pdf version of the entry Imre Lakatos https://plato.stanford.edu/archives/win2016/entries/lakatos/ Imre Lakatos from the Winter 2016 Edition of the First published Mon Apr 4, 2016
Stanford Encyclopedia Imre Lakatos (1922–1974) was a Hungarian-born philosopher of mathematics and science who rose to prominence in Britain, having fled of Philosophy his native land in 1956 when the Hungarian Uprising was suppressed by Soviet tanks. He was notable for his anti-formalist philosophy of mathematics (where “formalism” is not just the philosophy of Hilbert and his followers but also comprises logicism and intuitionism) and for his “Methodology of Scientific Research Programmes” or MSRP, a radical
Edward N. Zalta Uri Nodelman Colin Allen R. Lanier Anderson revision of Popper’s Demarcation Criterion between science and non- Principal Editor Senior Editor Associate Editor Faculty Sponsor science which gave rise to a novel theory of scientific rationality. Editorial Board https://plato.stanford.edu/board.html Although he lived and worked in London, rising to the post of Professor of
Library of Congress Catalog Data Logic at the London School of Economics (LSE), Lakatos never became a ISSN: 1095-5054 British citizen, but died a stateless person. Despite the star-studded array of academic lords and knights who were willing to testify on his behalf, Notice: This PDF version was distributed by request to mem- neither MI5 nor the Special Branch seem to have trusted him, and no less bers of the Friends of the SEP Society and by courtesy to SEP content contributors. It is solely for their fair use. Unauthorized a person than Roy Jenkins, the then Home Secretary, signed off on the distribution is prohibited. To learn how to join the Friends of the refusal to naturalize him. (See Bandy 2009: ch. 16, which includes the SEP Society and obtain authorized PDF versions of SEP entries, transcripts of successive interrogations.) please visit https://leibniz.stanford.edu/friends/ . Nonetheless, Lakatos’s influence, particularly in the philosophy of science, Stanford Encyclopedia of Philosophy has been immense. According to Google Scholar, by the 25th of January Copyright c 2016 by the publisher 2015, that is, just twenty-five days into the new year, thirty-three papers The Metaphysics Research Lab Center for the Study of Language and Information had been published citing Lakatos in that year alone, a citation rate of Stanford University, Stanford, CA 94305 over one paper per day. Introductory texts on the Philosophy of Science Imre Lakatos typically include substantial sections on Lakatos, some admiring, some Copyright c 2016 by the authors
Alan Musgrave and Charles Pigden critical, and many an admixture of the two (see for example Chalmers All rights reserved. 2013 and Godfrey-Smith 2003). The premier prize for the best book in the Copyright policy: https://leibniz.stanford.edu/friends/info/copyright/ Philosophy of Science (funded by the foundation of a wealthy and
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academically distinguished disciple, Spiro Latsis) is named in his honour. interpretation. We are more ambivalent with respect to the Philosophy of Moreover, Lakatos is one of those philosophers whose influence extends Mathematics.) well beyond the confines of academic philosophy. Of the thirty-three papers citing Lakatos published in the first twenty-five days of 2015, at Secondly we discuss Lakatos’s big ideas, the two contributions that most ten qualify as straight philosophy. The rest are devoted to such topics constitute his chief claims to fame as a philosopher, before moving on as educational theory, international relations, public policy research (with (thirdly) to a more detailed discussion of some of his principal papers. We special reference to the development of technology), informatics, design conclude with a section on the Feyerabend/Lakatos Debate. Lakatos was a science, religious studies, clinical psychology, social economics, political provocative and combative thinker, and it falsifies his thought to present it economy, mathematics, the history of physics and the sociology of the as less controversial (and perhaps less outrageous) than it actually was. family. Thus Imre Lakatos was very much more than a philosophers’ Note: In referring to Lakatos’s chief works (and to a couple of Popper’s) philosopher. we have employed a set of acronyms rather than the name/date system, First, we discuss Lakatos’s life in relation to his works. Lakatos’s hoping that this will be more perspicuous to readers. The acronyms are Hungarian career has now become a big issue in the critical literature. This explained in the Bibliography. is partly because of disturbing facts about Lakatos’s early life that have 1. Life only come to light in the West since his death, and partly because of a 1.1 A Tale of Two Lakatoses dispute between the “Hungarian” and the “English” interpreters of 1.2 Life and Works: The Third World and the Second Lakatos’s thought, between those writers (not all of them Magyars) who 