A hypothesis for the scientific integration between economics, psychology, sociology

  • «Per metter ordine al gran disordine.» [To put order to the great disorder] (Domenico Cimarosa, Il matrimonio segreto)

  • «If you want truly to understand something, try to change it.» (Kurt Lewin)

  • «For every complex problem there is a solution that is concise, clear, simple, and wrong.» (Henry Louis Mencken)

  • «That the intention of the Holy Ghost is to teach us how one goes to heaven, not how heaven goes.» (Galileo)

  • «For the rest, here I am in your hands; do as you please.» (Galileo)

  • «It is a windy sort of glory to seek fame from the mere glitter of words.» (Francesco Petrarca)

  • «I gathered up the scattered twigs and leaves.» (Dante)

  • «Thou shouldst not have been old before thou hadst been wise.» (William Shakespeare)

  • «Men learn while they teach.» (Lucio Anneo Seneca)

  • «The seas I sail were never crossed before.» (Dante)

  • «When the gods wish to punish us they answer our prayers.» (Oscar Wilde)

  • «Experience alone does not create knowledge.» (Kurt Lewin)

  • «There is nothing so practical as a good theory.» (Kurt Lewin)

  • «Who controls the past controls the future: who controls the present controls the past.» (George Orwell)

To address the challenges of the times from subjective standpoints is typical of the traditional cultural models of all ages. The diverse subjective positions give rise to a variety of non-scientific “scenarios” that correspond to as many sets of social roles. Individuals who exercise these roles act as modern haruspices, who do not back out of making predictions and forecasts based on information so partial that the applied subjective probabilities turn out to be simply unreliable. However, the functions attributed to these roles contribute to defining models and guidelines for future behaviors, which are essential to direct social dynamics. This fact is a crucial point of human culture throughout history. It always results in disputes and discussions, which end in conflicts on the social and political levels. These situations are widespread in every social field and sphere, from the educational context to the very democratic system. In short, we could say that human history is a set of subjective positions, presumably oriented to the primary aim of social control. In the different historical periods, some subjective positions have established and strengthened themselves to a greater or lesser extent, influencing and expanding the political space. The latter has thus acquired a scope devoid of connotations related to behavior analysis based on scientific grounds.


Behavior Science with the Theory of Interests allows us to account for the incorrectness of those standpoints establishing the anomalous consideration of social (and political) reality in terms of opposition and conflict-an altered representation of social reality, so much so that it fails in explaining the very concept of “organization.” However, the fact remains that analyzing social reality also implies influencing that reality; in fact, perceiving social reality by human organisms results in human organisms adapting their behaviors to their perceiving (that is not the case with physical reality). Therefore, to be more specific, we could state that Behavior Science with the Theory of Interests makes it possible to bring out the real context of social interactions and explicate organizational issues in a dimension of scientific rationality.


In today’s globalized society, fostered primarily by scientific knowledge, the “in-my-opinion” perspective has so spread that it questions even the methodological paradigm on which science (also Behavior Science) rests. To question science means not considering that the new methodological paradigm, which Galileo first had the strength and the ability to realize, is the only innovative proposal in human history. It paved the way for modern society in a more general and profound meaning. Therefore, modern society should rest on the basic concept of scientific rationality. Without the innovative opening by Galileo, the further innovative opening by Darwin would not have been even conceivable. That said, Galileo’s and Darwin’s works turn out to be strictly and inseparably related.


From a perspective that overcomes common sense and redefines, on Galilean and Darwinian bases, the explicative methods human organisms can use to analyze and perform their own actions, the inconsistencies among the diverse methods whose foundations are different from the Galilean scientific method stand out. These inconsistencies affect the current social disciplines, all of which tend to define their boundaries in an exclusive mode based on research methods referring to a more general concept of methodological relativism. As a matter of fact, the scientific perspective (which assumes that the same research methodological criteria apply to both the study of human organisms and the natural sciences) does not admit the segmentation of human organisms into diverse and specific types of behavior, autonomous and disjointed, accordingly to the boundaries characterizing (at the present state of knowledge) the various social disciplines (such as economics, law, psychology, social network theory, sociology) also preventing any integration among them.


