“Not merely a uniquely insightful account of the life and work of one of this century’s most original philosophers, this book provides a glimpse of a vanished intellectual world, that of Middle Europe before the catastrophes. Finding Georg Lukacs and Hegel in Lakatos does more than elucidate Lakatos’s thought; it provides us with an entry to a whole different intellectual style. As interpreted by Kadvany, Lakatos functions as a sort of Rosetta Stone to that brilliant but now quite foreign intellectual culture. A brilliant tour de force.”–Jerome Ravetz, author of “Scientific Knowledge and Its Social Problems”

John Kadvany is a Principal at the mangement consulting firm Policy and Decision Science. He has published essays on Lakatos, the philosophy of mathematics, risk, and environmental policy.

Imre Lakatos & Guises-CL

By John David Kadvany

Duke University Press

Copyright © 2001 John David Kadvany
All right reserved.
ISBN: 9780822326601

Chapter One

The Mathematical Present as History

It is remarkable the extent to which the nineteenth century was a time of error for mathematics: not trivial oversights or amateurish confusions but fundamental mistakes in the understanding of mathematical concepts and the formulation of mathematical proofs. These mistakes were not restricted to unknown mathematicians but occurred in the works of great mathematicians such as Joseph Fourier, Augustin Cauchy, and Denis Poisson. Equally notable was the criticism of these errors, which provided the impetus for some of the deepest conceptual reformulations of the century whose influence can be felt down to the present-not only in the specific mathematical fields in which the errors originated but in the foundations of mathematics as well.

In Proofs, Lakatos turns this historical observation into philosophical conjectures about mathematical criticism and changes to mathematical method occurring during the nineteenth century. In particular, the philosophy of mathematics represented by Proofs demonstrates the historicity of several important features of contemporary methods of mathematical proof. Proof here means informal proof-the proofs of journals, notebooks, and chalkboards -including twentieth-century logic and metamathematics as special cases. What Lakatos calls the method of proofs and refutations expresses different ways in which a theorem and proof can be improved by incorporating attributes of counterexamples or exceptions to a conjectured theorem into the conditions, definitions, or lemmas needed to make a proof meet criteria of rigor. Lakatos’s historical thesis is that the method of proofs and refutations became significant to mathematics only after about 1850. His philosophical thesis is that this method is best characterized through an antifoundational and antiformalist philosophy of mathematics based on a historical theory of mathematical criticism, fallibilism, learning, and error. This outlook derives, in part, from Lakatos’s adoption of Popper’s philosophy of conjectures and refutations from the natural sciences to mathematics. But much of what is philosophically novel in Proofs comes from the work’s ingenious historiography, the basis of what is entirely Hegelian.

The welter of topics and narratives in Proofs tends to amaze and bewilder, so that it is difficult to say at first glance just what kind of philosophical or historical work it is. Overflowing with historical detail, it can appear to have no fixed historical topic. The text consists mainly of intricate historical and methodological analyses of numerous proofs, developed during the nineteenth century, for a relatively simple geometrical result about polyhedra, such as cubes and pyramids, known as Euler’s theorem. Yet while Euler’s theorem was a prominent theorem of nineteenth-century mathematics, it was not central to the controversies pervading mathematics at that time; other theorems or concepts, such as the representation of functions by Fourier series or the function concept itself, could easily be used to narrate major changes in nineteenth-century mathematics. There is much in Proofs on formalized theories and languages that initially is familiar to analytically minded philosophers, yet there is also much philosophical development of mathematical pedagogy and theorem-proving heuristics, topics not normally included in an inventory of problems in the philosophy of mathematics. Lakatos polemicizes at length about the dangers of elitism in mathematics and what may be called epistemological aristocratism; while this polemic is naturally allied with his thinking about mathematical heuristic, without further explanation it would not seem to be directly part of Proofs‘s thoroughgoing historicism.

Years after Proofs, Lakatos’s controversial “rational reconstructions,” being his historical vignettes of scientific change written strictly from the perspective of his own methodology of scientific research programmes, would become his signature method in the philosophy of science. These historical rewrites would end up driving some readers to exclaim about “Lakatos’ absurd historiography” or his “historical parody that makes one’s hair stand on end.” The history found in Proofs, while generally as rigorous as could be demanded by any historian, already uses a similar narrative method, but applied to the history of mathematics rather than the history of science.

