What can the Social Sciences Learn from the Process of Mathematization in the Natural Sciences Ladislav Kvasz, Pedagogical faculty, Charles University in Prague and Faculty of mathematics, physics and informatics, Comenius University in Bratislava Abstract. The paper tries to put the conflict of the natural and the human sciences into its historical context. It describes the changes in classification of scientific disciplines that accompany a scientific revolution, and offers an alternative to Kuhn’s theory. Instead of a conflict between the proponents and opponents of the new paradigm it interprets the revolution as a conflict between the mixed disciplines and the metaphorical realm of the old paradigm. 1 Introduction For almost two centuries there has been a tension between the natural and the social sciences. As Thomas S. Kuhn writes in The Structure of Scientific Revolutions,1 it was this tension that led him to the creation of the notion of a paradigm. According to Kuhn the difference between natural and social sciences consists in the fact that while in natural sciences we have to do with research in the framework of normal science based on a widely accepted paradigm, in social sciences there is nothing comparable to paradigms and so scholars again and again question the foundations of their disciplines. Kuhn thus drew attention to an important difference between these two areas. Nevertheless, according to Kuhn this difference does not create a gap between them: “I’m aware of no principle that bars the possibility that one or another part of some human science might find a paradigm capable of supporting normal, puzzle-solving research. … Very probably the transition I’m suggesting is already under way in some current specialties within the human sciences. My impression is that in parts of economics and psychology, the case might already be made.”2 If we want to understand this problem it is expedient to look at the tension between the natural and social sciences in a broader historical perspective. The first thing which we probably notice after turning to a broader historical perspective is that the conflict between natural and social sciences is not as old as it might 1 seem. In the Classical era there was no conflict between the way how people understood human and social phenomena on the one hand and how they approached nature on the other. This, of course, does not mean that in the Classical era the whole knowledge would form a harmonic whole. Also in Greek science there was a conflict that in many respects resembles the tension between the natural and the social sciences that we encounter in modern times. The border, along which the tension manifested itself, nevertheless, ran elsewhere. It did not separate knowledge of nature from the knowledge of human and social phenomena but rather it separated the mathematical knowledge (based on the deductive method and using categories such as number, proportion, and shape) from the “organic” realm (based on causal explanation and using categories such as purpose, goal, and action). In this second realm we could find biological as well as social disciplines, i.e. disciplines which according our classification lie on the opposite sides of the barricade that separates the natural from the social sciences. Ancient Greeks approached in a similar way the study of the “generation of animals” and the study of “the psyche” or politics. Starting from the 17th century onwards the study of the “generation of animals” was gradually incorporated into the realm of the newly constituted natural science, while the study of “the psyche” became one of the crystallization cores of the emerging social sciences. Therefore one of the first aims of the present paper is to propose a framework for the reconstruction of the shifts in the classification of scientific disciplines. 2 Classification of scientific disciplines according to their relation to the paradigm In order to be able to understand the transitions of scientific disciplines between the categories of “hard” and “soft” sciences it is useful to form a more differentiated image of the “topography of the scientific landscape” that lies between these two poles. As a first move we suggest to abandon the terminology of dividing the scientific disciplines into “hard” and “soft”. Instead let us call the “hard” disciplines paradigmatic disciplines. In contemporary science the paradigm is formed by physics and so the paradigmatic disciplines are all those disciplines in which the methods of quantification and measurement lead to success. For a more precise characterization of a particular area of “soft” disciplines I suggest to introduce the term elusive region of the paradigm. It comprises those disciplines where the methods and approaches of the particular paradigm cannot be employed. Besides these two kinds of scientific disciplines we introduce two other kinds which lie somewhere between the paradigmatic region and the elusive region of the paradigm.3 The first sort of scientific disciplines that lie between the paradigmatic and the elusive 2 region are the mixed disciplines. This term is used by historians to describe a remarkable set of disciplines from late Antiquity, such as