It is arduous to synthesize, at this point, the historical processes whose occurrence relates to how some disciplines have strengthened to the detriment of others. Such an effort would imply revisiting all processes that marked the course of the entire human culture.


What appears to a behavior scientist to be at least strange is the various social disciplines being so defined within their own boundaries and separate from one another that they do not recognize the irrefutable dimension of behavior-a dimension which pertains to behavior as a biological event. Even in the modern world, this situation is so pervasive as to result in harsh opposition to Behavior Science. It also emerges from a tendency to eliminate the very names of Galileo and Darwin from today’s “lexical vulgate.”


On that ground, it is necessary to handle the objects of study of the social disciplines within a partitioning of the most extensive set characterized by the predicate “behavior” (be it individual or social behavior). Namely, social disciplines’ languages must be made compatible with a consistent system of scientifically sound axioms to which all propositions belonging to those languages can be logically brought back. The meaning of the integration between social disciplines lies in this reconstruction of the corresponding fields of knowledge, which should be proposed and carried out in principle from an explicative perspective on human organisms founded on Galilean and Darwinian bases. This concept of integration is a radical departure from the concept of interdisciplinary, which is in line, on the contrary, with the commonsense research methods establishing the current boundaries of the social disciplines.

THE THEORY OF INTERESTS AND THE STRUCTURE OF SOCIAL INTERACTION

An attempt at clarification oriented to define the current state of knowledge about human organisms and society may seem superfluous to advocates of a pragmatism end in itself. However, an effort to understand the methodologies and objectives of social disciplines, and the continuing backwardness of their field, is indeed indispensable. As a matter of fact, knowledge is a factor that directly influences (or should influence) all of our behaviors, making each of them rational. Nonetheless, the postulate of rationality, which is now accepted and generalized when referring to natural phenomena, is still viewed with distrust of social phenomena. That marks the different degrees of development and awareness of human behaviors depending on whether they refer to natural phenomena or social phenomena. The Theory of Interests expresses the different types of social interaction in terms of sciential rationality, thereby responding to the methodological need to make the languages of the social disciplines compatible with a common system of axioms conforming to the syntax and semantics of the scientific language.

The Theory of Interests expresses a reference paradigm on which the scientific integration of the social disciplines can be founded. This theory is formulated using a language apt to axiomatization, which is characterized (like any language) by a syntactic aspect and a semantic interpretation referring to that same syntax. Note

The syntactic and semantic aspects of the Theory of Interests conform to the criteria typical of scientific language: the syntax expresses the logical-formal structure of social interaction in terms of relations between interests; the semantic interpretation entails that the concept of interest is connected with the behavior through a correspondence relation.

The correspondence between behavior and interest allows for the foundation of the Theory of Interests on an experimental dimension. The Theory of Interests and the experimental analysis of behavior explicate (i.e., they explain scientifically) the invariants and the regularities characterizing the multiplicity of forms and features of the living organisms’ behaviors, emerging throughout the encephalization process that occurs in a strictly evolutionary context.

In a nutshell, the structural character and the dynamic character of the Theory of Interests allow for any case of social interaction to be explicated in scientific terms by bringing it back to two primary forms of social interaction:

the positive involvement between two interests (behaviors) of two subjects, and the corresponding conjunct interrelation (which reformulates the commonsense concept of cooperation);

the negative involvement between two interests (behaviors) of two subjects, and the corresponding disjunct interrelation (which reformulates the commonsense concept of conflict).

By further logical specifications, it is possible to define situations of social interaction increasingly complex and articulate:

the organization;
the power (typical of institutions);
the exchange (typical of the market);
every phenomenon typical of the social structures, such as pluralism and democracy (and its dysfunctions, including the appropriation-expropriation of public functions), social classes, deviant power, and the various forms of deviation.


