As a participant in the influential philosophy-of-science debates of the 1960s and 1970s, mostly surrounding Thomas Kuhn's The Structure of Scientific Revolutions, Imre Lakatos (1922–1974) made pedagogy and critical method the dual focus of his historicist philosophy.
At Cambridge as a refuge from the 1956 Hungarian Revolution, Lakatos wrote the English language PhD thesis edited and published posthumously as Proofs and Refutations: The Logic of Mathematical Discovery. Influenced by his countryman, the mathematician and pedagogue George Pólya, Lakatos made mathematical heuristic—meaning informal methods of mathematical discovery, innovation, and proof—a central philosophical idea.
The book takes the form of a pedagogical dialogue between a teacher and 18 characters, named Alpha, Beta, and so forth, who debate and improve a theorem and proof of polyhedra by the 18th-century Swiss mathematician Leonhard Euler. The theorem states that for any polyhedron, such as a cube, the number of vertices minus the number of edges plus the number of faces equals two: V − E + F = 2 (try it for a cube: 8 − 12 + 6 = 2). Pólya in his many books on problem solving, such as How to Solve It, emphasized heuristics for solving certain kinds of equations, integrals, probability calculations, and others. The emphasis was on conceptual understanding, and trial and error tested on special cases or variations, and reflected a disconnect with the mathematical logic made famous through Gottlob Frege, Bertrand Russell, Kurt Gödel, and many afterward. Logic was useful as another branch of mathematics, but how good was it at characterizing the practices that mathematicians used to create new ideas, methods, theorems, and proofs, including those of modern mathematical logic itself?
Lakatos made heuristic into his philosophical workhorse, extending Pólya's pedagogical perspective to the development of mathematics during the 19th century and modern conceptions of proof, theorems, and logic itself. The approach was thoroughly historical—the dialogue about Euler's theorem is not quite fiction. The characters represent historical innovations of 19th-century mathematics and how the modern idea of proof changed and improved throughout the century. Through that history, Lakatos explains that these were heuristic innovations in how to conceptualize what a theorem and its proof is about and how they work together to constitute logical rigor. For Euler's theorem, this means dealing with possible counterexamples, such as a cylinder (no vertices?) or a picture frame (hidden edges on those faces?). Here is where the fallible, “conjectures and refutations” philosophy of science of Lakatos's mentor in England, Karl Popper, is brought into mathematics. Theorems, in their periods of growth and development, can be informally “refuted,” much like a scientific hypothesis. For Lakatos, 19th-century mathematicians showed how that idea was internalized into methods of proof, from identifying relevant domains (e.g., polyhedra), to finessing a theorem, via what Lakatos called the method of proofs and refutations, so that potential counterexamples (a cylinder or a picture frame) were either carefully excluded from a theorem's scope or reinterpreted to neutralize its contradictory status. These are the heuristic methods Lakatos claimed drove creative mathematics and were explored in detail in his historical study. That history is not just colorful window dressing. Lakatos argues, through the dialogue, that mathematical theorems embody the history that gave rise to them, wedding pedagogy and history inexorably. That applies even to modern mathematical logic as yet another informal mathematical subject, whose topic just happens to be mathematics itself.
Proofs and Refutations is a classic of 20th-century philosophy, its specialized subject matter notwithstanding. Lakatos elevated heuristic to its deserved philosophical status decades before Daniel Kahneman and Amos Tversky would use psychological experiments to critique cognitive assumptions of economic models, in large part by showing the role of heuristics in reasoning about uncertainty. The engaging dialogue format of Proofs and Refutations, combined with its mathematical and historical rigor, helped popularize the book's pedagogical messages. That includes a critique of what Lakatos calls the “Euclidean style” of many textbooks, meaning the overused definition–theorem–proof presentation of mathematical knowledge. That style correctly delivers a logical basis, but often disguises a proof's “logic of discovery,” or the informal interpretation of how a proof “works,” known by experts but a mystery to students. The antidote to the Euclidean style is more history of the problems motivating solutions, thus reversing the pedagogical priority given to the “logic of justification.”
Following his graduate work, as a professor at the London School of Economics, Lakatos turned to the philosophy of science. Education and history would again play star roles, but now in a more critical spirit. Lakatos's contribution here was his “methodology of scientific research programs” organized as a kind of historiographic toolkit. The tools make up a flexible framework for interpreting historical progress in any science, and after Lakatos's death, the methods were applied to historical episodes in physics, chemistry, economics, geology, and even developmental psychology.
Lakatos argued that too much focus on isolated theories in science was a historical and methodological mistake. The relevant “unit” of appraisal and progress (or decline) was rather a competing series of theories, unified by some central tenets that are exploited and defended through an array of evolving models with more concrete verifications or refutations of varying quality. There can be creative reinterpretations of evidence, changing observational theories, with progress occurring in a “sea of anomalies,” even formal contradictions, as long as innovations allowed new, successful predictions to be made over time, and always relative to the competition. Given that, there will be ad hoc defenses, reversals of fortune, and ultimately winners and losers. What matters in modern science, for Lakatos, is less “verisimilitude” with some unknown underlying reality but mastery of a constantly expanding horizon of facts.
