Foreword

Roger Buis, professor emeritus at the University of Toulouse (INP - Institut National Polytechnique) - already the author of a considerable amount of research into the biomathematics of growth1 - delivers here a true panorama of the relationships between biology and mathematics over time, and in particular over the course of the last century, accompanied by a series of profound epistemological thoughts, thereby creating a book of great rarity and value.

As we know, mathematics is ancient, just like the interest taken in living things. But the word "biology" only appeared at the beginning of the 19th Century and, whilst E. Kant has already confirmed - and as Roger Buis rightly reminds us - a piece of knowledge is scientific insofar as mathematics has been integrated into it, the explicit idea of applying mathematics to biology is found only with C. Bernard2 - one of the great references in the book.

In contrast with physics, biology resists mathematization, for understandable reasons: the variability of living things, their dependence on time and on the environment, diversity and the complexity of biological processes, the diffuse aspect of causality (sometimes circular) and the difficulty of mastering the operational conditions of experiments have made obtaining consistencies problematic. Hence, the overall appraisal which may be interpreted as disappointing: apart from some specific sectors (in particular genetics), few laws are proven in biology, and even fewer that express themselves in mathematical language.

Roger Buis comments on this, but, going further than the "epistemological obstacles" - coined by G. Bachelard - he takes up the challenge. Even though the description in vernacular language (important in both natural history and Husserlian phenomenology), will always remain the most important in biology, in this discipline we more readily use "modeling" than "demonstration". Nevertheless, mathematics has transferable applications to biology: beyond the savings made by the move to symbols, using mathematical language is not simply using a "language", but a true instrument of thinking, of a remarkable tool of intelligibility which, whilst allowing hypotheses to be clearly laid down, will verify the conclusions by the same amount. Because - let us not doubt it - in biology and elsewhere, the scientific approach is always hypothetico-deductive. Whilst certain preliminary conjectures are less significant here than in physics - for example, the choice of a reference frame (dominant, certainly, in factor analysis, but hardly relevant, in general, elsewhere in biology) - others like approximations or simplifications that we will cautiously allow ourselves to use (linearization, or even quasi-stationarity of certain processes) are essential in this and necessarily lead to significant consequences. At least an advantage is drawn from this: modeling allows controlled experimentation. Thus, the modification of a parameter in a model that is elsewhere structurally stable is going to be possible at will. Mathematics, as a result, does not only provide symbolisms. It also contributes concepts and operating modes that allow real life to be simulated3.

By examining history, we also realize that, to use the expression of Roger Buis, mathematics has "sculpted" biology. From the point of view of the continuous, geometry, since ancient times, has given rise to the consideration of symmetries and continuous transformations, implicitly presented by Aristotle with the ago-antagonistic couple of power and action. The first separate formalizations appeared from the medieval period onwards, with the famous Fibonacci sequence, which, founded on strong hypotheses, provided the first model of the growth of a population (as it happened, rabbits). Then, in the Classical epoch, there was the era of the first phyllotaxic "laws" relating to the growth speeds of stems or leaves of plants, as well as to the mechanics of wood and its constraints (G.-L. Buffon, L. Euler). Finally, during the 19th Century and especially during the 20th Century, calculation of probabilities was based on development of Mendelian genetics, then of the genetics of populations, and subsequently on statistical biometrics (R. Fisher). Whereas C. Bernard highlighted the stationarity of the interior environment of living organisms, principles of optimality, underpinned by the calculation of variations (seeking the extremums of a functionality), are going to become dominant in biology, in particular in plant biology. Then formalisms from system theory (L. von Bertalanffy) encouraged A. J. Lotka and V. Volterra to model the dynamics of interactions between species (prey-predator systems, parasitism) with differential equations. At around the same time, projective geometry or geometry of transformations of coordinates will allow the morphology, shape and growth of living things to be summarized. The large project of a universal morphology, inaugurated by J.W. Goethe on the subject of plants4, began to be mathematized by D'Arcy Thomson, whilst awaiting the development of differential topology. With A. Turing and his reaction-diffusion systems, the mechanics of gradient - of which R. Thom later made great use in his famous "theory of catastrophes" - began to be introduced into the theory of morphogenesis5. Soon, the theory of automatons by J. Von Neuman took its turn, and cybernetics by N. Wiener with his command theory and retroaction loops, which were used amongst others in the description of hormone mechanisms. In addition, Roger Buis does not leave aside the formal grammar from N. Chomsky, at the origin of L-systems by A. Lindenmayer (useful to formalize the growth of certain algae), networks from Petri, well-suited to the logical representation of certain plant morphogenesis, the direct or indirect input from quantum physicians (such as N. Bohr, M. Delbrück or E. Schrödinger) to molecular biology, the influence, also on this, of information technology, with the notion of "program", of linguistics (R. Jakobson) and of the theory of information (C. Shannon) with the notion of "code". He also collects in great detail all the inputs of structuralist thinking in mathematics which, from the theory of Eilenberg-Mac Lane categories to that of graphs and networks, have allowed a systematic and relational biology to develop, in which the notions of self-organization, emergence, complexity6, scale invariance, order and disorder have become dominant today, leading in fine to the construction of biomimetic automatons (artificial life) and the development of an entire bio-informatics approach relating to simulation.

Obviously, given the spontaneous interaction of the disciplines, the existence of these empirical developments does not constitute a justification in itself. They therefore deserve to be revisited and for us to ask of them: when and under what conditions mathematics is really productive in biology? What do we expect to gain from applying it? Which mathematics should we use, where and why? Roger Buis, in the last chapter of his book, broaches these questions with courage and answers them very precisely, underlining each time the benefit that mathematics brings to biologists. If not predicting, is it about describing or explaining? Do we aim for architectures or processes? In contrast to modern philosophy, which often restricts itself, in the manner of Heraclitus, to dwell on the influence of difference, science - Roger Buis demonstrates this forcefully and epistemologists can but approve it - has the objective of finding invariants. The important thing is not that something changes. The important thing is to consider what does not change within things that change - because it is an invariant only in its connections to transformations. And we find some in biology and in physiology, as well as in physics. F. Cuvier, É. Geoffroy Saint-Hilaire and E. Haeckel already explained some. Today, we see them in metabolic cycles, macromolecules (DNA and RNA), genetic code (to the nearest few exceptions) and cell theory (J. Monod). It remains that in biomathematics, they must be linked to time and space. Since then, the use of continuous, or discrete, formalisms, of spatialized models of random or kinetic regulation processes - models such as those that Roger Buis studies competently and in detail - will contribute to their appearance. In conclusion, the singularity of living things must not be seen as an obstacle, and even though precautions and a certain modesty is required - because the model is not reality, it is, at best, only an isomorphic representation7 - there is no doubt about the usefulness of mathematics of living things in a well-defined conceptual framework, and that it allows us to achieve the objective of all well-understood science: making sense of what we are studying.

This eloquent pledge by Roger Buis in favor of biomathematics - a defense and illustration of rational models that are available - is much more than a simple memento or a catalogue. It is a true epistemological and scientific reflection, precise and nuanced, nourished in wide-reaching culture and which overcomes fractures and controversies. From Aristotle to G. Canguilhem, great names of Western thinking are found, which means that philosophy, and even the honesty of man, is not out of place. Going much further: the structure, which also reinforces the convictions of the...