THE SCIENTIFIC HERITAGE OF VITO VOLTERRA AND ALFRED J. LOTKA IN MATHEMATICAL BIOLOGY GIORGIO ISRAEL Dipartimento di Matematica Università di Roma "La Sapienza" P.le A. Moro, 2 00185 – ROMA

It is rather ironical that the term used to refer to what are perhaps the most famous equations in biomathematics, the "Volterra-Lotka equations", should bring together the names of two so different scientists, so radically far apart in their backgrounds, conceptions and scientific methods. Indeed, the very formulation of these equations was an occasion for a dispute over the relative priority of the two scientists which revealed the different nature of their viewpoints and the difficulty they had of even communicating. And yet modern biomathematics owes much to Vito Volterra and Alfred J. Lotka. Before them — that is, before the appearance within the span of a few years of several works laying the foundations of a new line of research — the applications of mathematics in biology did not amount to a great deal. We have above all Karl Pearson’s biometric research, Raymond Pearl’s, Lowell J. Reed’s and Pierre F. Verhulst’s studies on the logistic equation and the first mathematical model of matter elaborated by Sir Ronald Ross1. Despite the interest and importance of this research, the difference of approach displayed in the years 1924-26 in Lotka’s book Elements of Physical Biology2 and Volterra’s works3 (not to mention the book published by the latter in 19314) was striking. Mathematics was no longer an ancillary tool with the limited function of "assisting" the analysis of biological phenomena but actually provided the analysis itself with the necessary

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See S. E. Kingsland, Modeling Nature, Episodes in the History of Population Ecology, Chicago, The University of Chicago Press, 1985. 2 A. J. Lotka, Elements of Physical Biology, Baltimore, Williams & Wilkins, 1924 (reprint Elements of Mathematical Biology, New York, Dover, 1956). 3 V. Volterra, “Variazioni e fluttuazioni del numero d'individui in specie animali conviventi”, Memorie della R. Accademia dei Lincei, S. VI, II, 1926, pp. 31-113; “Fluctuations in the abundance of a species considered mathematically”, Nature, CXVIII, 1926 pp. 558-560; “Variazioni e fluttuazioni del numero d'individui in specie animali conviventi”, Memorie del R. Comitato Talassografico, Mem. CXXXI, 1927, p. 142. 4 V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, (redigées par Marcel Brélot), Paris, GauthierVillars, 1931 (reprint Paris, Gabay, 1990).

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language, concepts and methods. As early as 1901 Volterra had pointed out that the mere "use" of mathematics was only a first timid step: «the translation of natural events into the language of arithmetic or geometry means opening up to mathematics rather than an exercise of the analytical tool»5. It was necessary for mathematics to give proof of its effective capacity for allowing a conceptual and methodical approach to the description and prediction of significant classes of biological events. The work of Volterra and Lotka seemed at last to provide the necessary demonstration. There is thus some justification for calling the two decades between 1920 and 1940 "the Golden Age of theoretical biology", especially if we take into account the fact that, besides the work of Volterra and Lotka, also other branches such as population genetics and the mathematical theory of epidemics underwent rapid growth over the same period. And yet this period was followed by more than twenty years of stagnation and it was only in the 'seventies that a second, impetuous phase occurred in the development of biomathematics, that is, the phase that leads directly to the present day. In 1994, Simon A. Levin, in his introduction to the final volume in the series Lecture Notes in Biomathematics published by Springer, pointed out that in the ‘seventies «mathematical biology was peripheral to many subdisciplines of biology», while the «mathematical biologists, who once found in each other the most eager listeners and constructive critics, now see their work guiding experiments, and shaping the conceptual foundations of almost all areas of biology.»6 If it is true that the historical development of mathematical biology is thus characterized by these two distinct phases it is only natural to speculate as to what might be the heritage of the "Golden Age" that has been incorporated into the present phase; and in particular, what has been preserved of the work of two of its main figures such as Volterra and Lotka. At first sight a lot. This heritage seems to be substantial, at least in the field of population dynamics, and partly also in that of the mathematical theory of epidemics. This is clearly shown by the large number of citations of the works of these two scientists to be found in recent literature. For example, the above-mentioned volume of Lecture Notes in Biomathematics contains many of these citations and what is impressive is above all the repeated acknowledgement of the fact that «the use of mathematical models to study population growth flourished only after the "Golden Age" … and 5

