WHOʼS GOT A PARADIGM? SOME REFLECTIONS ON KUHN, WITH EXAMPLES FROM SPACE PLASMA PHYSICS Gerald T. Davidson December 22, 2004 INTRODUCTION...When we learn these facts, we improve our restorations and thus record a genuine gain in knowledge. A common mythology, much promoted professionals for self-serving benefit, argues that science always proceeds in this way - intrinsically and uniquely. The history of changing views should therefore record a simple progress to greater knowledge, mediated by our application of that infallible guide to empirical truth: the scientific method. Yet, on the other hand, science must proceed in a social context and must be done by human beings enmeshed in the constraints of their culture, the throes of surrounding politics, and the hopes and dreams of their social and psychological construction. We scientists tend to be minimally aware of these human influences because the mythology of our profession proclaims that changing views are driven by universal reasoning applied to an accumulating arsenal of observations. But all scientific change is a complex and inseparable mixture of increasing knowledge and altered social circumstances. Stephen Jay Gould In recent years the philosophy of science has taken a strange turn, following the appearance of Thomas Kuhn's revolutionary book The Structure of Scientific Revolutions. Kuhn seemed to offer a fresh view of the process by which alterations in our picture of the universe come about, but the outcome has been rather a muddle. His ideas have been largely resisted in the physical and mathematical sciences, but they created quite a stir in other places - places far distant from his original direction. I first became aware of the muddle when I became interested in historical evidence of climate change; whereupon I found myself thrown in with a diverse bunch of historians and geographers.
The social sciences seem to have gleefully picked up Kuhn s theses, and turned them quite upside down. Some philosophers and practitioners of the New Social Theories have gone so far as to claim that objective reality, which most of us accept as one of the constants of the physical world, may not exist. The strangest vision of the physical sciences (from which physicists have not been entirely immune) is that they are merely "social constructs," and that such notions as Newton's Laws are merely inventions, which might have been formulated quite differently on another world. Newton s Laws might not even have been necessary in an alternate Physics. This weird view is claimed to follow directly from Kuhn's attempt to describe the social interactions that accompany revolutions in science. Kuhn has described the research process as the casting away of old paradigms and their replacement by new paradigms. The new paradigms may or may not necessarily contain the elements of the old. If that is so, is it meaningful to speak of progress in the physical sciences, or in any academic field? Is there a net accumulation of knowledge? Of course distorted views of the role of paradigms in any field only only follow from the most simplistic reading of Kuhn s thesis. But how can we get back on the track? Is it possible to interpret the philosophy of science in a way that might be agreeable both to physicists and to philosophers, and perhaps even to social scientists? I d like to suggest an approach that appeals to me, as a physicist. I don t pretend to any great rigor, but perhaps our friends the philosophers could develop these ideas with the appropriate logical rigor. WHAT IS A THEORY? A large part of the confusion engendered by Kuhn s thesis seems to stem from the difference between theories and paradigms. At the risk of engendering boredom, let me try to get to the bottom of what Kuhn was talking about. Most of us think we know what a theory is. Of course most scientists are dismayed at the distorted picture that has been injected into the debates about the teaching of evolution. We regard the Principle of Evolution as a well established fact, and Darwin's Theory of Evolution as the best model for the workings of evolution. The theory is not the fact. Darwin s Theory is much more than just a hypothesis concocted to explain a limited set of observed facts; it comprises an entire body of knowledge. A theory in the modern world is assumed to have several essential qualities: (1) it must embody measurable or observable properties of the universe or some small part of the universe (2) it must be extensible - that is, it must lead to new formulations and to new observable properties of nature, and (3) it must be testable. The second condition is vital in distinguishing a theory from a mere hypothesis: a theory, if it is to be useful must have the power of explaining things beyond
