf. Bishop, E., Foundations of Constructive Analysis (New York: McGraw-Hill Publishing Co., 1967)Google Scholar, esp. p. viii, and ch. 1 (‘A Constructivist Manifesto’), pp. 1–10.
16 Cf. Halmos, P. R., I Want to Be a Mathematician: An Automathography (New York: Springer-Verlag, 1985), pp. 161–162CrossRefGoogle Scholar. See also Bishop, op. cit., p. 2: ‘In the words of Kronecker, the positive integers were created by God. Kroneckerwould have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should know how to find it. If God has mathematics of his own that needs to be done, let him do it himself.’
17 It is not without a certain irony that Collingwood bases his argument upon the Pythagorean Theorem, which is unquestionably the most proved theorem in history. It should be noted, moreover, that he does not tell us which of the many hundreds of the theorem's proofs he has in mind. In the popular view, clearly, any proof will do as well as any other.
18 Another example is the proof by Nachman Aronszajn and Kennan Smith in 1954 that every compact operator from a Hilbert space onto itself admits a non-trivial invariant subspace. Cf. Halmos, op. tit., p. 320: ‘smith pointed out, I might almost say complained, that the proof was “tight”. It left no room for modifications and generalizations; it proved exactly what it was designed to prove, no more Aronszajn taught me the proof on a restaurant napkin several months before the paper appeared. I understood it, I cherished it, and along with many others I kept trying to “loosen” it so as to be able to apply it more broadly — but all to no avail.’ In 1966, however, Abby Robinson and Allen Bernstein published a new proof of the theorem, ‘the main accomplishment [of which] was the sought-for “loosening” they showed that the Aronszajn-Smith technique was not as narrow as we feared’.
19 Among the most important tools in mathematics research are those journals, such as Mathematical Reviews, which report, withmitproofs, many thousands of theorems which never otherwise appear in the technical literature.
20 For a generalization of this point, cf. Polanyi, Personal Knowledge, p. 163.
21 Non-trivial public errors on the part of professional mathematicians are exceedingly rare and, accordingly, always come as a profound surprise. Out of the many mathematical colloquia which I have attended I can recall only one instance of a genuine impasse in the course of an argument; and even in that case the theorem in question was later demonstrated to be (what mathematicians call) ‘a true fact’.
22 Cf. Polanyi, op. rit., pp. 58f; and Puddefoot, , ‘Indwelling: Formal and Non-Formal Elements in Faith and Life’, in Torrance, , ed., Belief in Science, pp. 30f.Google Scholar
23 Polanyi, , The Tacit Dimension (Gloucester, Mass: Peter Smith, 1983), p. 17.Google Scholar
24 In his delightful book, How to Solve It: A New Aspect of Mathematical Method (2nd ed.; Princeton University Press, 1971)Google Scholar, George Polya introduces the character whom he calls ‘the traditional mathematics professor’, one of whose aphorisms is as follows: ‘What is the difference between method and device? A method is a device which you use twice’ (p. 208).
25 On this point, cf. Puddefoot, , Logic and Affirmation, p. 70.Google Scholar
26 Cf. Lonergan, B. J. F., Method in Theology (2nd ed.; London: Darton, Longman, & Todd, 1973), p. 22Google Scholar: ‘Every inquiry aims at transforming some unknown into a known. Inquiry itself, then, is something between ignorance, for it makes ignorance manifest and strives to replace it with knowledge. This intermediary between ignorance and knowing is an intending, and what is intended is an unknown that is to be known.’ For Lonergan's discussion of ‘heuristic structures’ in mathematics, see also, Insight: A Study of Human Understanding (London: Darton, Longman & Todd, 1983), pp. 33–46, 309–15.Google Scholar
27 Paul Halmos is a good example of such a mathematician. For his own illuminating discussion of the process of questioning in mathematics, cf. Halmos, op. cit., pp. 116, 323, 402–3; and esp. p. 324: ‘Where do the good questions, the research problems, come from? They probably come from the same hidden cave where authors find their plots and composers their tunes — and no one knows where that is or can even remember where it was after luckily stumbling into it once or twice.’
28 Cf. Plato, Symposium, 215d–216a, where Alcibiades describes the ‘sacred rage’ which the Socratic inquiry arouses in him; and Republic, 515e–516a, where the emphasis is upon the violence suffered by the prisoner who is ‘drawn out’ of the Cave by the philosopher.
29 Cf. Polanyi, , ‘The Unaccountable Element in Science’, in Knowing and Being: Essays by Michael Polanyi, Grene, M., ed. (Chicago: The University of Chicago Press, 1969), p. 117Google Scholar: ‘Problems are the goad and the guide of all intellectual effort, which harass and beguile us into the search for an ever deeper understanding of things. The knowledge of a true problem is indeed a paradigm of all knowing. For knowing is always a tension alerted by largely unspecifiable clues and directed by them towards a focus at which we sense the presence of a thing — a thing that, like a problem, embodies the clues on which we rely for attending to it.’