1.3 From Stalinist Revolutionary to Methodologist of Science take the later Lakatos to be much more of a Hegelian (and perhaps much 2. Lakatos’s Big Ideas more of a disciple of György Lukács) than he liked to let on, and those 2.1 Against Formalism in Mathematics who take his Hegelianism to be an increasingly residual affair, not much 2.2 Improving on Popper in the Philosophy of Science more, in the end, than a habit of “coquetting” with Hegelian expressions 3. Works (Marx, Capital: 103). Just as there are analytic Marxists who think that 3.1 Proofs and Refutations (1963–4, 1976) Marx’s thought can be rationally reconstructed without the Hegelian 3.2 “Regress” and “Renaissance” coquetry and dialectical Marxists who think that it cannot, so also there 3.3 “Changes in the Problem of Inductive Logic” (1968) are analytic Lakatosians who think that Lakatos’s thought can be largely 3.4 “Falsification and the Methodology of Scientific Research reconstructed without the Hegelian coquetry and dialectical Lakatosians Programmes” (1970) who think that it cannot (see for instance Kadvany 2001 and Larvor 1998). 3.5 “The History of Science and Its Rational Reconstructions” Obviously, we cannot settle the matter in an Encyclopedia entry but we (1971) hope to say enough to illuminate the issue. (Spoiler alert: so far as the 3.6 “Popper on Demarcation and Induction” (1974) Philosophy of Science is concerned, we tend to favor the English 3.7 “Why Did Copernicus’s Research Programme Supersede
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Ptolemy’s?” (1976) fallibilist, the young Lakatos displayed a cocksure self-confidence in his 4. Mincemeat Unmade: Lakatos versus Feyerabend grasp of the historical situation, enough to exclude any alternative solution Bibliography to the admittedly appalling problems that this group of young and mostly Works by Lakatos Jewish communists were facing in Nazi-occupied Hungary. (“Is there no Secondary Literature other way?” the young comrade asked. The answer, apparently, was “No”; Academic Tools Long 2002: 267.) After the Soviet victory, during the late 1940s, he was an Other Internet Resources eager co-conspirator in the creation of a Stalinist state, in which the Related Entries denunciation of deviationists was the order of the day (Bandy 2009: ch. 9). Lakatos was something close to a thought policeman himself, with a powerful job in the Ministry of Education, vetting university teachers for 1. Life their political reliability (Bandy 2009: ch. 8; Long 2002: 272–3; Congden 1997). Later on, after falling afoul of the regime that he had helped to 1.1 A Tale of Two Lakatoses establish and doing time in a gulag at Recsk, he served the ÁVH, the Hungarian secret police, as an informant by keeping tabs on his friends Imre Lakatos was a warm and witty friend and a charismatic and inspiring and comrades (Bandy 2009: ch. 14; Long 2002). And he took a prominent teacher (see Feyerabend 1975a). He was also a fallibilist, and a professed part, as a Stalinist student radical, in trying to purge the University of foe of elitism and authoritarianism, taking a dim view of what he Debrecen of “reactionary” professors and students and in undermining the described as the Wittgensteinian “thought police” (owing to the Orwellian prestigious but unduly independent Eötvös College, arguing passionately tendency on the part of some Wittgensteinians to suppress dissent by against the depoliticized (but covertly bourgeois) scholarship that Eötvös constricting the language, dismissing the stuff that they did not like as allegedly stood for (Bandy 2009: chs. 4 and 9; Long 1998 and 2002). inherently meaningless) (UT: 225 and 228–36). In the later (and British) phase of his career he was a dedicated opponent of Marxism who played a 1.2 Life and Works: The Second World and the Third prominent part in opposing the socialist student radicals at the LSE in 1968, arguing passionately against the politicization of scholarship (LTD; To the many that knew and loved the later Lakatos, some of these facts are Congden 2002). difficult to digest. But how relevant are they to assessing his philosophy, which was largely the product of his British years? This is an important But in the earlier and Hungarian phase of his life, Lakatos was a Stalinist question as Lakatos was wont to draw a Popperian distinction between revolutionary, the leader of a communist cell who persuaded a young World 3—the world of theories, propositions and arguments—and World 2 comrade that it was her duty to the revolution to commit suicide, since —the psychological world of beliefs, decisions and desires. And he was otherwise she was likely to be arrested by the Nazis and coerced into sometimes inclined to suggest that in assessing a philosopher’s work we betraying the valuable young cadres who constituted the group (Bandy should confine ourselves to World 3 considerations, leaving the 2009: ch. 5; Long 1998 and 2002; Congden 1997). So far from being a subjectivities of World 2 to one side (see, for instance, F&AM: 140).