the set of observations for which it was formulated. While my emphasis is on the role of theories in the physical and biological fields, similarly structured theories also exist in the social sciences and economics. The beauty of theories in the physical sciences is that a proper test can lead to a definite yes/no answer. Theories become fuzzier and more resistant to empirical tests the deeper we delve into the so called softer sciences. Karl Popper suggests that a useful theory must contain within it certain factors that render it subject to being disproved if it does not accord truly with facts. It must contain elements that could be subjected to rigorous experimental tests in order to verify its integrity. It must contain the seeds of its own negation. Ideally those elements must be spelled out explicitly. Popper insists that a theory can never be proven true; bad theories can, however, be proven false. Kuhn s bold step was to introduce the notion of a paradigm, which subsumes together with the theories, all the assumptions and methods of a science, along with the social conventions that prescribe how science is to be carried out. A paradigm is a big messy bundle of stuff, and it s not surprising that few practitioners of science or philosophy take the trouble to describe or enumerate the actual ingredients. Kuhn was rather vague in his definition of a scientific paradigm, which led to much confusion that he later tried to correct. A paradigm, in whatever field, is not just a single new theory, but an entire way of thinking and acting, based on a specific world picture. It includes not just the theory, but the social actions and conventions that lead to the acceptance of that theory. For example, the tradition of compelling naive young students to present anxious summaries of their work at grand convocation of the experts and graybeards is a part of the paradigm. The acceptance of 2+2 = 4 as a universal truth belongs to the deeper roots of the paradigm. When I assert that a mathematical truth is really a convention, I am invoking the ideas of Kurt Gödel who demonstrated that all mathematical or logical systems must include some untestable assumptions. At this point Popper and Kuhn seem to be offering contradictory views. I don t think this is necessary, if we keep in mind that Popper was talking mainly about the theories, while Kuhn was really talking about something much more inclusive (even though his exposition may seem to imply the narrower view). One can prove a theory false, but a paradigm is never proven or disproven. The paradigm may be discarded when the theoretical underpinnings are disproven or found wanting, but it is abandoned as a matter of choice rather than necessity. Most of us feel no qualms about resorting to the supposedly obsolete paradigm of Newtonian physics when we deal with everyday things - things we can actually touch. To get a grasp of the notion of a paradigm, it may be easier to return to a world of long ago, where we are not distracted by things we have supposedly learned to be essential truths. Consider the world picture of Duns Scotus and Thomas Aquinus. Their thought was embedded in a
medieval paradigm that we have difficulty accepting or recognizing today. The social element was much more explicit than in the modern paradigms. Any study of natural phenomena had to be reconcilable with holy scriptures, and must acknowledge the existence of a prime mover. Moreover, to safely practice any sort of philosophical enquiry, one almost certainly had to be a monk or ecclesiastic in one of the centers authorized by the church. Some of the deep roots of the paradigm pertaining to laws of physics were similar to the modern paradigms; but the fundamental basis of the medieval paradigm embodied concepts, such as the notion of a prime mover that are far removed from the mechanistic modern picture in which everything acts and moves by universal laws. A HIERARCHICAL ORDERING OF THEORIES, METHODS, AND PARADIGMS We seem to have here several different kinds of systems, one of which may include the other. To be precise, we should think of a paradigm as a system of assumptions, beliefs, procedures, and social structures built around one or more theoretical concepts. The procedural elements are an essential part of the paradigm. The procedures involve accepted ways by which the theories are developed, promulgated, extended, and tested. Without procedures we would have only unverifiable speculations. In the physical sciences the procedures consist mainly of standard laboratory techniques, that have been agreed on by the majority of scientists. In the social sciences, such as history, the procedures must involve standard ways of gathering and analyzing data. We can say that theories, methods, and paradigms are all different meta-languages, all of which are employed in the development of science and philosophy. It may be appropriate to consider the work of Alfred Tarski who suggested that many of puzzles and paradoxes of mathematical logic can be resolved by constructing hierarchies of metalanguages. Consider the statement "This sentence is false." If taken at face value it seems to present a discomforting paradox. Any statement is usually supposed to be either true or false, with no intermediate position. So if we try to test it by assuming it to be true, we are presented with a lie. But it cannot be false, because then it would then be true! There are innumerable such paradoxes, such as Russell's Paradox, which concerns a set of all sets that obey certain conditions. One thing that logical paradoxes have in common is self-reference. Can a statement proclaim itself to be true or false? Most practitioners of logic think such self-referential statements are meaningless, and cannot contain useful information; in other words such statements are only sequences of logically disconnected words. But it is obvious that such paradoxes exist and cannot be dealt with by simply denying their existence. Tarski gets around the difficulty by posing a hierarchy of meta-languages. For the meta-language which contains "This sentence is false." or
