Bergson, Mathematics, and Creativity

The following article appeared in Process Studies, pp. 268-288, Vol. 28, Number 3-4, Fall-Winter, 1999. Process Studies is published quarterly by the Center for Process Studies, 1325 N. College Ave., Claremont, CA 91711. Used by permission. This material was prepared for Religion Online by Ted and Winnie Brock.

SUMMARY

Dr. Gunter corrects two misunderstandings of Henri Bergson: 1. That his philosophy is “irrationalist.” 2. That his philosophy is “literary.” The author’s basic goal is to explain Dr. Bergson’s concept of the calculus.

Preliminary Consideration

I would like to make a couple of points concerning two popular misunderstandings of Bergson. The first is the impression that this philosophy is “irrationalist. The second, often incongruously conjoined with the first, is the suggestion that his philosophy is — for lack of a better word – “literary.” Though this essay is primarily an effort to situate and explain Bergson’s use of fundamental concepts of the calculus, one of its goals will be, in the process, to correct these two errors.

That Bergson is an “irrationalist” is often claimed: explicitly by Benedetto Croce, Leonard M. Marsak, Walter J. Slatoff, Gerhardt Lehmann, Egon Friedall, Raymond Bayer, Walter Bruning, implicitly by Bertrand Russell and George Santayana. ((Benedetto Croce, What is Living and What is Dead in the Philosophy of Hegel? Trans. D. Ainslie. (London: MacMillan, 1915), 213-215. The author argues that Bergson’s Philosophy requires “….. the renunciation of thought” Leonard M. Marsak, Editor French Philosophers from Descartes to Sartre (New York: World Publishing Co., 1961). The author takes an interesting tack, arguing that while Bergson’s intuition is not irrationalist, his universe is irrational. For a similar view, cf. A. Tymieniecka, Why is there Something Rather than Nothing? (Assen: Van Gorcum, 1966), 10-11. Walter J. Slatoff, Quest for Failure (Ithaca, New York: Cornell University Press, 1960), 242-248. The author finds “irrationalism” to be common and basic to both Bergson and W. Faulkner. Gerhardt Lehmann, Geschichte der Philosophie. Die Philosophie im ersten Drittel des XX Jabbrbunderts, vol.10 (Berlin: de Gruyter 1957), 128. Egon Friedell, A Cultural History of the Modern Age, vol.3, trans. C. F. Atkinson (New York: Knopf. 1954), 372-373. Raymond Bayer Epistemologie et logique depuis Kant jusqu’a nos jours (Paris: Presses Universitaires de France, 1954), 95. Walter Brunting, “La filosofia irraciotialista de la historia en la actualidad” in Revista de Filosofia, 5,2 (1958), 3-17. The author treats Nietzsche. Ludwig Klages and Bergson as asserting irrationalist philosophies of history. Bertrand Russell, “The Philosophy of Bergson,” Monist, 22/3 (1912), 321-347. George Santayana, Winds of Doctrine (New York: Scribner’s, 1913), 58-109.)) I will spare the reader further citations; they are easy to find in the literature. This term, as I hope to show, is radically imprecise (slipshod in fact) as applied to Bergson, whose duration has “form,” whose intuition is “reflection:” whose goal is renewed conceptual creativity. My basic objection to it, however, is not so much its inaccuracy as its noetic destructiveness. One can hardly comprehend a philosophy if one begins with the assumption that it is intrinsically incomprehensible. The ascription of the term “irrationalist” prevents both oneself and others from gaining understanding, where understanding is to be had.

As for Bergson’s “literary” aura, one seeks in vain for adequate synonyms to explicate it. “Superficial,” “facile,” “trendy,” “poetic” help but do not go far enough. Will Durant states that Bergson’s lecture rooms became the salon of splendid ladies “happy to have their hearts’ desire upheld.” ((Will Durant, The Story of Philosophy: The Lives and Opinions of the Great Philosophers of the Western World, 2nd ed. (New York: Simon and Schuster 1961), 349. Cf. also 343.)) Biologist Jacques Monod dismisses his philosophy as “. . . a metaphysical dialectic bare of knowledge but not of poetry.” ((Jacques Monod, Chance and necessity: An Essay on the Natural Philosophy of Modern Biology (New York: Vintage Books, 1972), 26.)) Monod not only condemns Bergson to the hinterland of the poetic, he also, like Russell and Bochenski, types Bergson as irrational (or anti-intellectual, a kindred classification) for his “rebellion against the rational”(27). André Gide complains: “What I dislike in Bergson’s doctrine is all I ever thought without his saying it, and everything that is flattering, even caressing to the mind . . . he himself belongs to the epoch and constantly yields to the trend.” ((André Gidé, The Journals of André Gidé, vol.2. trans. J. O’Brien (New York: Knopf, 1948), 348.)) Bergson, we recall, won the Nobel Prize for literature. In some quarters this is tacitly assumed as evidence that he could not think straight. I hope to show that he could.

Since he is not widely read in English-speaking philosophical circles, it should prove helpful, in sections two and three, to outline some fundamental features of Bergson’s philosophy. This will help us to understand some of the basic problems to be examined below. Since most of these problems involve his philosophy of mathematics, and since the mathematics with ‘which he is concerned is primarily the calculus, it will help also, in section four, to sketch, briefly, a number of its fundamental features. This will make it possible, in section five, to analyze the centrality of the calculus to Bergson’s philosophical method (hence to both his epistemology and his metaphysics).

Bergson A Refresher Course

Bergson’s philosophy sprang, he tells us, from an analysis of Herbert Spencer’s First Principles, undertaken to update and clarify Spencer’s physics — particularly his concepts of space and time. ((Henri Bergson, The Creative Mind trans. Mabelle L. Andison (New York: Philosophical Library, 1946), 221. All further references to this work will he cited in the text as CM.)) The young philosopher was astonished to discover, through this analysis, that there is a yawning chasm between the time physicists use and the time human beings (including physicists) experience.