Euclidean optics, Archimedean theory of the lever, the theory of simple machines, or Ptolemaic astronomy.4 These disciplines have in common the use of exact mathematical language in the description of situations which according to the ancient understanding of science should not be described using mathematics because matter plays a substantial role in them. These disciplines cannot be fully deductive and therefore they do not fulfill the standards of mathematics. On the other hand these disciplines do not use explanations based on the notions of aim and purpose (final cause), that Aristotle considered being the explanation of phenomena that belong to the elusive region of the ancient paradigm. Thus in a whole range of cases the practice of ancient science did not follow the standards laid down by Aristotle and it formed disciplines the methodological status of which was rather unclear. The fact that a lever, a mirror, or a pulley are material objects, but in spite of this, in their description scholars use mathematics, is from the ancient point of view inconsistent. The mixed disciplines played an important role during the scientific revolution of the 17th century. Galileo made important discoveries in the theory of simple machines, while Fermat and Descartes created theories of refraction of light. We may say that it were the mixed disciplines where the fundamental notions of the paradigm of modern science were born. The second category of disciplines lying between the paradigmatic and the elusive region can be called the metaphorical region of the paradigm. It forms a counterpart to the mixed disciplines. While in the case of the mixed disciplines the notions and methods of the paradigm are used in a precise and unambiguous way, and the problem is only that they are being used outside the area where their use can be justified by the paradigm’s methodology, in the metaphorical region the fundamental notions of the paradigm are used with a transferred, distorted and stretched meaning. As a representative of the metaphorical region of the ancient paradigm we can consider Aristotle’s theory of local motions, according to which heavy bodies fall downwards while light bodies float upwards. In a paper on Cartesian physics5 I argued that the Aristotelian theory of local motions is a geometrical theory. It is based on the image of a geometrically ordered universe and it understands motion as a transition between different places of this geometrical order. Nevertheless, geometry (the paradigmatic discipline of ancient science) is used here in a different manner from that in the mixed disciplines. Geometry does not enter the Aristotelian view of the order of the cosmos in an explicit way as a set of exact notions and methods for making constructions and proving theorems (as it enters the Archimedean theory of the lever), but only implicitly, as a set of metaphors, by means of which we can discern order and meaning in the phenomena. Thus 3 even though Aristotle’s understanding of motion is biological (or organic) and therefore belongs into the elusive region of the ancient paradigm, a fraction of it—the theory of local motions—is based on geometrical metaphors. We see that besides the paradigmatic region, i.e. the realm of disciplines that use the notions and methods of the paradigm in accordance with the methodological standards of the paradigm, and the elusive region, i.e. the realm which defies the use of the notions and methods of the paradigm, there are at least two other areas of scientific disciplines that are constructed using the means of the paradigm. On the one hand the paradigm offers the technical tools for the formation of the region of the mixed disciplines, i.e. disciplines that use the notions and methods of the paradigm in a precise and correct manner but apply them to phenomena which were not foreseen by the creators of the paradigm and where it is not possible to fully comply with the methodological standards dictated by the paradigm. Further, the paradigm leads to the formation of the metaphorical realm of the paradigm, which comprises those phenomena that are too complex, and so a precise technical use of the notions and methods of the paradigm is not possible. Nevertheless, the paradigm offers a whole range of metaphors that make it possible to understand these phenomena at least in a qualitative manner and so to incorporate them into the rational discourse created by the paradigm. If we wish to understand the relation of natural sciences (forming the paradigmatic region of contemporary science) and humanities (lying to a great extent in the elusive region of that paradigm), it seems reasonable to replace the opposition of the “hard” and “soft” sciences by the following scheme:6 METAPHORICAL REGION OF THE PARADIGM REGION OF PARADIGMATIC DISCIPLINES ELUSIVE REGION OF THE PARADIGM REGION OF MIXED DISCIPLINES 3 An outline of a reconstruction of the scientific revolution The above scheme makes it possible to soften the contrast between the paradigmatic region that is formed by the “hard” disciplines and the elusive region of the paradigm that is formed by the “soft” disciplines and so opens a new possibility for a rational reconstruction of the scientific