its logical equivalent there is a higher language that can be used to discuss issues of verity in all the meta-languages that exist below it in the hierarchy. Similarly, in the languages we know as the languages of science, we implicitly use several levels of less formal language that consist of the rules for manipulating the lower languages. For example in classical physics the appropriate language consists of mathematical symbols and equations, such as those in which Newton's laws are generally posed. But concepts such as mass are not defined within Newton's theory. When we try to understand the implications of the equation F = ma or Force = mass x acceleration we could easily get bogged down in circular reasoning, where force is defined as the quantity that produces a certain acceleration of a certain mass, but mass is just the quantity that is accelerated by the given amount when subjected to a given force. Such confusion is avoided even to this day by invoking more abstract ideas to understand what mass and force are and how they are to be represented in mathematical manipulations. For every mathematical formula of physics there is a higher-level language that includes commonly accepted conventions defining mass, and other physical observables. Above that are other higher meta-languages that include various prescriptions for measurement, such as the use of clocks and meter-sticks. Tarski s prescription has been implicit in the practice of the physical sciences since the time of Newton. So far we have nothing really new except the formalism. Kuhn's achievement was to insert another language above the lower languages I have described above. What Kuhn has done is to construct a "meta-theory" of scientific theories. This language is even more abstract, comprising the invisible social interactions and formalities by which science is practiced. He thus formulates a non-deterministic meta-theory that applies at all scales (this should not be confused with quantum mechanics, where the non-deterministic element is an explicit part of the microscopic physics). This I suggest is the paradigmatic level or language, beneath which there may be several layers of languages and meta-languages. The example below is intended solely to illustrate how number of meta-languages grows as we delve deeper into the logical structure of a science. I make no claims that it represents an ideal classification of the hierarchic levels. Example: Several Layers in the Hierarchy of Physical Theory Level Rank Statement 1 Paradigm Newton s laws apply to the realm of ordinary experience; neither too small nor too large... 2 Procedure Acceleration is to be measured by observing the motion of bodies under the influence of what we call forces, and is the second derivative of the position with respect to time...
3 Conceptual The symbols F, m, and a refer to measurable properties of the motion of a material body... 4 Model F = ma 5 Mathematics The symbol = signifies that the quantities on the left are equivalent in magnitude to those on the right... 6 Symbolic Latin and Greek characters, plus a set of special symbols are to represent physical entities or quantities... The confusion between theories and paradigms is easily disposed of. Theories and physical laws occupy ranks several levels below the paradigm. A single paradigm may incorporate a multitude of separate but related theories. The role of procedures has not received the attention it deserves. Procedures are vital to a paradigm, and may change with time, along with the theoretical framework. They occupy a sort of middle ground, containing a rather substantial - but often unrecognized - dose of social conventions. Procedures apply not just to the laboratory, but to the theorist s desk. In physics we take for granted that we will develop theories that constitute mechanistic models describing various aspects of the physical or organic universe. Before the Galilean-Newtonian revolution philosophers were free to inquire about ultimate causes and "explanations" for physical phenomena. No longer. Even though philosophers before Galileo and Newton had been developing mathematical tools for describing the world, Galileo and Newton imposed an entirely new way of looking at the world. The aim of science was no longer to find ultimate causes and explanations, but to develop models of how things work. And, in physics, those models soon came to be dominated by mathematical models. The profundity of this paradigm shift is not always appreciated. In physics we now take for granted that we will develop theories that constitute mechanistic models consisting entirely of mathematical formulae and symbols representing measurable quantities. Up to this point the description could apply to any of the sciences, or to other "softer" fields, such as history. For the mathematical sciences, particularly physics, the hierarchy continues downward for several more steps. After we have agreed on the terms of argument, we must agree on certain fundamental concepts. For that reason I have called the next level the "Conceptual" level; though that designation is a bit awkward, and some might prefer something like "Fundamental" level. Most scientists agree on the meanings of mass, force, energy, and electrical charge; though we don't really know how to define them in a non-circular way. For most purposes operational definitions, such as Newton's laws suffice.
