A GANDHIAN CRITICISM TO MODERN SCIENCE (Part 1)

COMMENTARY ARCHIVES, 15 Nov 2009

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

Contents:

1. Is Western progress a truly human development, or a misleading spiritual direction?  

2. A radical criticism by authoritative non-violent figures of the dominant scientific and technological    progress;  

3. Any conflict within science?

4. The birth of conflict and pluralism in science during the French revolution;

5. Formally qualifying the conflict in science;

6. A verification: pluralism in stating the inertia principle;  

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

8. A new relationship between ethics and science.
 
Abstract: The paper begins by asking a crucial question posed by Gandhi in his Hind Swaraj: Is Western progress a truly human development, or is it leading us astray spiritually and ethically? This question is typical of the non-violent tradition, from Tolstoy to the Western Gandhi’s disciple Lanza del Vasto, who improved Gandhi’s criticism from within Western culture, with reference in particular to Christian theology and political science. But Western civilisation disregards such criticism of its progress, criticism that does not directly concern the core of such progress, i.e. modern science. In order to include science in the criticism, it is first pointed out that, during the French revolution, new scientific theories were born each of which was essentially mutually incompatible with the previous one – e.g. Chemistry with respect to Newton’s mechanics -and therefore constituted the beginning of a pluralism in science. By examining the characteristic features of their basic differences, this pluralism is formally qualified; it is then verified even in the scientific principle which was the basis of the modern science, i.e. the inertia principle, which scientists actually stated in at least two different versions; which in turn prove to be in agreement with the above qualified pluralism. As a social consequence, a person and a community who live within such pluralism have to decide by means of their ethics which scientific outlook he wants to follow. Thus, ethics now precedes science and progress. This was precisely the target of Gandhi’s criticism a century ago.

1. Is Western progress a truly development for mankind, or does it do violence to the spiritual life?

In the 17th Century the powerful Catholic Church had accused science, in particular Galileo, of being against both religion and spirituality. However, in the 20th Century, the attitude of the Catholic Church changed. Although there was no official document, during the Vatican Council 2, there was an ascendancy of “incarnationist” over “eschatologist” theologians; in other words, in the present attitude of the most authoritative theologians, the will to be involved in even the contradictions of the world prevails over the will to emphasise a detachment of spiritual life from the evils of the society. Science was accepted as an inevitable reality. In conclusion, what had previously been the enemy, i.e. Catholic Church, was thus conquered by the previously severely condemned Science.

At present, theology itself has taken its place among the other Western sciences,  mimicking the techniques and aims of the sciences that are closest to it.  In particular, Catholic theology no longer conceived of peace as either a personal achievement or a metaphysical "gift from God", but rather took up the social slogan: "[scientific] progress is the new name of peace".1

Not only the Catholic Church, but almost all the Christian Churches shared a favourable attitude towards Western scientific progress, from its beginnings. Moreover, a pro-progress ideology of this kind penetrated to the common people and was brought to the Third World by priests and missionaries, who believed that scientific progress would give human dignity to the primitive. In this sense, the expansion of science and technology, which brought with it increased welfare, appeared to the naïve to be a spiritual blessing. Indeed, most people embraced an ideology according to which science is a modern salvation not only materially (let us recall epidemics or the work of slaves), but even spiritually, in that it eliminates both social and psychic evils.

In fact, in the ’30s the sociologist R.K. Merton characterized the underlying ideology of the West, i.e. Science, as an individualist, Anglo-Saxon and mainly Puritan phenomenon.2 The best representative of this was the chemist R. Boyle, owing to his rigorous curriculum of studies, personal goodwill, his spirit of self-denial in his devotion to the discovery of nature’s secrets, the universalistic passion for mankind’s welfare; in short, he interpreted a modern way of living a monk’s life; while the architectonic expression of this kind of scientific phenomenon was constituted by the University colleges, which were built on the model of the old Roman or Gothic convents and, moreover, were usually named after Saints or even the Holy Trinity.

As a consequence, in the last century, society in general formed a favourable conception of science’s relationship with spiritual life (even in an atheistic sense3). In the end, the scientific conception of the World as suggested by Science seemed to be the only one possible. Never in the history of mankind was a cultural phenomenon so pervasive and so dominant among the people of the World (we find something approaching it in the Roman Law during the Roman Empire and in Christianity in medieval Europe).

