Prof. Antonino Drago - Univ. of Pisa and Univ. of Florence

6. A verification: Pluralism in stating the inertia principle

The clearest demonstration that science as a whole diverges with regard to its formal foundations is obtained by an examination of the inertia principle, which, being the starting pointing the most important theory of traditional science, Newton’s mechanics, represents the beginning of modern science.

Descartes-Newton’s version is: “Every body perseveres in its state of being at rest, or of moving uniformly forward in a straight line, except insofar as it is compelled to change its state by a force acting on it”.38 An alternative version was suggested by (again!) L. Carnot: “Once a body is at rest, it will not move by itself; once it is in motion, it will not change either its speed or its direction”39 (where changing and moving are the negation of “rest”, the only situation which does not require scientific explanations).40 It is worth noting that L. Carnot’s doubly negated sentence (e.g. not move) does not have a corresponding positive word in science; in fact, in order to be able to express the same idea positively, Newton makes use of the verb “to persevere” (or sometimes “to continue”), which is clearly a moral and animistic word. Here we have a drastic alternative about which kind of logic, either classical or non-classical, shapes a theory. Being a basic principle, the version of the inertia principle determines the entire organisation of the subsequent development of the theory; Descartes-Newton’s version is an axiom-based organisation of mechanics, whereas L. Carnot’s version a problem-based organisation.  

In addition, it is worth noting that Newton wrote: “Every body…”: these two words include even the bodies that we will discover in the future; here we recognise an infinity in action. Which he also appeals to when he wants to establish with total accuracy – i.e. an accuracy which implies the actual infinity – when a force is impressed upon the body or not, if the body is absolutely at rest or not, if the motion is perfectly rectilinear or not, and perfectly uniform or not; and if the distance that the body covers is infinite or not.41 All these qualifications require such accuracy as to sever the null value of each of the above magnitudes from any other value, however little; they require not an unlimited infinity, but an actual infinity. All the above qualifications are avoided by Carnot’s version of the inertia principle, which instead includes only the typical properties detectable by experimental physics, i.e. the only ones that are operational and calculable; that is, those which do not use actual infinity.    

Being a basic principle, the inertia principle establishes the kind of mathematics of the subsequent development of the theory; Descartes-Newton’s version mathematics with actual infinity and L. Carnot’s version mathematics with the potential infinity.

In the history of mechanics this kind of alternative theory of mechanics had already been put forward by Leibniz.42 He moreover added two basic ideas. First, in the human mind there exist “two labyrinths of human reason”. One is about infinity, either actual or simply potential. We recognise that in our times the first labyrinth was formalized by the option concerning the kind of mathematics, either the classical or the constructive. The other dilemma is between “law” (i.e. to behave according to some a priori principles) and “free will” (i.e. to investigate heuristically); we recognise that at the present time this second labyrinth is formalised by the option concerning the way of organizing a theory, either by using a few abstract principles from which all laws may be rigorously derived, as theorems, by means of classical logic; or organizing a theory to search  inductively for a new method for solving a general problem.

Leibniz suggested also that there are two basic principles of the human mind: the principle of non-contradiction and the principle of sufficient reason; the latter was stated by him with the following words: “Nothing is without reason”;43 really, a doubly negated sentence. We recognise that he was suggesting the two basic principles of the two different kinds of logic, respectively the classical and the non-classical. In short, the two dilemmas that Leibniz cleverly recognised, represent, although in no more than philosophical terms (i.e. infinity and organization), the two above-illustrated basic options, which at the present time are well formalized in, respectively, mat
hematics and logic.

Every theory chooses one of these two options.44 Being two independent dimensions, when we cross them we divide the space of all theories in four quadrants and each may be considered as representing a particular model for scientific theories.45 Being severed one from the other by mutually conflictual choices, these four models represent a well-rooted pluralism in science. Moreover, the two options provide the human mind with the cardinal points of a compass by which it is oriented among the innumerable theories of modern world. In such a way one obtains an answer to the problem put by Lanza del Vasto (see §. 5); a person can obtain a comprehensive knowledge of science.

