Biology and Mathematics

History and Challenges

Roger Buis

• Publisher: Wiley-ISTE
• Year: 2019
• ISBN:
  9781119663515
• Edition: 1
• Topic: Medicine
• Subtopic: Biostatistics

To formalize the dynamics of living things is to search for invariants in a system that contains an irreducible aspect of "fuzziness", because biological processes are characterized by their large statistical variability, and strong dependence on temporal and environmental factors. What is essential is the identification of what remains stable in a "living being" that is highly fluctuating. The use of mathematics is not limited to the use of calculating tools to simulate and predict results. It also allows us to adopt a way of thinking that is founded on concepts and hypotheses, leading to their discussion and validation. Instruments of mathematical intelligibility and coherence have gradually "fashioned" the view we now have of biological systems. Teaching and research, fundamental or applied, are now dependent on this new order known as Integrative Biology or Systems Biology.

1
On the Status of Biology: On the Definition of Life

Stating that biology is the “study of living things”, or processes that take place in the latter, comfortably sidelines what is meant by the “life” part of this term, which is supposedly their sui generis characteristic in comparison to “non-living things”. Without doubt we can make the observation that the definition of life can only be an ideal point of view, otherwise seen by some as useless speculation. However, insistence on the question demonstrates the originality of biology in comparison to other disciplines that do not appear to have this type of requirement. Physics, for example, does not need to “define” matter or energy in order to study the phenomena in which they intervene, giving them instead the role of “variables” that can be manipulated, whether theoretically or experimentally. Let us recall the basics of the fundamental concept of force, introduced in Newton’s second law, using which we can deduce a satisfactory definition of it by multiplying the mass and the acceleration of a movement¹.

Let us return to C. Bernard, who provides an explanation of the notion of life. Stating that “physiology is an experimental science [that] has no place giving a priori definitions”, he considered that “it is illusory and irrational, contrary to the very spirit of science which is to seek an absolute definition”². In fact, we can state that biology ignores the notion of life, whereas this is paradoxically the purpose of its field of study. In summary, he goes as far as saying that the field refers to objects that common sense describes as “living”. But this does not prevent the definition of “living things” (as opposed to “life” itself) – moreover, we prefer to talk about singularity rather than look for a true definition of it – remaining a theme of reflection, the subject of university PhDs, stimulated by the latest developments in research on the initial conditions of appearance of life, as well as sometimes by metaphysical considerations that we do not need to include here.

Unfortunately, this pragmatism can only sideline, without resolving, the remaining reason for the existence of the debate. There is, in fact, a debate to be had if we note the diversity of the points of view, ranging, for example, from the statement by the biochemist Szent-Györgyi that “life as such does not exist”, to attempts at a definition, if not about the word “life” designating a specified entity, then at least about the nature of what makes an organism “living”. Although Aristotle proposed a fundamental definition of it, based on a property of autonomy and self-reproduction (see Chapter 3), modern biology considers that life is an abstract entity that it is unable to characterize. François Jacob expresses this in the following way:

“Life is a process, an organization of matter. […] We can therefore study the process or the organization, but not the abstract idea of life. We can attempt to describe and we can attempt to define what a living and a non-living organism is. But there is no ‘living matter’”³.

Let us agree that the designation “living matter”, refused in the above, is simply a convenience of language, like the famous “vital force” advocated by vitalists. And, since it is necessary to take a side, let us admit the following position concerning our subject, which will be illustrated several times over in the following.

DEFINITION.– Any living object is a precise assembly of interactive “elements” (meaning matter, energy, information), and a system of this kind always faces a risk of instability. It is therefore necessary to search for the structural characteristics and dynamic properties that ensure correct maintenance, development and reproduction of it.

Faced with this issue, mathematics, which is itself a science of structures and transformations, steps up to participate in this study and, better still, to establish a coherence of dynamic representation, in part invariant and in part fluctuating (adaptation).

With this in mind, we can refer to the principle stated by H. Atlan (and independent of its position with regard to the deterministic/random debate): “The only specificity of living things relates to the complexity of their organization and the activities that accompany them”⁴. Obviously, we need to specify this notion of complexity and distinguish it from what it is in non-living systems (Chapter 3). In the meantime, let us say that the notion of life, understood in this way, consists of admitting the following property.