When driven by historical scruples back to the philosophy of mathematics, we do not find in Proofs any of its traditional problems, such as the definition of number, set-theoretic “existence,” the extent of infinite cardinal numbers, or the interpretation of the classical paradoxes. These important though conventional topics provide no starting point here. The solution to understanding the apparent eclecticism of Proofs-a combination of history and error, elitism and learning, philosophy and historical parody, and mathematical formalism traced back a hundred years before its time-is to see how a specific historical sensibility is set to work in a characteristically ahistorical subject. As philosophical history, Proofs is even described by Lakatos’s colleague and arch critic Feyerabend as “the best and most detailed presentation and analysis of conceptual problems in the entire history of ideas,” one that “removes the last Aristotelian element, the element of necessity, from modern science.” But just how is one to think of mathematical necessity as historically specific and variable? And if history is to be more than novelty or illustration, its use should be tied to some significant mathematical problem or ideas.

The introduction to Proofs places it as a challenge to the complete identification of mathematics with some formal, metamathematical representation, a view that in some form inhabits nearly every key philosophical position on mathematics since the turn of the twentieth century, and descends from the work of Gottlob Frege, Giuseppe Peano, Bertrand Russell, David Hilbert, Kurt Godel, and other great logicians. Formalism, in Lakatos’s sense, demarcates a broad range of approaches to the philosophy of mathematics, centered on the idea that some largely mathematical theory explains most of what is philosophically important about mathematics: a formalized proof calculus, the output of an abstract computer such as the Turing machine, or a theory of sets interpreted as an “ontology” for numbers, functions, and geometric and other higher-order mathematical objects. Much ordinary mathematical logic, while not necessarily interpreted for its philosophical content, falls under formalism, as one uses specific mathematical conceptions of proof, truth, computation, axiom, and so on to create mathematical theories of mathematics, and these tools have been successfully used to formulate and solve many fundamental problems related to the structure and limits of mathematics. What Lakatos wants to contest is assuming that all methodological problems about mathematics are adequately addressed only by continuing the mathematicization of mathematics. Proofs does not reject mathematical formalism but rather recognizes metamathematics and mathematical logic themselves as informal mathematical theories, whose objects of study are not topological spaces, probability measures, or solutions to differential equations but proofs, theorems, formal languages, and their syntax and semantics.

Lakatos’s topic is the temporal and historical process of proving theorems and creating new mathematical concepts instead of the formal analysis of existing mathematical theory. Even right in his introduction, Lakatos notes that Godel’s famous incompleteness theorems, to which we will return in chapter 4, are informal mathematical theorems about the limits of formal systems. The heart of Lakatos’s challenge is his claim that formalist philosophies of mathematics can say nothing about the historical growth of mathematics or the role of central conceptual changes in many mathematical theories. Formalism, therefore, leaves unexplained changes in patterns of mathematical reasoning, in particular the emergence of nineteenth-century mathematical criticism and its role in creating the mathematical formalisms that are familiar today. An important goal of Proofs, then, is to demonstrate the inherently informal character of the main methods of contemporary mathematical proof and their changes through time. As such, Proofs is a historical criticism of the limits of modern logical theory in representing its own origins and describing its own rationality. If Lakatos’s account is even roughly correct, then by historicizing what has been taken by many to be the standard of a priori or nearly a priori reasoning, he turns centuries of received wisdom about the nature of mathematical knowledge on its head, replacing the traditional quest for certainty or some kind of specialized truth immune to change with an account of mathematical proof itself as a historical phenomenon. Instead of a single, idealized notion of mathematical proof and truth, Proofs presents different conceptions of proof changing over time; instead of an explanation of contemporary mathematics in terms of some formalism, Proofs provides a historical account of formalism itself.