V. Volterra, “Sui tentativi di applicazione delle Matematiche alle scienze biologiche e sociali”, Discorso inaugurale, Annuario della R. Università di Roma, 1901, pp. 3-28; also in V. Volterra Opere Matematiche, Roma, Accademia dei Lincei, Vol. III, pp. 14-29 (republ. in Giornale degli Economisti, s. II, XXIII, 1901, pp. 436-458, in French in La Revue du Mois, 1906, pp. 1-20; in Archivio di Fisiologia, vol. III, 1906, pp. 175-191 and in V. Volterra, Saggi scientifici, Bologna, Zanichelli, 1920 (reprint 1990). 6 S. A. Levin, Frontiers in Mathematical Biology, Berlin-New York, Springer-Verlag, 1994.

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the resulting explosion in research […] resulted in theoretical support for many basic tenets of population dynamics and theoretical ecology that are still central to much of the thinking in these subjects today»7. Clearly, the Volterra and Lotka achievements still represent the starting point for many models that can be as sophisticated as you like. Suffice it to note that the entire analysis of the relationship between stability and complexity developed by Robert M. May in the 'seventies is based on a general form of Volterra’s equations for the n-dimensional case. The notion of "collision" or "encounter" in population dynamics and mathematical epidemiology has now become something natural, like using syntactical rules when speaking, although it must not be overlooked that the "collision theory"— that is, the representation of certain biological interactions among the elements of a population by applying a scheme based on the kinetic theory of gases – was introduced for the first time in Volterra’s and Lotka’s work. We could go on and illustrate how Volterra’s use of integro-differential equations introduced the consideration of systems having a memory, namely "hereditary" systems — to use his terminology — in which "hysteresis" phenomena are present and the evolution of the system is a consequence of all the past states and not just the present one. In general it is clear than many physical and mechanical analogies that still apply in mathematical biology and that have shaped many of its concepts, derive from the work of Volterra and Lotka. But there are also a number of empirical results, such as Volterra’s so called third law – which states that the undifferentiated withdrawal of the species ("fishing") favours the prey and penalizes the predators — still represent a point of reference in ecology. Nevertheless a number of points must be made clear. If we remain at the level of the preceding considerations — number of citations, explicit references to mathematical models and techniques, the use of certain mechanical and physiological analogies, empirical results — the weight of the heritage left by Volterra and that of Lotka appears to be completely different. It is not a matter only of the number of citations and references, which is clearly seen to be in Volterra’s favour. It is a matter that may be encapsulated in the term style — scientific style, of course. From the point of view of style — that is, of the structure of the models, of the mathematical languages, of the problems tackled in the treatment — Volterra’s work, although distant from us by 60-70 years and thus inevitably "archaeological", is nevertheless completely in line with current biomathematical research. The impression you get from reading such a well known book as Elements of Physical Biology — subsequently republished and now known under the title of Elements of Mathematical Biology — is completely different. This is a famous book that no one 7

