Abstract
In several of his works, Alfred North Whitehead (1861–1947) presents mathematics as a way of learning about general ideas that increase our understanding of the universe. The danger is that students get bogged down in its technical operations. He argues that mathematics should be an integral part of a new kind of liberal education, incorporating science, the humanities, and “technical education” (making things with one’s hands), thereby integrating “head-work and hand-work.” In order to appreciate the role mathematics plays in modern science, students should understand its diverse history which is capable of bringing abstract ideas to life. Moreover, mathematics can discern the alternating rhythms of repetition and difference in nature constituting the periodicity of life. Since these same rhythms are to be found in his theory of learning as growth, there appears to be a pattern linking Whitehead’s approach to mathematics and his educational philosophy.
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Notes
Jean-Pascal Alcantara (2009) claims that during the Renaissance a revival of Greek and Latin humanities combined with the new conception of human beings as free from the traditions of the past created a conception of liberal education which had competing streams to it (pp. 128–130).
Elsewhere, I have shown there to be a relationship between the rhythmic cycles of growth and the “characteristics of life” in Whitehead’s philosophy of organism (Woodhouse 1995, pp. 348–353).
This quotation anticipates to some extent the Fallacy of Misplaced Concreteness and its critique of “the high abstractions” underpinning 17th century mechanistic materialism and their exclusion of human experience.
In a lecture delivered to the Harvard Business School and later published in Adventures of Ideas (1961) Whitehead claimed that Foresight, based on philosophical understanding and generalization, is the capacity best suited for understanding the rapidly changing modern world of commerce (pp. 97–8).
These changes in the approach to liberal education in the early 20th century are documented by Paul Axelrod (2002, pp. 22–4). An alternative account to liberal education as exhibiting the value of “useless knowledge” was given by Bertrand Russell (2006). For a commentary on Russell’s views, see Woodhouse (2006).
At the same time, Whitehead recognized that in practice science advances despite imprecise calculations and laws that are open to question—“so, after all, our inaccurate laws may be good enough” (1958, p. 16). In Process and Reality (1929a), he reasserted that it is the “general success” of first principles, not their “peculiar certainty or initial clarity,” which is the goal of rational thought, since “even in mathematics the statement of the ultimate logical principles is beset with difficulties, as yet insuperable” (p. 11). While metaphysical first principles can always be questioned, they are open to human experience: “There is no first principle which is in itself unknowable, not to be captured by a flash of insight … The elucidation of immediate experience is the sole justification for any thought; and the starting point for thought is the analytic observation of components of this experience.” As a result, “metaphysical first principles can never fail of exemplification. We can never catch the actual world taking a holiday from their sway” (p. 11). Winchester (2000) observes that Whitehead was engaging in a critique of the limitations of the empirical or scientific method: “Because of the omnipresence of the metaphysical there are limits to the employment of the empirical method here. Thus any account of meaning that limits thought to the empirical is bound to break down just as it approaches the metaphysical” (p. 297). For a critique of Newton’s attempt to exclude hypotheses from “experimental philosophy,” see Carey (2012).
Ernest (2000), like de Berg (1992), underlines the ways in which mathematics has been vital in the development of quantum mechanics, as does Aczel (2002, p. 252). Aczel also shows how the history of the process of entanglement proposed by mathematical physicists is an unfolding story which, as chance would have it, also illustrates the importance of internal relations and feelings in Whitehead’s ontological sense (2002, pp. xiii, xv, 1, 250–2).
According to one historian, “the intellectual renaissance of the Latin West in the 12th and 13th centuries would not have been possible without the advances made by the scholars of Islam. Their translations, commentaries, and original speculation on a wide variety of subjects were among the cornerstones of the European curriculum” (Domonkos 1977, p. 9, cited in Axelrod 2002, p. 14). See also, Haskins (1960, pp. 4–5).
Current research utilizing the same approach of MCC is also being conducted with the Sami people in Norway (Fyhn et al. 2011). As my colleague Ed Thompson has pointed out in conversation, there are two ways in which the study of indigenous mathematics may prove valuable. First, to provide new insights into both the theoretical framework and teaching of mathematics; second, to enable the well established mathematician to engage in philosophical reflection about the question, “What is it that all of us mathematicians are actually doing?” For examples of mathematics education in a Whiteheadian vein, see Brumbaugh (1992), Pelton (1995), and Guest (2010).