The conceptual level is perhaps where we can expect the most momentous shifts. The relativity revolution showed that mass and energy are not immutable and separate, but are completely entangled. (I use the term "entangled" deliberately, to emphasize that both relativity and quantum mechanics have thoroughly confused the concepts of Newtonian physics.) Space and time have been particularly troublesome concepts. Relativity showed that they were not distinct, and some recent theoretical ideas have thoroughly shaken up the picture by adding unseen spatial dimensions. Once the fundamental concepts were in place, physicists could construct an ever more elaborate picture of how our world works, delving deeper and deeper into the structure of matter, and wider and wider into the structure of the universe. I have deliberately avoided the word "theory" to designate the next level; "Model" seems to be the least cumbersome designation within the framework of post-medieval physics. This is because it's difficult to define what actually constitutes a theory; it may embrace several of the lowest levels of the diagram. Alternatively one might label this level "Physical Law," which implies that the theoretical concepts have a universal validity. As we go further down the hierarchy, we come to the basic notions of mathematics, and how we manipulate symbols to describe the universe. All of these levels can be considered as conventions agreed on by almost all the practitioners. Of course Gödel's proof is a hidden time bomb embedded in the paradigm, suggesting that every level remains subject to question, and a paradigm shift can begin anywhere in the hierarchy. As the structure of a paradigm is developed over time, it acquires numerous patches, which are expected to be repaired eventually. The patches mostly appear to be minor blemishes on the edifice, so they don't at first cause any trouble. Some of the patches are indeed repaired as more knowledge is acquired. Some of them can be concealed by moving them to a less noticeable part of the structure. Others prove more difficult. To thoroughly mix the metaphor, we must also recognize that the patches can occur on any level of the hierarchy of meta languages. Kuhn sees this process as being a natural part of science. This is a consequence we can accept if we also acknowledge that we are nowhere near an ultimate understanding of the universe. Usually the experienced practitioners of science pay little attention to the patches, for they can go about their business of adding gables and dormers to the edifice with little regard for some unsightly blemishes. But the patches eventually become so numerous that the youngest generation of scientists, who must begin at the ground level and struggle to the top, increasingly find their upward progress obstructed by patch upon patch. The younger generation probably still has faith in the inductive method, so some of them begin to find new ways to build the paradigm. A small number of them begin erecting a new structure that better fits the facts of measurement and observation. Eventually as the older generation passes away, the old paradigm collapses and everyone turns to the new paradigm. This point is
the "Paradigm Shift." The new paradigm will usually be the one that is successful in modeling the greatest number of observations that will satisfy the greatest number of researchers. The new paradigm almost certainly has a broader scope than the old; thus there is a net augmentation of knowledge. It can be argued that progress is an underlying tenet of Kuhn s thesis. WHAT DOES THIS HAVE TO DO WITH THE CURRENT PRACTICE OF PHYSICS? The notions of inductive and deductive reasoning have caused discord in the social disciplines, particularly history. Practitioners of the New Social Theories have often expressed a profound disdain for traditional narrative history. This is largely because of its lack of theoretical underpinnings and the exclusive use of non-inductive methods. But to assail non-inductive methods as unscientific is to demonstrate a lack of knowledge of the workings of the sciences. Perhaps we have been misled by elementary textbooks, which presented the inductive method as the only logically satisfying method to be applied in post-medieval science. We were told that scientists develop theories induced from elementary facts and reasoning. Those theories then predict new phenomena, which can be tested by laboratory experiments. Such a "pure" inductive method is actually so rare that it may not actually exist in either the physical or social sciences. Physicists in the early twentieth century were quite convinced that theirs was the supreme science, because they adhered to the purest form of the inductive method. All great discoveries of the nineteenth and early twentieth centuries were professed to follow from theoretical reasoning. Alas, the traditional picture of disinterested scientists finding the rules of nature by a direct and orderly process just isn't true. In some outstanding cases, like Einstein's theories of special and general relativity the inductive method may have served well. At other times, however, serendipity and directed searches turned up unexpected new pieces of knowledge. It would be difficult (perhaps impossible) to come up with an example where the inductive method has been strictly followed in either science or history. Actual scientists operate in three modes: a mode that approaches the idealized inductive mode, a mode that includes a large dose of deductive reasoning, and a mode that might best be called an exploratory mode (others have called it an empirical mode). The exploratory mode contains elements of both induction and deduction.  