2. A radical criticism of the dominant scientific and technological progress  by the  authoritative representatives of non-violence

However, a balanced evaluation of scientific progress and its benefits for mankind must also include some terrible achievements. Science also gave mankind the ability to commit collective suicidal through, for instance, nuclear weapons.

This fact raises the following question: Is it possible to object to science?

Let us remark that the history of the 20th century left two legacies; one, a bloody list of scientifically performed slaughters (wars) occurring mainly in Western countries; the other, a marvellous achievement, obtained in a non-Western country: Gandhi’s renewal of the people’s ability to solve conflicts – even extreme conflicts such as anti-colonialist struggles and rebellions against dictatorships are  – with nonviolent means, i.e. without weapons that threaten the survival of the adversary.

His example led to the birth of a non-violent movement also in Western countries. Owing to the ever more colossal carnage of wars, it did not share the State’s belief in scientific progress as applied to war; it radically opposed it, the arms-race and all social structures supporting them. Owing to the strong link between the arms-race and social progress, this movement concluded that the dominant progress itself needed to be contested, including the most monumental product of Western thought over the centuries: Science.

The great teachers of non-violence (Tolstoy, Gandhi and Lanza del Vasto) radically criticized Western science. They shared the view that science represents the wrong direction taken by the human soul that has gone astray. The non-violent Tolstoy started a radical attack on Western science by asking the question: “Science can give answers to all but the important question ‘What is life for?’ ”;4 that is, Science is separate from our life in the sense that it lacks an ethical dimension.

In addition to social achievements, Gandhi promoted a new way of thinking with respect to the Western tradition, i.e. a non-violent political theory.5 Twenty years after Tolstoy, i.e. just a century ago, Gandhi wrote the “red-book” of the Indian revolution: Hind Swaraj;6  and in it he radically questioned, from the viewpoint of ethics and non-violence, all the areas of Western progress, one after the other. He also suggested, at the cost of being accused of a backward attitude, how to re-build them on a clear ethical basis. He also criticised Western science.7

The above criticisms by Gandhi have been considered too crude even by some of the  politicians who followed him. However, fifty years later, one of his Western disciples, Lanza del Vasto, improved on them. He based them upon two sacred texts of the Western tradition, Genesis,  and the  Apocalypse.

He interpreted original sin (Genesis 38) as an inversion of human knowledge from the loving contemplation of the world to that knowledge-calculation of good and evil which is used for utilitarian purposes.9 He attributed this sin not only to the first two human beings, but to all men throughout history. Such an exploitative attitude regards not only nature, but also people. Hence, such original sin is not at the beginning of the times – as the traditional interpretations suggest -, but it is the origin of our life and of social life. Within social relationships intellectual activity aimed at exploiting others covers up selfishness by means of the common formalities; above all the social formalities, i.e. laws, which, growing in number, end up shaping society as a pyramidal structure of social power. When the formalities reach the highest degree by means of a formal language and a formal world, namely in Science, the collective aims to exploit nature for the benefit of all, hides any number of malicious political aims of dominant political groups. Conversely, a person wanting to exploit some scientific advantages for his own welfare, is so interested in his happiness to ignore his responsibility for controlling Science, which is a so wide initiative to result to be uncontrollable by whatsoever State; hence, whereas people see their happiness Science progresses in an independent way from mankind will. In conclusion, original sin is essentially a structural sin, in both a person and in a society.

By hiding the attitude of domination of the few over the many, the above formal institutions may grow until they completely dominate the people, as an impersonal dictatorship. According to Lanza del Vasto, this extreme social situation is described in the Bible by the second sacred text, Apocalypse 13, through the image of a Beast rising from the sea and dominating the world. He interpreted it as modern Science, because "The irreparable lack of modern science is that it lacks someone who knows it entirely";10 which means that at present we are subordinated to the super-human project constituted by scientific progress, so called. Then Apocalypse 13 describes another Beast rising from the earth, whose authority depends on the power of the former Beast. Lanza del Vasto interpreted it as the Machine, or the State-Machine; which, by dispensing numerous facilities and conveniences, leads to a false kind of development, where even the wisest seek personal profit rather than cooperative fairness; so that social life becomes based on such a degree of alienation as to become entirely subjugated by the two Beasts (the front numbers are the no.s of the verses):  

“16And he shall make all, both little and great, rich and poor, freemen and bondmen, to have a character in their right hand or on their foreheads: 17And that no man might buy or sell, but he that hath the character, or the name of the beast, or the number of his name.”