The Platonic belief in “disinterested intellectual appreciation” as the goal of education should be replaced by an emphasis on “action and our implication in the transition of events amid the inevitable bond of cause to effect” (Whitehead 1957b, p. 47). Students learn to bring about change by creating objects of beauty through a combination of thought (head-work) and action (hand-work). They thereby come to appreciate the importance of “causal efficacy,” or “the ‘withness’ of the body … that makes the starting point for our knowledge of the circumambient world” (Whitehead 1957a, p. 81). The bodily feelings expressed in the unity of mental and manual labour provide a direct epistemological connection between the learner and reality.
For a more detailed analysis of the importance of art and craft in overcoming alienated labour in Whitehead’s educational philosophy, see Adam Scarfe and Woodhouse (2008, pp. 195–6).
G.H. Hardy remarked that “A mathematician, like a painter or a poet, is a maker of patterns … The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test … (1967, pp. 84–5; cited in Hallez 1992, p. 313). As Ronny Desmet points out: “According to Whitehead, mathematics is the study of relational structures or patterns. The emergence of mathematics involves the direct pattern recognition that is proper to our sense perception … [which] is inseparably tied to our feeling of space, and because our space-intuition is so essential an aid to the study of geometry, it seems as if geometry cannot be part of pure mathematics” (2010a, p. 369). Desmet goes further, arguing that, “his [Whitehead’s] ultimate drive was … to unify the mathematical structures underlying the analogical reasonings that constitute the art of physics, an art which his Cambridge training impressed on him” (2010b, p. 121). Whitehead himself (1958) calls attention to the importance of spatial intuitions despite their logical independence from the mathematical science of geometry (pp. 180–1).
Dewey (2005) sounds very much like Whitehead when writing: “What is not so generally perceived is that every uniformity and regularity of change in nature is a rhythm. The terms ‘natural law’ and ‘natural rhythm’ are synonymous… Mathematics are the most generalized statements conceivable corresponding to the most universally obtaining rhythms. The one, two, three, four, of counting, the construction of lines and angles into geometric patterns, the highest flights of vector analysis, are means of recording or of imposing rhythm” (p. 155). Neville (2009) analyzes biological rhythms in terms of monthly, daily, and ultradian phases (pp. 63–4).
The concept of a periodic function is Whitehead (1958a, b) preferred method of expressing these formal abstractions on the basis of “a sum of sines … called the ‘harmonic analysis’ of the function … [which produces a] process of gradual approximation” capable of analyzing such events as the relationship between “the tide-generating influences of one ‘period’ to the height of the tide at any instant” (pp. 142, 143). Later, when writing of the development of science in the 16th and 17th centuries (1953), he states that: “The birth of modern physics depended upon the application of the abstract idea of periodicity to a variety of concrete instances. But this would have been impossible, unless mathematicians had already worked out in the abstract the various abstract ideas which cluster round the notions of periodicity … Then, under the influence of the newly discovered mathematical science of the analysis of functions, it [trigonometry] broadened out into the study of the simple abstract periodic functions which these ratios exemplify” (p. 31). Commenting on Whitehead’s use of periodic functions in expressing events like day and night, the seasons as well as tides, Ian Winchester writes: “Both sines and cosines can be portrayed as an undulating line above and below passing regularly through a straight line, usually a horizontal one. It is not obvious to me that all of these periodic regularities are best portrayed this way, but in general they can be so portrayed. It depends on what one plots. Day and night, for example, have to do with the amount of sunlight daily which could be so portrayed, though once one has passed the "no sunlight" point one would have to think of darkness, I suppose, which is a bit of a strain. In northern and southern regions the sine (or cosine) curves would be shifted up or down relative to the line. The seasons are related to the relation of the sun relative to the axis of the earth, and at the equator the daily rhythm of day and night is exactly 12 h, and varies north and south from there, while the seasonal variation, which disappears at the equator picks up on both sides until one has maximum variation at the poles. So again a kind of sine or cosine function would be possible to portray it annually. The tides would, I suppose, be a matter of portraying the height of the tide hourly using some neutral base point so that there would be high tide and low tide, high above the line where the slope of the sine curve is zero and the low time similarly below the line where the slope of the sine curve is also zero.” (Email to author, 11 April 2012.)