30 Cf., e.g., Berberian, op. cit., p. 253.
31 The hypothesis of commutativity in the Gel‘fand-Naimark Theorm, among other things, is the basis of the theorem's applicability to the ‘rational functional calculus’ in spectral theory.
32 Cf. Davis, P. J. and Hersh, R., The Mathematical Experience (Boston: Birkhauser, 1981), p. 151Google Scholar: ‘Proof, in its best instances, increases understanding by revealing the heart of the matter. Proof suggests new mathematics. The novice who studies proofs gets closer to the creation of new mathematics. Proof is mathematical power, the electric voltage of the subject which vitalizes the static assertions of the theorems.’
33 Polya, , Mathematics and Plausible Reasoning (London: Oxford University Press, 1954), vol. I: Induction and Analogy in Mathematics, p. 76.Google Scholar
34 Ibid., p. 76, n. 1.
35 Polanyi, , The Tacit Dimension, p. 89.Google Scholar
36 Cf. Polanyi, M. and Prosch, H., Meaning (Chicago: The University of Chicago Press, 1975), pp. 173–181.Google Scholar
37 Cf. also Polya's description of ‘plausible inference’ (or ‘heuristic syllogism’) as a ‘force’ vector, having ‘direction’ and ‘strength’ (Mathematics and Plausible Reasoning, vol. II: Patterns of Plausible Inference, pp. 114f.).
38 Cf. Polya, How to Solve It, p. 185–86: ‘The main advantage of the exceptionally talented may consist in a sort of extraordinary mental sensibility. With exquisite sensibility, he feels subtle signs of progress or notices their absence where the less talented are unable to perceive a difference.’
39 Cf. Ibid., pp. 70, 207.
40 I have prefaced many of my own lectures in mathematics with the opening lines of Lincoln's famous ‘House Divided’ speech (in Angle, P. M., ed., Created Equal?: The Complete Lincoln-Douglas Debates of 1858 (Chicago: The University of Chicago Press, 1958), p. 1)Google Scholar: ‘If we could first know where we are, and whither we are tending, we could then better judge what to do and how to do it’ (emphasis added).
41 A good example is the classic text, Riesz, R. and Nagy, B. Sz., Functional Analysis, Boron, L. F., trans. (New York: Ungar Publishing Co., 1955).Google Scholar
42 There are few occasions, mercifully, when a mathematician will be faced with an argument which is literally ‘full’. The most detailed arguments in a modern treatise of which I am aware are to be found in Prugovecki, E., Quantum Mechanics in Hilbert Space (New York: Academic Press, 1971)Google Scholar. It is no accident that this volume is also among the most absolutely unreadable texts of which I am aware. The more common experience is that described in James, W., The Principles of Psychology, Burkhardt, F. H., Bowers, F., and Skrupskelis, I. K., eds. (Cambridge: Harvard University Press, 1981), vol. II, p. 992Google Scholar: ‘Bowditch, who translated and annotated Laplace's Mechaniqut celeste, said that whenever his author prefaced a proposition by the words “it is evident” he knew that many hours of hard study lay before him.’
43 Concluding Unscientific Postscript, Swenson, D. F. and Lowrie, W., trans. (Princeton: Princeton University Press, 1941), p. 79.Google Scholar
44 Ibid.
45 On this point, cf. M. W. Sinnett, ‘The Primacy of Relation in Paul Tillich's Theology of Correlation: A Reply to the Critique of Charles Hartshorne’, (unpublished manuscript).
46 Tillich, P., Systematic Theology, vol. I (Chicago: The University of Chicago Press. 1951), p. 60.Google Scholar
47 Ibid., p. 61.
48 Ibid., p. 62. Tillich's insistence upon the pre-theoretical reality of correlation is especially clear in his essay on ‘The Problem of Theological Method’, Journal of Religion, vol. 27 (1947), pp. 16–27.CrossRefGoogle Scholar
49 Systematic Theology, 1, p. 61 (emphasis added). Cf. also Voegelin, , ‘The Gospel and Culture,’ in The Collected Works of Eric Voegelin, vol. XII: Published Essays, 1966–1985, Sandoz, E., ed. (Baton Rouge: Lousiana State University Press, 1990), p. 175Google Scholar: ‘Question and answer are intimately related one toward the other; the search moves in the metaxy, as Plato has called it, in the In-Between of poverty and wealth, of human and divine; the question is knowing, but its knowledge is yet the trembling of a question that may reach the true answer or miss it. This luminous search in which the Finding of the true answer depends on asking the true question, and the asking of the true question on the spiritual apprehension of the true answer, is the life of reason.’
50 This is by way of stark contrast with the symbolism of the ‘mysteries’ (teletai), whose initiates were regarded as the ‘complete’ or ‘perfect ones’ (teleioi). Cf. Beare, F. W., A Commentary on the Epistle to the Philippians (London: A. & C. Black, 1959), pp. 128–131.Google Scholar