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So does a philosopher’s life have any bearing on his works? We take our Mendelian genetics disappeared in Soviet Russia in the 1950s” [the reason cue from the writings of Lakatos himself. Of course, there were facts being that Lysenko, a Stalin favourite, acquired hegemonic status within about his early career that Lakatos would not have wanted to be widely the world of Soviet biology and persecuted the Mendelians] (HS&IRR: known, and which he managed to keep concealed from his Western friends 114). and colleagues during his lifetime. But what does his official philosophy have to say about the relevance of biographical data to intellectual history? (Perhaps this marks an important departure from Hegel. For a true Hegelian, everything can, in the last analysis, be seen as rationally In “The History of Science and its Rational Reconstructions” (HS&IRR) required for the self-realization of the Absolute. Hence all history is Lakatos develops a theory of how to do the history of science, which, with “internal” in something like Lakatos’s sense, since the “cunning of reason” some adjustments, can be blown up into an account of how to do ensures that apparently irrational impulses are subordinated to the ultimate intellectual history in general. For Lakatos, the default assumption in the goal of history.) history of science is that the scientists in question are engaged in a more- or-less rational effort to solve a set of (relatively) “pure” problems (such Is there, so to speak, an “internal” history of Lakatos’s intellectual as “How to explain the apparent motions of the heavenly bodies development that can be displayed as rational? Or must it be partly consistently with a plausible mechanics?”). A “rational reconstruction” in explained in terms of “external” influences? The answer depends on the the history of science, employs a theory of (scientific) rationality in account of rationality that we adopt and the problem situation that we take conjunction with an account of the problems as they appeared to the him to have been addressing. scientists in question to display some intellectual episode as a series of Whether or not a particular theoretical (or practical) choice is susceptible rational responses to the problem-situation. On the whole, it is a plus for a to an internal explanation depends, in part, on the actor’s problem. theory of [scientific] rationality if it can display the history of science as a Consider, for example, Descartes’ theory of the vortices, namely that the relatively rational affair and a strike against it if it cannot. Thus in planets are whirled round the sun by a fluid medium which itself contains Lakatos’s opinion, naïve versions of Popper’s falsificationism are in a little whirlpools in which the individual planets are swimming. Descartes’ sense falsified by the history of science, since they represent too much of it theory of the vortices, is fairly rational if we take it as an attempt (in the as an irrational affair with too many scientists hanging on to hypotheses light of what was then known) to explain the motion of the heavenly that they ought to have recognized as refuted. If the rational reconstruction bodies in a way that is consistent with Copernican astronomy. But it is a succeeds—that is if we can display some intellectual development as a lot more rational if we take to be an attempt to explain the motion of the rational response to the problem situation—then we have an “internal” heavenly bodies in a way that is consistent with Copernican astronomy history of the developments in question. If not, then the “rational without formally contradicting the Church’s teaching that the earth does reconstruction of history needs to be supplemented by an empirical (socio- not move. (The earth goes round the sun but it does not move with respect psychological) ‘external history’” (HS&IRR: 102). Non-rational or to the fluid medium that whirls it round the sun, and, for Descartes, motion “external” factors sometimes interfere with the rational development of is defined as motion with respect to the contiguous matter.) So do we read science. “No rationality theory will ever solve problems like why
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Descartes’ theory as a fairly rational attempt to solve one problem which is kingdom without a king ruled by an admiral without a navy, the “Regent” distorted by an external factor or as a very rational attempt solve a related Admiral Horthy, who had gained his naval rank in the service of the then- but more complex problem? Well the answer is not clear, but if we want to defunct Austro-Hungarian Empire. The regime was authoritarian, a sort of understand Descartes intellectual development we need to know that it fascism-lite. After a brilliant school career, during which he won was an important constraint on his theorizing that his views should be mathematics competitions and a multitude of prizes, Lakatos entered formally consistent with the doctrines of the Church. Debrecen University in 1940. Lakatos graduated in Physics, Mathematics, and Philosophy in 1944. During his time at Debrecen he became a Similarly, it is important in understanding Lakatos’s theorizing to realize committed communist, attending illegal underground communist meetings (for example) that in later life he wanted to develop a demarcation and, in 1943, starting his own illegal study group. criterion between science and non-science that left Soviet Marxism (though not perhaps all forms of Marxism) on the non-scientific side of the No-one who attended Imre’s groups has forgotten the intensity and divide. And this holds whether we regard this constraint as a non-rational brilliance of the atmosphere. “He opened the world to me!” a