J. M. Cushing, "Structured Population Dynamics", in S. A. Levin, Frontiers in Mathematical Biology, cit. pp. 280295.

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actually reads any more. And it is not hard to see why. What scholar of biomathematics would today accept of coping with a preliminary reading of 50 pages on the concept of evolution, the definition of life, of living matter and of the vital force before coming to grips with the actual subject? Or would he accept to dwell on the «origin of subjective sense of direction-in-time»? Or, again, would he read a chapter on the «energy relations of consciousness»? Let us add that the uncontrolled mingling of mathematical, physical, chemical, biological, economic, social and philosophical considerations is too far removed from our tendency towards specialization — the same tendency that, as Simon A. Levin pointed out, has led to the «bittersweet conclusion» of having to close down the Springer’s biomathematics series for lack of readers, owing to the «decreased demand for broad gauged work», and because «the theoretical biologist has shifted his or her reading towards narrower, more disciplinary pursuits». From this point of view, Volterra, although still an eighteenth century style scientist and thus highly receptive to the need for interdisciplinarity and to "natural philosophy" considerations, paid much more attention to disciplinary boundaries and was thus closer to our contemporary point of view, however contemptible this may be considered. The curious fact is that, if you examine the index to Lotka’s book, you find that it is divided into four sections: the first is dedicated to «general principles», the second to «kinetics», the third to «statics» and the fourth to «dynamics». Disregarding the inversion of kinetics and statics – which has an almost Aristotelian flavour – it would seem to be the scholastic layout of a handbook on conventional mechanics. This structure suggests the almost slavish application of a mechanistic schema. But this is only a superficial impression and does not correspond to actual reality. The explanation of why Lotka should include in such a conventional schema such an unconventional collection of boldly mingled topics is perhaps to be found in the epigraph to chapter two: «Nature must be considered as a whole if she is to be understood in detail». This is the opposite of the key principle of reductionism, according to which "the whole is the sum of the parts". In a single stroke Lotka clearly distances himself from the idea that the reconstruction of the global behaviour of a system must be performed through the aggregation of the description of the behaviours of its elementary components. He states that he intends to pursue the opposite approach: only by considering nature as a whole can one achieve an understanding of the behaviour of the several parts. Nothing could be further from Volterra’s viewpoint, which is completely extraneous to any such holistic considerations and instead proposes the application to biology of a rigidly reductionistic and mechanistic approach. Although not indulging in the somewhat scholastic

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classifications of Lotka, Volterra is the true mechanist of the two: he is the one who believes it is possible to construct a mathematical biology based entirely on the model of mechanics. That Volterra was inspired by such a project was already quite clear from his famous 1901 lecture "On the attempts to apply mathematics to the biological and social sciences"8. Here he set off biology against economics, saying that mathematical applications to the former were still at an embryonic stage as they were limited to the use of statistical and probabilistic techniques, while mathematical economics was pointing in the true direction to be followed. And he quite clearly indicates which direction this was: «The notion of homo œconomicus which led to much debate and created so many difficulties, and which some people are still loath to accept, appears so easy to our mechanical scientist that he is taken aback at other people's diffident surprise at this ideal, schematic being. He considers the concept of homo œconomicus as analogous to those which are so familiar to him as the result of long habitual use. He is accustomed to idealising surfaces, considering them to be frictionless, accepting lines to be non-extendible and solid bodies to be indeformable, and he is used to replacing natural fluids with perfect liquids and gases. Not only is this second nature to him: he also knows the advantages that derive from these concepts. If the mechanical scholar pursues this study, he will see that both in his own science and in economics everything can be reduced to an interplay of trends and constraints — the latter restricting the former that in turn react by generating tensions. It is from this interplay that equilibrium or motion stems in both these sciences, one concept being static and the other one dynamic. We have already referred to the vicissitudes of the idea of force in the history of mechanics: from the peaks of metaphysics, we descended into the sphere of measurable things. In economics, for example, we no longer speak as Jevons did of the mathematical expression of non-measurable quantities. […] Lastly, our mechanical scientist sees in the logical process intended to determine the conditions of economic equilibrium the same reasoning that he is accustomed to use when establishing the principle of virtual work, and when he comes across the differential equations of economics he feels the urge to apply to them the integration methods whose performance he knows so well.» This clearly illustrates what Volterra had in mind when, in 1925, he began his research on population dynamics: to construct a mechanics of biological associations that was perfectly parallel to the mechanics of inanimate material bodies. This is particularly apparent in the three-tier structure of his research programme which he developed over a period of fifteen years. The first phase was that of the rational mechanics of biological associations, similar to the rational 8

See note 5.