On one side, there is mathematical (“clock”) time, composed of instants. each of which is entirely distinct from all others, none of which, clearly conceived, can move, transform, or change itself. Or, we have units (minutes, hours, etc.) each of which is entirely homogeneous within its boundaries and identical with all units of equal length. This is a strange sort of time indeed, made up of static parts, all the same, each entirely distinct from the others. Rather than call this real time, Bergson concluded, it would be more accurate to term it a “fourth dimension of space” (TFW 109).

Meanwhile there is actually experienced time which, Bergson found, looks less like mathematical time the more one explores it. “Lived time,” far from being made up of instants, is dynamic throughout. No two “segments” of it — if there are segments — are identical; all differ qualitatively. And, far from being separate, successive states of consciousness merge into each other without sharp boundaries. Bergson calls this inner time “duration” to distinguish it from mathematical time and to stress the fundamental endurance of each of its moments into the next.

To treat duration and its continuity as a kind of calibrated spatial coordinate — to spatialize and spatially segment it — is, Bergson agrees, extremely useful. Without spatialization we would not only not have modem physics: we would not have clocks, calendars, and the kinds of organized societies they make possible. But it is a mistake to raise distinctions made for pragmatic purposes to the level of fundamental theoretical postulates without first submitting them to a critique. In this case a critique demonstrates the real time — real duration — is neither static nor homogeneous nor internally discrete. Real duration is dynamic, heterogeneous, and (qualitatively) continuous.

This fundamental distinction, between lived time and dock time, and the insight into the character of experienced time on which it depends, mark the starting-point of Bergson’s philosophy From this base, his thought is continually broadened to include embodied human consciousness (Matter and Memory, 1896), biological evolution and physical cosmology (Creative Evolution 1907), and human history and prehistory (The Two Sources of Morality, and Religion, 1932).

In each of these works one finds a tension, and a resulting conflict between two contrary tendencies: one creative, expansive, dynamic, the other conservative, repetitive, static. In Time and Free Will this conflict juxtaposes an “inner self” capable of singular free, spontaneous acts with an outer self which is both habitual and superficial. In Matter and Memory, mind (conceived primarily as memory) is contrasted with body (brain, nervous system, motor and perceptual organs). The body, Bergson concludes, acts in two respects as a kind of “choke filter.” The brain functions to exclude the bulk of our accumulated memories, allowing the emergence of only those which help us to recognize and cope with present situations.

Analogously, sense organs filter out of the welter of our environment all those influences that would keep us from responding effectively to our surroundings. The result is a stable, pictorial world. These joint constraints, however, make possible our focused consciousness. They allow us to deal with our ordinary affairs in ordinary ways. They also make possible those exceptional acts which, Bergson states — echoing the fundamental conclusion of Time and Free Will — express our freedom. ((Henri Bergson, Matter and Memory auth. trans. Nancy Margaret Paul and W. Scott Palmer (London; George Allen and Unwin Ltd.; New York: The Macmillan Company. 1911), 339. All future references to this work will be cited in the text as MM.))

Creative Evolution is an extrapolation of the mind-matter duality of Matter and Memory. It opposes not individual mind, but “life,” to the downward drift of a partly-reconceived matter. Life, Bergson argues, is a tendency towards increasing flexibility, organization, consciousness. Matter, inversely, is a drift towards dispersion, loss of form, degradation of potential energy. The tension between matter and life results in the endlessly renewed creativity of life, vectored not towards a single preestablished goal but towards a multiple branching of diverse forms. ((Henri Bergson, Creative Evolution, auth. trans. Arthur Mitchell, intro. Pete A. Y. Gunter (Lanham, Maryland: University Press of America, 1983), 301. All future references to this work will he cited in the text as CE.))

In The Two Sources of Morality and Religion a similar tension (‘with its potential for conflict) is explored in terms of human anthropology, sociology, and history.’ Here Bergson contrasts two root impulses in human nature: one creative, outgoing, free; the other conservative, constraining, quasi mechanical. The first leads to the open society — or to the continual opening-out of societies; the second leads to the closed society — or to societies primarily closed and (therefore) arrayed against each other, The first would lead to the brotherhood (and sisterhood) of humankind; the second is forever vectored towards war, and defense, and conquest.

This sketch of the development of Bergson’s philosophy conveys neither the depth and suggestiveness of this thought nor the seriousness of the conceptual problems to which it leads. Though an extensive analysis of this sketch cannot be attempted here, some amplification of it is needed. And since the present essay concerns Bergson’s philosophy of mathematics, what follows will emphasize his concept of matter: its status, and its aptness for mathematical prediction and description.

Structures of Duration

Matter and Mind

In Time and Free Will, duration possesses only continuity a “fluid” character (TFW 135), with “interpenetrating” contents (TFW 104, 162), whose moments “melt” (TFW 100, 127, 128. 138, 163) into their successors. Such a continuity, though qualitatively heterogeneous, lacks structure. It appears amorphous. Any “cut” in its perpetual flow would appear to be arbitrary. This in itself is a problem. But is a problem that creates further problems. For example, Bergson describes his exceptional free act as gradually elaborating itself and then suddenly emerging into behavior (TFW 169-170, 176). But how, on the terms available in Time and Free Will could one meaningfully distinguish the length or slowness of this elaboration from the brevity of its expression? On Bergson’s terms, in this work there is only amorphous becoming, having nothing in common with tempos or proportions . . . or, of course, numbers of any kind.

This problem is rendered especially acute in Time and Free Will through a radical bifurcation of self and world. Within us, Bergson states, “. . . there is succession without mutual externality; outside the ego, in space, mutual externality without succession,“ (TFW 108, italics mine; cf. also TFW 116, 227), If Descartes’ dualism is famous for the problems to which it leads, how many more will result from Bergson’s joint assumption of an unextended durational mind and a geometrical and static world?