7. Rejecting science’s claim to have a monopoly on truth

The general conclusion is that, despite the changes brought about by the French Revolution, for two hundred years the scientific community refused to consider the internal conflicts in science. Scientists tenaciously presented Science as a monolithic construction with no possible alternatives, that implied the lack of any conflict in scientific activity in general, i.e. that it was “at peace”. By claiming to be the only peaceful activity in a society, science was presented as the greatest rational effort possible to solve the conflicts arising in any human activity. Hence, science was applied to pacify all social conflict: for example, in the early years of the 20th Century, Science was applied to solve social conflicts in the factory system by applying Taylor’s scientific principles to the equitable evaluation of human labour; between the ‘50s and the ‘80s, science was applied to reconcile the East-West conflict through scientists’ superior formulae on disarmament. In the ‘70s science was applied to solve the problem of energy planning by imposing nuclear power as the only way of guaranteeing mankind universal welfare and therefore peace.

Let us recall that Galtung’s important distinction between three types of violence: personal, cultural and structural.46 We see that the dominant science falls within cultural violence, not only because it justifies structural violence but also because it monopolizes the truth by means of its results, which are obtained regardless of human life, presenting itself as the only, unquestioned solution to human problems. The violence of science consists, more than in justifying structural violence and war, in its claim to monopolise the truth on any subject, including wars. All of which was dictated by the motto (which parallels the old Catholic Church’s motto: Nulla salus extra hanc Ecclesiam; No salvation outside this Church; this motto claims the monopoly on souls): Nulla ratio extra hanc scientiam (No reason outside this science); this motto claims the monopoly on human reason.47

And indeed, notwithstanding the scientists’ formulae, the factory conflicts, the East-West clash and the energy problem have persisted, showing that historically the initiatives of modern science look like a huge, terrible deception, even a form of subservience to a super-human power, as Lanza del Vasto suggested.

What I have shown above regarding the foundations of science leads precisely to the opposite  conclusion to that of the belief in the peaceful science, i.e. the fundamental nature of science is conflict, owing to the options regarding its foundations. In the previous sections I argued that at least through the different versions of the inertia principle, science does not have a monopoly of the truth; every single scientific theory (even mechanics) is divided in formally alternative formulations.48

But even at the present time the dominant science hides such a conflictual nature by presenting one truth only, which actually is just the truth of the dominant model of scientific theory, which in turn corresponds to the dominant power in society. Thus it is necessary to dethrone the cultural violence which is operated by science which monopolizes truth and claims, in a  pre-conceived manner, to bring peace. In order to understand how to achieve peace we need to find a new scientific approach which will generalize the solutions to conflicts concerning the foundations of science; i.e. we have to replace the paradigm of a science that has a monopoly on the truth with the pluralism of the four models of scientific theory.49

8. A new relationship between ethics and science

As a consequence, there is a new relationship between science and ethics. No longer is science an absolute value, to which ethics is subordinate. When a scientist constructs a scientific theory, at very start he makes two basic choices, respectively on the kind of infinity and the kind of organisation; owing to these choices found the theory, ethics comes first, science second. As a consequence, Tolstoy’s question is answered; the traditional science claiming to come before ethics is dethroned, and science is subordinate to ethics.

Let us remark that the dominant Western view of science requires the belief in its Unity. This belief never will be verified, since it refers to all times to come; it is an absolute belief. In comparison, the belief in the Unity of mankind, which should be applied to conflict resolution, is more suited to the life of humanity; in short, it is a more valid value for mankind.

The same conclusion is reached when we compare the costs of the two beliefs. With the former the citizen is required simply to delegate to scientific experts, allowing them bring about the scientific destruction of an indeterminate number of human beings, while with the latter, the  citizen, doubting the absolute value of mankind’s intellectual constructions, involves his/her personal life in finding the best  solutions to collective conflicts.   

As general conclusion, previous analysis on modern science supports Gandhi’s basic criticism illustrated by Hind Swaraj; i.e., in its main activities Western civilisation subordinated ethics to progress defined quantitatively, which was supported and incessantly implemented by modern science; the social emphasis on the consequent advantages only attributed to science the role of the highest value of society; being science built according to a unique paradigm, it claimed an absolute truth with respect to all life values so that it dominated both intellectual and spiritual life. Only a socially    renewed ethic, i.e. Gandhian non-violence, could contrast this historical trend.