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mechanics of the material point and systems. The second phase was that of the analytical mechanics of these associations, similar to the Lagrangian and Hamiltonian mechanics of material systems and based on a variational principle analogous to Hamilton’s principle and expressed in a canonical form. The third phase was called by Volterra the applied phase and was aimed at the empirical verification of the theory. The 1931 book Leçons sur la théorie mathématique de la lutte pour la vie9 was dedicated to the rational phase. The biological subject was similar to the material point in mechanics and the first purpose was to derive from collision theory equations of motion similar to Newton’s equations in mechanics. Volterra’s immediate aim was to introduce a concept analogous to that of energy in mechanics in order to introduce a distinction between conservative and dissipative systems and to derive theorems referring to the behaviour of these two types of system. The final part of the work was dedicated to the introduction of "hereditary" actions, that is, systems whose evolution is determined by their entire past and not just their present state. The final chapter of the book bears the significant title of: "On comparative hereditary actions in biology and mechanics". In the closing years of his life, Volterra devoted considerable efforts to the construction of a canonical and Hamiltonian form of the mechanics of biological associations10. In the first instance, the problem was to rewrite the equations of motion as second-order differential equations and then to find a function that could play the role of Hamilton function. He succeeded in this construction, finding an equivalent for all the typical forms of analytical mechanics, including a form of the Hamilton-Jacobi equation. His efforts culminated in the discovery of a principle similar to Maupertuis’ least action principle, to which he gave the name of principle of least vital action. In his emphatic comment on the role of this principle he stated that it «was perhaps more important than all the laws we have dealt with», and that it was linked to a «supreme principle of nature, the least action principle according to which nature acts in such a way as to save as much as possible». And he added: «Fermat had perceived it intuitively as the basis of the propagation of light, Maupertuis as the foundation of mechanics and, as it evolves, after Hamilton, Jacobi and other scientists, it is penetrating all the fields of natural philosophy».

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See note 4. V. Volterra, “Les équations des fluctuations biologiques et le calcul des variations”, Comptes rendus de l'Académie des Sciences, CCII, 1936, pp. 1953-1957; “Les équations canoniques des fluctuations biologiques”, Comptes rendus de l'Académie des Sciences, CCII, 1936, pp. 2023-2026; “Sur l'intégration des équations des fluctuations biologiques”, Comptes rendus de l'Académie des Sciences, CCII, 1936, pp. 2113-2116; “Le principe de la moindre action en biologie”, Comptes rendus de l'Académie des Sciences, CCIII, 1936, pp. 417-421; “Sur la moindre action vitale”, Comptes rendus de l'Académie des Sciences, CCIII, 1936, pp. 480-481; “Principes de biologie mathématique”, Acta Biotheoretica (Leiden), III, P.I, 1937, pp. 6-39. 10

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Also of great significance were Volterra’s efforts to demonstrate the empirical validity of his results. The sound “classical” scientist that he was, he was convinced that an applied mathematical theory had no value unless a satisfactory empirical proof was available. What would have been the value of classical mechanics if it had not given such good results in so many fields of terrestrial and celestial mechanics? The negative results and doubts, the abundant criticism received from numerous biologists of his time in no way deterred him and indeed spurred him on in his efforts to come up with significant and convincing proof. The vast number of contacts that Volterra had with the biologists of his time — as proved by his correspondence — shows the determination with which he dedicated himself to this task, which he deemed of crucial importance and which essentially lay outside his province of theoretical mathematical physics11. In our approach to the evaluation of the heritage of this scientific programme as viewed in its specific scientific results, our conclusions are necessarily quite different from those in which we must acknowledge the undeniable influence that Volterra’s work had at the methodological level. It must be admitted that very little of this programme can be saved and that, in any case, very little of it has been incorporated in contemporary mathematical biology. It is easy to understand Volterra’s eagerness to find a first integral that was analogous to mechanical energy. However, the problem lies in the fact that this first integral in no way shares the nice properties that ensure that energy plays such a special role in mechanics. In other words, there is no "energy" in the mechanics of biological associations. Consequently, the presentation of the distinction between associations based on exponential growth and those based on logistic curve growth as a distinction between “conservative” and “dissipative” associations is purely nominal and does not correspond to any useful and truly discriminative concept as in mechanics. This is particularly evident in the structure of Volterra’s analytical mechanics of biological associations. In seeking a canonical and variational formulation, Volterra is obliged to construct a "Hamiltonian function" that is quite unrelated to the "energy function" introduced in his 1931 book to distinguish conservative systems from dissipative systems12. All these drawbacks are related to the fact that the equations referring to biological associations are not posited "naturally" as second-order equations but, in order to get them to resemble Newton’s or Lagrange’s equations, Volterra is compelled to differentiate them and thus perform an artificial operation. What is worse is that the principle of least vital action (analogous to 11