The Theory of Interests and Behavior Science have been developed with different modalities and knowledge objectives. The Theory of Interest has been developed to explain all human behavior’s characterizations in a unified way in the attempt to define a consistent set based on the primary concepts of “positive involvement” and “negative involvement.” Behavior Science has been developed to take behavior as an element susceptible to rigorous experimental analysis. Only later did it become clear the connection between the two paradigms (the Theory of Interests and Behavior Science), which, as said, complete each other through a correspondence relation


At the present state of knowledge, the science of behavior allows us to take for granted the following knowledge acquisitions:

Every behavior is conditioned by the relation between organism and environmental (natural and social) stimuli.

Every (operant) behavior is learned (conditioned), including economics, ethics, social norms, legal rules, and scientific regularities.

The behaviors belonging to the past history of any organism are repeated, necessarily, under similar stimuli or stimuli belonging to the same class (set) of original stimuli.

Punishment does not cause the removal of a given (illegal or unethical) behavior from the repertoire of learned behavior, although all current educational and organizational processes are based on punishment.

Organisms are more reinforced by getting smaller rewards in a short time rather than greater rewards in a long time (unless they learned otherwise). These results also explain the reversal of the “preference” order expressed by the hyperbolic discount functions.

Economics is the most developed social discipline at the syntactic level because mathematics (and, in the most modern version, Debreu’s axiomatization in algebraic and topological terms) guarantees the consistency of the economic language. However, the semantic interpretation of the economic language is not equally guaranteed because there is no experimental verification making it univocal, despite the many attempts that have been made and are still being made.

We can say the same for the social network analysis elaborating formal (mathematical and stochastic) models about behaviors yet giving them a semantic interpretation through commonsense exemplifications.

Behavior science shows that also experiments based on game theory carried out in the economic field realize (through behavior repetition) trial-and-error learning processes, which are hindered or facilitated by factors such as the subjects’ past history and the particular social interactions (external behavioral stimuli) reinforcing the subjects’ operant behaviors.

One can recognize only a normative purport to economic theory as things stand. It means economics must be learned so that it can be reflected in the social context.


A discourse is scientific when its theoretical context corresponds to the experimental context. In this way, we can analyze those hidden variables we would not know otherwise. Behavior science allows us to find hidden variables, just as natural science investigates the field of hidden variables in physics. Therefore, we should assume that by applying the scientific method to social issues, we should be able to approach those issues radically differently from common sense.


The crucial contradiction of the current historical period (strangely unnoticed by the present common culture) is that two antithetical tendencies coexist in human behavior. The first is specified in a set of operations conforming to dictates of experimental science, now attested in many fundamental fields of our existence. The second is specified, in fields equally important, in those actions using in an irrational and distorted manner the technologies (and technical knowledge) made available to us by the science or not using them at all.

In brief, this is a particular situation of “dissonance” between (scientific) knowledge of nature and (a-scientific) knowledge of human organisms that the present culture does not seem to sense.

Behavior Science is currently in this situation: it could make a decisive contribution to solving many problems of our societies, but human organisms ignore and reject it to leave room to value judgments, that is, opinions by definition subjective, which are constantly being applied in the field of individual and collective choices.

Giulio Bolacchi presenta una ipotesi di integrazione delle discipline sociali fondata sulla scienza del comportamento

In general, opinions have no sciential purport because, by definition, every opinion has the same value as its negation, whereas science is such because, by definition, no scientific proposition can have the same sciential purport of its negation, that is, no scientific proposition can be replaced by its contrary.

Scientific language is univocal and consistent, whereas value judgments are not univocal and can be contradicted.

Some issues about Behavior Science and the Theory of Interests and their application to explain different models of social organizations are discussed in a video interview published on the History of Behavior Analysis channel.