external factor or as a constituent of his problem situation and hence participant said. Even those who were later disillusioned with internal to a rational reconstruction of his intellectual development. communism or ashamed of acts they committed remember the Biographical facts can be relevant to understanding a thinker’s ideas since sense of inspiration, clear thinking and hope for a new society they they can help to illuminate the problem situation to which they were felt in Imre’s secret seminars. (Long 2002: 265) addressed. However, in Lakatos’s group the emphasis was on preparing the young Furthermore, the big issue with respect to Lakatos’s development is how cadres for the coming communist revolution, rather than engaging in much of the old Hegelian-Marxist remained in the later post-Popperian public propaganda or antifascist resistance activities (Bandy 2009: ch. 3). philosopher, and how much of his philosophy was a reaction against his earlier self. To answer this question we need to know something about that In March 1944 the Germans invaded Hungary to forestall its attempts to earlier self—either the self that secretly persisted or the self that the later negotiate a separate peace. (The Hungarian government had allied with the Lakatos was reacting against. Axis powers, in the hopes of recovering some of the territories lost at the Treaty of Trianon in 1920. By 1944 they had begun to realize that this was 1.3 From Stalinist Revolutionary to Methodologist of Science a mistake.) Admiral Horthy, whose anti-Semitism was a more gentlemanly affair than that of the Nazis (he was fine with systematic discrimination Imre Lakatos was born Imre Lipsitz in Debrecen, eastern Hungary, on but apparently drew the line at mass-murder), was forced to accept a November 9, 1922, the only child of Jewish parents, Jacob Marton Lipsitz collaborationist government led by Döme Sztójay as prime minister. The and Margit Herczfeld. Lakatos’s parents parted when he was very young new regime had none of Horthy’s humanitarian scruples and began a and he was largely brought up by his grandmother and his mother who policy of enthusiastic and systematic cooperation with the Nazi genocide worked as a beautician. The Hungary into which Lakatos was born was a program. In May, Lakatos’s mother, grandmother and other relatives were
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forced into the Debrecen ghetto, thence to die in Auschwitz—the fate of Cheka (the forerunner of the KGB). Popular in Hungary, it encouraged a about 600,000 Hungarian Jews. Lakatos’s father, a wine merchant, romantic cult of revolutionary ruthlessness and sacrifice in its (mostly) managed to get away and survived the war, eventually ending up in youthful readers. As one of Lakatos’s contemporaries, György Magosh put Australia. A little earlier, in March, Lakatos himself had managed to it, escape from Debrecen to Nagváryad (now Oradea in Romania) with false papers under the name of Molnár. Later, a Hungarian friend, Vilma Balázs, How that book inspired us. How we longed to be professional recalled that revolutionaries who could be hanged several times a day in the interest of the working class and of the great Soviet Union. (Bandy Imre [had been] very close to his mother and they were quite poor. 2009: 31) He often blamed himself for her death and wondered if he could have saved her. (Bandy 2009: 32) It was in that spirit, that the ardent young Marxist, Éva Izsák, could be persuaded that it was her duty to kill herself for the sake of the cause. As In Nagváryad Lakatos restarted his Marxist group. The co-leader was his for Lakatos himself, a chance remark in his most famous paper suggests then-girlfriend and subsequent wife, Éva Révész. In May, the group was something about his attitude. joined by Éva Izsák, a 19-year-old Jewish antifascist activist who needed lodgings with a non-Jewish family. Lakatos decided that there was a risk One has to appreciate the dare-devil attitude of our methodological that she would be captured and forced to betray them, hence her duty both falsificationist [or perhaps as he would have said in an earlier to the group and to the cause was to commit suicide. A member of the phase of his career, the conscientious Leninist]. He feels himself to group took her across country to Debrecen and gave her cyanide (Congden be a hero who, faced with two catastrophic alternatives, dares to 1997, Long 2002, Bandy 2009, ch. 5). To lovers of Russian literature, the reflect coolly on their relative merits and [to] choose the lesser evil. episode recalls Dostoevsky’s The Possessed/Demons (based in part on the (FMSRP: 28) real life Nechaev affair). In Dostoevsky’s novel the anti-Tsarist If you admire the hero who has the courage to make the tough choice revolutionary, Pyotr Verkhovensky, posing as the representative of a large between two catastrophic alternatives, isn’t there a temptation to revolutionary organization, tries to solidify the provincial cell of which he manufacture catastrophic alternatives so that you can heroically choose is the chief by getting the rest of group to share in the murder of a between them? dissident member who supposedly poses a threat to the group. (It does not work for the fictional Pytor Verkhovensky and it did work for the real-life Late in 1944, following a Soviet victory, Lakatos returned to Debrecen, Sergei Nechaev.) Hence the title of Congden’s 1997 exposé “Possessed: and changed his name from the Germanic Jewish Lipsitz to the Hungarian Imre Lakatos’s Road to 1956”. But to communists or former communists proletarian Lakatos (meaning “locksmith”). He became active in the now of Lakatos’s generation, it recalled a different book: Chocolate, by the legal Communist Party and in two leftist youth and student organizations, Bolshevik writer Aleksandr Tarasov-Rodianov. This is a stirring tale of the Hungarian Democratic Youth Federation (MADISZ) and the Debrecen revolutionary self-sacrifice in which the hero is the chief of the local University Circle (DEK). As one of the leaders of the DEK, Lakatos
