Olivier Rey is a researcher at the CNRS, professor of philosophy at the University of Paris I and mathematician. He is a member of the Institute for the History and Philosophy of Science and Technology. He is the guest of the seminar "Changer d'échelle - Changer de nature" for his book "Une question de taille" published in 2014.

Olivier Rey's background as a mathematician has stimulated his interest in size issues, the subject of today's lecture. He has worked on nonlinear partial differential equations. In short, in the non-linear, when the change in size occurs to the point of crossing thresholds, these thresholds induce drastic qualitative changes. A great British mathematician, famous for having decoded the German Enigma machine during the war, Alan Turing was interested in morphogenesis, which is linked to the differentiation phenomena caused by size. Morphogenesis is the set of laws that determine the shape and structure of tissues, organs and organisms. During embryogenesis, differentiation phenomena will appear. They do not originate directly from the genes, as these are all the same in the cells. The growth of the embryo determines the differentiation of the cells, a growth that leads to bifurcations and causes the cells to differentiate from each other.

He imagines spontaneous differentiation mechanisms and assumes that cells differentiate through a cellular reaction-diffusion process. In this model, two molecules act in concert in certain chemical reactions: the first acts as an activator but also stimulates a second molecule acting as an inhibitor that diffuses more rapidly into the cells. In an embryo, this double reaction will create chemically different cell patterns that are the basis for the formation of distinct tissues.

Turing's additional hypothesis is that the activator diffuses into the medium but less quickly than the inhibitor. So what happens? At this point, due to catalysis, there are more and more activators and inhibitors. As the activator spreads slower than the inhibitor, there will be a peak of activators and a peak of inhibitors in different places. There will be a small zone around which the activators, surrounded by a ring of inhibitors, will dominate. In other words, the reaction will take place in one area and not in another. A randomly produced reaction will cause spontaneous differentiation. Everything ends up being equalized. This type of pattern can only occur when the area is large enough. The spontaneous differentiation system imagined by Turing can only occur at a certain size.  

One of Alan Turing's favourite models was the starfish because it is round in shape when it is small. The circular symmetry of the starfish will be broken at a certain stage of its growth and the branches grow spontaneously. This mechanism described here explains only some of the spontaneous differentiation phenomena.

Modern thought is not comfortable with these phenomena of non-linearity and questions of size.   Within the infinite Euclidean space that will come into play at the time of the advent of modern science, the world is no longer closed, as the title of Alexandre Koiré's book "Du passage du monde clos à l'univers infini" indicates. The idea of infinite space goes hand in hand with the idea of absolute measurements because absolute reference elements do not exist as previously the size of the entire cosmos.

In one of his fragments, Blaise Pascal asks his reader to think of a ciron, a tiny mite, the smallest animal known at the time. He asks them to imagine its legs and joints, then its veins down to the blood that irrigates them, and to push the vision to the smallest the reader can conceive. He develops a kind of fractal conception of the universe, i.e. one that has a similar structure at all scales. Pascal's reasoning is essentially rhetorical because he wishes to make man feel this disproportion, a "nothingness with respect to the infinite and a whole with respect to nothingness". This mental experience would not have been possible before the advent of the thought of the infinite.

Galileo's last work, Discourses on Two New Sciences, which deals with material science and what would later be called rational mechanics, returns to these questions of non-linearity. The text begins with a visit to the arsenals of Venice by three characters: Sagredo, Salviati and Simplicio. Around a very large ship waiting to be launched, a support apparatus is erected, much larger in proportion than that surrounding the smaller ships. An old man in the dockyard says that this is done with ships of this size to prevent them from breaking up, crushed by their weight. Sagredo, the open and intelligent character in Galileo's book, criticises the popular view that projects on a small scale cannot succeed on a larger scale. For Sagredo, mechanical demonstrations based on geometry and geometrical properties are the same at all scales, so a large machine must work just as well as a small one, provided that the proportions are respected. Salviati, Galileo's spokesman, then enters the scene, criticises popular common sense but says: "Do not believe any longer with many of those who have studied mechanics that machines made of the same materials scrupulously reproducing the same proportions between their parts must be proportionally apt to resist or yield to assaults and shocks from the outside, for it can be demonstrated geometrically that the larger ones are always less resistant than the smaller ones. So that in the end all machines and constructions, whether artificial or natural, have a necessary and prescribed limit which neither art nor nature can exceed. Provided that the propositions and materials remain the same". Galileo states very clearly that the world cannot be invariant by change of scale. In general, the different quantities do not vary proportionally with respect to each other. This results in changes in appearance when sizes vary and in what are called bifurcations.

Abrupt changes occur when thresholds are reached, which is called a break in strength of materials. The example given by Galileo of giants is developed by Olivier Rey. If all the dimensions of the giant are increased tenfold, its volume and weight are multiplied by a thousand (a man of 2 metres and 100 kilos becomes a giant of 20 metres and 100,000 kilos). As a result, the forces exerted per unit area of the section of a bone such as the femur, when the giant is standing, are multiplied by ten, and at the first step he breaks his leg.