Parte prima dell'intervista sulle problematiche della scienza del comportamento Parte seconda dell'intervista sulle problematiche della scienza del comportamento Parte terza dell'intervista sulle problematiche della scienza del comportamento Parte quarta dell'intervista sulle problematiche della scienza del comportamento

  • «Nature had been generous to Galileo. It had given him not only a good mind but also skillful hands. What his acute mind could design, his hands could build. Thus he built a telescope which extended the limited powers of human senses in ways that no one had foreseen. He saw a universe “enlarged a hundred and thousand times from what the wise men of all past ages had thought”; he saw new and extraordinary phenomena in the sky. The thorough study of these phenomena led him gradually to conclusion which went against the accepted principles of his time. To defend his conclusions and with them the freedom of thought in science, he engaged in a long battle which brought persecution down on him and forced him to spend the last years of his life in seclusion. But, as Max Born, a scientist of our times, has said: “The scientific attitude and methods of experimental and theoretical research have been the same through the centuries since Galileo and will remain so.” (Max Born, Nobel Prize 1954, Physics in My Generation, Pergamon Press, 1956).»

    Laura Fermi e Gilberto Bernardini, Galileo and the Scientific Revolution, 1961/2003
  • «True knowledge is ignorant of values, but it cannot be grounded elsewhere than upon a value judgment, or rather upon an axiomatic value. It is obvious that the positing of the principle of objectivity as the condition of true knowledge constitutes an ethical choice and not a judgment arrived at from knowledge, since, according to the postulate’s own terms, there cannot have been any “true” knowledge prior to this arbitrary choice. In order to establish the norm for knowledge the objectivity principle defines a value: that value is objectivity knowledge itself. Thus, assenting to the principle of objectivity one announces one’s adherence to the basic statement of an ethical system, one asserts the ethic of knowledge
    Jacques Monod,  Chance and Necessity, 1972
  • «While not disparaging the insights that have been gained, we feel that in these areas [growth models] the point of diminishing returns may have been reached. Nothing is easier than to ring the changes on more and more complicated models, without bringing in any really new ideas and without bringing the theory any nearer to casting light on the causes of the wealth of nations. The problem posed may well have intellectual fascination. But it is essentially a frivolous occupation to take a chain with links of very uneven strength and devote one’s energies to strengthening and polishing the link that are already relatively strong.»
    Frank H. Hahn & Robert C.O. Matthews, The Theory of Economic Growth: A Survey, 1964
  • «Most Serene Grand Duke:

    Though the difference between man and the other animals is enormous, yet one might say reasonably that it is little less than the difference among men themselves. What is the ratio of one to a thousand? Yet it is proverbial that one man is worth a thousand where a thousand are of less value than a single one. Such differences depend upon diverse mental abilities, and I reduce them to the difference between being or not being a philosopher; for philosophy, as the proper nutriment of those who can feed upon it, does in fact distinguish that single man from the common herd in a greater or less degree of merit according as his diet varies. He who looks the higher is the more highly distinguished, and turning over the great book of nature (which is the proper object of philosophy) is the way to elevate one’s gaze. And though whatever we read in that book is the creation of the omnipotent Craftsman, and is accordingly excellently proportioned, nevertheless that part is most suitable and most worthy which makes His work and His craftsmanship most evident to our view. The constitution of the universe I believe may be set in first place among all natural things that can be known, for coming before all others in grandeur by reason of its universal content, it must also stand above them all in nobility as their rule and standard. […]