In retrospect, notice that Gandhi’s only Western disciple, Lanza del Vasto, in his book Les Quatre Fléaux , developed Gandhi’s negative analysis of both Western society and its science just after fifty years after the first edition of Hind Swaraj.50 He developed Gandhi’s appraisal of Western civilisation to produce a new, deeper critique, from within the Western spiritual tradition, science and technology. He extended this criticism to the main Western social theories – anthropology, sociology, economics and politics and he concluded that the nuclear bomb was the natural consequence of Western progress. Which he contrasted this progress with the development of the Gandhian communities that he had founded, beginning in 1948, in France (the Ark Community) according a non-violent theory of social life. In conclusion, he developed Gandhi’s ethical viewpoint of Hind Swaraj into an intellectual system which he applied to what in Gandhian philosophy constitutes the basic social cell of an alternative society.51  

Notes and Bibliography

1    It is the title of the “Conclusion” of Pope Paul VI, Populorum progressio, 1968.

2 Robert K. Merton, "Science, Technology and Society in Seventeenth Century England", Osiris 4 (1938) p. 167-171.

3 See the investigation on 60,000 academic professors reported by Rodney Stark and Roger Finke, Acts of Faith: Explaining the Human Side of Religion (London: University of California Press, 2000) and the more recent investigation by Elaine H. Ecklund, “Religion among academic scientists: distinctions, disciplines, and demographics”, Social Problems, 54, 2 (2007) p. 289-307.

4 Leon Tolstoy, My Confession (1882) (London: Bradde Books, 1963). Max Weber,  Intellectual Work as a Profession (1919), reiterates this question as one of the most important ones in European culture.  

5 Beyond the several books by authoritative non-violent thinkers, see my paper “The Birth of Non-Violence as a Political Theory”, Gandhi Marg, 29, 3 (Oct.-Nov. 2007) p. 275-295.

6     Mohandas K. Gandhi, Hind Swaraj, or Indian Home Rule (1909) (Amhedabad: Navajivan Publ. House, 200013); PDF version in: http://www.soilandhealth.org/03sov/0303critic/hind%20swaraj.pdf#search=’Indian%20Home%20Rule (accessed  July 20th 2009), ch.s VI and XIX. Let us recall that Indian tradition qualifies the present epoch as the Kali Yuga, the Dark Age.

7     Usually the anthologies of Gandhi’s writings (an endless number of short articles comprising more than a hundred volumes) miss these criticisms. In Mohandas K. Gandhi, The Writings of M.K. Gandhi (R.N. Iyer, ed.) (Oxford: Oxford U.P., 1986) §§. 108 and 110, are quoted the more mildly ones.

8 Genesis is a sacred text belonging to Hebrew, Christian and Islamic traditions. It tells the story of God creating the world and then the first two people, Adam and Eve. He placed them in the Garden of Eden, but He warned them not to eat the fruit of one tree, the tree of the knowledge of good and evil. A snake induced them to eat the fruit of this tree, which resulted in their expulsion from Eden and a life of suffering. This story actually gives an answer to theodicean question arising in monotheistic religions; i.e. how it is possible for an entirely good God to create a world in which there is evil and, in particular, the human suffering. Over the millennia several interpretations of this text have been suggested; i.e. the first sin in time, disobedience, disobedience to God’ authority, the of carnality, the unclearness of birth, concupiscence, the seeking after status, pride, the allegory of the sum of all sins. Lanza del Vasto presents these interpretations and discusses them in chapter 3 of his book: La Montée des Ames Vivantes (Paris: Denoel, 1968), in which he sets out to interpret the entire book of Genesis. A current interpretation (see, for instance, Erich Fromm, “Disobedience as a psychological and moral problem” (1963), in Disobedience and other Essays (London: Routledge and Kegan, 1984) p. 1-12, in http://eqi.org/fromm.htm#On Disobedience, accessed 10th july 2009) does not investigate the nature of this original sin, but emphasises that by means of this act of disobedience the two people, and subsequently all mankind, acquired full knowledge, i.e. they have been endowed with reason. By “breaking the ties” with God, the human race was set free to make choices and fend for itself. This defiance is again brought up in the Greek myth of Prometheus who stole fire from the Gods. As a result of such types of defiance, humans have evolved intellectually and spiritually. In other words, full awareness is correlated with separation from God and others. Notice that this interpretation disagrees with both Gandhi’s teachings and Gospel teachings on social disobedience.