On these topics see A. Millán Gasca, "Mathematical theories versus biological facts: A debate on mathematical population dynamics in the 1930", Historical Studies in the Physical and Biological Sciences, 26, no. 2, 1996, pp. 347403.

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Maupertuis’ principle) can in no way be considered equivalent to the equations referring to biological associations. There is an interesting correspondence between Volterra and Levi-Civita in which the latter, whom Volterra was consulting in relation to the value of his results, still in manuscript form, politely but firmly pointed out that: «With reference to the various forms you attribute to the stationary action theorem, I should like to remark […] that they express an equivalent number of important properties of your fundamental equations […]. Moreover, if I am not mistaken, none of them is an integral equivalent of them.»13 Nevertheless, this asymmetry, which Volterra acknowledges in his replies to Levi-Civita, is not mentioned at all in his published works, in which he actually emphatically states that the principle of least vital action plays a similar role in biology to that of Maupertuis’s principle in mechanics. From this it is clear that Volterra’s attachment to the mechanistic approach predominates. The question of Volterra’s attempts to produce convincing empirical proof of his results is of considerable interest. It is also too complex to be dealt with fully here14 and we shall only remark that, after a period during which the introduction of the new biomathematics was greeted with surprise and interest by many biologists, scepticism and criticism ultimately prevailed. Also Georgii F. Gause, who had originally followed this research with interest, remarked in his book Vérifications expérimentales de la lutte pour la vie15 published in 1935, in connection with the principle of encounters: «Although Lotka and Volterra introduced this principle in the equations of their mathematical theory, they were fully aware that it could not be justified in practice. Lotka thus wrote in 1925 that many, and perhaps all, organisms possess some power of selection and are to some extent independent of pure chance. This introduces an extremely peculiar complication into the mechanism governing living beings, a complication which does not exist in the statistical dynamics of molecular physics. Not only is the living organism capable of accomplishing at the macroscopic level achievements similar to those that in the world of molecules are allowed to figments in the imagination, such as Maxwell's demon, but this power "to beat chance", as it were, is possessed to varying degrees by the different living organisms, and the mechanisms underlying the systems containing living matter must necessarily take into account, in addition to this capacity, also the

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On these topics see G. Israel, “Volterra's Analytical Mechanics of Biological Associations” (two parts), Archives Internationales d'Histoire des Sciences, 41, N. 126, Juin 1991, pp. 57-104; N. 127, Décembre 1991, pp. 307-352. 13 See the paper quoted in note 12. 14 A full account could be found in the paper quoted in note 11. 15 G. F. Gause, Vérifications expérimentales de la lutte pour la vie, Paris, Hermann, 1935. See also G. F. Gause, The struggle for existence, Baltimore, 1934.