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agitated for the dismissal of reactionary professors from Debrecen and the to the wolves (Bandy 2009: ch. 12; Long 2002). He was arrested in April exclusion of reactionary students. 1950 on charges of revisionism and, after a period in the cellars of the secret police (including, of course, torture), he was condemned to the We are aware that this move on our part is incompatible with the prison camp at Recsk. traditional and often voiced “autonomy” of the university [Lakatos stated], but respect for autonomy, in our view, cannot mean that we However Lakatos was probably doomed anyway. In later life Lakatos was have to tolerate the strengthening of fascism and reaction. (Bandy big admirer of Orwell’s Nineteen Eighty-Four. Perhaps he recognized 2009: 59 and 61) himself in Orwell’s description of the Party intellectual (and expert on Newspeak) Syme: Lakatos moved to Budapest in 1946. He became a graduate student at Budapest University, but spent much of his time working towards the Unquestionably Syme will be vaporized, Winston thought again. communist takeover of Hungary. This was a slow-motion affair, He thought it with a kind of sadness, although well knowing that characterized by the infamous “salami tactics” of the Communist leader Syme…was fully capable of denouncing him as a thought-criminal Mátyás Rákosi. Lakatos worked chiefly in the Ministry of Education, if he saw any reason for doing so. There was something subtly evaluating the credentials of university teachers and making lists of those wrong with Syme. There was something that he lacked: discretion, who should be dismissed as untrustworthy once the communists took over aloofness, a sort of saving stupidity. You could not say that he was (Bandy 2009: ch. 8). He was also a student at Eötvös College, but attacked unorthodox. He believed in the principles of Ingsoc, he venerated it publicly as an elitist and bourgeois institution. The College, and others Big Brother, he rejoiced over victories, he hated heretics…. Yet a like it, was closed in 1950 after the communist takeover. In 1947 Lakatos faint air of disreputability always clung to him. He said things that gained his doctorate from Debrecen University for a thesis entitled “On would have been better unsaid, he had read too many books…. the Sociology of Concept Formation in the Natural Sciences”. In 1948, (Orwell 2008 [1949]: 58) after the communist takeover was substantially complete, he gained a scholarship to undertake further study in Moscow. An instance of Lakatos’s Syme-like behaviour is his 1947 denunciation of the literary critic and philosopher György Lukács, one of the intellectual Lakatos flew to Moscow in January 1949, only to be recalled for “un- luminaries of the communist movement. Lukács represented the Party-like” behaviour in July. What these “un-party-like” activities were is academically respectable face of communism, and favoured a gradual and something of a mystery but even more of a mystery is why, having democratic transition to the dictatorship of the proletariat. Lakatos returned from Moscow under a cloud, he seemed so cool, calm and organized an “anti-Lukács meeting…held under the aegis of the Valóság collected. Lakatos’s biographers, Long and Bandy, speculate that he was Circle” to critique Lukács’s foot-dragging and “Weimarism” (Bandy 2009: being held in reserve to prepare a case against the communist education 110). Once the regime was firmly in control, Lukács was indeed censured chief, József Révai, who was scheduled to appear in a new show trial. But for his undue concessions to bourgeois democracy, and he spent the early when Rákosi decided not to prosecute Révai after all, Lakatos was thrown fifties under a cloud. But in 1947, Lakatos’s criticisms were deemed
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premature and he got into trouble because of his un-Party-like activities. against believing one’s own senses. This was the struggle against (Lukács himself referred to the episode as a “cliquish kaffe klatsch”.) In empiricism [Laughter and applause]. (Bandy 2009: 221. Bandy Communist Hungary it was important not to be “one pamphlet behind” the quotes the transcripts which seem to differ slightly from the Party line (Bandy 2009: 92). Lakatos was the sort of over-zealous prepared text in the Lakatos archives, reprinted in F&AM) communist who was sometimes a couple of pamphlets ahead. But Lakatos was not just explicitly repudiating Stalinism. He was also After his release from Recsk in September 1953 (minus several teeth), implicitly criticizing another prominent member of the Petőfi Circle who Lakatos remained for a while, a loyal Stalinist. He eked out a living in the had been a big influence on his first PhD, namely György Lukács (see Mathematics Institute of the Hungarian Academy of Science, reading, Ropolyi 2002 for the early influence). For Lukács’s work is pervaded by researching and translating (including a translation into Hungarian of just the kind of hostility towards empiricism and disdain for facts that George Pólya’s How to Solve It). During this time he was informing on Lakatos is denouncing in his speech, as well as an arts-sider’s contempt friends and colleagues to the ÁVH., the Hungarian secret police, though for the natural sciences, all of which would have been anathema to the he subsequently claimed that he did not pass on anything incriminating later Lakatos. Indeed Lukács was notorious for the view that that (Long, 2002: 290 ). It was whilst working at the Mathematics Institute that he first gained access to the works of Popper. Gradually he turned against even if the development of science had proved all Marx’s the Stalinist Marxism that had been his creed. He married (as his second assertions to be false…we could accept this scientific criticism wife) Éva Pap and lived at her parents’ house (his father-in-law being the without demur and still remain Marxists—as long as we adhered to distinguished agronomist, Endre Pap). In 1956 he joined the revisionist the Marxist method Petőfi Circle and delivered a stirring speech on “On Rearing Scholars” and that which at least burnt his bridges with Stalinism: the orthodox Marxist who realizes that…the time has come for the The very foundation of scholarly education is to foster in students expropriation of the exploiters, will respond to the vulgar-Marxist and postgrads a respect for facts, for the necessity of thinking litany of “facts” which contradict this process with the words of precisely, and to demand proof. Stalinism, however, branded this Fichte, one of the greatest of classical German philosophers: “So as bourgeois objectivism. Under the banner of partinost [Party- much the worse for the facts”. (Lukács 2014 [1919]: ch. 3) like] science and scholarship, we saw a vast experiment to create a science without facts, without proofs. Thus the Stalinist Lakatos of 1947 had explicitly denounced Lukács for not being Stalinist enough, but the revisionist Lakatos of 1956 was … a basic aspect of the rearing of scholars must be an endeavour implicitly denouncing Lukács for being methodologically too much of a to promote independent thought, individual judgment, and to Stalinist. For the later Lakatos, what was wrong with “orthodox Marxism” develop conscience and a sense of justice. Recent years have seen was chiefly that its novel factual predictions had been systematically an entire ideological campaign against independent thinking and falsified (see §3.2 below). But that was pretty much the complaint of early
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revisionists such as Bernstein (see Kolakowski 1978: ch. 4, and it was 2. Lakatos’s Big Ideas against that kind of revisionism that Lukács’s Bolshevik writings were a protest (see Lukács 1971 [1923] and 2014 [1919]). Though factual Imre Lakatos has two chief claims to fame. “refutations” of a research programme are not always decisive, a Lukács- like indifference to the facts is, for Lakatos, the mark of a fundamentally 2.1 Against Formalism in Mathematics unscientific attitude. In our opinion, this puts paid to Ropolyi’s opinion The first is his Philosophy of Mathematics, especially as set forth in that Lukács continued to be a major influence on the later Lakatos. “Proofs and Refutations” (1963–64) a series of four articles, based on his Lakatos left Hungary in November 1956 after the Soviet Union crushed PhD thesis, and written in the form of a many-sided dialogue. These were the short-lived Hungarian revolution. He walked across the border into subsequently combined in a posthumous book and published, with Austria with his wife and her parents. Within two months he was at King’s additions, in 1976. The title is an allusion to a famous paper of Popper’s, College Cambridge, with a Rockefeller Fellowship to write a PhD under “Conjectures and Refutations” (the signature essay of his best-known the supervision of R.B. Braithwaite, which he completed in 1959 under the collection), in which Popper outlines his philosophy of science. Lakatos’s title “Essays in the Logic of Mathematical Discovery”. If we set aside his point is that the development of mathematics is much more like the romantic adventures, the story of Lakatos’s life thereafter is largely the development of science as portrayed by Popper than is commonly story of his work, though we should not forget his activities as an supposed, and indeed much more like the development of science as academic politician. Even his friendship with Feyerabend and his portrayed by Popper than Popper himself supposed. friendship and subsequent bust-up with Popper were very much work- What Lakatos does not make so much of (though he does not conceal it related. In Britain his academic career was meteoric. In 1960 he was either) is that in his view the development of mathematics is also much appointed Assistant Lecturer in Karl Popper’s department at the London more like the development of thought in general as analysed by Hegel School of Economics. By 1969 he was Professor of Logic, with a than Hegel himself supposed. There is thesis, antithesis and synthesis, worldwide reputation as a philosopher of science. During the student “Hegelian language, which [Lakatos thinks would], generally be capable revolts of the 1960s, which in Britain were centred on the LSE, Lakatos of describing the various developments in mathematics” (P&R: 146). Thus became an establishment figure. He wrote a “Letter to the Director of the there is a certain sense in which Lakatos out-Hegels Hegel, giving a London School of Economics” defending academic freedom and academic dialectical analysis of a discipline (mathematics) that Hegel himself autonomy, which was widely circulated. It denounces the student radicals despised as insufficiently dialectical (see Larvor 1998, 1999, 2001). Hence for allegedly trying to do what he himself had done at Debrecen and Feyerabend’s gibe (which Lakatos took in good part) that Lakatos was a Eötvös (though he is careful to conceal the parallel, citing Nazi and Pop-Hegelian, the bastard child of Popperian father and a Hegelian mother Muscovite precedents instead) (LTD: 247). (F&AM: 184–185). Lakatos died suddenly in 1974 of a heart attack at the height of his powers. He was 51.