9 Lanza del Vasto, Les Quatre Fléaux (Paris: Denoël, 1959) ch. 1. This interpretation was summarised in three lectures in the English language which he gave in Gujarat Vidyapith in 1977; see the site  http://www.wikilivres.info/wiki/Pilgrimage_to_Non-violence (accessed July 10th 2009). A similar interpretation of original sin has been already suggested by Arnold Toynbee, Christianity and Civilization. From Civilization on Trial (Oxford: Oxford University Press, 1948). Incidentally, notice the following statement by Toynbee on religious violence: “A church is in danger of lapsing into… idolatry insofar as she lapses into believing herself to be, not merely a depository of truth, but the sole depository of the whole truth in a complete and definite revelation." This statement applies to Science by merely replacing the term “church”.

10 Lanza del Vasto, Les Quatre Fléaux, op. cit, p. 240.  

11 In the history the past interpretation of Apocalypse 13, the one above is the first interpretation of a structural kind, i.e. it sees the actors in terms of social structures, instead of some individuals or even abstract ideas. As a consequence, it involves a conversion not only at the personal, but also at the collective level, by means e.g. the conscientious objection. the foundation of a communitarian life as an instance of the alternative society (Lanza del Vasto founded, first in France (1948) and then in some other countries, the Ark communities, which are similar to Gandhian communities) and struggle to change both evil institutions and negative society.

12 Some decades later, Hans Jonas: The Imperative of Responsibility (1979) (Chicago: U. Chicago P., 1984), started a renewal of social ethics by supporting the view that we have to be responsible with respect to both  mankind’s survival and the welfare of the future generations.

13 For instance, there exist several celebrated books on the relationship between modern science and Eastern philosophies; e.g. Fritjof Capra, The Tao of Physics (London: Collins/Fontana, 19763) (translations in 23 languages); Gary Zukov, The Dancing Wu Li Masters – An Overview of the New Physics (New York: Bantam New Age books, 1983),  etc.. But they compare intuitively scientific notions with those of Eastern philosophies, without examining the formal notions of science.

14 A. Drago, "A historical critique tof Western science", Int. Peace Res. Newsletter, 16 (1978), p. 32-37; “What science for Peace?”, Gandhi Marg, 7 (1986), p. 733-742.

15 The present viewpoint will be qualified in philosophical terms in sect. 5.

16 Alexandre Koyré, From the Closed World to the Infinite Universe (Baltimore: Univ. Maryland), 1957.

17 This shows that science includes a philosophy. Already a century ago one scholar concluded his investigation into the foundations of science by the following insight: "Metaphysics they [the scientists] tended more and more to avoid, so far as they could avoid it; so far as not, it became an instrument for their further mathematical conquest of the world". Edwin A. Burtt, The Metaphysical Foundations of Modern Science (London: Routledge and Kegan, 1924), p. 303.

18 Thomas S. Kuhn, The Structure of the Scientific Revolutions (Chicago: Chicago U.P., 1969).

19 Ilia Prigogine and Isabelle Stengers, Order out of Chaos: Man’s New Dialogue with Nature (London: Flamingo, 1984).

20 Thomas S. Kuhn, The structure…, op. cit, ch. 9: “The large body of eighteenth-century literature on chemical affinities and replacement series also derives from this supra-mechanical aspect of Newtonianism. Chemists who believed in these differential attractions between the various chemical species set up previously unimagined experiments and searched for new sorts of reactions. Without the data and the chemical concepts developed in that process, the later work of Lavoisier and, more particularly, of Dalton would be incomprehensible.8[this footnote refers to the historian Metzger] Changes in the standards governing permissible problems, concepts, and explanations can transform a science.” Here it is apparent that Kuhn wants to attribute Lavoisier’s foundation of classical chemistry to a “supra-mechanical aspect of Newtonianism”. Hence, he does not see any alternative to Newton’s mechanics. Otherwise, his conception of the scientific conflict as a conflict between a paradigm and the subsequent one produced a paradoxical result; the new theory of classical chemistry was to be considered the new paradigm, succeeding in the subsequent theoretical physics to Newtonian paradigm.