PROPERTY.– The most characteristic processes of any living thing, which can be described as sui generis, are autopoiesis (continuity of an autonomous production of oneself) and compliant reproduction from one generation to the next.

Without spending too much time on this question, we at least need to agree on what and how biology presents a demonstrated specificity with respect to other disciplines, leading us to say, in the words of the biologist E. Mayr, that it is “a science unlike any other”⁵. We call on some past considerations in order to do this.

Let us first recall that the word “biology” is relatively belated, dating back only to 1802, proposed jointly by J.-B. de Lamarck in France and G.R. Treviranus in Germany. For G.R. Treviranus, “biology or philosophy of living nature” (“Biologie oder Philosophie der lebenden Natur”) must aim to study “various phenomena and forms of life, the conditions and laws that dictate its existence and the causes that determine its activity”⁶. For his part, J.-B de Lamarck entitled one of his lessons “Biology or considerations on nature, faculties, developments and the origin of living bodies” (1812). Moreover, he specified:

“The name of living bodies has been given to these singular and truly admirable bodies… They effectively offer, in themselves and in the various phenomena that they present, the materials of a particular science that has not yet been founded […] and that I shall name biology”⁷.

With this new terminology, the objective appears first as the desire to group together all the studies, scattered to a greater or lesser extent, that relate to living things, going beyond the observation and classification of living things whose morphology was the basis for recognition, naming and classification. This first requirement did not prevent a reasoned practice of the use of living things developing in parallel within this natural history, duly distinguished by their nature and their state of development (pharmacopeia, feeding, clothing, housing). This implied a technical experimentation that should not at all be considered negligible (harvesting and hunting, then agriculture and domestication). But the most important point in this “profession of faith”, namely biology, is now to consider life as a singular phenomenon whose two aspects are specified by it as: its generality and its diversity. A two-fold task is therefore assigned to this new discipline: (i) looking for properties or characteristic laws that have a certain universal value and (ii) studying the diversity of how these are manifested in reality, including the truly historical aspects⁸.

The eruption of the word “biology” is often (wrongly) considered as the declaration of the status of this discipline as a true experimental science. At least, this was written by F. Magendie (1783–1855), in reference to Galileo, writing in his era in his Principe élémentaire de physiologie [An Elementary Treatise on Human Physiology] (1833): “in order to know nature […] it was necessary to observe and above all interrogate it via experiments”. This position was taken up and developed by C. Bernard, who studied the lessons given by Magendie, and whose successor he became at the Collège de France.

In fact, it must not be forgotten that experimentation on living things has been around for a long time. Let us recall a few notable examples. In Antiquity, we have Hippocrates and his experimental studies on the development of a hen’s egg. Then the 17th Century provides us with remarkable research by W. Harper (1578–1657) on blood circulation. An important point is that the originality of these studies was the methodical implementation of quantitative measures of blood flow under different conditions (application of a tourniquet, then its release). Let us note also in this era the tests by L. Spallanzani (1729–1799) that led to the disproving of the theory of spontaneous generation and the highlighting of the role of gastric juices in digestion.

Natural history therefore realised very early on that the anatomical description of structures must precede research into their functioning, without remaining limited to questions of compared morphology and classification, which led to progressive labeling of the reality of various fields of study of phenomena involving living things.

1.1. Causality in biology

On the sidelines of this fundamental topic, it is necessary to mention an old question that has been around in the field of biology for a long time, in particular in embryology, in reference to whether the development of an organism consists of deployment of pre-existing structures in the egg, or, on the contrary, whether ontogenesis takes place according to a spatially and temporally organized sequence of generating processes. This was the famous dilemma of “preformation versus epigenesis”. We all have in mind certain famous figures from the past in anthropology, illustrated representations of a miniature being (homunculus) that is thought to be housed either in the head of the spermatozoid, or in the ovum. This debate was already alive in the era of Aristotle, who was a supporter of epigenesis or the progressive formation of a series of structures. For a long period of time, this position remained up to date, fed by various staunch positions in which naturalists and philosophers played a part, as well as physicians such as Maupertuis who, sup...