It is easy to see in this description the dictatorships that infested advanced European countries some decades ago.  Thus, modern civilisation, by relying upon the worldwide expansion of Western science, is seen by Lanza del Vasto as the greatest renewal of Original sin.11

At present this negative attitude towards modern science goes against the present attitudes of Christian churches. It is on this issue that there is greatest divergence between the non-violent attitude and the attitude of Western Churches, otherwise very sympathetic to non-violence. But at present this critical vision of Science is shared, at least in part, by some political movements, e.g. the radical ecological movement.

According to the above non-violent teachers, the meaning of peace is the opposite to  that attributed to peace by the traditional scientific vision for which its meaning is abstracted from the person’s life, while the former, by trying to solve conflicts through interpersonal relationships, relies heavily on the personal witness of the kind of life one chooses. Moreover, peace is understood not just at an individual level; i.e., a new ethics is actively sought at the political level of society as a whole. Let us recall that Gandhi’s life united Indian and Western culture through the notion of “law”; which in the West is juridical law (of which Gandhi, as a lawyer, was a representative) and in the East is “inner law” (“the little inner voice"). Therefore, in the wide arena of all social relationships peace is achieved by promoting a new kind of social ethics which relies on co-responsibility,12 the egalitarianism, sharing, justice, community, in short, an ethics relying on trust in man and therefore anti-Machiavellian.  

3. Any conflict within Science?

But, if the non-violent position of the great teachers is correct, i.e. that Science represents the breeding ground of the present cultural violence, then should we reject Western Science?

Indeed, the above criticisms of Science come from outside Science. They may be the  result of pre-conceptions, insistently maintained by some who are nostalgic for the past, as well as by those  resistant to change.13 However, I have taken these criticisms seriously, especially those of Lanza del Vasto, and I have devoted thirty years of my scientific life trying to clarify the problem.14 I asked: Does an alternative Science exist? Does a non-killing, non-violent Science exist? First of all, is there a conflict between two ways of producing Science?

Let us start to explore science on the basis of the above questions avoiding what cannot be fully grasped by laymen, i.e. the technicalities or a philosophical debate.15 We will look at the historical development of the relationship between science and conflict; and then at the conflicts within science.

Two historians of science introduced the subject of the conflict into their illustrations of past science. Fifty years ago, A. Koyré16 cleverly interpreted the birth of modern science as determined by the use of the notion of infinity. Remarkably, at that time some scientists (Huygens, Descartes, etc.) supported potential infinity (whose best instance is a counting of natural numbers, i.e. an unlimited process which lacks a final number), whereas other scientists (Newton) supported actual infinity (whose two best instances are the final end, i.e. the point at infinity, of a straight line and the infinitesimal, which is defined as a number which is less than all real numbers).17 Hence, Koyré highlighted a basic conflict at the birth of modern science. (Notice that in this dispute it was  Newton who finally won. But I remark that a century and half later, an entire physical theory, thermodynamics, was born by making use of a mathematics that lacked actual infinity).

The historian T.S. Kuhn18 also suggested a conflictual vision of the history of science, which in his case concerned the development of classical physics as a whole. He thought that science proceeded by constantly applying a paradigm that is shared by the scientists that make up the scientific community. But it may occur that a specific case-study (such as the black-body theory in theoretical physics at the end of the 19th Century) halts the successful applications of this paradigm; such a case-study constitutes a theoretical anomaly, which brings about a scientific revolution (in the above case-study, the quanta revolution), leading to the replacement, through a Gestalt phenomenon in the minds of the entire scientific community, by a new paradigm (the corpuscle-wave complementarity) of the previous paradigm (the continuous vision of reality). Owing to the nature of a Gestalt change, the new paradigm proves to be incommensurable with the previous one, with the risk of untranslatability, and even incommunicability, between the two paradigms. In other terms, according to Kuhn, science suffers conflicts between successive paradigms.

However, neither Koyré’s (on the philosophical notions of infinity) nor Kuhn’s (on some the philosophical and historical notions, i.e. paradigm, anomaly, revolutions) interpretations on history of science relied upon the basic notions of science itself. Their analyses are very appealing because they are cleverly supported by suggestive historical cases; but they did not achieve scientific proof of the validity of their interpretations. Therefore, both constitute no more than philosophical analyses, not directly linked to the core of science.