21 Robert Fox, “Les Réflexions sur la puissance motrice du feu de Sadi Carnot et la leçon de leur édition critique”, La Vie des Sciences, C.R. Acad. France, 5, 4 (Ju-Aug. 1988), p. 283-301, offers a final appraisal on the research carried out according to the dominant attitude among the historians of this case-study, i.e. interpreting S. Carnot’s exceptional theoretical novelties by means of historical factors of technological nature.

22 Charles C. Gillispie, "The Encyclopédie and the Jacobinian Philosophy of Science", in M. Clagett, ed., Critical Problems in the History of Science (Madison: Wisconsin U.P., 1962), p. 255-269.

23 Antonino Drago, "The alternative science of the Enlightenment”, Studies on Voltaire and the Eighteenth Century 348, Trans. Ninth Int. Congr. on the Enlightenment, Münster 1995, 1997, p. 1081-1805; “History of the relationships Chemistry-Mathematics”, Fresenius J. Anal. Chem., 337 (1990), p. 220-224; Erratum, ibidem, 340  (1991), p. 787; “A new appraisal of old formulations of mechanics”, Am. J. Phys., 72, 3 (2004), p. 407-9; “Sadi Carnot e la nascita di una nuova scienza", in F. Bevilacqua and Alfio Russo, eds., Atti del III Congresso Nazionale di Storia della Fisica (Palermo, 1982), p. 460-465; “The introduction of the Cycle Method in Thermodynamics”, in K. Martinas, L. Ropolyi, P. Szegedi, eds., Thermodynamics: History and Philosophy. Facts, Trend, Debates (Singapore: World Scientific, 1991), p. 36-41 (with A. Della Selva).

24 Adolf P. Yushkevitch, "French Mathematics in Russia", in USSR Acad. Sci., Science and Society, Nauka, Moscow, 1989, p. 212-228. Antonino Drago, “The beginnings of a pluralist history of mathematics: L. Carnot and Lobachevsky”, In Memoriam N. I. Lobachevskii, 3, pt. 2 (1995), p. 134-144; Salvatore Cicenia and Antonino Drago, “The organizational structures of geometry in Euclid, L. Carnot and Lobachevsky. An analysis of Lobachevsky’ s works”, ibidem, p. 116-124.

25 The works by Lazare Carnot referred to are the following ones: Essai sur les machines en général (Dijon: Defay, 1783); Principes fondamentales de l’équilibre et du mouvement (Paris: Deterville, 1803); Réflexions sur la metaphysique du calcul infinitesimal (Paris: Courcier, 1813); Géométrie de position (Paris: Duprat, 1803). A first comprehensive study of L. Carnot’s work is Charles C. Gillispie, Lazare Carnot Savant (Princeton: Princeton U.P., 1971). About the scientific relationship between the two Carnot’s see ch. III D. Notice that L. Carnot’s theory (which tackles an extremely complex situation, constituted by a machine composed of an unlimited number of levers, wheels and impacting parts), and even more so S. Carnot’s theory (which tackles the complexity of a gas, where there is a jumble of millions of billions of billions of molecules mutually impacting), show that a complex situation may be easily solved in scientific terms when the appropriate theoretical parameters are recognised. In fact, the aforesaid theories have abandoned the analytical attitude (a typical feature of Newtonian mechanistic physics) of examining the single parts, or molecules, composing a system, and instead proceed to assess the situation using global parameters such as energy, volume, temperature and gas pressure. These theories were the beginning of a conflict with Newtonian theory, hence a conflict between the various physical theories. Notice that nothing is more complex than a conflict, because it is always changeable and unforeseeable in all its implications. Hence, the birth of complexity theory, underlining  the complex phenomena which have to be formalised by a non-local, non-analytical attitude, may be seen as the first approach to notion of conflicts between scientific theories; in my opinion, such complexity is more relevant than complexity in reality. The weakness of present complexity theory appears also when one considers that it does not make a clear choice between the analytical and the global attitude.