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Proofs and Refutations is a critique of “formalist” philosophies of really refutations either, since something rather like the “refuted” thesis mathematics (including formalism proper, logicism and intuitionism), often survives the refutation and arises refreshed and invigorated from the which, in Lakatos’s view, radically misrepresent the nature of mathematics dialectical process. as an intellectual enterprise. For Lakatos, the development of mathematics should not be construed as series of Euclidean deductions where the This becomes apparent early on in the dialogue, when the Popperian contents of the relevant concepts has been carefully specified in advance Gamma protests at the Teacher’s insouciance with respect to refutation, a so as to preclude equivocation. Rather, these water-tight deductions from counterexample to Euler’s thesis (and therefore to Cauchy’s proof) that, well-defined premises are the (perhaps temporary) end-points of an for all regular polyhedra, the number of vertices, minus the number of evolutionary, and indeed a dialectical, process in which the constituent edges, plus the number of faces equals two (V − E + F = 2). The concepts are initially ill-defined, open-ended or ambiguous but become counterexample is a solid bounded by a pair of nested cubes, one of which sharper and more precise in the context of a protracted debate. The proofs is inside, but does not touch the other: are refined in conjunction with the concepts (hence “proof-generated concepts”) whilst “refutations” in the form of counterexamples play a prominent part in the process. [One might almost say, paraphrasing Hegel, that in Lakatos’s view “when Euclidean demonstrations paint their grey in grey, then has a shape of mathematical life grown old…The owl of the formalist Minerva begins its flight only with the falling of dusk” (Hegel 2008 [1820/21]: 16).] For this hollow cube, (including both the inner and the outer Lakatos is also keen to display the development of mathematics as a V − E + F ones) . According to Gamma, this simply refutes Euler’s conjecture rational affair even though the proofs (to begin with) are often lacking in = 4 and disproves Cauchy’s proof: logical rigour and the key concepts are often open-ended and unclear GAMMA: Sir, your composure baffles me. A single The idea—expressed so clearly by Seidel [and clearly endorsed by counterexample refutes a conjecture as effectively as ten. The Lakatos himself]—that a proof can be respectable without being conjecture and its proof have completely misfired. Hands up! You flawless, was a revolutionary one in 1847, and, unfortunately, still have to surrender. Scrap the false conjecture, forget about it and try sounds revolutionary today. (P&R: 139) a radically new approach. A corollary of this is that in mathematics many of the “proofs” are not TEACHER: I agree with you that the conjecture has received a really proofs in the full sense of the word (that is, demonstrations that severe criticism by Alpha’s counterexample. But it is untrue that proceed deductively from apodictic premises via unquestionable rules of the proof has “completely misfired”. If, for the time being, you inference to certain conclusions) and that many of the “refutations” are not
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agree to my earlier proposal to use the word “proof” for a to a novel theory of scientific rationality. This is arguably a lot more “thought-experiment which leads to decomposition of the original realistic than the Popperian theory it was designed to supplant (or, in conjecture into subconjectures”, instead of using it in the sense of a earlier formulations, the Popperian theory that it was designed to amend). “ guarantee of certain truth”, you need not draw this conclusion. For Popper, a theory is only scientific if is empirically falsifiable, that is if My proof certainly proved Euler’s conjecture in the first sense, but it is possible to specify observation statements which would prove it not necessarily in the second. You are interested only in proofs wrong. A theory is good science, the sort of theory you should stick with which “prove” what they have set out to prove. I am interested in (though not the sort of thing you should believe as Popper did not believe proofs even if they do not accomplish their intended task. in belief), if it is refutable, risky, and problem-solving and has stood up to Columbus did not reach India but he discovered something quite successive attempts at refutation. It must be highly falsifiable, well-tested interesting. but (thus far) unfalsified.
Thus even in his earlier work, when he is still a professed disciple of Lakatos objects that although there is something to be said for Popper’s Popper, Lakatos is already a rather dissident Popperian. Firstly, there are criterion, it is far too restrictive, since it would rule out too much of the hat-tips to Hegel as well as to Popper that crop up from time to time in everyday scientific practice (not to mention the value-judgments of the Proofs and Refutations including the passage where he praises (and scientific elite) as unscientific and irrational. For scientists often persist— condemns) them both in the same breath. (“Hegel and Popper represent and, it seems, rationally persist—with theories, such as Newtonian the only fallibilist traditions in modem philosophy, but even they both celestial mechanics that by Popper’s standards they ought to have rejected made the mistake of reserving a privileged infallible status for as “refuted”, that is theories that (in conjunction with other assumptions) mathematics”. P&R: 139n.1.) Given that Hegel was anathema to Popper have led to falsified predictions. A key example for Lakatos is the (witness his famous or notorious anti-Hegel “scherzo” in The Open “Precession of Mercury” that is, the anomalous behaviour of the Society and Its Enemies, (1945 [1966])) this strongly suggests that Lakatos perihelion of Mercury, which shifts around the Sun in a way that it ought took his Popper with a large pinch of salt. Secondly, for Popper himself a not to do if Newton’s mechanics were correct and there were no other proof is a proof and a refutation is supposed to kill a scientific conjecture sizable body influencing its orbit. The problem is that there seems to be no stone-dead. Thus non-demonstrative proofs and non-refuting refutations such body. The difficulty was well known for decades but it did not cause mark a major departure from Popperian orthodoxy. astronomers to collectively give up on Newton until Einstein’s theory came along. Lakatos thought that the astronomers were right not to 2.2 Improving on Popper in the Philosophy of Science abandon Newton even though Newton eventually turned out to be wrong and Einstein turned out to be right. The dissidence continues with Lakatos’s second major contribution to philosophy, his “Methodology of Scientific Research Programmes” or Again, Copernican heliocentric astronomy was born “refuted” because of MSRP (developed in detail in in his FMSRP), a radical revision of the apparent non-existence of stellar parallax. If the earth goes round the Popper’s Demarcation Criterion between science and non-science, leading sun then the apparent position of at least some of the fixed stars (namely