Matter and Memory is an effort to outflank both the famous Cartesian dualism and Bergson’s own, even more stringent, bifurcation. This is done in three ways: First, by describing matter, like mind, as a kind of duration. Second, by conceiving both sorts of durations as quasi-epochal (i.e., consisting of “rhythms”). Third, by rejecting two insidious dogmas of Cartesianism: that the mind is entirely unextended and, inversely, that matter is pure (geometrical) extension. These three moves jointly establish a conceptual context in which interrelations between mind and matter become intelligible, and in which also there is an ample realistic basis for the mathematization of the real.

The fundamental problem, Bergson argues, is our spatial notion of the relations between mind and world: “Questions relating to subject and object, to their distinction and their union, should be put in terms of time, rather than of space” (MM 77). Throughout the eighteenth and nineteenth centuries matter had been conceived as comprised of passive, simply-located particles whose most fundamental character is their imperviousness to change. Bergson takes a contrasting view, conceiving matter as: “. . modifications, perturbations, changes of tension or of energy and nothing else” (MM 266). That is: matter is a kind of duration, a succession of “rhythms,” a present which is “always beginning again.” Relative to human duration ‘with its directed spontaneity, matter is inert. In itself it “lives and vibrates” (MM 270).

The rhythms of human consciousness are given an elaborate analysis in Matter and Memory. The highest psychological rhythms (a term treated as synonymous with “tensions”) here have the greatest breadth, the lowest the greatest brevity (MM 279). Even the briefest rhythms of consciousness, however, are far broader than, and extend over, those of matter. Bergson notes that in one second red light goes through 400 billion vibrations: “Now the smallest interval of empty time which we can detect equals, according to Exner 1/500 of a second; and it is even doubtful whether we can perceive in successions several intervals as short as this. Let us admit, however, that we can go on doing so indefinitely” (MM 272). If this were possible, then to watch each of 400 billion vibrations separated by 1/500 of a second from the next would require the passage of 25,000 years.

There is a certain artifice involved in setting up this simple calculation, Bergson admits. It is necessary to “separate the vibrations” sufficiently to allow us to count them (which suggests as will be shown below, in section 51, that some distancing between them is already present). The result is unmistakable, however; the arithmetic relationship at with Bergson arrives here is presented by him not as illusory but as having an objective basis. Between the rhythms of consciousness and those of matter there is an element of quantitative commensurability. Indeed, there is the unmistakable suggestion that it is the indivisibility of the rhythms, of both sorts, which grounds the possibility of quantitative relations.

The theory of differing breadths of duration, capable of extending over each other, and the notion of matter ‘which results from it, are further developed in Creative Evolution through reflections on thermodynamics. The constant production of entropy and the increasing unavailability of energy to perform work which thermodynamics depicts suggest, Bergson speculates, that the physical world may have possessed characteristics in the past that it does not now have, and that characteristics it now has may be lost in the future. The direction of all these changes, however, is constant Matter possesses a kind of history But it is the history of an unbecoming “. . . . changes that are visible and heterogeneous will be more and more diluted into changes that are invisible and homogeneous, and . . . the instability to which we owe the richness and variety of the changes taking place in our solar system will gradually give way to the relative stability of vibrations continually and perpetually repeated” (CE 243). This process results in a continual extension of matter into space (CE 203), a process never completed. There are, therefore, “degrees of spatiality” (CE 205) in nature. As matter descends towards space, it becomes more homogeneous, its successive moments become less mutually continuous, its stability increases. But if matter “stretches itself out in the direction of space” (CE 207), it never reaches this limit.

Thus, in Creative Edition Bergson develops a theory already proposed in MM: namely, that matter is (a term proposed by William James) “extensive.” It does not have pure spatial extension: it is neither as extended as geometrical space nor, simply, extra-spatial. “Extensity” is used to described not only matter, but mind. Bergson denies that mind is entirely unextended: thus denying a Cartesian dogma which has dominated modern philosophy. Rather, Bergson argues, mind is dipolar. Pure memory the “mental” pole, approximates Cartesian nonextension. Pure perception, the “physical” side, approximates the extensity of matter (MM 74). Hence our perception is capable of participating in matter, and it becomes comprehensible (at least it ceases to be incomprehensible) that mind could act on matter (and vice versa).

This universal “movement” of matter not only involves an expansion of the physical universe. It also helps to explain why there is an “approximately mathematical” order in nature which science “approaches in proportion to its progress” (CE 218). When a physicist validly marks an event as starting at time t there is a real almost instantaneous event: a rhythm, to which this variable refers. When a physicist describes a motion from a to b there is, first of all, real motion, and second, an approximately — geometrical a location and b location.

Life and Matter

The theory of matter developed in Creative Evolution is an exact inverse of the theory of life developed there. But where Bergson’s theory of matter is close in many respects to contemporary cosmology and physics, his biological theory appears not only far removed from present biochemistry, taxonomy and paleontology, but devoid of content. What is élan vital except a myth? Bergson does provide conceptual content with which his biology can be understood, however. It is found in his notion of the durations (the rhythms) of life and of matter, and their interrelations.

That these concepts are present in Creative Evolution is a fact easily overlooked. In fact, they are omnipresent. The universe, Bergson states, consists of two movements, of ascent and descent The ascending movement: “. . . which corresponds to an inner ripening or creating: endures essentially and imposes its rhythm on the first, which is inseparable from it” (CE 11).

The resulting “progress,” with is multiple branching, is continuous from beings that vibrate almost in unison with the oscillations of the ether, up to those that embrace trillions of these in their shortest perceptions” (CE 201). The “proportionality” between consciousness and the durations of matter outlined in Matter and Memory is thus retained in Creative Evolution. Here the proportionality becomes part of a non-reductionist biology sustaining the thesis that life proceeds by creating organisms whose temporalities increasingly broadened over those of matter — whose fundamental rhythms are both greater in breadth than those of matter, and richer in content. Evolution, a temporal process, endeavors to create broader temporalities.