26 See the lucid presentation of the alternative in the organization of a scientific theory, although he qualified as “empirical” the problem-based organization.  Lazare Carnot, Essai, op. cit. , p. 101-103 and Principes, op. cit. pp. xii-xix. Antonino Drago, “A new appraisal…”, op. cit. Independently both Henri Poincaré and Albert Einstein arrived at the same result. Henri Poincaré, La science et l’Hypothèse (Paris, Flammarion, 1903), ch. "Optique et Electricité"; La Valeur de la Science (Paris: Hermann, 1905), ch. vii.; Albert Einstein, Ideas and Opinions (New York: Crown, 1982). Arthur I. Miller, A. Einstein’s special theory of relativity (New York: Addison-Wesley, 1981), p. 123-142.

27 Antonino Drago, “The beginnings of a pluralist history of mathematics: L. Carnot and Lobachevsky”, In Memoriam N. I. Lobachevskii, 3, pt. 2 (1995) p. 134-144; Salvatore Cicenia and Antonion Drago, “The organizational structures of geometry in Euclid, L. Carnot and Lobachevsky. An analysis of Lobachevsky’ s works”, ibidem, p. 116-124. Antonino Drago and Anna Perno, "La teoria geometrica delle parallele impostata coerentemente su un problema", Per. Matem., 4 (2004), p. 41-52. Valentin A. Bazhanov and Antonino Drago, “A Logical Analysis of Lobachevsky’s Geometrical Theory”, submitted to Journal of Applied of Non-Classical Logic.

28 For a general view, see John U. Nef, War and Human Progress (Cambridge: Harvard U.P., 1952). A relevant exception was C. Huygens who wanted to exploit cannon  powder for building an engine.

29 Giovanni Salio, "S. Carnot, la nascita della termodinamica e le tecnologie belliche", Atti del III Congresso Nazionale di Storia della Fisica, op. cit., p. 236-241.

30 Indeed, the Restoration institutionalized academic science according to a number of authoritarian constraints: 1) the setting up of scientific academies with rigid professional roles; 2) "rigorous" procedures to communicate and accept scientific results; 3) embedding science in a sophisticated (mathematical) language which acted as a barrier against those who wished to discuss fundamental problems; 4) splitting up scientific work in several fields, that are sharply separated one from another (e.g. economics from physics, in particular thermodynamics; mathematics from computing machines, etc.); 5) maintaining scientificity as the final criterion also for solving social issues; that is, a monolithic science set above all other social values. See Joseph Ben-David, The Scientist’s Role in Society (Chicago: U. Chicago Press, 1974).

31 Three decades ago an alternative within scientific theories was suggested by an important social problem, i.e,  the energy crisis which recalled the scientific alternative of one century and half earlier. Owing to the oil crisis of 1973, the Western world discovered that as a society it had never taken into account energy consumption and energy waste. In reaction, the dominant scientific attitude foresaw the same rate of progress as in previous years, i.e. an exponential growth of energy consumption; as a consequence, society had to produce a huge amount of energy (mainly electrical). It seemed that nuclear power, developed thanks to most advanced modern scientific theory, i. e. nuclear physics, could guarantee such levels of production. It was presented as the only viable solution and its opponents were not credited with rationality. Yet surprisingly, the second principle of the older theory of thermodynamics contradicted the development of nuclear power. The American Physical Society discovered that, strangely enough, for over  one hundred and fifty years Western society had not applied the specific scientific theory of energy, i.e. thermodynamics; whose central idea is that in any energy transformation the optimum yield is given by a S. Carnot cycle, whose efficiency depends on the difference between the temperatures of the heat source and the temperature of the final use; hence, it would be wise to choose that energy source whose temperature is as close as possible to the temperature of the final use. By disregarding this principle, the present social organization systematically leads to an enormous waste of energy. APS Study Group, “Efficient use of Energy”, Phys. Today, 26 (Aug. 1976), p. 23-33. The alternative energy planing chooses  low temperature, renewable sources of energy, because they are the more suited to the final use of energy at local level. Hence, the question: "How much energy?" was followed by the question: "What kind of energy?" The debate made it clear that there exists a distinction between two radically different ways of producing energy for a society. US Senate, Long-term energy planning, 1975, Washington; Amory Lovins, Soft Energy Paths: Toward a Durable Peace  (Harmondsworth: Penguin Books, 1977). One may trace back the internal conflict within technology to S. Carnot who began his booklet on thermodynamic discussing energy planning for a society; moreover, he warned of energy crises and foresaw the great change in future society brought about by the widespread use of heat engines; and even more important, he suggested the criteria for achieving the greatest efficiency in energy transformations.