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degree to which this capacity is possessed, as it plays an important part in the determination of the place occupied by the various biological species in the scale of evolution.» In the same year, even Volterra’s son-in-law, Umberto D'Ancona — who had played a fundamental role in suggesting the first empirical basis for Volterra’s theories, the study of Upper Adriatic fishery — distanced himself from his father-in-law’s attempts to achieve a complete empirical proof. «Of course — he said — my observations on fishery in the Upper Adriatic should provide stronger support for your theory […] Unfortunately, however, my statistical observations may be interpreted in the way I did, but also in a different way, as Pearson, Bodenheimer and Gause think. Therefore, I too acknowledge that there is some truth in the criticism. […] You should in no way think that I want to diminish the experimental research on which your theory rests. However, I think one must be very cautious about accepting this experimental research as a proof. If it is accepted too readily it could be refuted by someone else. In all these issues your theory is by no way affected. It is a theory that follows a logical and plausible approach, and is supported by a large number of known and plausible data. Therefore, it can be considered a working hypothesis that can be used as a source of new investigations and that holds even though it is not supported by empirical data. Of course, the latter could confer greater authority upon it. However, one must be cautious about accepting this evidence, and make sure that it is reliable and conclusive. Otherwise it is better not to link your theory to an experimental basis, which is certainly less solid than the theory itself»16. This was in fact a proposal for a modelling approach that was very similar to the modern conception but also very distant from Volterra’s point of view. He indeed was visibly annoyed by his son-in-law’s advice and persisted with his experimental tests, rejecting the idea that a mathematical theory could be considered as a mere plausible hypothesis. The fundamental difficulty in Volterra’s approach consisted in the fact that he claimed to be seeking an experimental verification, at the same time displaying relatively little sensitivity to the specific nature of the biological problems involved in the belief that these could be solved by identifying analogies with those of mechanics. From this standpoint the difference with Lotka’s views is quite apparent. We have already pointed out how his book’s structure, seemingly modelled on mechanics, is mere appearance and actually reflects quite different contents. Lotka does not stop short at considering the subject of biological associations as a "material point" and – as Gause’s words tell us – views the principle of encounters with a critical and cautious mind.

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In 1926, Volterra and Lotka had to confront each other on the question of who was the first to formulate the equations now referred to with their joint names17. I shall examine this matter solely to extract the content of several letters written by Lotka to Volterra which quite clearly reveals the complexity of the former’s point of view and the various interrelated mechanical, chemical, energetic and economic considerations employed in his attempt to provide a non banal definition of the biological subject as an "energy transformer". In November 1926 he wrote to Volterra: « […] it seems that the main interest does not lie so much in the kinetics as in the dynamics of systems composed of a large number of energy transformers having certain characteristic properties. In the final part of my book I have endeavoured to develop this branch of science. Here the discussion leads to a study of the apparatus by means of which these transformers manage to capture the free energy they need in order to function. This brings in certain partial derivatives ri/Xj that I have related back to the "economic value". On the basis of the principles of Spencer and Jevons according to which the feeling being tends to maximize his pleasure, at the same time as he is acting in the interest of the species, there is found to be a relationship between the economic value and the partial derivatives ri/Xj. This relationship would be perfect in a perfectly adapted species. It differs to varying degrees from this ideal in the present species. These reflections automatically spring to mind when I read your article. I should be very happy to believe that they are of some interest to you.» And in a subsequent letter he brought the matter up again in the following terms: «I have, of course, found in your longer monograph many things of prime importance. I will not now enter into a discussion of the matter; only one point I might speak of in passing, namely, that I have been particularly interested in Section 4 in which you broach the question of encounters between individuals of the two species. I believe that here is the starting point for some further developments. The problems of the statistical mechanics of systems of energy transformers (for that is what animals and plants are), hinge just on this question of the collisions or encounters between the individual organisms. All these encounters are determined, not, as in the comparatively simple case of gaseous molecules, merely by velocities and positions, but by an elaborate mechanism comprising sense organs, motor organs and adjustor faculties. If you have been turning over the 16

This letter is cited and its content is discussed in the paper quoted in note 11 and in G. Israel, “The Emergence of Biomathematics and the Case of Population Dynamics: A Revival of Mechanical Reductionism and Darwinism”, Science in Context, 6, no. 2, 1993, pp. 469-509. 17 See G. Israel, “Le equazioni di Volterra e Lotka: una questione di priorità” , in Atti del Convegno su "La Storia delle Matematiche in Italia", Cagliari 29-30 Settembre-1° Ottobre 1982, (a cura di O. Montaldo e L. Grugnetti), Università di Cagliari, Istituti di Matematica della Facoltà di Scienze e Ingegneria, 1982, pp. 495-502; “On the contribution of