It was not only historians of science that introduced conflicts into science; a more accurate inspection of past science reveals that some scientists did so too. Already at the end of the 19th Century, Haeckel proposed a new science, i.e. ecology. It originated from a new, global scientific vision of reality (oikos = home), as opposed to the local, analytic vision of dominant science. It was moreover based on the notion of the cycle of life rather than on either ideal notions (absolute space in Newtonian mechanics) or functional relationships (the field in electromagnetism). Haeckel’s theory was almost ignored by the scientific community. But after a century, it was realised that the various kinds of pollution, resulting from ecological ignorance of cycles in nature, constituted a threat to human life on the planet. Although reluctantly, the academic world had to inaugurate a specific University curriculum on ecology; however, it was not the global scientific method that Haeckel had suggested, but the sum of a series of distinct analytical techniques

More recently, I. Prigogine19 charged Newtonian mechanics with having led to the catastrophic exploitation of Nature; in order to establish “a new Alliance” with nature it is necessary to understand life, for the first time, scientifically; to this end, he proposed thermodynamics as the more appropriate theory for starting to establish a new, harmonious alliance between mankind and Nature. Such a proposal adds the mathematical theory of chaos, which claims to go beyond the deterministic conception of Newtonian mechanics and hence radically changed the previous scientific conception of the world. In addition to the theory of chaos, Prigogine, together with several other scientists, proposes the mathematical theory of complexity as the new direction of scientific research. In other words, through a new scientific attitude he supported a program of scientific research which would achieve a new kind of scientific development.

Hence, ecology, Prigogine’s program and complexity theory propose for the future a vision of scientific progress that will renew that derived from mechanistic science. But  they do not clarify the nature of their opposition  to traditional science, i.e. whether it is merely cultural and therefore collateral or complementary to traditional science; or whether they are proposing a truly alternative science.

Although they are unable to recognise at what point in the foundations of science there exists a conflict and what its nature is, all the above scientific proposals suggest however some philosophical distinctions, e.g. analytical and global, deterministic or chaotic, simple and complex, etc..

Let us now consider hat the above implies for the notion of peace. Both scientific programs, Haeckel’s and Prigogine’s, involving respect for life and hence outlawing the very ideas of war, enemy and destructive solutions to conflicts, suggest an active process for achieving peace. They imply therefore positive peace, rather than the passive peace suggested by the dominant science.

This radical change in the meaning of peace is in agreement with the non-violent meaning of peace. Such an agreement encourages us to proceed in search of a non-violent, non-killing science. However, non-violence adds to the previous meaning by specifying the global method with which one searches positively for peace; non-violence suggests that in the process of achieving peace in an alternative way to war, it is necessary focus  attention not only on the aim, however positive it may be, but above all tools employed, which have to be non-violent if they are to be adequate for achieving the  positive aim.

4. The birth of conflict and pluralism in science during the French revolution

A more accurate analysis of Kuhn’s history of science does not support one crucial point of his interpretation, i.e. the birth of classical chemistry, which was not determined by any “supra-mechanical aspect”;20 rather, it is well-known that it was the result of a cultural battle against the Newtonian tradition of interpreting chemical affinity through gravitational force. Moreover, a similar analysis does not support Prigogine’s thesis that the birth of  thermodynamics was no more than “an abortion” of the alternative that he is searching. Rather, past historians have been unable to understand the genius of the founder, Sadi Carnot;21 moreover, one has to remark that thermodynamics seems at a first glance to be an alternative theory to Newtonian science because it was formalised without actual infinity and all its variables are global in nature. A more accurate historical appraisal is therefore necessary of the origins of these two scientific theories, and, more in general, of the corresponding period of the history of science.

The French Revolution wanted to reform Newton’s science, accusing it of being mythical in nature.22 Lavoisier is known for having done so in chemistry by rejecting Newton’s notion of affinity as gravitational force). He intentionally published his main book in the same year as the  French revolution, 1789; in the introduction he wrote that he sought to bring about a “scientific revolution”. Moreover, during this period all scientific theories were founded anew: geometry (Monge, L. Carnot, Poncelet), infinitesimal calculus (L. Carnot, Lagrange), mathematized mechanics (L. Carnot, Lagrange, Navier, Poisson) and, in addition, thermodynamics was born (S. Carnot).23 Moreover, historians evaluate the revolution in geometry, i.e. Lobachevsky’s invention of non-Euclidean geometry in the remote Kazan University, as a long-term consequence of the French revolution.24