32 Everett Bishop, Constructive Analysis (New York: Mc Graw-Hill, 1967). Notice that the dominant mathematics, the so-called “rigorous” mathematics which was developed by both Cauchy and Weierstrass in the 19th Century, includes actual infinity even in the basic notion of limit. See Ervand G. Kogbetlianz: Fundamentals of Mathematics from an Advanced Point of View (New York: Gordon and Breach, 1968), App. 2.  

33 James R. Newmann, ed., The World of Mathematics (New York: Simon & Schuster1956) vol. 4. Anatol Rapoport, Strategy and Conscience (New York: Harper, 1964). A celebrated application of Rapoport’s cleverly describes the arms’ race, carried on by the two super-powers, through the game of prisoner’s dilemma.

34 Michael Dummett, Elements of Intuitionism (Oxford: Claredon, 1977). Dag Prawitz and Peer-E. Melmnaas, “A survey of some connections between classical intuitionistic and minimal logic”, in H. Arnold Schmidt, Kurt Schuette and Helmut-J. Thiele eds., Contributions to Mathematical Logic  (Amsterdam: North-Holland, 1968), p. 215-229. In the following I underline the negative words in a doubly negated statement in order to show its nature.

35 Sigmund Freud, “Die Verneinung”, Imago, 11, 3 (1925) p. 217-221.

36 It is well-known that also the theoretician of social conflicts, i.e. Marx, tried, by turning up it Hegel’s metaphysical dialectics, to obtain a new logical way of arguing; but unsuccessfully, although he made use of many double negations.

37 We have already remarked that in the energy debate, involving essentially scientific principles, there were two different and irreconcilable positions, of equal scientific validity; i.e.  nuclear energy planning and soft-energy planning. In fact, a similar division occurred in each applied scientific sector. A similar division is clear in agriculture, between chemical-industrial agriculture on the one hand, and organic, or biodynamic, or permacultural agriculture etc., on the other. A similar division also exists in the health sector, between the dominant bio-chemical medicine and homeopathy, or acupuncture, or herbal medicine, etc.. In general terms, “alternative technologies” were invented and were claimed to be independent of dominant technologies. There is no easy definition of these alternatives; however some instances are the bicycle instead of the motor car, wooden instead of concrete houses, solar panels instead of the electricity for heating water.

38 Isaac Newton, Philosophiae Naturalis Principia Mathematica (London 1687), p. 12.

39 Lazare Carnot, Principes fondamentaux de l’équilibre et du mouvement (Paris: Deterville, 1803), p. 49.

40 This remark was made by Norwood R. Hanson ("Newton’s first Law. A Philosopher’s door in Natural Philosophy", in Robert G. Colodny, ed., Beyond the edge of certainty (Englewood Cliffs (NJ): Prentice-Hall, 1965), 6-28) who ingeniously produced an almost exhaustive analysis of the inertia principle. See also Antonino Drago, "A Characterization of the Newtonian Paradigm", in Paul B. Scheurer, Guy Debrock, eds., Newton’s Scientific and Philosophical Legacy (Dordrect: Kluwer Acad. P., 1988), p. 239-252.

41 Norwood R. Hanson, "Newton’s first Law… “op. cit.

42 Antonino Drago, “The birth of an alternative mechanics: Leibniz’ principle of sufficient reason”, in Hans Poser et al., eds., Leibniz-Kongress. Nihil Sine Ratione, 2001, Berlin, vol. I, 322-330; La riforma della dinamica di G.W. Leibniz, Hevelius, Benevento, 2003. In retrospect, Leibniz’ mechanics lacks two theoretical improvements; the introduction of the index of elasticity and the principle of virtual velocities (which was formulated by Bernoulli one year after Leibniz’s death).