The theories of Creative Evolution thus become more comprehensible if viewed through the lenses of chronobiology. Each species, on Bergson’s terms, has a unique rhythm, a distinctive temporality. This rhythm is correlated with the briefer rhythms which constitute its materiality. Interestingly — and, I think, unexpectedly — it will be Bergson’s understanding of the role that the calculus plays in effecting these correlations that will help explain them.

Notes on the Calculus

The “infinitesimal calculus” was developed by Isaac Newton and Gottfried Leibniz, independently, in the late seventeenth century. Their achievement scarcely took place in a noetic vacuum. Newton was right to protest that if he saw so far it was because he stood on the shoulders of giants. Both he and Leibniz were deeply indebted to a large set of mathematicians: Archimedes, Cavalieri, Wallis, Descartes, Barrow, and Fermat among them. Nor were Newton’s and Leibniz’ achievements complete as they stood. Extensions, corrections, reorganizations of the calculus continued long after their work, leading after more than two centuries to its present, presumably rigorous, foundations.

From its beginnings the calculus has consisted of two contrasting parts, each designed to solve different sorts of problems. The differential calculus was developed to deal with motion — velocity at a point and acceleration being fundamental concepts. Quantities representing velocity are termed first derivatives, those dealing with acceleration, second derivatives. The integral calculus, by contrast, was developed to deal with areas (“areas under curves”) and, by extension, volumes.

Both parts of the calculus have been extended beyond their original scope. The differential calculus, though designed to deal with states of motion, can be used in fields far removed from planetary orbits or falling bodies. In the words of Edward Kasner and James Newman: “Structural engineers, concerned with the elasticity of beams, the strength columns, and any phase of construction where there is shear and stress, find first, second, third, and fourth derivatives indispensable . . .” ((Edward Kasner and James Newman, Mathematics and the Imagination (New York: Simon and Schuster, 1940), 329. Beyond the second derivative, simple physical or geometric interpretations of third, fourth and higher derivatives, the authors point out, do not exist. Even if such concepts, applied to nature, are in this sense conventional, they are no less useful.)) As will be noted shortly there is no limit to the number of a derivative.

The integral calculus, similarly, is more complex than it at first appears. This complexity is summed up in the elegant Fundamental Theorem of the Calculus, which establishes both that there are two sorts of integrals with precise equivalences and — a quite different idea — that there is an inverse relation between the integral calculus and the differential calculus.

The two fundamental integrals are the definite and the indefinite integral. The indefinite integral is a number, while the indefinite integral is a function. In the words of David Berlinski:

Integration, from the vantage point of the indefinite integral, involves following the generation of a curve up to a point, thus finding a “changing quantity” from its rate of change.

The Fundamental Theorem also demonstrates that it is possible to generate a definite integral (a number) from any indefinite integral (a function). The simplicity of this computation turns out to be a boon to mathematicians, who otherwise, to get their definite integrals, would have to go through laborious computations based on Riemannian sums.

It is both interesting and important to note that in the calculus one can pass from functions to numbers, and simply. It is equally important to see that one can also pass from integrals to derivatives (from the integral calculus to the differential calculus) and then back again. The creators of the calculus knew this, That is, they understood that integration and differentiation are the “inverse” of each other, in the way that addition and subtraction stand in inverse relations.

What is meant by such inverse relations can be conveyed by very elementary symbolism. By differentiating y=f(x) we get dy/dx (a derivative); by integrating dy/dx we get y=f(x) again (the original integral). The relationships entailed by the Fundamental Theorem stretch well beyond this simple one-step-forward, one-step-back progression, however. Chains of differentiations and integrations are also possible, reaching from the most local of derivatives, by single steps, though the most global of integrals; and from the most global of integrals it is possible, inversely, by single steps to find to the most local of derivatives.

This chaining relationship can be outlined using the language of first, second, third, and, in general, nth derivatives. From a third derivative we can move, by integration, to a second derivative, which will be its (the third derivative’s) integral. From a second derivative we can generate a first derivative; which will be the second derivative’s integral. From the first derivative, we can by the same calculation, arrive at the “zero-eth” derivative. The inverse procedure allows us, given any integral, to generate its derivative, and so on, ad indefinitum. Integration and differentiation thus form chains with some resemblance to hierarchies. If these “chaining” relationships are stressed here, it is not only because of their intrinsic interest I will argue later that they are treated by Bergson as parallel with his concept of hierarchies of duration.

Originally the calculus was established by using, not the concept of a function but that of a limit, and of the notion of “convergence” to a limit. To the beginner the notion of convergence to a limit, with its sense of endless approximation, may appear clumsy or counterintuitive, as if one were being required to “corner” velocity or acceleration or area by endlessly patching together ever smaller (or, in the case of the integral, larger) bits of time or space. In fact, if the “bits” are understood not as ad hoc increments but as parts of a series, and if the series are allowed to converge, the sought-for quantities suddenly appear as if by magic. In the calculus the student discovers a new way of thinking, one which is both dynamic and supple.

It is interesting, and a bit puzzling, that this new way of thinking did not emerge until the seventeenth century. Greek mathematicians had worked out many of the concepts necessary to it. Yet they seem to have halted at the entrance. One major problem lay in their unwillingness to accept change on its own terms. According to Morris Kline, the problem was both mathematical and cultural:

In the terms of another historian of mathematics, Salomon Bochner, there is simply an immense gap between modern “analytical variability” and “Greek stationarity.” Bergson could have added a second “static” limitation of Aristotle’s thought in this regard: his refusal to countenance kineseos kinesis (motion of motion).” Motion of motion, is of course, the definition of acceleration. Bochner asserts “Aristotle’s statement . . . ‘There cannot be motion of motion or becoming of becoming or in general change of change’ is a devastating self-indictment of Greek rational thinking at its root” (Bochner 168). The reference here is to Chapter 2, Book V of Aristotle’s Physics. Cf. J. P Anton, Aristotle’s Theory of Contrariety [New York: Humanities Press, 1957], 219-221).