A leading figure of this renewal of science was L. Carnot. In opposition to celestial mechanics (the best application of Newton’s mechanics, which relies upon the metaphysical notions of absolute space and absolute time), he founded terrestrial mechanics (dealing with the impacts of bodies; and more precisely, the mechanics of machines; notice that each of them is a complex aggregate of bodies, which was considered by L. Carnot globally); he founded the theory on the practical concept of work and not on the metaphysical one of force-cause. Moreover, he re-formulated both the mathematical theories of his times, i.e. geometry and infinitesimal calculus, in an alternative way; furthermore, he suggested to his son Sadi the key ideas that gave rise to thermodynamics, whose theoretical structure is very different from that of Newton’s theory.25

L. Carnot’s main scientific achievement was to suggest an alternative to the dominant organization of a scientific theory. Instead of the pyramidal organization (which we find by Aristotle and then in both Euclid and Newton; at the top it puts “evident” principles, from which all laws are deductively drawn; we will call it axiom-based organisation).  L. Carnot’s new kind of organisation (we will call it problem-based organisation) is centred on a general problem (in mechanics: that of  finding the invariant quantities during a phenomenon of an impact), to which problem the development of the theory finds a general solution.26

Also S. Carnot founded thermodynamics by posing a central problem (maximum efficiency in energy transformations); and by then finding a new method (Carnot cycles) that solves this problem.

The discovery of two ways of organizing a scientific theory suggested to L. Carnot a pluralistic attitude towards the foundations of science. He clarified it in infinitesimal analysis. In this theory he accepted and supported all the various foundations of analysis on the basis of a pluralistic attitude. His book received wide popular acclaim, but was then dismissed by the “war-like” attitude of the academic world of the subsequent age, according to which in any scientific theory proposed – if only for didactic reasons – there was only one foundation which cancelled out all others.

Soon after the French revolution in a remote town in Russia, Kazan, Lobacevsky (who had studied French books) was able to propose a new kind of geometry. He did not just change a single postulate (the fifth), but posed the problem of how many parallel lines there are and put forward an original method for solving it. He thus changed the entire theoretical framework of Euclidean geometry.27 A few decades after the failure of the French Revolution, the labour movement (unfortunately ignoring the new scientific theories) wanted to start an alternative theory in social sciences. Marx’s theory tackled the central problem of how to overcome capitalism in the history of mankind; first he studied the relationships between factory owner and workers, rather than that between buying and selling in the market; then through his studies he sought a new political method, based on scientific principles, for bringing about the social revolution.

I would also point out that the both Carnot’s and Lobachevsky’s theories are alternative not only in their organization, but also in their use of mathematics; instead of Newton’s (metaphysical) infinitesimal calculus, which includes actual infinity (or its inverse, the infinitesimal dx), they make use of a mathematics that is appropriate for operative calculations, i.e. it relies on potential infinity only. We might conclude that the French revolution gave rise historically to pluralism in scientific theories.

What was the relationship in this period between science and conflict (war)? Over the centuries, science has always been exploited for war purposes.28 However, an alternative attitude came into being during the French revolution. The military devoted itself to improving civil society. In other words. at that time there was a process of conversion of  those working in the military to civil purposes. Most of the new scientific theories of the French revolution were the work of military scientists: Monge, L. Carnot, Poncelet, Navier, Poisson; in particular thermodynamics was born almost entirely when the former soldier S. Carnot turned his attention from cannons, mythologised as having almost unlimited power, to civil machines, which he studied from the point of view of maximizing their efficiency.29

On the other hand, during the French revolution civil society wanted to apply human reason to social life as a whole, in particular to creating an alternative State to the old absolute, metaphysical State (recall the blue blood of the kings!).

In fact, the French revolution succeeded (notice, before Napoleon) in reforming the State’s military sector. It turned the mythical military structure of the aristocracy, which aimed at the kingdom’s expansion, into an institution that was an expression of the people’s will simply to defend civil society. Indeed in 1793, when the European monarchies united against revolutionary France, a military structure was rapidly re-built by means of the first great “levée en masse”; it was launched by the supreme head of the French army, Lazare Carnot. With a military background, he before 1789 had theorized the new strategic theory of total (popular) defence (as opposed to the ideology of ‘total war’ that had just come into being). In 1793 he successfully applied this strategy to defending democracy. The French people, although weaker in destructive weapons, achieved  “Victoire”.