Lakatos's historiographic views were quite radical, not to be matched until continental philosophers of history from Michel Foucault onward. Lakatos recognized that the sea change in philosophy was to bring history in as the yardstick against which philippics of science are to be judged, a view shared by Kuhn and Paul Feyerabend, the third member of the historicist vanguard. Lakatos frankly put it that there was no methodology outside of history, that methodology of science is but a “rational reconstruction” of science's past. Philosophy of science and its history were inseparable and were without any absolute criterion of “verisimilitude” or “scientific reality.” Science learns through its own experience, including how to be scientifically rigorous itself. Those standards then become the theory by which a scientific past is understood, and so too the present.
That critical learning process is the motivation behind the methodology of scientific research programs. Notably, research programs have a “historical character,” they are series of theories joined together in time by shared program goals and assumptions. Programs were in competition, such as the wave and particle theories of light, or classical and relativistic physics, so Lakatos provides methods for judging both (relative) progress and degeneration. The pedagogy here is meant to be thoroughly exoteric, means by which the necessary expertise of scientific specialization is made transparent through research program spectacles. The slow accumulation of confirmations associated with climate change over the past decade is a sad (because risky) but salient example. There are not really any “crucial experiments,” Lakatos argues, even when they are proclaimed as such. One of Lakatos's best historical analyses was to show that the Michelson-Morley experiments on the speed of light had almost no role in early relativity theory; it was learned only in “hindsight,” as Lakatos puts it, that the experiments were part of classical physics’ concluding chapter.
Lakatos's historiographic methods for interpreting change were also for him critical for assessing the state of play across competing programs. Feyerabend appreciated the usefulness of Lakatos's toolkit for understanding all kinds of “normal science,” Kuhn's notion of the more mundane working out of theory and models between more revolutionary “paradigm shifts.” But there's always a chance for a weak program to recover (the atomic theory around 1905) or a strong one to falter (the one-way “dogma” that DNA [deoxyribonucleic acid] creates RNA [ribonucleic acid] creates protein). Scientific change, as against Kuhn's revolutionary changes, is often a slow and irregular critical battle. Hence, criticism from philosophy, Feyerabend saw, is limited until the history is complete and philosophers can reflect on that change. Nonetheless, Lakatos's approach, illustrated again by his carefully staged but telling historical reconstructions, form a usefully exoteric and critical pedagogy, often making arcane science and its closed debates transparent to outsiders. As put by Ian Hacking, the role of heuristic and history in Proofs and Refutations was “forward looking” and creative. In the philosophy of science, Lakatos's methods are critical, explicitly historiographic, and so “backward looking.” Common to both is a vision of philosophy suffused by a remarkable pedagogical spirit, consistent with Lakatos's influential role as a philosopher and educator promulgating ideas. In both approaches, philosophy is a descriptive means for historical reconstruction, hence also for the teaching of mathematics or science through their past. There is also a normative vocabulary for explaining historical progress without a supreme goal of matching reality or truth in itself.
Lakatos's convolution of history and pedagogy, whether taken as creative or critical, is wholly original in English language philosophy. It is also wholly the product of Lakatos's innovative use of the Hegelian and Marxist philosophy he learned in Hungary, particularly from his mentor and role model Georg Lukács. In a nutshell, Lakatos equals Lukács in Hungary plus Popper in England. The common denominator is learning, literally philosophical pedagogy, which for Lukács is epitomized by the classical idea of Bildung. In German philosophy, especially in Hegel, Bildung connotes both individual and cultural learning through error and hence is the basis for modern conceptions of self and society that are ultimately secular and reinventing. Bildung is necessarily a historical, because constructive, concept, being equally useful to writers and social scientists from Goethe, author of the first Bildungsroman, to Karl Marx, who conceived the Bildung of modern capital society. What Lukács saw, and likely taught Lakatos, was that before Marx, Hegel was the historicist philosopher par excellence, but with history as Bildungsprozess, not metaphysical demiurge. Hegel's Phenomenology of Spirit has the explicit pedagogical goal of organizing dozens of past philosophical ideas into its own stylized history of the philosophical present. Proofs and Refutations is a mini Phenomenology, but targeted to 19th-century mathematics. The caricatured “gestalts” of mathematical method presented in Lakatos's dialogue become our history of the mathematical present, just as Hegel rewrote history to serve his philosophical pedagogy. The methodology of scientific research programs is a critical philosophy of science whose modus operandi is the reconstruction of the history of science using contemporary critical categories of method, a way of writing histories of the scientific, rather than economic, present. Like Marx, Lakatos reinvented Hegel for his own purposes, and as shown by Lukács, knowing how to artfully dissemble that influence. The latter is yet another means by which Lakatos educates his readers in the power of historical thought and the transmission of ideas.
See also Bildung; Hegel, Georg Wilhelm Friedrich; Kuhn, Thomas S.; Marx, Karl; Popper, Karl
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