    Your Most Serene Highness’s most humble

    and most devoted servant and subject,

    Galileo Galilei»
    Dialogue concerning the Two Chief World Systems - Dialogo sopra i due massimi sistemi del mondo, 1632 (translation by Stillman Drake, 1953/1967)
  • «Exit and voice, that is, market and non-market forces, that is, economic and political mechanisms, have been introduced as two principal actors of strictly equal rank and importance. In developing my play on that basis I hope to demonstrate to political scientists the usefulness of economic concepts and to economists the usefulness of political concepts. This reciprocity has been lacking in recent interdisciplinary work as economists have claimed that concepts developed for the purpose of analyzing phenomena of scarcity and resource allocation can be successfully used for explaining political phenomena as diverse as power, democracy, and nationalism. They have thus succeeded in occupying large portions of the neighboring discipline while political scientists – whose inferiority complex vis-à-vis the tool-rich economist is equaled only by that of economists vis-à-vis the physicist – have shown themselves quite eager to be colonized and have often actively joined the invaders.»
    Albert O. Hirschman, Exit, Voice, and Loyalty, 1970
  • «Some years ago, as Your Serene Highness well knows, I discovered in the heavens many things that had not been seen before our own age. The novelty of these things, as well as some consequences which followed from them in contradiction to the physical notions commonly held among academic philosophers, stirred up against me no small number of professors— as if I had placed these things in the sky with my own hands in order to upset nature and overturn the sciences. They seemed to forget that the increase of known truths stimulates the investigation, establishment, and growth of the arts; not their diminution or destruction. Showing a greater fondness for their own opinions than for truth, they sought to deny and disprove the new things which, if they had cared to look for themselves, their own senses would have demonstrated to them. To this end they hurled various charges and published numerous writings filled with vain arguments, and they made the grave mistake of sprinkling these with passages taken from places in the Bible which they had failed to understand properly, and which were ill suited to their purposes. […]

    This being granted, I think that in discussions of physical problems we ought to begin not from the authority of scriptural passages, but from sense-experiences and necessary demonstrations; for the holy Bible and the phenomena of nature proceed alike from the divine Word, the former as the dictate of the Holy Ghost and the latter as the observant executrix of God’s commands. It is necessary for the Bible, in order to be accommodated to the understanding of every man, to speak many things which appear to differ from the absolute truth so far as the bare meaning of the words is concerned. But Nature, on the other hand, is inexorable and immutable; she never transgresses the laws imposed upon her, or cares a whit whether her abstruse reasons and methods of operation are understandable to men. For that reason it appears that nothing physical which sense-experience sets before our eyes, or which necessary demonstrations prove to us, ought to be called in question (much less condemned) upon the testimony of biblical passages which may have some different meaning beneath their words. For the Bible is not chained in every expression to conditions as strict as those which govern all physical effects; nor is God any less excellently revealed in Nature’s actions than in the sacred statements of the Bible.»

    Galileo, Letter to Madame Christina of Lorrain Grand Duchess of Tuscany, 1615 (translation by Stillman Drake, 1957)
  • «One problem with sociology that I noticed particularly through my contacts with the Harvard Sociology Department, where I knew a number of people, was that no two people seemed to be doing the same thing. […] It was just as though every sociologist was starting the subject from the beginning. I hope I am not too cruel here, but it didn’t seem as if all these people were building on something. They would bring together some quantitative methods; they would make a model; or they would just do straight empirical work; or whatever it was. But it didn’t seem to build on any previously established generalities of sociology, as far as I could see.»
    Kenneth J. Arrow,  in: R. Swedberg, Economics and Sociology, 1990
  • «Among the Greeks and the Scholastics the dominant view was one of a unified, completely integrated universe. In such a scheme, the behavior of any single object was regarded as determined by its relation to the rest of the cosmos, by the necessary role it must play in the whole drama. It was difficult, therefore, to think of events in isolation, for example, to concentrate attention on the behavior of a single object in a partiucular region rather than on events occurring in the universe as a whole.