Exactly two centuries later, in 1989 the peoples that freed themselves from the dictatorships of Eastern Europe reiterated this policy of people’s defence and defeated a super-power which was ready for the greatest destructive confrontation in mankind’s history. The French Revolution had therefore anticipated the only possible alternative we have today to the mythical and disastrous arms’ race, i.e.  collective defence only; and moreover a defence that is not entrusted to the mythical destructive power of an enormous military arsenal, but to the solidarity of a population wanting to protect both itself and its democratic institutions. Hence, in national defence there exists an historical tradition which constitutes an alternative to merely destructive defence, of which nuclear defence is an example.

More in general, in the history of the relationship between science and war, the link between the dominant science and the development of ever more destructive weapons is clear. However, the French revolution established a new, alternative link; even extreme conflicts are solved in the wisest way, as it was exemplified by Gandhi first and in the 20th Century by many other peoples. What is extraordinary in the French revolution is that the new notion of defence was developed by individuals from the military.  

But in the following period, the policy of the Restoration was to present science as it had been before French revolution, i.e. without internal conflicts, and to outlaw many scientific theories. After 1850, when the bourgeoisie took the social power, most of them were rehabilitated; but some of the previous theories have never been accepted;30; in particular, Marx’s theory; but also some “revolutionary” scientific theories (e.g. those of L. Carnot). On the other hand, Lavoisier’s chemical theory survived despite academic opposition, because it was supported by chemists and chemical engineers, who were indispensable to contemporary society. 31

5. Formally qualifying the conflicts within Science

We have considered some conflicts concerning the history and the philosophy of science; there have even more decisive conflicts within science after an acute crisis in the first years of the 20th Century, through studies investigating the internal structure of science, that is, the foundations of both mathematics and logic.

The studies of the foundations of mathematics recognised an essential conflict between two kinds of mathematics; i.e. the dominant mathematics that is taught in scientific Faculties and includes actual infinity, and the mathematics that makes use of potential infinity only; the latter mathematics is closely approximated by the mathematics that represents the working of the computer. Four decades ago this conflict was formally founded.32

An indication of the significance of this conflict was the idea, up to the 20th Century, producing hard science meant applying mathematics using actual infinity; thus, classical chemistry, which makes use of only rational numbers, was not considered (nor by Kant) a scientific theory, while thermodynamics, which was created without infinitesimal analysis, was considered an immature theory. Using actual infinity in several specific notions, such as infinitesimals, the mathematics of the past was idealistic in nature, so that two centuries and half after the birth of infinitesimal analysis, it still did not dealt with the subject of social conflicts. During the World Wars some scientists succeeded in doing so when they discovered that two coupled either difference equations or differential equations describe phenomena of mutual competition, including the subject of the arms’ race. The founders of infinitesimal analysis, i.e. both Leibniz and Newton, could have developed this same theory two centuries earlier, if they had not been prejudiced by the idealistic nature of this kind of mathematics.

Between the two World Wars game theory was born; it analyses in detail the aspects of a conflict by means of few integer numbers; the mathematical technique is so elementary that even Archimedes or Galileo had the technical capabilities to develop it.33

As an important consequence, game theory inaugurated a new mathematical relationship with reality which is alternative to the relationship established by Newton’s theoretical physics; instead of the metaphysical mathematics of the infinitesimals, it makes use of the more elementary theory of constructive mathematics, i.e. the theory of integer numbers.

It should be noted that almost in the same period of the birth of game theory, theoretical physics too had to admit that all reality is constituted, in a ‘complementary’ way to waves, by quanta which require the mathematics of integer numbers; and soon after game theory, theoretical biology also came about in association with discrete mathematics (e.g., a neuron as a two-state switch, the constitution of DNA by an integer number of bases, etc.) all outside continuous mathematics and even more outside the actual infinity. Since that time a conflict was apparent between the new scientific theories and traditional science linked to the idealised mathematical continuum (including actual infinity; for instance, the notion of infinitesimals).

In the above we have already seen that this novelty was anticipated by science during the French revolution. Chemistry was born from the mathematics of integer numbers; and more in general both L. Carnot’s mechanics and S. Carnot’s thermodynamics made use of the mathematics of the potential infinity only.

At the end of the 19th Century logic was confident that, having been mathematized, it had achieved an absolute nature. Nevertheless, at the beginning of 20th Century a conflict also arose in mathematical logic; in addition to classical logic, several kinds of different mathematical logics were discovered.