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leaves of my book, you may have found that I have a little to say on the question of collisions on page 359. What I have then said I would like to modify only in one minor particular. The discussion is there conducted as if the notion of the organism extended over the twenty-four hours of the day. In actual fact, of course, only certain fractions K of the twenty-four hours are so employed, and this fraction should enter in obvious manner into the discussion. The discussion will further have to take into account the kind of thing that is discussed on pages 346 to 358. I am at present giving my thoughts to this phase of the subject and I am hoping that perhaps in a little while I may have something in readiness for publication. There is something unsatisfactory in the rather extreme degree of idealization, of conventionalization, which is necessary to introduce into the discussion at the present stage of this very new science. I believe that it is necessary for us to deliberately overcome a certain repugnance which one feels towards such extreme conventionalization and to proceed with the work in the hope that the first crude steps may turn out in time to have been necessary preliminaries for a more perfect treatment of the subject. Owing to the immense complexities of the physical systems presented to us in living nature it is necessary, I believe, in order to be able to progress at all, to begin with modest and greatly simplified examples. From your monograph it appears to me that you entertain somewhat similar thoughts. Naturally, I should be greatly interested in an expression of your views.» A dialogue among the deaf, or rather a monologue. Volterra’s sole answer was an icy comment that may even sound somewhat mocking: «In your letter you speak of the most important issue: the dynamics and energy question. It obviously comes before all the others. I have given much thought to it and will continue to do so. In view of your wide-ranging ideas and the broad scope of your studies, you should obtain some very significant results.» In any case this was not the first time that poor Lotka came to find himself in such a situation. A figure completely extraneous to the academic world, from which he received only humiliation and rejection, he worked as supervisor at the Statistical Bureau of the Metropolitan Life Insurance Company of New York and at the same time cultivated numerous different interests – ranging from population statistics and chemical kinetics: (the sectors in which he had the greatest competence), to the theory of evolution, Jevons’ economic theories, Spencer’s theories, two figures who had strongly influenced his background. The list of Lotka’s publications gives the impression of an eclecticism that was unlikely to arouse indulgent feelings in those holding conventional disciplinary views.

Volterra and Lotka to the development of modern biomathematics”, History and Philosophy of Life Sciences, 10, No. 1, 1988, pp. 37-49.

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The Russian biochemist Vladimir Kostitzin, a great friend of Volterra’s who collaborated closely with him in the development of his theories, actually reproached him for his attitude to Lotka18. In a letter written in 1939 this is what he said to Volterra: «How do you expect him [Lotka] not to be somewhat annoyed when he sees his works, although very recent, neglected and unjustly passed over? You are not involved in it, but it is true nonetheless. Lotka is perfectly well aware that he deserves better than this, and he is right. His book Elements of Physical Biology, and his notes of the same period contain many interesting thoughts, remarks and considerations, as well as the germs of future research. Even if he was not the first to introduce this form of equation, he has made a nice study of them for the case of two species which contains insights into more general problems. His considerations on the absence of contradiction between the existence of living matter and the law of entropy are still not sufficiently well appreciated by biologists and natural philosophers.» Doubtless Kostitizin’s background led him instinctively to appreciate Lotka’s holistic approach. A member of the Moscow School and holding similar views to Vernadsky, Kostitzin shared his conception of the biological world (or biosphere) as a whole, a system that cannot be reduced to a mechanical aggregate of particles. In acknowledging Lotka’s "seminal work" he also traced out the subsequent fate of this work. This is the same issue as we saw earlier: the structure of Lotka’s work was too complex, heterogeneous, extensive and at the same time pretentious for his results to be accepted as a positive and decisive breakthrough rather than as mere brilliant suggestions. With all the indulgence justly invoked by Kostitzin it is often difficult not to feel somewhat irritated by the flippancy with which Lotka liquidates age-old philosophic issues, such as that of the freedom of the living subject or the relationship between consciousness and physical phenomena. His book is a vast and poorly organized image of the living world in which the latter is considered as a coherent whole. Nevertheless Kostitzin also correctly emphasized that Lotka had shown numerous «germs of future research». Oddly enough Lotka’s work was destined to have a strong influence only after being filtered through the work of several of his enthusiatic readers, such as Norbert Wiener and Nicholas Rashevsky. It is difficult to imagine Ludwig von Bertalanffy’s systems theory without taking into account the influence of the ideas and suggestions contained in Lotka’s book. And the same may be said for the work of Ilya Prigogine. Anyone involved in the thermodynamics of living systems, in the systems theory point of view, in holistic or integrated approaches, and above all in complexity 18