    From the early 17th century on, scientific thinking tended more and more toward a narrow limitation of attention in a given experimental situation, concentrating upon those few factors that seemed most relevant and decisive. The great success on this approach became evident in Galileo's work on mechanics. To investigate the behavior of a body freed "from its weight", he asked us to consider a ball placed upon a frictionless horizontal plane [...] In Newton's first law of motion the conception of the isolated system is most evident, for the postulate allows us to think about the behavior of a totally isolated object, one which in thought is removed altogether from the influence of other objects.»
    Gerald Holton & Duane H.D. Roller, Foundations of Modern Physical Science, 1965
  • «Human behavior is distinguished by its complexity, its variety, and its greater accomplishments, but the basic processes are not therefore necessarily different. Science advances from the simple to the complex; it is constantly concerned with whether the processes and laws discovered at one stage are adequate for the next. It would be rash to assert at this point that there is no essential difference between human behavior and the behavior of lower species; but until an attempt has been made to deal with both in the same terms, it would be equally rash to assert that there is. A discussion of human embryology makes considerable use of research on the embryos of chicks, pigs, and other animals. Treatises on digestion, respiration, circulation, endocrine secretion, and other physiological processes deal with rats, hamsters, rabbits, and so on, even though the interest is primarily in human beings. The study of behavior has much to gain from the same practice. We study the behavior of animals because it is simpler. Basic processes are revealed more easily and can be recorded over longer periods of time. Our observations are not complicated by the social relation between subject and experimenter. Conditions may be better controlled. We may arrange genetic histories to control certain variables and special life histories to control others [...] We are also able to control current circumstances to an extent not easily realized in human behavior–for example, we can vary states of deprivation over wide ranges. These are advantages which should not be dismissed on the a priori contention that human behavior is inevitably set apart as a separate field.»
    B. F. Skinner, Science and Human Behavior, 1956
  • «We are apt to take it for granted that economic and political associations will quickly arise wherever technical conditions and natural resources permit. If the state of the technical arts is such that large gains are possible by concerting the activity of many people, capital and organizing skill will appear from somewhere, and organizations will spring up and grow. This is the comfortable assumption that is often made. The assumption is wrong because it overlooks the crucial importance of culture.»
    Edward C. Banfield, The Moral Basis of a Backward Society, 1958
  • «Taken all in all, in the departments of the arts and sciences development has certainly been in the direction of an increase in the importance attached to experimental methods; but the evidence in favour of such an evolution is not so good if we turn to the fields of politics and social organization. It is significant that simple combinations foreign to scientific experience are far from having disappeared from modern social life; in fact, they persist in great numbers, thriving in prosperous exuberance. […] The combinations of experimental science have been vastly expanding all the way down to our own times, but for the most part they have occupied territory formerly held by the combinations of trial-and-error empiricism, magic, theology, and metaphysics. […] Many social scientists who are supremely great in the natural sciences, where they use logico-experimental principles exclusively or almost so, forget them entirely when they venture into the social sciences. »
    Vilfredo Pareto, The Mind and Society, 1935 (Edited by Arthur Livingstone, translated by Andrew Bongiorno, Arthur Livingstone, James Harvey Rogers)
  • «If we are asked to name the man who did most to start physical science on the triumphant course which lasted for nearly three hundred years, we must answer: Galileo Galilei (1564-1642). Leonardo, Copernicus and Gilbert, each in his different way, foreshadowed the coming revolution, but Galileo went farther, and in his writings we recognize for the first time the authentic modern touch. He brought the theory of Copernicus to the practical test of the telescope, but above all in his work on dynamics he combined observation and induction with mathematical deduction tested by experiment, and thus inaugurated the true method of physical research. In his work there is no reliance on an authoritative and rational scheme as in Scholasticism; each problem is faced in isolation, each fact is accepted as it stands with no desire to make it fit into a universal pre-ordained whole; concordance, if possible at all, comes slowly and partially; mediaeval Scholasticism was rational, modern science is in essence empirical, accepting brute facts whether they seem reasonable or not.»