This was not the only difficulty. The very logical rigor which made Greek mathematics exemplary also stood in the way of introducing any idea which could not be rigorously defined. Carl B. Boyer explains:

It is arguable that breakthroughs in the sciences require, first of all, imagination and audacity Later it will be possible to formulate the original intuition with sharp precision.

This sketch of some basic concepts of the calculus and of factors involved in its creation is intended neither a complete outline nor as a technical account It is intended, rather, to set the stage for Bergson’s appropriation of the calculus as fundamental to his metaphysics and epistemology; Bergson believed that the calculus represented (and furthered) a profound shift in understanding, one which made modern science possible. This shift involved both increased linguistic precision (mathematical precision) and a deepening sense of change — of mobility of all kinds. A renewed scientific and philosophical effort is necessary, he believed, to take account of it.

An Introduction to Intuition

1903 saw the publication of An Introduction to Metaphysics, in which the term “intuition” is introduced for the first time and contrasted with its contrary “analysis.” Many approaches might fruitfully be taken to this brief essay. It could, for example, be usefully interpreted in terms of musical form. But interestingly, in its culminating sections it is the calculus which is singled out not only as decisive for the natural sciences but as a key to the understanding of intuition and to the development of metaphysics. The reader of this article will probably not, by now, be surprised at this. But in the massive Bergson literature very little has been written about it. ((Jean Milet, Bergson et le calcul infinitèsimal, ou La raison et le temps, prèf Jean Ullmo (Paris: Presses Universitaires de France, 1974, 1. Jean Milet, “Bergsonian Epistemology and its Origins in Mathematical Thought,” in Bergson and Modern thought: Towards a Unified Science, eds. A. C. Papanicolaou and Pete A. Y. Gunter (New York: Harwood Academic Publishers, 1987), 29-38. I am indebted to these excellent studies, but disagree with the author over the extent to which Bergson accepted the calculus as a quantitative science as a description of the real.))

A thorough analysis of An Introduction to Metaphysics cannot be attempted here. What follows is an exposition and interpretation of some of its key passages. I hope this approach will dispel some of the confusions surrounding Bergson’s position in this work. I hope also, it will, by shedding further light on the place of mathematics in Bergson’s universe, allow us to make sense of the otherwise murky “vitalism” of Creative Evolution, and of the possible heuristic value of Bergson’s approach to biology.

Our usual way of thinking, Bergson observes, starts from static concepts (points, instants, lines, etc.) “in order to grasp by their means the flowing reality” (CM 224). But, he insists, we are capable of the reverse procedure, of working away from our usual concepts and assumptions towards the “flowing reality” This will upset our categories. But it may help us to arrive at fluid concepts “capable of following reality in all its windings” (CM 224). The procedure does not end here, however. From intuition, with its fluid concepts, it is possible on Bergson’s terms to return again to analysis. But this return may involve new conceptual systems and new symbols: new modes of analysis. An intuition is, thus, a “generative concept” (CM 225); transcending analysis through an inversion of our usual modes of thought, it nonetheless can enrich analysis and transform it.

Bergson proposes, therefore, that metaphysics adopt the “generative idea” behind the calculus and extend this to reality in general. Hence he concludes that one of the aims of metaphysics is to “operate qualitative differentiations and integrations” (CM 226).

The notion of qualitative differentiations and integrations, nonetheless, will seem strange to many thinkers. Possibly it will become clearer and less arcane if Bergson’s uses of it are explored. What follows will be an examination of three of these uses: (i)differentiation and the limit concept, as applied to matter (ii) integration and the relation of wholes to parts; (iii) temporal hierarchy and the interrelations of differentiation and integration.

Limit Concepts and Qualitative Rhythms

A rigorous limit concept was not introduced into calculus until the work of Augustin-Louis Cauchy in the early nineteenth century. Until then mathematics worked — often uneasily — with concepts like “infinitesimal” and “motion at a point.” Bergson’s qualitative calculus is not intended to replace rigorous quantitative thinking. Rather, it is intended to provide, among other things, a realistic basis not only for more mundane practical activities but also for the natural sciences, particularly astronomy and physics. His qualitative “differentiation” of matter terminates, at the limit, in rhythms which, however brief; are still rhythms, and dynamic.

The extent to which physical reality can retain its durational, modal character for Bergson and yet present characteristics which justify mathematical representations can be seen by examining a passage from Duration and Simultaneity. Suppose, he states, we try to imagine adding one instant to another to produce real process. In doing so we try to:

There is thus, in material reality a minimal but real connection between successive events and a “forgetting” by them of their predecessors. (DS 48) One thus has a quasi-epochal theory of material duration. ((This account of the duration of matter is also given, in almost exactly the same terms, in “Life and Consciousness”, the Huxley lecture, given in 1911 at the University of Birmingham, and published in Mind Energy: Lectures and Essays, auth. trans. H. Wildon Carr (New York: Henry Holt and Company, 1920), 3-36. Bergson states on page 8: “A consciousness unable to conserve its past, forgetting itself unceasingly, would be a consciousness perishing and having to be reborn at each moment and what is this but unconsciousness? When Leibniz said of matter that it is a ‘momentary mind,’ did he not declare it, whether he would or not; insensible?” Cf. 22,41. On this point cf. also David A. Sipfle “Henri Bergson and the Epochal Theory of Time” in Bergson and the Evolution of Physics, ed. Pete A. Y. Gunner (Knoxville: University of Tennessee Press, 1969), 275-.294. I am in essential agreement with professor Sipfle on this point, but prefer to say that Bergson has a quasi-epochal theory of the duration of matter, to distinguish it from the epochal theory of Process and Reality with its absolutely distinct units of becoming.)) Prom the viewpoint of applied mathematics this theory has obvious advantages. As noted above, it gives an objective basis to counting (since there is a real aspect of discreteness to the epochs or rhythms). It also provides such a basis for measuring temporal breadths (e.g., frequencies), so long as these are taken not as absolutes but as quasi-discrete. Temporal boundaries would have also been taken as approximative, that is, as limits. The same would hold for spatial location and extent, and for motion. But for Bergson, these limits are strictly ideal. Taken in themselves (as mathematical points and instants) they do not exist.

