He published Le énième nombre premier comme valeur asymptotique d'une fonction déduite de la fonction ζ(s) de Riemann Ⓣ (1925) and then in the 1930s he published papers on analytic functionals of analytic functions of a single variable. This work was based on the idea of a functional which had been introduced by Vito Volterra. What Fantappiè did was take the theory which had been developed for analytical functions and generalise this theory to the case of functionals. This led to what Fantappiè called analytic functionals, and he developed this theory over almost twenty years through the 1920s and 1930s. He published Nuovi fondamenti della teoria dei funzionali analitici Ⓣ in 1942 which extended some of his earlier results on analytic functionals of analytic functions of a single variable to analytic functionals of analytic functions of n variables. Then in 1943 he published the monograph Teoría de los Funcionales Analíticos y sus Aplicaciones Ⓣ which was reviewed by F J Murray:-
This memoir presents a study of the functionals on a space of analytic functions, topologized by a Hausdorff topology in which neighborhoods are set up in terms of the maximum difference of functions on regions. ... Linear functionals are studied ... Bilinear functionals are also introduced. The next step is the consideration of nonlinear analytic functionals in general. For these the author establishes the usual concepts of the differential calculus such as the derivative, with the customary properties and rules and the series expansion of such a function. The corresponding questions for functions of more than one variable are also considered but the results are not as complete. However, these results contain a Cauchy integral formula for the many-variable case, which permits an operational calculus of a somewhat restricted sort. As an application the Cauchy theory of partial differential equations is considered. In certain cases, the Cauchy problem is solved. The geometrical significance in abstract spaces of such notions as characteristic strips and singular solutions is given. A similar discussion of the notions of hyperbolic, parabolic and elliptic partial differential equations which pertain to the case of two independent variables also appears.Fantappiè's work on analytic functionals had led to him receiving a number of awards. For example the Italian Society for Sciences awarded him their Gold Medal in Mathematics in 1929, while two years later he received the Accademia dei Lincei's Royal Prize in Mathematics and the Volta Prize from the National Academy of Sciences of Italy. In 1933 Fantappiè left Italy and went to the University of Sao Paulo in Brazil where he founded the Mathematics Department and was head of the new department from 1933 to 1939. He returned to Italy at the outbreak of World War II in 1939 when he was offered the Chair of Higher Analysis in the University of Rome, a position he held for the rest of his life.
Although Fantappiè continued to undertake research on analytic functionals, he began to study a new area in 1941. Let us quote his own words regarding his new idea of syntropy which is dual to the concept of entropy [5]:-
I have no doubts about the date when I discovered the law of syntropy. It was in the days just before Christmas 1941, when, as a consequence of conversations with two colleagues, a physicist and a biologist, I was suddenly projected in a new panorama, which radically changed the vision of science and of the Universe which I had inherited from my teachers, and which I had always considered the strong and certain ground on which to base my scientific investigations. Suddenly I saw the possibility of interpreting a wide range of solutions (the anticipated potentials) of the wave equation which can be considered the fundamental law of the Universe. These solutions had been always rejected as "impossible", but suddenly they appeared "possible", and they explained a new category of phenomena which I later named "syntropic", totally different from the entropic ones, of the mechanical, physical and chemical laws, which obey only the principle of classical causation and the law of entropy. Syntropic phenomena, which are instead represented by those strange solutions of the "anticipated potentials", should obey two opposite principles of finality (moved by a final cause placed in the future, and not by a cause which is placed in the past): differentiation and non-causable in a laboratory. This last characteristic explained why this type of phenomena had never been reproduced in a laboratory, and its finalistic properties justified the refusal among scientists, who accepted without any doubt the assumption that finalism is a "metaphysical" principle, outside Science and Nature. This assumption obstructed the way to a calm investigation of the real existence of this second type of phenomena; an investigation which I accepted to carry out, even though I felt as if I were falling in a abyss, with incredible consequences and conclusions. It suddenly seemed as if the sky were falling apart, or at least the certainties on which mechanical science had based its assumptions. It appeared to me clear that these "syntropic", finalistic phenomena which lead to differentiation and could not be reproduced in a laboratory, were real, and existed in nature, as I could recognize them in the living systems. The properties of this new law, opened consequences which were just incredible and which could deeply change the biological, medical, psychological, and social sciences.On 30 October 1942 Fantappiè presented his theory of syntropy to the Accademia d'Italia with a work entitled The Unified Theory of the Physical and Biological World (published in Spanish in 1943). His finding were that syntropic phenomena invert the second law of thermodynamics and [3]:-
- a reduction in entropy and an increase in differentiation is observed;
- converging waves attract in smaller places energy and matter;
- concentration of matter and energy cannot be indefinite and entropic processes are needed to compensate syntropic concentration;
- in nature, syntropy and entropy interact constantly;
- scientific finalism, final causes, are introduced;
- a new scientific methodology is needed since the experimental method can only study causes located in the past.