revolution of the 17th century. It turns out that it were the mixed disciplines and their conflict with the metaphorical realm of the paradigm which were the driving force of 4 that revolution. Newtonian physics was created not inside the paradigmatic region of the old paradigm. The paradigmatic region of ancient science was mathematics. The birth of Newtonian physics stimulated the creation of several new mathematical disciplines, but despite of this, we cannot say that inside of mathematics there occurred some massive refutation of the previous research (which would be a case if a revolution occurred in this region). Also the elusive region of the old paradigm (the realm of the organic) did not undergo radical changes. Biology was during the scientific revolution of the 17th century on the fringe of the scientific interest. It came into the center of interest towards the end of the 18th century when the scientific revolution already reached its consummation. It is fair to say that the scientific revolution of the 17th century took place on the contact of the mixed disciplines of the ancient paradigm (astronomy, optics, the theory of simple machines) and the metaphorical region of that paradigm (the geocentric view of the cosmos). And this is rather natural. In the paradigmatic region of ancient science, i.e. mathematics, the methodological standards are so strict and well founded that a refutation of the overall picture is improbable. On the other hand the elusive region of the paradigm (i.e. the realm of biology) is not sufficiently stable and therefore changes happen there too often to be able to cause some deeper considerations. It is precisely the mixed sciences where the methods of the paradigm offer sufficiently effective means of research so that their progress is intensive. On the other hand the application of the paradigmatic methods to unintended areas of phenomena increases the probability of the discovery of something radically new and unexpected, something that will be in sharp contrast with all that we are used to expect in the paradigmatic region. The metaphorical region of the paradigm is important for another reason. There the research is carried out on the fringe of what the paradigm allows to thematize and therefore the metaphorical region is often the place for the basic cultural projections with the emotional charge that accompanies such projections. The mixed disciplines alone would probably never have led to a revolution. Had Galileo accepted the suggestions of the Church and discussed the Copernican system only as a hypothesis, i.e. if he had restricted himself to the technical realm of the mixed disciplines and had not confronted this system with the geocentric worldview, it is probable that he Church would have succeeded in keeping the new astronomical discoveries on the periphery of the interest of the public as an incomprehensible, innocuous technical hypotheses. The dynamic of the scientific revolution of the 17th century was driven precisely by the conflict of the mixed disciplines with the metaphorical region when not absolutely sure results of scientific inquiry got into conflict with metaphors by means of which we articulate our place in the universe. 5 METAPHORICAL REGION PARADIGMATIC DISCIPLINS NEW PARADIGM ELUSIVE REGION MIXED DISCIPLINES In this scheme paradigmatic disciplines are the paradigmatic disciplines of the old paradigm, and the same holds for the mixed disciplines as well as for the metaphorical and the elusive region. If we restrict ourselves to the scientific revolution of the 17th century, the above scheme expresses the fact that the paradigm of modern physics originated neither in the paradigmatic nor in the elusive region of the ancient paradigm, but in the area between them. The paradigmatic region of the ancient science was mathematics, which during the 17th century underwent a dramatic development (creation of the analytic geometry and of the calculus), but this development occurred in the framework of normal science. In mathematics of the 17th century nobody seriously questioned the results of the past. The elusive region of the ancient science was the realm of the organic, and the founders of modern physics almost completely avoided the discussion of questions of biology. Galileo marginally discussed the question of size of organisms and Descartes made occasional dissections. It is important to realize that the new paradigm rises from a conflict between the mixed disciplines and the metaphorical region of the old paradigm. This indicates where to look for the source of revolutionary changes in the contemporary social sciences. The paradigm of the modern science is physics while its elusive region is the realm of the subjective (the Cartesian res cogitans), i.e. the area of social sciences. The above scheme shows that all those who were waiting for “Newton of the social sciences”, waited at the wrong door. Social sciences form the elusive region of the physical paradigm. The elusive region is inaccessible to scientific methods and therefore it will not play any important role in the contemporary revolutionary changes. The next fundamental change in science will take place not in the elusive region of the physical paradigm, i.e. in the social sciences, but on the border of the mixed disciplines and the metaphorical region of physics. So let us have a closer look at this border. 