A successful explorer does not set out aimlessly; he is guided by an abstract notion of what he is trying to find. He needs, as a start, a generalized picture of the landscape. There is almost always a background of theoretical reasoning that provides that generalized map, but it does not necessarily provide specific predictions of what will be found along the way.
The exploratory method has played a significant and widely publicized role in the field of astronomy, which many consider a sub-branch of physics. Some discoveries, like black holes, generated a tremendous amount of excitement simply because they had been first predicted by theory. But a major part of our recent knowledge of the universe comes from a concentrated exploration for new phenomena. Gradually astronomers have mapped out the skies, and found many things that could be understood only after the fact of their existence became known. After establishing a framework for organizing the search; theory was then invoked to limit the range of plausible explanations. The methods of narrative historians may perhaps be likened to the methods of astronomers. Astronomers use specialized apparatus to survey thousands of objects in the skies; few of their great discoveries were first predicted by theory. Theory was used mainly to limit the range of plausible explanations for newly discovered phenomena. Historians, on the other hand, employ mainly their own eyes to survey thousands of documents. Historians are limited only in having weaker theories with which to classify their data and map out the terrain to be explored. The progress of science is actually a very messy affair, involving induction, deduction, and sometimes just laborious searches for something new, with lots of mis-starts, wrong turns, and blind alleys. Serendipity was not even considered by Bacon, Newton, and the early philosophers of science, but it plays a role that is more important than usually acknowledged. AN EXAMPLE OF THE CONFUSION BETWEEN PHYSICAL PROCESSES AND ARTIFICIAL CONCEPTS If I were some sort of omniscient being I might stand somewhere above Kuhn's theories, and see where science stands now, and where it is going. This is really quite impossible, particularly when we realize how many layers of meta-languages are involved. And recent developments, such as string theory, have thoroughly jumbled all the ranks of the hierarchy. But permit me a brief digression that shows how the elements of social conventions can become wrapped up in our thinking processes. I select an example from solar and plasma physics, not just because it's one with which I am familiar, but because it is such an intriguing example. Present-day physics seems to contain a multitude of paradigms, of which relativity (special and general) and quantum mechanics are pre-eminent. However, it is quite amazing how many "black boxes" clutter up the various branches of physics. These are, indeed, some of Kuhn's "patches. Seldom does anyone notice them until someone begins to look inside them to find out what makes them work. Some of these serve a beneficial purpose, in allowing scientists to by-
pass some tedious mathematics describing processes which are well enough understood that one can accurately deduce the output from a given input. Other black boxes really just social constructs that came into being because we really don't understand the detailed physics. But they have become so embedded in our thinking that they prevent us from seeing important elements of the picture. In some fields that have only recently begun to approach maturity, such as solar physics and plasma physics, the current paradigm is not fully developed. That paradigm seems to have something to do with the motions of energetic charged particles and magnetic field lines. The paradigm thus embraces a questionable concept: it can be argued that magnetic field lines have no reality, but are only a "short-hand" way for describing one element of the motion of charged particles. Solar and space physicists talk about such mind-numbing concepts as "magnetic reconnection," "magnetic field emergence," and even "continuous magnetic reconnection rates." Both field lines and magnetic reconnection are black boxes, which physicists accept only