This account of systematically approximative character of mathematical descriptions will doubtless appear familiar to readers of Whitehead. Both Bergson and the early Whitehead conclude that, to quote Whitehead, “. . . an abstractive set as we pass along it converges to the ideal of all nature with no temporal extension, namely, to the ideal of all nature at an instant. But this ideal is in fact the ideal of a nonentity.” ((Alfred North Whitehead, The Concept of Nature (Cambridge: Cambridge University Press, 1955 [1919]), 61.))

Though Bergson did not develop an elaborate theory of extensive abstraction, it is not surprising that he should have viewed Whitehead’s The Concept of Nature, in which this theory is given its classic formulation, as “one of the most profound (works) ever “written on the philosophy of nature” (DS 62n).

Integration: Real Parts in Real (Dynamic) Wholes

Philosophy bristles with theories of “wholes” and “parts,” theories which reach from extreme atomisms (in which there are only parts, and wholes are at best mere aggregates) to extreme monisms (in which putative parts lose their identities to the whole). Bergson’s language, which stresses the wholeness of duration against the static fragmentariness of space, leads to the suspicion that his philosophy (as is argued in a recent essay in Process Studies”) falls into the latter camp, dissolving parts back into the whole, (and us, apparently with them).

Here, as is true more than once in this study, it is not possible to deal with an important question in depth. What can be done here is to point out the language which Bergson uses to talk about one self: that is, about parts (as opposed to mere “elements”) and about “integral experience.” The one-many view ‘which Bergson takes of the self will be applied by him both to biological evolution and to social organization.

The self Bergson states, is mathematically neither one nor many. Seen in itself; it is both (CM 198-199, 206-207, 218-219). In it “real parts” must be distinguished from partial expressions and partial notions (CM 202); lust as they must be distinguished from “elements,” which are abstractions (CM 199, 201, 206); just as actual parts must be distinguished from “fragmented parts” (CM 198) or from parts that are merely “juxtaposed” (CM 196). Equally, parts of movement must be distinguished from points of space. (CM 213, 215). It goes without saying that this is not the language of extreme monism. Nor is it the language of atomism. Speaking of states of the self; Bergson insists:

The Bergsonian self resists fragmentation into distinct, juxtaposed parts (as is attempted in, for example, associationist psychology). Yet even though the self must be conceived as a whole, it contains (consists of real parts which “encroach upon one another” (CM 198), which “dovetail.” This is not a unity which ablates its parts, it is a unity of parts. The parts, constituted by their mutual relations, constitute a whole which, in turn, influences its parts.

This is extremely important in understanding Bergson’s position, because the knowledge of the self by the self, is, he insists, the “privileged case” (CM 236) of his philosophy. It provides a model in terms of which other levels of reality are to be understood. Interestingly, he characterizes this kind of knowledge as “integral experience” (CM 237). Nothing could appear more qualitative than the Bergsonian self; nothing could seem more out of place with regard to it than metaphors taken from the calculus. Yet his language is clear. Intuition — even in the case of the self — is for him a qualitative Integration. A timeworn textbook example may be helpful here. If we are to find the area of a circle, we can circumscribe it with regular polygons with indefinitely increasing numbers of sides. By increasing the number of sides we never reach the exact area. But if integration is introduced, there is an almost-magical effect The limit of the series appears, the area is attained. So with the knowledge of the self by the self. As the patchwork of partially misleading “views” of analyses which misrepresent (yet circumscribe) it disappear, the fundamental insight is given. Without comparing many psychological analyses, Bergson insists, we cannot achieve such as insight (CM 236). But this insight, and the higher degree of temporality it achieves, follows from the integration of the parts, not from their obliteration (CE 152).

This sketch is of interest as it stands. It is of equally great interest in terms of its applications (which Bergson intends to carry out) to the rest of his philosophy. A biological organism is a whole for Bergson; yet he points out that each cell is itself an organism (CE 41-42, 162) hence an organism is a whole comprised of real parts. A society, he points out in The Two Sources of Morality and Religion, is a whole, and can be compared to an organism. Yet he is extremely careful to point out that it is a whole comprised of individuals, each of which has free will. (TSMR 3,73,107, 109) Evidently — certainly, with regard to these two cases — integral experience dissolves neither the knower nor the known.

Temporal Hierarchy and the “Fundamental Theorem”

We have noted, in Matter and Memory, the importance of Bergson’s idea that longer durations (of human awareness) can extend over far briefer physical durations. It is interesting that this notion is extended in the same work to include a hypothetical ordering of living organisms in terms of a hierarchy of durations: “In reality there is no rhythm of duration; it is possible to imagine many different rhythms which, slower or faster, measure the degree of tension or relaxation of different kinds of consciousness, and thereby fix their respective places on the scale of being” (MM 275). This speculation occurs in the context of a discussion of Bergson’s theory of psychological duration. In subsequent passage, however, he adds that it is possible to conceive “an infinite number of degrees” between matter and fully developed spirit: “Each of these successive degrees, which measures a growing intensity of life, corresponds to a higher tension of duration and is made manifest externally by a greater development of the sensory-motor system” (MM 296, Cf. also MM 332). Thus a full-blown temporal hierarchy is postulated, from the rhythms of human consciousness, through those of countless species of living organisms, to those of matter — longer durations extending over briefer durations, these extending over briefer durations still.