Others, including W. V. O. Quine, have developed holistic views of mathematics and science in which mathematical logic might potentially change through some fundamental reorganization of physics or other empirical study. At one time, similarly, some argued that quantum physics might be better formulated using a “three-valued” logic, allowing statements to be true, false, or indeterminate. But Lakatos is not providing an exotic possibility based on many-valued logic or an abstract holism. He gives a specific history of nineteenth-century mathematics and changes in mathematical practice, following them to the emergence of modern logic-what Lakatos calls the “dominant theory” for contemporary mathematics as a whole. The philosophical source is not a hypothetical empirical world moving us to change our logical theory but the specific historical reality that gave rise to it. Lakatos would likely be open to holistic inferences cast from physics all the way into mathematics or logic. But mathematics is more than a handmaiden to science, and more than a set of theorems and its results. In Proofs, mathematics is its results together with their becoming. This historical becoming is just the production of mathematical proofs and the activity of theorem proving in journals, letters, and notebooks, as far as needed to make these part of the intellectual content of mathematics, and leaving out the material, ideological, or other “external” conditions relevant to their causal history. The history of mathematics is taken in Proofs not as what mathematics is about-say, numbers, structures, or sets, or which particular sets-but as what mathematicians do, which is to create, criticize, and improve proofs, create new mathematical concepts, and resolve mathematical conjectures and pose new ones. The history in Proofs is neither that of a zeitgeist nor paradigms, nor occult historical forces, nor even people, but history as different ways of creating collections of statements called mathematical proofs. That activity, according to Lakatos, has changed greatly since 1800, and so, in turn, have conceptions of mathematical rigor and truth. Lakatos’s ambitious goal is to explicate and enact much of that remarkable historical process, especially the intricate relationships that have developed between mathematical innovation in creating proofs and their modern logical justification: the method of proofs and refutations is supposed to be the engine for that process, in which the competing demands set by mathematical creativity and rigor are traded off to achieve progress. A powerful historical methodology is needed, then, to see mathematics not as the ultimately static and ahistorical subject but as a dynamic historical process. How is this to be attained?

As a case study in the history of mathematics, Proofs challenges the one-sided identification of mathematics with its metamathematical representations by showing, as Lakatos says, that “the history of mathematics and the logic of mathematical discovery, i.e., the phylogenesis and the ontogenesis of mathematical thought, cannot be developed without the criticism and ultimate rejection of formalism” (P & R 4). So Proofs provides a certain kind of history, one associated with “phylogeny” and “ontogeny,” Ernst Haeckel’s correlative terms for biological development in species and the life of an individual, respectively. Lakatos cites Henri Poincare and George Polya as mathematicians who have proposed that Haeckel’s now-discredited “biogenetic law” be adapted for mathematical pedagogy. The motto of the biogenetic law is that “ontogeny recapitulates phylogeny,” meaning that the biological growth of the individual repeats in abbreviated and caricatured form the historical development of the species, thus telescoping millennia of historical change into the physical development of a single life. As a narrative technique, and not a historical or even psychological “law,” the approach means that the “embryonic” development of a student’s knowledge should recapitulate in brief the history of mathematics itself, or at least those parts relevant to the mathematics being taught. The biogenetic law is not mentioned again in Proofs but it defines the architectonic of the whole work, and Lakatos’s single remark characterizes the book’s fantastic historiography.

The entity whose “ontogeny” is followed in Proofs is nothing mental or mindlike, but rather a fully externalized discursive object: it is simply Euler’s theorem itself, stating that the number of vertices plus the number of faces of a polyhedron exceeds the number of edges by two: V – E + F = 2. The “phylogeny” against which this individual history is laid out is mostly that of nineteenth-century theorems and their proofs-many explicitly falsified, some not-of Euler’s classic result. Lakatos develops this history using two techniques executed with great virtuosity: a dialogue form in which Euler’s theorem is continually debated and reproven; and a running commentary in the footnotes associating the dialogue’s timeless and analytic content with events mostly from nineteenth-century mathematics, but including contemporary and ancient sources as well. Both of these narrative methods combine to give Proofs its specific philosophical-historical form.

Continues…
Excerpted from Imre Lakatos & Guises-CLby John David Kadvany Copyright © 2001 by John David Kadvany. Excerpted by permission.
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