4 The revolution in biology If we want to form a clearer idea about the scientific revolution that is happening in 6 contemporary science, it is useful to turn to a scheme, which would contain the paradigmatic, mixed, metaphorical, and elusive regions not only of the ancient science (that we analyzed in the previous section), but also of the paradigm of the contemporary science. As we already mentioned, the elusive region of the ancient paradigm, representing the realm of the organic phenomena, did not play any important role in the formation of the Newtonian paradigm. Nevertheless, the elusive region underwent a radical change. The elusive region of the ancient science became the center of the area between the metaphorical region and the region of the mixed disciplines of the Newtonian paradigm. METAPHORICAL REGION OF ANCIENT SCIENCE METAPHORICAL REGION OF MODERN SCIENCE PARADIGM OF ANCIENT SCIENCE PARADIGM OF MODERN SCIENCE ELUSIVE REGION OF ANCIENT SCIENCE ELUSIVE REGION OF MODERN SCIENCE MIXED DISCIPLINES OFANCIENT SCIENCE MIXED DISCIPLINES OF MODERN SCIENCE Because scientific revolutions happen in the area between the region of the mixed sciences and the metaphorical region, it is reasonable to conclude that the contemporary scientific revolution is taking place in biology, in the science of the organic. The new biological paradigm will emerge from the conflict between the mixed disciplines and the metaphorical region of the physical paradigm. So let us analyze these regions more thoroughly. 4.1 The mixed disciplines of modern science The mixed disciplines of modern science use the technical and theoretical tools of physics (its experimental methods and laboratory equipments, its theoretical notions and mathematical formalism) in the study of nonphysical systems. For the mixed disciplines it is characteristic that they use these tools in an exact and methodologically correct way; the only problem is that they use these tools in the study of biological systems, i.e. systems where strict repeatability of experiments is impossible. Despite these difficulties we witness a spectacular progress of experimental techniques leading from the discovery of the microscope through the Roentgen apparatus to the computer tomography. Among the recent developments are 7 magnetic resonance imaging and positron emission tomography which make it possible to visualize the brain activity during cognitive processes.7 The discovery of a new physical method of registration of data leads to a new breakthrough in biology and medicine. A similarly spectacular development has also occurred in the area of chemical analysis of living matter, leading from the first artificial synthesis of uric acid through the understanding of the structure of hemoglobin to the decipherment of the human genome. Therefore I suggest including disciplines such as biochemistry, molecular biology, or neurophysiology among the mixed disciplines, the methodological status of which is analogous to Euclidean optics or Archimedean theory of the lever in the antiquity. This region of the landscape of science does not present any serious problems, except that in the philosophy of science these disciplines do not get an adequate attention. 4.2 The metaphorical region of modern science In contrast with the mixed disciplines, the interpretation of the metaphorical region of the paradigm is problematic. The elusive region of the physical paradigm is the realm of the subjective. It is not important whether we define it metaphysically as Descartes did at the dawn of the physical paradigm, or epistemologically as did Dilthey, who witnessed its climax at the end of the 19th century. What is important is to realize the elusive nature of the subjective, i.e. the fact that it cannot be dealt with by means of the physical paradigm. From this elusive region of the physical paradigm gradually a small part separated itself in a similar way as from the Aristotelian organic theory of motion the theory of local motions was separated. It is the part that makes use of the metaphors of the paradigmatic disciplines. As an example I would like to mention the association psychology, developing the ideas of David Hume, the economic theory of the circulation of capital initiated by Francois Quesnay, or classical sociology initiated by August Comte.8 All these disciplines use notions like process, dynamics, speed, intensity, increase or force. Nevertheless, the use of notions as ‘mental process’, ‘intensity of emotional experience’, ‘speed of associations’ has very the same epistemological status as the use of notions ‘upwards’ and ‘downwards’ in the Aristotelian theory of local motions. The point is that these notions are used not in their strict meaning, determined by the physical paradigm. Physical processes take place in space and the metric structure of this space enables us to speak about their velocity. Associations do not happen in any space that would have a straightforward metric structure and so the term ‘process’ is being used here only in a metaphorical way. Similarly, Comte