Interdisciplinary Science Reviews, 12, B. B. Mandelbrot: 1987, 117-127 It occupies a critical position along the tortuous path that eventually led to fractals. This, and the fact that this text appears in a Towards a second stage journal called Interdisciplinary Science of indeterminism in Reviews, seems to call for a few philosophical and autobiographical comments. My research science career, which must be described as “improbable,” was triggered by a casual side interest in diverse isolated empirical Benoit Mandelbrot regularities that everyone else viewed as of little consequence. As I look back, my life divides into three well-separated periods. When the International Congress for Logic, A period of gestation started with my PhD Methodology, and the Philosophy of Science thesis in 1952 and lasted until 1964. was held in Jerusalem, in September 1964, I Jumping ahead, the third period that delivered an invited address titled “The started in 1975, witnessed consolidation and Epistemology of Chance in Certain Newer increasingly broad, rapid, and smooth Sciences.” But I hardly tried to prepare a development, marked by books that do seem text for the Proceedings, and for many years to involve an effective mix of technique and I kept resisting friendly suggestions – philosophy. Fractal geometry has the special notably by the Berkeley molecular biologist charm of allowing uninterrupted interplay Gunther S. Stent – that the draft be between concrete fields (ranging from widely reworked, completed and printed. One practiced ones to the very obscure) and reason was that success kept eluding sophisticated pure mathematics. It has been repeated attempts to state a technical point, successful as mathematics. In fact, it has while also making clear its philosophical shamed the iconoclastic tradition that ran implications. But it is good to see the old from Laplace to Bourbaki by stimulating or text published at long last. It has been reviving several mathematical theories; it substantially edited for style and shortened, has become a widely used tool in the but not otherwise modified, and it is description of nature and in the wide search preceded by a few pages of miscellaneous for order in chaos; and finally, fractal art is observations, which have been recast in the now becoming widely admired as art, form of a dialogue. References were updated irrespective of its unusual origin. in 2002. The middle period lasted from 1964 to 1975. From the viewpoint of fractals' development, it was in many ways the most 1. Reflections from the interesting but from a personal viewpoint it perspective of 1987 on a was most frustrating. This period was premature fractal manifesto punctuated by successive fractal manifestos, the most notable ones having been a 1972 written in 1964 lecture at the Collège de France in Paris, which followed a Trumbull Lecture at Yale in While the word fractal did not appear until 1970, and the even earlier 1964 Jerusalem 1975, this 1964 draft was important in the lecture with which we deal here. evolution of fractal geometry, an When chance or duty makes me reread interdisciplinary enterprise I conceived in this and other unpublished texts of the 1964, then developed. I have devoted to it middle period, I am surprised at the almost all my creative life. precision and clarity given to many ideas that were not fully worked out until much later in Question: my life. But my style failed to encourage the Why should this old text be of historical reader to plow through papers that had interest today? already acquired the reputation of advancing very disturbing ideas. It is useful, therefore, weather and of prices were “wild.” I used to to state at this point one basic idea of use “erratic,” an ill-chosen Latin word that fractals. did not last. My work invited the sciences to Why is school geometry so often described move on to a second stage of indeterminism. as “cold” and “dry?” One reason is this How was this invitation received? geometry’s inability to tell what shape a Certainly not to my satisfaction! On the one cloud is, or a mountain, or a coastline. hand, many influential scholars considered Clouds are not spheres, mountains are not my discoveries to be potentially important, cones, coastlines are not circles, and more and offered me a series of renowned pulpits generally, man’s oldest questions concerning from which to present them. Yet, until 1975, the shape of this world were left unanswered they were called controversial. In fact, they by Euclid and his successors, who concerned provoked little discussion, pro or con, to themselves exclusively with an unrealistically justify them being so called controversial. orderly universe. In order to achieve a They failed to affect the work of numerous, handle on nature, a radically different diverse, distinguished and often well- geometry is needed, one that must disposed people who heard me. To use a