43 Gottfried W. Leibniz, “Letter” to Arnaud, 14/7/1686. As an improvement of Leibniz’ philosophy of science, see Antonino Drago, “The modern fulfilment of Leibniz’ program for a Scientia generalis”, in Hans Breger, ed., VI Int. Kongress: Leibniz und Europa, Hannover, 1994, p. 185-195. In particular, Popper’s celebrated philosophy of science is interpreted as a new attitude inasmuch as it first made an implicit use of non-classical logic: Antonino Drago and Antonio Venezia, “Popper’s falsificationism interpreted by non-classical Logic”, Epistemologia, 30 (2007) p. 235-264.

44 One may re-construct the entire history of theoretical Physics. For instance, the history of quantum mechanics is illustrated according to the pluralist view that first suggested Hermann Weyl. Antonino Drago, “Which kind of mathematics for quantum mechanics? The relevance of H. Weyl’s program of research”, in Claudio Garola, Arcangelo Rossi, eds., Foundations of Quantum Mechanics. Historical Analysis and Open Questions (Singapore: World Scientific, 2000), p. 167-193.

45 See Antonino Drago, "Mathematics and alternative theoretical physics: The method for linking them together", Epistemologia, 19 (1996) p. 33-50. A crucial philosophical notion proves to be the incommensurability between two theories. Antonino Drago, ”Incommensurability as a bound of hermeneutics in science”, in M. Fehér, O. Kiss, L. Ropolyi, eds., Hermeneutics and Science (Dordrecht: Kluwer Acad. P., 1999), p. 135-155. “Mathematics and alternative theoretical physics”, op. cit. Nowadays many think of science as a variety of “scientific models” by means of which one sketches reality. In the present paper the word “model” has a more precise meaning; here, there are four models only, each having its own peculiar features, which can be traced back to a pair of choices regarding the two options, which constitute the foundations of science.

46 Johan Galtung, “Cultural Violence”,  Journal of Peace Research, 27, 3 (1990) p. 291-305.

47 My motto sums up the paper by Paul K. Feyerabend,: "Philosophy of Science 2001", in Robert S. Cohen, Marx W. Wartofsky, eds., Methodology, Metaphysics and the History of Science (Boston: Reidel, 1984), p. 137-147.

48 Of course, the alternative science does not concern the experimental laws, but only the foundations of a scientific theory; i.e. the mathematical techniques for formalising the experimental laws, the theoretical principle for understanding them systematically, the organization of them and the logic for arguing about them.

49 It is by attributing to science a monopoly on the truth that some scientists want answer by means of scientific tenets the question whether the nature of man requires wars (Unfortunately, in 1986 Unesco replied to these scientists by claiming a higher monopoly, on the scientific methodology; see Sevilla Declaration in http://www.unesco.org/webworld/peace-library/UNESCO/HRIGHTS/293-303.htm; accessed july 10th 2009). The same hold true for the question of the scientific validity of religion.

50 Lanza del Vasto, Les Quatre Fléaux, op. cit..

51 Recently, Lanza del Vasto’s thinking has been studied in an accurate way. Donatella Abignente and Sergio Tanzarella, eds., Tra Cristo e Gandhi. L’insegnamento di Lanza del Vasto alle radici della nonviolenza (Milano: San Paolo, 2003). Città di San Vito dei Normanni, ed., Lanza del Vasto e le Arti Visive (Fasano (BR): Schena, 2007). Daniel Vigne, La Relation infinie. La Philosophie de Lanza del Vasto. I. Les Arts et les Sciences (Paris: du Cerf, 2008). Antonino Drago and Paolo Trianni, eds., La Filosofia di Lanza del Vasto. Un Ponte tra Occidente e Oriente (Milano: Il Grande Vetro / Jaca Book, 2009).


Drago, Antonino is retired associated professor in History of Physics at Naples University “Federico II”; presently is teaching as contract professor at Florence University and Pisa University subjects concerning non-violence; he is a follower of Lanza del Vasto; whose legacy recently he organised two conferences; the proceedings have been edited by respectively Jaca Book (Milan) and the Pozzo di Giacobbe (Palermo). His last book is on the People non-violent defence (Turin: EGA, 2006).


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