used the terms ‘social statics’ and ‘social dynamics’ in a metaphorical way. In the strict sense, i.e. the sense fixed by the 8 physical paradigm, the term statics refers to the science studying the equilibria of forces. Force is a physical quantity that is measured in unequivocally defined units (kg.m.s -2). On the other hand “forces” acting in society have no units in which we could measure them and so we can speak about equilibria in the social context only in a metaphorical sense. Similarly metaphorical is the notion of labor force in economics. Labor force is not a real force in the physical sense of the term force; it cannot be measured by means of the physical units by means of which we measure gravitational or electric forces. Similarly the use of the notion of work in economics is a metaphorical use of the physical notion of work, which is defined as a path integral of force. In economics it is not clear what forces we have to integrate along what path. The metaphors used in these disciplines cannot be conceptually clarified: from the metaphor of social forces or labor forces it is impossible to create a notion that would be at least approximately as clear and unambiguous as the notion of force in classical physics. It is precisely due to this vagueness and ambiguity of the basic notions of sociology and economics why I suggest including them into the metaphorical region of the physical paradigm. I would like to suggest that disciplines such as association psychology, political economy, or classical sociology are trying to understand their particular subject matter using metaphors coming from physics. At the same time the phenomena to which these disciplines apply their metaphors come from the elusive region of the paradigm, i.e. from the region to which the concepts and methods of the paradigm cannot be unambiguously applied. That is the reason why these disciplines have a problematic status (when compared with the paradigmatic disciplines), but on the other hand, just like the Aristotelian theory of local motion, these disciplines are the place for basic cultural projections and have a great potential for a radical transformation in the course of the next scientific revolution. 5 The biological revolution and social sciences The main weakness of all discussions about the differences between natural and social sciences is the dominance of physics and ignorance of biology. All ruminations on the alleged different character of the social sciences can be seen as an articulation of the fact that the social sciences have their origin in the elusive region of the physical paradigm. This may be correct but the example of biology shows that the paradigmatic disciplines of physics do not exhaust the entire region of natural sciences. Between the physical disciplines, which are usually taken as paradigmatic examples of science, and the biological sciences there are many deep differences in the nature of their empirical basis, epistemological status of fundamental 9 categories as well as logical structure of the whole theory.9 Biological data (say in ecology or in the theory of evolution) are often qualitative; the theory often contains notions of different levels of complexity. If we extrapolate the scheme presented in the previous chapter one step further, it seems probable that a breakthrough in the area of social sciences will occur only when the biological paradigm matures, so that it will develop its own mixed disciplines and own metaphorical region. The social sciences forming at present the elusive region of the physical paradigm will then be clinched between the mixed disciplines and the metaphorical region of the biological paradigm. Biology will thus lead to a fundamental change of the social sciences, similar to that which physics brought about in the sphere of the organic.10 Aristotelian theory of local motion, which was initially close to the elusive region of the ancient paradigm, was in the course of the scientific revolution of the 17th century shifted into the very center of the newly emerging mathematical physics. It is probable that during the biological revolution a similar shift awaits also the metaphorical region of the physical paradigm, namely psychology, economics, and sociology. These disciplines will be shifted from the elusive region of the physical paradigm (from the realm of social sciences) to the very center of the new paradigm of biology. Nevertheless, this will at the same time transform biology as well. Similarly as Newtonian physics was no longer physics in the Aristotelian sense of this word. It was not based on the four Aristotelian causes. It is probable that biology, after it absorbs psychology, economics and sociology, will be not the same science as we know it now. It will be not the theory of living systems (i.e. a discipline defined in the contraposition to the theory of non-living systems which are the subject matter of physics) but rather it will be the theory of systems with biological information (i.e. information understood as a code—in contrast to theories of information understood as symbol). From an informational point of view a cognitive scheme, the price of a commodity or a social hierarchy is similar to