Fantappiè ended this work by stating how he believed syntropy was the essence of life [3]:-
Let us conclude by looking at what we can say about life. What makes life different is the presence of syntropic qualities: finalities, goals, and attractors. Now as we consider causality the essence of the entropic world, it is natural to consider finality the essence of the syntropic world. It is therefore possible to say that the essence of life are final causes, syntropy. Living means tending to attractors ... the law of life is not the law of mechanical causes; this is the law of non-life, the law of death, the law of entropy; the law which dominates life is the law of finalities, the law of syntropy.There was another major idea introduced by Fantappiè, namely a cosmological theory based on a geometry arising from a group which, in some sense, generalises the Lorentz group. For example in Deduzione autonoma dell'equazione generalizzata di Schrödinger, nella teoria di relatività finale Ⓣ (1955) Fantappiè deduces the Klein-Gordon equation in quantum mechanics as a limit, as the radius of the universe tends to infinity, of a classical (non-quantized) equation in his extension of relativity based on a simple (pseudo-orthogonal) group having the Lorentz group as a type of limit. Fantappiè was still working on this theory at the time of his death in 1956 at the age of only 54 years. The manuscript he left was edited by G Arcidiacono, M Carafa and D Del Pasqua and published as the sixty-page paper Sui fondamentali gruppali della fisica Ⓣ in 1959.
Finally let us look briefly at some of the papers which Fantappiè published in the last seven years of his life: Costruzione effettiva di prodotti funzionali relativisticamente invarianti Ⓣ (1949) constructs functional scalar products of two functions, as required in quantum mechanics, which are relativistically invariant; Caratterizzazione analitica delle grandezze della meccanica quantica Ⓣ (1952) gives conditions on an hermitian operator that he claims are necessary and sufficient for it to satisfy to represent a physically real observable; Determinazione di tutte le grandezze fisiche possibili in un universo quantico Ⓣ (1952) discusses aspects of group invariance of wave equations; Gli operatori funzionali vettoriali e tensoriali, covarianti rispetto a un gruppo qualunque Ⓣ (1953) discusses the role of operators and Lie groups in a quantum-mechanical universe; Deduzione della legge di gravitazione di Newton dalle proprietà del gruppo di Galilei Ⓣ (1955) shows that the inverse square law is a necessary consequence if certain specific assumptions are made; Les nouvelles méthodes d'intégration, en termes finis, des équations aux dérivées partielles Ⓣ (1955) applies analytic functionals to find explicit solutions of partial differential equations; and Sur les méthodes nouvelles d'intégration des équations aux dérivées partielles au moyen des fonctionnelles analytiques Ⓣ (1956) gives a new method for the solution of Cauchy's problem.
To honour Fantappiè, a congress entitled Frontiers of Biology and Contemporary Physics was held in his home town of Viterbo from 13 to 16 August 2009.
Article by: J J O'Connor and E F Robertson