    William Cecil Dampier, A Shorter History of Science, 1944
  • «Some scientists and philosopher feel a strong disinclination against all abstractions or schematizations. They demand that any methodological or even logical analysis of science should never lose sight of the actual behavior of scientists both in the laboratory and at the desk. They warn against neglecting any of the factors which a good scientist takes into consideration in inventing and testing his hypotheses; they emphasize that the complex judgment on the acceptability of a hypothesis cannot be based on just one number, the degree of confirmation. I think that this view contains a correct and important idea. Whenever we make an abstraction, we certainly ought to be fully aware of what we are doing and not to forget that we leave aside certain features of the real processes and that these features from which we abstract at the moment must not be entirely overlooked but must be given their rightful place at some point in the full investigation of science. On the other hand, if some authors exaggerate this valid requirement into a wholesale rejection of all abstractions and schematizations, an attitude which sometimes develops into a veritable abstractophobia, then they deprive science of some of its fruitful methods.»
    Rudolf Carnap,  Logical Foundations of Probability, 1950
  • «Twenty-five hundred years ago it might have been said that man understood himself as well as any other part of his world. Today he is the thing he understands least. Physics and biology have come a long way, but there has been no comparable development of anything like a science of human behaviour. Greek physics and biology are now of historical interest only (no modern physicist or biologist would turn to Aristotle for help), but the dialogues of Plato are still assigned to students and cited as if they threw light on human behaviour. Aristotle could not have understood a page of modern physics or biology, but Socrates and his friends would have little trouble in following most current discussions of human affairs.»
    B. F. Skinner, Beyond Freedom and Dignity, 1971
  • «However important set theory may be now, when it began some scholars considered it a disease from which, it was to be hoped, mathematics would soon recover. For this reason many set-theoretic considerations were called pathological, and the word leaves on in mathematical usage; it often refers to something the speaker does not like. The explicit definition of an ordered pair ((a,b) = {{a},{a,b}}) is frequently relegated to pathological set theory. For the benefit of those who think that in this case the name is deserved, we note that the definition has served its purpose [...]. We need to know that ordered pairs are determined by and uniquely determine their first and second coordinates, that Cartesian products can be formed, and that every set of ordered pairs is a subset of some Cartesian product; which particular approach is used to achieve these ends is immaterial. It is easy to locate the source of the mistrust and suspicion that many mathematicians feel toward the explicit definition of ordered pair given above. The trouble is not that there is anything wrong or anything missing; the relevant properties of the concept we have defined are all correct (that is, in accord with the demands of intuition) and all the correct properties are present. The trouble is that the concept has some irrelevant properties that are accidental and distracting. The theorem that (a,b) = (x,y) if and only if a=x and b=y is the sort of thing we expect to learn about ordered pairs. The fact that {a,b} ∈ (a,b), on the other end, seems accidental; it is a freak property of the definition rather than an intrinsic property of the concept.

    The charge of artificiality is true, but it is not too high a price to pay for conceptual economy. The concept of an ordered pair could have been introduced as an additional primitive, axiomatically endowed with just the right properties, no more and no less. In some theory this is done. The mathematician’s choice is between having to remember a few more axioms and having to forget a few accidental facts; the choice is pretty clearly a matter of taste. Similar choices occur frequently in mathematics [...].

    Paul R. Halmos, Naive Set Theory, Springer, 1974, pp. 24-25
  • «The regularities in the phenomena which physical science endeavors to uncover are called the laws of nature. The name is actually very appropriate. Just as legal laws regulate actions and behavior under certain conditions, but do not try to regulate all actions and behaviors, the laws of physics also determine the behavior of its objects of interest only under certain well defined conditions, but leave much freedom otherwise. The elements of the behavior which are not specified by the laws of nature are called initial conditions. These, then, together with the laws of nature, specify the behavior as far as it can be specified at all: if a further specification were possible, this specification would be considered as an added initial condition ... The surprising discovery of Newton’s age is just the clear separation of laws of nature on the one hand and initial conditions on the other. The former are precise beyond anything reasonable; we know virtually nothing about the latter.»
    Eugene P. Wigner, Events, Laws of Nature, and Invariance Principles (Nobel Lecture), 1963
  • «In Sarsi I seem to discern the firm belief that in philosophizing one must support oneself upon the opinion of some celebrated author, as if our minds ought to remain completely sterile and barren unless wedded to the reasoning of some other person. Possibly he thinks that philosophy is a book of fiction by some writer, like the Iliad or Orlando Furioso, productions in which the least important thing is whether what is written there is true. Well, Sarsi, that is not how matters stand. Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.»
    Galileo, The Assayer - Il Saggiatore, 1623 (translation by Stillman Drake, 1957)
error: Protetto © Giulio Bolacchi