In particular, it was discovered that it is not the law of the excluded middle (either ‘A is true’ or ‘not-A is true’), but the law of double negation (“Two negatives affirm”); this distinction constitutes the borderline between classical logic and almost all the kinds of non-classical logic; in the latter kinds of logic two negations do not affirm (for an example: “Absolved owing to the lack of evidence of guilt” does not mean that the accused person is clean-handed, but that the court had insufficient evidence to establish whether he was guilty or not). Hence, mathematical logic is split into (at least) two incompatible branches.34

Again one can trace back the use of non-classical logic to some centuries before, in particular to some scientists of the period of French revolution. In their original scientific works one finds several sentences which are doubly negated statements of non-classical logic: "We call element what we could not yet decompose" (Lavoisier); "A never ending motion is impossible" (L. Carnot and S. Carnot); “This hypothesis [of two parallel lines to a given one] does not lead to any contradiction” (Lobachevsky); "These two postulates [constancy of the light speed and relativity] are only apparently irreconcilable" (Einstein); "One cannot simultaneously measure an object’s position and speed with absolute [= not relative] accuracy" (Heisenberg). Each of them play a fundamental role in the respective scientific theory.

It is precisely on this logical point that the enormous experience of Freud, who founded the theory of inner conflicts, was based. He explained his method in a paper of a few pages.35 He points out that the analyst asks the patient to speak freely about say, what he dreamt. The patient tells a dream; he met his mother; but a dispute arose and he, in a fit of rage, nearly killed her; but then he urges: “I did not want to kill her”. The analyst must notice this negation and, in turn, has to add one more negation: “It is not true that the patient did not want to kill his mother”.

The doubly negated sentence provides the clue to recognising the trauma that the patient has repressed in the past (i.e. denied in his inner self), which, however, emerges again and again. This enables the analyst to recognise the repressed part of the patient and hence to start the healing process.36 Let us remark that Freud’s whole theory is in agreement with the problem-based organisation; he poses the problem of the patient’s healing, then solves it through the invention of a new method, which interprets the dialogue inductively through doubly negated sentences constructed upon the patient’s negated sentences.

Non-classical logic plays a fundamental role also in conflict resolution when it is considered in general terms. Let us recall that the great discovery of the 20th Century was the non-violent method. In fact, the very term non-violence is a double negation (violence being a negation of life); notice that it does not have a positive equivalent (notwithstanding Gandhi’s efforts to substitute for it the affirmative word “satyagraha”); thus, the two negations do not affirm. On the contrary, the military way  of theorising the resolution of a conflict in the barracks makes use of classical logic, enforcing absolute certainties: “The enemy of my enemy is my friend” where the two negations affirm; and also of the equivalent logical law of the excluded middle: “Either friend or foe”, “Either patriot or stranger”, “Either obedience or disobedience”, etc..

Hence, unlike the classical logic of the military, the word “non-violence” introduces an entirely new way of reasoning with respect to the dominant one. This fact is also apparent in logical terms; indeed, classical logic guarantees rigorous deductions, whereas non-classical logic is the basis of inductive arguing.

Since both logic and mathematics are the foundations of all branches of science, from the above two kinds of conflicts it follows that there is a fundamental division within science as a whole, giving rise to intellectual conflict.37

Such a division within both logic and mathematics generates divisions within each scientific theory through both the plurality of its formulations and the radical variations in meanings of its basic notions when changing the formulation of the theory and even more when changing the theory itself. For instance in geometry, a straight line may be conceived of either as (Euclid and Lobachevsky) an infinitely prolonged segment or as (Hilbert) one single object including its two end-points; in theoretical physics, the space may be conceived of either as (Newton’s mechanics) absolute space or (in L. Carnot’s mechanics, and even more in special relativity) relative space; the notion of time may be conceived of as continuous (Newton’s mechanics) or as (thermodynamics) before and after; the two notions of classical physics, wave and corpuscle, play complementary roles in quantum mechanics, etc.

Notice that the two different logical worlds are mutually incompatible in their basic tenets. But, each doubly negated sentence is an open sentence; hence, non-classical logic is not exclusive in nature – as is classical logic; let us recall military logic -; it allows mutual dialogue and co-existence, that is, it introduces a fundamental pluralism.

(THE PAPER CONTINUES ON PART 2 IN THIS SECTION)

This article originally appeared on Transcend Media Service (TMS) on 15 Nov 2009.

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