The concept of temporal hierarchy first stated in Matter and Memory is introduced again in An Introduction to Metaphysics, this time in extremely general terms:

This passage is two-sided. It is epistemological, describing different sorts of intuition, each correlative to the level of understanding on which it operates: “an indefinite series of acts, all doubtless of the same genus but each one of a very particular species” (CM 217). There is thus no one intuition, to which this term refers. There are many sorts of intuition, each appropriate to the rhythms and the qualities of its particular object It is metaphysical, since the different sorts of intuition are said to correspond to the different “degrees of being” (CM 217; cf. also CM 218).

So far this study has isolated two factors which are fundamental to Bergson’s thought: temporal hierarchy and the calculus. It is very likely that these two factors will be interrelated in some manner. But how? The reader will recall the brief but terminologically dense sketch, above, of the Fundamental Theorem of the Calculus, with its chaining of inverse relationships between differentiation and integration. It is a central thesis of this paper that the hierarchy involved in this chaining is understood by Bergson as congruent with the hierarchy which he describes as comprised of broader and briefer rhythms of duration.

That there should be a precise parallel between Bergson’s hierarchy of durations and the indefinite chaining of derivatives and integrals may seem unlikely. But closer inspection will render it plausible. From the vantage-point of his metaphysics, the goals will be, as noted above, to operative qualitative differentiations and integrations. But on these same terms, what else can there be to integrate/ differentiate in his universe except different “levels” of duration, broader or briefer. In the last analysis, for Bergson such durations are all that exist. To integrate will be to move towards broader, more inclusive durations: that is, towards durations which include the briefer durations as real parts.

Èlan Vital as Mathematics

To explore these conclusions it will be necessary to take a second look at Creative Evolution. Not only will this best known of Bergson’s works turn out to be a compendium of mathematical metaphors, but its central actor, the vital impetus (élan vital), will appear to have something intelligible to do: to create increasingly broad rhythms of duration. The evolution of life will be understood as the integration of lesser into higher temporalities, with an accompanying rise in the behavioral independence of the living organism relative to its environment. This will be the central meaning of Bergson’s “vitalistic” biology.

Life, Bergson proclaims, is not reducible to physico-chemistry But if he rejects mechanistic explanations of life, he also denies another kind of reductionism, which equates life with the simple step-by-step filling in of a plan. Both mechanism and finalism presuppose that “All is given,” the first through efficient causes, the second through a kind of cosmic blueprint present at the beginning (CE 39- 41,45,46) In rejecting both, Bergson insists that evolution operates within inherent limitations. Matter, with its entropic drift towards increasing disorder (CE 245-246) constitutes an obstacle. The ‘vital impetus, in turn, is not omnipotent. (CE 125, 141-142, 149) It is thus not surprising that the course of biological evolution, in its innumerable branchings, should exhibit so many dead ends (CE 107,116,129), halts (CE 104, 113, 125n, 132, 134), and regressions (CE 40,50-51, 100, 127, 131), or that is progress should be accompanied by increasing conflict (CE 103).

Life on this planet has, in the context of its limitations, managed a threefold success: plant and animal life, and among the animals, arthropods (including social insects) and vertebrates (including humankind. Plants, Bergson asserts, are “societies” (CE 16; 12) whose evolution may not require a vital principle. With animals, Bergson concludes, something more than mutation and natural selection is required.

An Introduction to Metaphysics precedes Creative Evolution by four years. The joint emphasis of the former on temporal hierarchy and the calculus emerge, not surprisingly, in the latter.

The change in outlook which led to modern physics and its mathematics thus might, extended to biology, lead to a new, more flexible, more temporalist paradigm (and perhaps to a new mathematical understanding of biology).

The creative action by which a new species is formed would involve a saltation which raises the species to a higher temporality. It would involve an integration from which, however, it would be possible to derive a derivative:

He adds that just as an infinity of functions may have the same differential (functions which differ from each other by a constant — the so-called “constant of integration”) so the integration of physico-chemical elements (a “summation” which, inversely, proceeds from the derivative to the function) would determine the vital action only in part. A part would be “left to indetermination” (CE 33).

A detailed discussion of the questions raised by these ideas would require a second article, longer than the present one. Perhaps the following comments will help explain what is meant. First, Bergson’s speculations here are not conceived by him as imaginary That is, they are understood by him in terms of the actual chemicals, energies, physical principles without which life could not exist. Second, they are understood by him as putative extensions into biolog, of fundamental insights which he believes are at the root of the paradigm shift which lead to modern physics and its mathematics. It would be consistent for him to hold that another such “inversion” could lead to a more dynamic, less reductionist scientific biology, one which could avoid strict deterministic explanations.

In what sense are we to understand these speculations? Bergson believes that the chemicals, energies, and physical principles without which life could not exist are misconstrued by us as being perfectly spatial, not as possessing degrees of spatiality (i.e, as characterized by their “extensity”). It is thus easy to understand the extent to which these factors, extensive and durational, could be believed by Bergson to be brought together into forms possessing broader durations. The qualitative calculus of life can be taken as having done so on our planet — to use one of William James’ favorite phrases — so far forth. But can a quantitative calculus, of any kind, succeed in fully understanding evolution and living organisms? As noted above, Bergson denies — in spite of the overwhelming success of the Newtonian physics of his time — that there will be a “theory of everything” in physics. (There is not, one notices, one at the present time.) So he denies that the utopian integration he suggests can be “more than dreamed of ” (CE 33).