Winchester analyzes this complex rhythm as follows: “In fact the various chambers work in a sequence and the detailed story is rather complicated, but if one concentrates on systole and diastole then one can picture the filling up or expansion phase of the heart (i.e. diastole), when it takes in blood from the wider circulatory system that has been returned to it through the venus drainage, as one of the semi-periodic movements (say above the horizontal line) and the contraction phase or systole, when it expels the blood that it has received to the rest of the body (including, in fact, itself through the coronary arteries) might be seen as the movement below the line … I have cut out the interesting details in order to see how Whitehead might have used the sine curve as a metaphor for the overall process… But the heart does function with a large major expansion and a large major contraction which enables the sine curve to plot the overall story of the pulse.” (Emails to the author April 13 and 14, 2012.)
This same process also provides the basis for freedom of action, provided that “individuality” is balanced with a further recognition that “coordination … of community life” is a significant aspect of students’ environment (Whitehead 1961, p. 67). Only where the individual strives to integrate the polarities of “self-development” and “the complex pattern of community life,” can the “art of life” take hold (1957b, p. 39; 1961, p. 67). While this kind of balance is a major goal of liberal education for Whitehead, it should also include enjoyment of the emotions, as he notes in Modes of Thought (1966): “Life is the enjoyment of emotion, derived from the past and aimed at the future. It is the enjoyment of emotion which was then, which is now, and which will be then. This vector character is of the essence of such entertainment” (p. 167). For an analysis of this vectoral character of emotions, see Woodhouse (2012).
Only where students appreciate the beauty in nature and human artifacts, and the panoply of changing values inherent in both, will they learn the art of life, “(i) to live, (ii) to live well, (iii) to live better.” Art and aesthetic appreciation enable human beings to lead civilized lives in which they strive “towards the attainment of an end realized in imagination but not in fact” (1958, pp. 4, 8). At the same time, art brings the potentiality of the imagination into the actuality of everyday life.
At the same time, Whitehead warned that: “We shall ruin mathematical education if we use it merely to impress general truths. The general ideas are the means of connecting particular results. After all, it is the concrete special cases which are important. Thus in the handling of mathematics in your results you cannot be too concrete, and in your methods you cannot be too general” (1957b, p. 53).
Hallez (1992) shows how the history of mathematics enables students to appreciate the intellectual contributions of both men and women (1992, p. 327).
Matthews (1992) chronicles the rapprochement between history, philosophy, and science teaching with special reference to the proposals of the British National Curriculum in Science and the American Association for the Advancement of Science (pp. 13–14). Proceedings of the History and Philosophy of Science and Science Teaching Society have appeared in two volumes, edited by Skip Hills (1992a, b), and special issues of Interchange (1993 and 1997) edited by Matthews and Winchester. Special issues of Interchange edited by Winchester on creativity and mathematical proof (1990) (with Gil Hanna), stories in science teaching (2010), and circumpolar indigenous issues in relation to mathematics and science education (2011) show a growing interest in the field. Hallez (1992) also mentions the group MATH at Paris V11 University IREM as an organization concerned with an integrated teaching of mathematics (p. 314).
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Acknowledgments
An earlier version of this article was presented to the Centre for Logic and Philosophy of Science, Free University of Brussels (VUB) on 16th May 2012. Dr Ronny Desmet, Research Fellow at the Centre for Logic and Philosophy of Science, was kind enough to invite me and chair the session, Dr Michel Weber, Centre de philosophie pratique et Chromatiques whiteheadiennes, made the initial arrangements for my lecture, and the audience posed interesting questions. Dr Ian Winchester (Dean Emeritus of Education, University of Calgary), Dr Michael Collins (Professor Emeritus, Department of Educational Foundations, University of Saskatchewan), and Viola Woodhouse (Professor, Department of Philosophy, St Thomas More College, University of Saskatchewan) all made helpful comments on earlier drafts, as did Vincent Cable and Murray Guest, both high school teachers of mathematics and physics. I am also indebted to my colleagues in the University of Saskatchewan Process Philosophy Research Unit—Professors Mark Flynn, Bob Regnier, Ed Thompson, and Adam Scarfe—for their ongoing support over the years.
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Woodhouse, H. Mathematics as Liberal Education: Whitehead and the Rhythm of Life. Interchange 43, 1–23 (2012). https://doi.org/10.1007/s10780-012-9169-4
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DOI: https://doi.org/10.1007/s10780-012-9169-4