because of the difficulty of visualizing the complex motions of particles inside the box. But when physicists talk of the motions of the field lines comprising the Earth's magnetic field, are they implying that the field lines also move in the Earth's lower (and neutral) atmosphere? And what does it really mean when we talk of motions of magnetic field lines? And why don't we ever hear about the motions of electric field lines? The concepts having to do with motions of field lines are pertinent only to certain conditions relating to density, temperature, and sources of energy. They stem from a familiar result of elementary plasma physics: the magnetic field and charged particles of a fully ionized plasma act as if they are "frozen" together, and every particle moves as if it were firmly tied to a single field line. Obviously there are complications such as particle collisions that must be accounted for in many situations. But even where the conditions for "frozen-in fields" are fully satisfied, we have to wonder what the picture of moving field lines really represents. But in many instances the field-line picture poses conceptual problems, that could lead to highly entertaining problems for students. Think for a moment about the homopolar generator, that shows up so often in elementary textbooks. Any attempt to explain its workings with the aid of field lines and conductors quickly leads to logical difficulties. An explanation in terms of charged particles and current carriers moving in a magnetic field encounters no such difficulties. The motion of moving field lines is not usually objectionable, being simply a special representation of the motions of charged particles. However the common acceptance of this picture demonstrates the role of social conventions. The social convention - magnetic field lines and frozen plasma - operates at a level above the detailed microphysics of particle motions. The image has become so familiar that many physicists instinctively think about field lines as real objects. In many cases that image, rather than the microphysics, has guided theoretical developments.
What is magnetic reconnection? This is not a physical process, even though many practitioners speak of it as a process as real as, say, the orbiting of a planet around the sun. Reconnection might be better viewed as analogous to the language developed to visualize particle collisions at the atomic level. Quantum effects blur our observations, so we see only the end results rather than a continuous process. In the case of turbulent assemblies of magnetized particles, the discontinuity is imposed by the complexity of the motions of uncountable numbers of particles, and represented conceptually as reconnection. Thus the magnetic reconnection black box is analogous to the Heisenberg S matrix of quantum scattering theory, in which the microphysics is concealed. Let us imagine with a plasma that is initially in a relatively stable configuration. Something happens that results in a net flow of particles in a direction orthogonal to the magnetic field. This is seen as a bending or stretching of the magnetic field lines. At some point the system becomes unstable. The particle motions are re-arranged, and the system relaxes to a new configuration in which the configuration of magnetic field lines appears to have changed. There is nothing wrong with the picture, but it obscures the actual complex physical processes. The danger is that moving and reconnecting field lines might occasionally be invoked in ways that are not appropriate or valid. Usually plasma physicists are careful to verify that the microscopic model does indeed support the field-line picture. But, under peculiar conditions of temperature, resistivity, etc. the detailed physics might be too complex to visualize at the microscopic level. This is especially consequential when physicists view reconnection as a process, rather than two end points of an obscure physical process, and talk of meaningless concepts such as continuous reconnection. WRAPPING IT UP Kuhn has confined his work mainly to the philosophy and history of science. Perhaps because he was working outside the physical sciences, his ideas were quickly hijacked by a gang of young and not-so-young rebels in the social sciences. Clearly they were delighted to find someone who could assert in the most convincing language that indeterminism and human interactions played an important role in the development of theory. Some of them went so far as to assert that this demolished the physical scientists claims to objectivity. How could physicists claim to be objective in saying that F = ma when non-scientific issues played such a large role? The rebels found even easier going when they looked at quantum mechanics. Perhaps, they thought, scientists on another world might have constructed an effective model of the universe without ever invoking quantum mechanics. Some of those ideas have attracted much attention among philosophers and even among distinguished scientists. It is amazing that several re-