Yet it is clear that it is in this direction — towards a more temporalist biology, utilizing a mathematics more suited to the “sinuousities” (CE 212-213) of life — he believed biology could most profitably proceed. ((That there is for Bergson a hierarchy of durations, a scala naturae, resulting from the divergent developments of evolution, is dear. The question immediately arises as to whether such a hierarchy also exists within each organism. Bergson does not give an unqualified answer to this question, but it would seem that on his own terms some such hierarchy sub specie durationis within the living organism must exist If each organism has its one unique rhythm of existence, and each cell is an organism, then it follows (a straight forward syllogism) that each organism contains as many temporalities (rhythms and sub-rhythms) as it has cells. This is interesting as it stands, and entirely up-to-date. (Cf E. Pennisi, “Multiple Clocks Keep Time in My Tissues,” in Science,, 278, No.5343(1997 (1560-1561). But it does not answer the question of hierarchical organization.)) An adequate answer is suggested, however, by Bergson’s statement in Mailer and Memory (quoted above) that levels of consciousness (hence breadths of duration) in each organism are precisely commensurate with the organism’s capacity for movement — more precisely, for its capacity to employ a wide behavioral repertoire, coupled with necessity of having to choose between specific acts. This capacity and its imperatives are dependent on the organism’s neuromotor system. A brain and nervous system in a vat are not an organism. The organism is a brain and nervous system intimately connected with a sensorimotor system. A higher organism, Bergson states, is essentially a neuromotor system installed on systems of digestion, respiration, secretion, etc., “whose essential function is to cleanse and protect it. . . . .” (CE 124; cf. also 121, 123, 126, 252) A Bergsonian temporal hierarchy in a higher organism must then be essentially threefold: Consciousness, brain and nervous system, and motor system, with the rhythms of the first “extending over” the second and those of the second over the third. It is interesting to note, in this context the importance brain rhythms have assumed for contemporary neurophysiologists. What science might lose in certainty it might regain in understanding, and in practical results. It is instructive in this regard to cite Bergson’s letter to C. Lloyd Morgan of November 21, 1912, expressing his belief that science did not have to remain as mechanistic as it was in their time. ((This letter is in the archives of the University Library at the University of Bristol. Provisions in Bergson’s will make it impossible to publish his previously unpublished writings, including his letters. I believe that in citing his opinion in this way, without quotes, I am not transgressing any official prohibition.))

A Mechanism for “Vitalism”; Hierarchical Integrations

The present essay could halt here. Its basic contentions have been sketched out, at some length, and, I hope, in such a way as to be intelligible. But to have stressed his theses of the part which an increasing awareness of temporality has played in the genesis of modern science, and to have indicated the way in which he parallels his theory of levels of duration with the “chaining” of integration/differentiation leaves one fascinating problem unsolved: namely, that of how, given what is known of genetics, Bergson could have imagined that the “vital impetus” effects its evolutionary saltations. What follows is a speculation on Bergson’s ultimate speculation.

One advantage that a broader duration will have over the briefer durations with which it is contemporaneous is its capacity to sum up successive briefer duration; making them simultaneous. ((Another advantage of broader over briefer durations is the capacity of the former to constrain the latter. This is one of Bergson’s fundamental contentions. It has been introduced by H. H. Pattee as a factor without which the cohesiveness of the organism cannot be understood. Cf. H. H. Patter, “The Problem of Biological Hierarchy,” in Towards a theoretical Biology, vol. 3 (Chicago: Aldine, 1970), 117-135.)) We have already seen that this is Bergson’s explanation of the manner in which the successive “vibrations” of objects around us are transformed by us into stable, unchanging surfaces. We are not aware of the successive release and absorption of photons in physical objects around us. The transformation of such activity into simultaneity and passivity is a function for Bergson of human perceptive consciousness. A similar integration is used by him to suggest how mind (i.e., memory) can act on matter (i.e., the brain). There is, he speculates, an element of indeterminacy in matter. By summing up this spontaneity (making its successive moments simultaneous) sufficient energy could be accumulated and released to influence the “hair-trigger” behavioral mechanisms of the brain. ((This theory is introduced by Bergson “The Soul and the Body,” a lecture first given April 28, 1912, and published in Henri Bergson, Mind-Energy 37-74 (cf. 21-22. also 18-19, 44-45). On this point also cf. Milic Capec, “Bergson’s Theory of the Mind-Brain Relation,” in Bergson and Modern Thought: Towards a Unified Science,, esp. 139-143.)) A similar explanation is available to Bergson to account for the manner in which the vital impetus could influence evolution,

Bergson was anti-Lamarckian. He rejected the ‘view that the state of the environment can influence the genes. Like the majority of contemporary geneticists, he held the opposite view the genes influence the body, and only changes in the genotype can influence the phenotype. Obviously, given what has gone before, he offers a different explanation of how the genotype is altered (mutations) than do contemporary geneticists. The two kinds of synchronization sketched above are different in kind. The first (synchronization of the perceptual object) leads to static spatiality, the second (synchronization of successive indeterminacy) to summed spontaneity. Either or both could influence the DNA double helix, systematically.

It will be objected that this explanation is closer to science fiction than to science. It no doubt appears so. The main point of this essay is that this theory — though it could well be wrong – is intelligible. The vital impetus is construed here as destabilizing and respatializing the genetic material, by a twofold synchronization. But it is hardly enough for a thesis to be understandable. It is also important that it be believable. For this latter eventuality to transpire, a number of prior claims would have to be established. Is there an element, however infinitesimal, of indeterminism in matter? Is matter really only partly extended in (into) space? The concept of fractal dimension at least gives us a way to make this possibility intelligible. ((Benoit B. Mandelbrot, The Fractal Geometry of Nature, updated and augmented ed. (New York: W. H. Freeman and Company, 1983), 1, 15, 17-18, 29-31, 362-368. Fractal dimensions need not he unitary; they can he fractions (of a dimension). One thus has a way, taken from a mathematics, of understanding Bergson’s notion that matter and mind are not fully extended in space. However, this matter is complicated. No more is asserted here than an analogy.)) Is DNA really a static “spiral staircase,” or does it contain an unsuspected dynamic, rhythmic structure? Can we conceive of nature as essentially dipolar, as possessing both “bottom up” and “top down” causality, the former tending towards mechanism, the latter towards spontaneity? This essay does not try to answer such questions. But its author cannot see that it is wrong to speculate, so long as speculations are labeled as such.