See G. Israel, A. Millán Gasca, “La correspondencia entre Vladimir A. Kostitzin y Vito Volterra (1933-1962) y los inicios de la biomatematica”, LLULL, Revista de la Sociedad Española de Historia de las Ciencias, Vol. 16, No. 30,

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— one of the most fashionable terms in contemporary science — owes much, whether they know it or not, to the ideas put forward more than seventy years ago by Lotka. For these reasons, the works of the two great founders, Vito Volterra and Alfred Lotka, have followed odd, and in a sense opposite, destinies with reference to contemporary mathematical biology. The present structure of this discipline, in its highly mathematical and modelling form, is closer to the language and operating schemata of Volterra’s work, perhaps also because Volterra was a mathematician and biomathematics has since become increasingly mathematized. There is no doubt that the modelling approach was completely extraneous to Volterra’s conception and aims. But it is true that D'Ancona was a clear-sighted prophet: and, pruned of the aspects linking it to the classical reductionist conception, and after a modelling and almost exclusively mathematical reappraisal, it became possible to consider Volterra’s work as the "first" to open up the way to contemporary population dynamics. However, in its mathematical modelling form it is quite remote from and also completely ignores the specific issues of the mechanistic programme pursued by Volterra in his work. On the other hand, although Lotka’s language and approach have little or nothing in common with contemporary biomathematics — who would dream of including a section on industrial development in a work on contemporary biomathematics? — the topics and problems treated by him are present and active, albeit filtered through different contexts and languages. Volterra’s and Lotka’s scientific heritage thus lives on in modern biomathematics after being filtered through a strong mathematical modelling approach. This filter has stripped Volterra’s work of the rigidly mechanistic programme and the claim of a rigorous and accurate empirical verification, while retaining the mathematical treatment and the type of problem, as well as the attention focused on the issues of stability and the formulation in terms of dynamical systems. Lotka’s work has been pruned of the somewhat vague connection between mathematical representation and philosophical and methodological considerations drawn from all possible contexts, at the same time retaining the suggestions and stimuli and a number of forward-looking ideas. This clearly led to the abandoning of something that was of central importance to both scholars – namely a close, indeed very close, relationship between mathematical representation and empirical and experimental reality, despite the fact that they each held a quite different and even divergent conception of this relationship. One wonders whether this abandonment, to some extent necessary and inevitable, is not also a sign of weakness. One purpose of the history of science is to raise this kind of issue and not merely to indulge in archaeology of the past. Although Volterra’s 1993, pp. 159-224.

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strong inclination towards the topic of empirical verification was framed in questionable terms, it nevertheless expressed a correct general need that is perhaps neglected too much today. The recovery of Lotka’s issues in a strongly formalized approach may render less vital and perceptible the content of certain issues he raised and that lead to interrelations among disciplines that are perhaps abstruse but ineluctable. In other words, the destiny of Volterra’s and Lotka’s scientific heritage encourages us to meditate upon the troubled relationship between mathematical formalization and the empirical and conceptual content of the theories that is perhaps somewhat blurred in the mathematical modelling approach.

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