nowned scientists at the end of the 20th century could have expressed doubts about the nature of physical reality that went even further than the philosophers of previous centuries. Of course scientists generally reacted with great indignation to the notion that their motives were anything but objective and rigorous. How ironic that Kuhn's pronouncements - which were mainly aimed at the classical sciences such as physics and astronomy - were accepted most readily by scholars in the social sciences and humanities. I believe the issue of objective reality should present no problems when we view the mathematical physics and theory as only a lower-level meta-language. Our physical models constitute the formal structure that would probably be discovered in any world. I say discovered rather than invented, because the formalism and mathematical language might look different on another world, but its meaning would be the same. The higher level meta-languages only determine how we should go about discovering the rules of nature. Indeed, the process by which these rules are found could be quite different in two worlds, but eventually most of the phenomena would be described by similar models. Scientists should not have any great difficulty with Kuhn's concept of paradigms if they would fully grasp that Kuhn was speaking mainly of a meta-language involving not just the underlying theories and structures, but the social interactions which govern any field of endeavor.. Only the most naive scientist could attend a major international scientific meeting or read a scientific journal without perceiving some un-objective personality clashes. To some of us those human interactions are the best reason for attending scientific meetings; we learn as much from witnessing a hot debate as we might from reading a single-sided printed statement.  Kuhn definitely did not suggest that the value of the theories could have been in any way diminished by the role of social forces Though all scientists would probably claim to practice something called the "Scientific Method," there are innumerable ways of going about it. Most scientists struggle through their entire careers laboriously working out new variations on the standard theories and observations. Most of that work is uncontroversial, and will last as long as the theories on which it is based. Others strike out in all directions, putting up wild ideas that go against all the standard wisdom. Most of those wild ideas are quickly shot down, but the ones that survive could shove the old paradigm in the direction of toppling. Further Reading
Robert Lacey & Danny Danziger, The Year 1000, What Life Was Like at the Turn of the First Millennium, Little Brown and Co., 1999 An excellent introduction to the mental world of the middle ages, and the environment that molded their thoughts and actions. Jean Huizinga, The Waning of the Middle Ages, Doubleday Anchor, 1949. A classic on medieval modes of thought; one of the best introductions to the world view on which the paradigm of Duns Scotus and Thomas Aquinus was built. Jean Gimpel, The Medieval Machine, The Industrial Revolution of the Middle Ages, Henry Holt & Co., 1976. Another classic: an eye-opener on the technical/industrial world of the middle ages. E.J. Dijksterhuis, The Mechanization of the World Picture, Oxford University Press, transl. 1961. A rather rare and hard to find book, but one of the best explorations of the transition from the medieval physics paradigm to the pre-modern Galilean-Newtonian paradigm. This was written before Kuhn s ideas became well known, but Kuhn would probably have approved of the approach. Thomas S. Kuhn, The Structure of Scientific Revolutions, University of Chicago Press, 1962. A much quoted book, but many of the readers may not have grasped the context of Kuhn s thoughts. Kuhn doesn t get everything right, and his concept of a paradigm may be too narrow, but this was truly an overturning of the philosophy paradigm. Lee Smolin, The Trouble with Physics, Mariner Books, 2007. Smolin s intent was to voice his doubts on the present state of theoretical physics, but he may have unintentionally provided one of the best pictures of a scientific field in that state Kuhn describes as preceding a paradigm shift. Keith Windschuttle, The Killing of History, How Literary Critics and Social Theorists are Murdering Our Past, Encounter Books, 1996. A rather intemperate book, decrying the hash that has been made of history and the social sciences by theorists who misapply the ideas of Kuhn in dubious context. Enrest Nagel and James R. Newman, Gödel's Proof, New York University Press, 1960. A readable account of one of the most significant breakthroughs in mathematics in the twentieth century. Also consult the Wikipedia article. Tarski's work is probably most accessible in his popularized account: A. Tarski, Truth and Proof, Scientific American, June 1963