These words may seem trite, but it is not clear that modern thought has drawn all the consequences. Sagredo, the Galileo character, seems further from the truth than the old man full of traditional wisdom he met at the arsenal. The emancipation of thought from natural dimensions appears to be a sort of conquest of modernity. D'Arcy Thompson, a leading figure in theoretical biology in the first half of the twentieth century, cites the thoughts of two great scientists as examples: the physicist Oliver Heaviside and the astronomer John Hershel. The former used to say that "there is no absolute scale of size or universe, for the universe has no limit either in the immensely large or in the immensely small". The second said that "he who devotes himself to the study must renounce the distinction between large and small, which is totally erased in nature".

Subsequently, questions of scale are still not sufficiently taken into account. J.B.S. Haldane, the great biologist of the first half of the twentieth century and a specialist in genetics, remarked: "The most obvious differences between different animals are differences in size, but for some reason zoologists have paid singularly little attention to these. In a thick textbook I have in front of me I find no mention of the eagle being larger than the sparrow and the hippopotamus being larger than the hare, although some hints are given about the mouse or the whale. And yet it would be easy to show that a hare could not have the dimensions of a hippopotamus, nor a whale those of a herring. For every animal there is a suitable size, and a great variation in size necessarily leads to a change in shape".

Indeed, the first characteristic of an animal is its size. A giant gull ten times bigger than a normal gull would be frightening. The fear of this animal is determined by the size and the shape is linked to the size. D'Arcy Thompson, like J.B.S. Haldane after him, insisted that, for simple reasons of physics, such as those uncovered by Galileo, size cannot be considered as a secondary parameter in the characterisation of a living form: size determines, to a large extent, the type of organisation possible. The need for food is proportional to volume, while the possibilities of feeding are proportional to surface area. However, for a constant shape, the ratio of surface area to volume varies as the inverse of size: if the cell is too large, it can no longer feed itself and division becomes necessary to re-establish a viable surface area/volume ratio (without prejudging the actual mechanisms that control this division). Increasing size requires increasingly complex structures to be overcome, and "the most evolved animals are not larger than the least evolved because they are more complex, they are more complex because they are larger".

It should be noted that while size imposes a more complex organisation, it also has certain advantages: the number of potential predators is reduced, and temperature regulation, for homeothermic organisms, is facilitated by the reduction in the ratio of the external surface, through which heat loss occurs, to the volume from which heat must be conserved, whereas this is a constant challenge for small animals. For example, the oxygenation of insects by simple diffusion of air along the fine tubes that are the tracheae and tracheoles is only practicable over very short distances, with the result that the body thickness of an insect or arachnid can hardly exceed half a centimetre and that, in reality, the giant spiders in horror films, apart from never having been able to reach such a size, would die during the asphyxiation session. Crustaceans, whose anatomical plan resembles that of insects, can reach larger dimensions because they have a respiratory system.

As J.B.S. Haldane again points out, for every type of animal there is an optimal size. A man of three metres would be impossible. Only in the imagination of Swift and his Gulliver readers can Brobdingnags twelve times his size and Lilliputians ten times his size coexist: in reality, the same organic form could not be viable at such different scales.

It is remarkable that Galileo, while working to dismantle the traditional cosmos and seemingly paving the way for a purely relative conception of measurement, also provided the means to re-found the idea of absolute measurements on new grounds at the end of his life, by showing that the world cannot be invariant by changing scale. The opposition or disjunction between the quantitative and the qualitative is largely weakened, since quantity determines to a large extent the possible qualities, and qualities can only be realised within quantitative limits that are not very extensible.

It is no less remarkable how modern thinking seems to evade this relationship: the whole functioning of nature is perfectly adjusted and the size of each object is precisely adapted to this functioning. Our brains imagine linearity by extrapolating questions of size to what we know without taking into account changes in regimes and bifurcations because that is the easiest to imagine.

Another reason for the almost systematic underestimation of the importance of questions of size in the very definition of things is the gradual autonomisation, in the course of the modern period, of what was first called natural philosophy, leading in the 19th century to an almost total separation between science and literature. Mathematics was put on the side of science, in consideration of its essential involvement in modern physics. Philosophy was placed on the side of the humanities, by virtue of a persistent link with the humanities, and because it was dispossessed by the new science of its former scientific claims. Philosophical thought tended to show an increasingly marked indifference or even disdain for quantitative questions, which were left to scientists and their calculations.

Nevertheless, some sociologists have drawn attention to these questions of scale by pointing out that the first characteristic of a society is its size because its organisation depends on its size, some solutions are possible while others are not. Once our attention is opened to these questions of size, a whole universe of reflections opens up on questions that are always considered for their own sake, detached from quantitative horizons, even though these qualitative horizons determine these questions.

For more information : Entretien avec Olivier Rey, Le Philosophoire 2017   and Penser les limites

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