the genetic code. The information content can be interpreted as a code that represents the degree of adaptation of the system to its environment (the cognitive task, the market, the social, or the natural environment). The affinity of these disciplines is visible also from the increasing role which game theory plays in them.11 If we define biology by the means of description it uses rather than by the subject matter to which these means are applied, then psychology, economics and sociology will become biological disciplines, however strange this might sound. Acknowledgements I would like to thank Donald Gillies and Marek Tomecek for valuable comments. The paper 10 is a part of the project VEGA 1/3621/06 Historical and philosophical aspects of exact disciplines granted by the Slovak Grant Agency. Notes 1. Thomas S. Kuhn, The Structure of Scientific Revolutions. Chicago: University of Chicago Press 1962. 2. Thomas S. Kuhn, “The Natural and the Human Sciences”, in: Thomas S. Kuhn, The Road since Structure. Chicago: University of Chicago Press 2000, pp. 222-223. 3. Kuhn’s notion of paradigm had many meanings. Later Kuhn restricted the scope of this notion (see Thomas S. Kuhn, “Second Thoughts on Paradigms”, in: Thomas S. Kuhn, The Essential Tension: Selected Studies in Scientific Tradition and Change. Chicago: University of Chicago Press 1977, pp. 293-319). But it still remained rather broad. In Ladislav Kvasz, “On classification of scientific revolutions”, in: Journal for General Philosophy of Science 30, 1999, pp. 201-232. I suggested to distinguish three kinds of scientific revolutions and three kinds of paradigms: the paradigm of idealization, of representation and of objectification. For the present paper the paradigm idealization is the most relevant one and in the text that follows, by paradigm I will understand the paradigm of idealization. 4. I suggest (in contrast to Kuhn) to consider Euclid’s Elements as the paradigm of Ancient science. It may sound unusual to call Elements a paradigmatic theory. We understand paradigms as a part of science while for us mathematics does not belong to science. Nevertheless, it is problematic to use our contemporary classification of disciplines in interpreting antiquity. If we look at Ancient science not from our but from its own viewpoint, it is rather the Elements than the Almagest that had a paradigmatic status. Therefore the Ptolemaic astronomy that Kuhn characterized as paradigmatic I prefer to include among the mixed disciplines. 5. Ladislav Kvasz, “The Mathematisation of Nature and Cartesian Physics”, in: Philosophia Naturalis 40, 2003, pp. 157-182. 6. The scheme represents the topography of the scientific landscape. The horizontal arrows separate the strict use from the metaphorical use of the basic notions (in the paradigmatic region and in the mixed disciplines the notions of the paradigm are used in the strict sense, while in the metaphorical and in the elusive regions they are used in a distorted sense). The vertical arrows separate the intended area from the unintended one (in the paradigmatic region the methods, in the metaphoric region the metaphors are applied to those situations, for which they were introduced, while in the region of the mixed disciplines and in the elusive region the methods or the metaphors are applied to situations, for which they were originally not intended). 7. See Thomas Koenig and Dietrich Lehmann, “Microstates in language-related brain potential maps show noun-verb differences”, in: Brain and Language 53, 1996, pp. 169-182, or Naho Ikuta et all, Brain activation during the course of sentence comprehension. Brain and Language 97, pp. 154-161. 2006). 8. See David Lewisohn, “Mill and Comte on the methods of social science”, in: Journal of the History of Ideas 33, 1972, pp. 315-324. I could mention also historicism the view that in human history there are laws similar to physical laws. The discussion of historical explanation falls outside the scope of the present paper (see Eugen Zelenak, “On Explanatory Relata in Singular Causal Explanation”, in: Theoria 75, 2009, pp. 179-195). 11 9. See Allan Franklin, “The role of experiments in the natural sciences: Examples from physics and biology”, in: Theo Kuipers (Ed.), Handbook of the Philosophy of Science: General Philosophy of Science—Focal Issues. Amsterdam: Elsevier 2007, pp. 219-274, and William Bechtel and Andrew Hamilton, “Reduction, integration, and the unity of science: natural, behavioral, and social sciences and the humanities”, in: Theo Kuipers (Ed.), Handbook of the Philosophy of Science: General Philosophy of Science—Focal Issues. Amsterdam: Elsevier 2007, pp. 377-430. 10. This change is already on the way under the heading of the “Naturalist Turn” (see e.g. Wenceslao Gonzalez, “Trends and Problems in Philosophy of Social and Cultural Sciences: A European Perspective”, in: Friedrich Stadler (Ed.), The Present Situation in the Philosophy of Science. Vienna: Springer 2010, pp. 227-232. 11. See Wenceslao Gonzalez, “The role of Experiments in the Social Sciences: The Case of Economics”, in: Theo Kuipers (Ed.), Handbook of the Philosophy of Science: General Philosophy of Science—Focal Issues. Amsterdam: Elsevier 2007, pp. 292-294. 12