contradict many old ideas that have become term favored by Stent, my work suffered so familiar as to seem obvious and from being “premature.” universally valid. However, to negate these ideas completely would be self-defeating Question: because it would replace excessive order You have said that, in your work, a with utter chaos. Fractal geometry is a new growing role is played by sophisticated and very different broad area of order within graphics, dear to a geometer’s and an artist’s the domain of the old chaos. Some fractals eye. Could you elaborate? imitate the mountains and the clouds, while others are wild and wonderful new shapes. BBM: More generally, the new fractal world is in Being premature is particularly painful some cases hard to tell from the real one, when one’s whole scientific work has been and in other cases it is of fantastic and interdisciplinary. Thus, it is unusual indeed surprising beauty. that fractal geometry managed to survive and to become part of the mainstream, Question: without having to be first forgotten and later Is there any relation between second stage rediscovered by others, when its time came. indeterminism and chaotic fluctuations? Why did its time come after 1975, but not before? We cannot be sure, except that an BBM: essential role has clearly been played by The conventional wisdom has long been computer graphics – of which I became a that the study of the weather and of pioneer by necessity. economics is harder than the study of perfect Mention of Stent’s paper necessarily brings gases, but will eventually use the same forth a thought concerning the issue of means to achieve the same degree of uniqueness in scientific discovery. Indeed, perfection. To the contrary, my work Stent draws our attention to the (hostile) suggested a profound qualitative distinction review that the biochemist Erwin Chargaff between the underlying fluctuations, and as wrote of The Double Helix by James D. a result the theories of the corresponding Watson. In that review, we read that phenomena were bound to differ sharply. On “Timon of Athens could not have been the one hand, the fluctuations that written, Les Desmoiselles d’Avignon could characterize the theory of gases should be not have been painted, had Shakespeare and viewed as “mild,” and the first stage of Picasso not existed. But of how many indeterminism in science was comparatively scientific achievements can this be claimed? easy because of their being mild. On the One could almost say that, with very few other hand, the facts already established by exceptions, it is not the men that make 1964 indicated that the fluctuations of the science, it is science that makes the men. What A does today, B or C or D could surely that – for better or worse – were greatly do tomorrow.” affected by my peculiar life story. Would This may be true of many of the individual another individual, or some collectivity, have strands of fractal geometry. But fractal reached the same philosophy and built the geometry is not merely a juxtaposition of its same whole? A worthy question for the individual strands. It arose as an integrated future, assuming that this whole actually whole, ruled by a philosophy that was survives. conceived and developed under conditions 2.1. Differences in scientific 2. Text of the Premature development It is often asserted that differences in Fractal Manifesto of 1964 development between sciences are solely due to differences of “age” as measured from Since the turn of the century, acceptance the earliest systematic investigation of the of indeterministic stochastic theories in different topics. I disagree. Indeed, science has spread spectacularly. A new probability theory saw its first triumphs in epistemology has arisen as a result, physics, but first arose in the study of the superseding the epistemology built upon statistical problems raised by economic- deterministic causal theories. In certain psychological choice. In the hands of areas of physics, the new approach was Laplace (circa 1800), a probabilistic view of rapidly and strikingly successful, for example social science and an arch-deterministic view in the study of thermal fluctuations in gases of physics had reached a high point at the and in solids, and in quantum mechanics. same time. Even as late as 1912, statistical Elsewhere, progress has turned out to be social science could still be presented as a slow, and the fulfillment of high initial model to be followed by statistical physics. expectations is continually postponed. Such Similarly, in the works of Boussinesq (1872) is the case of meteorology and in most of and Osborne Reynolds (1895), the statistical economics. The present paper proposes to concept of turbulence in fluids was roughly trace this difference to the existence of a contemporary with Maxwell’s and deep qualitative contrast between the nature Boltzmann’s (1866) kinetic theories of gases.