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the closest ones) ought to vary with respect to the more distant ones as the Firstly scientists working within the programme are typically (and rightly) earth is moving with respect to them. Some parts of the night sky should reluctant to give up on the claims that constitute the hard core. look a little different at perihelion (when the earth is furthest from the sun) from the way that they look at aphelion (when the earth is at its nearest to Secondly the hard core theses by themselves are often devoid of empirical the sun, and hence at the other end of its orbit). But for nearly three consequences. For example, Newtonian mechanics by itself—the three centuries after the publication of Copernicus’ De Revolutionibus 1543, no laws of mechanics and the law of gravitation—won’t tell you what you such differences were observed. In fact, there is a very slight difference in will see in the night sky. To derive empirical predictions from Newtonian the apparent positions of the nearest stars depending on the earth’s mechanics you need a whole host of auxiliary hypotheses about the position in its orbit, but the difference is so very slight as to be almost positions, masses and relative velocities of the heavenly bodies, including undetectable. Indeed it was completely undetectable until 1838 when the earth. (This is related to Duhem’s thesis that, generally speaking, sufficiently powerful telescopes and measuring techniques were able to theoretical propositions—and indeed sets of theoretical propositions— detect it, by which time the heliocentric view had long been regarded as an cannot be conclusively falsified by experimental observations, since they established fact. Thus astronomers had not given up on either Copernicus only entail observation-statements in conjunction with auxiliary or his successors despite this apparent falsification. hypotheses. So when something goes wrong, and the observation statements that they entail turn out to be false, we have two intellectual But if scientists often persist with “refuted” theories, either the scientists options: modify the theoretical propositions or modify the auxiliary are being unscientific or Popper is wrong about what constitutes good hypotheses. See Ariew 2014.) For Lakatos an individual theory within a science, and hence about what scientists ought to do. Lakatos’s idea is to research programme typically consists of two components: the (more or construct a methodology of science, and with it a demarcation criterion, less) irrefutable hard core plus a set of auxiliary hypotheses. Together with whose precepts are more in accordance with scientific practice. the hard core these auxiliary hypotheses entail empirical predictions, thus making the theory as a whole—hard core plus auxiliary hypotheses—a How does it work? Well, falsifiability continues to play a part in Lakatos’s falsifiable affair. conception of science but its importance is somewhat diminished. Instead of an individual falsifiable theory which ought to be rejected as soon as it What happens when refutation strikes, that is when the hard core in is refuted, we have a sequence of falsifiable theories characterized by conjunction with the auxiliary hypotheses entails empirical predictions shared a hard core of central theses that are deemed irrefutable—or, at which turn out to be false? What we have essentially is a modus tollens least, refutation-resistant—by methodological fiat. This sequence of argument in which science supplies one of the premises and nature (plus theories constitutes a research programme. experiment and observation) supplies the other:
The shared hard core of this sequence of theories is often unfalsifiable in 1. If
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Therefore positive heuristic of the Newtonian programme bids us look for another heavenly body whose gravitational force might be distorting the first 3. Not
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examples of bad science or pseudo-science for Popperians], are all progressive, and degenerating if it is not” (FMSRP: 34). Thus a research research programmes, each with a characteristic hard core programme is degenerating if the successive theories do not deliver novel stubbornly defended, each with its more flexible protective belt and predictions or if the novel predictions that they deliver turn out to be false. each with its elaborate problem-solving machinery. Each of them, at any stage of its development, has unsolved problems and Novelty is, in part, a competitive notion. The novelty of a research undigested anomalies. All theories, in this sense, are born refuted programme’s predictions is defined with respect to its rivals. A prediction and die refuted. But are they [all] equally good? (S&P: 4–5) is novel if the theory not only predicts something not predicted by the previous theories in the sequence, but if the predicted observation is Lakatos, of course, thinks not. Some science is objectively better than predicted neither by any rival programme that might be in the offing nor other science and some science is so unscientific as to hardly qualify as by the conventional wisdom. A programme gets no brownie points by science at all. So how does he distinguish between “a scientific or predicting what everyone knows to be the case but only by predicting progressive programme” and a “pseudoscientific or degenerating one”? observations which come as some sort of a surprise. (There is some (S&P: 4–5) ambiguity here and some softening later on—see below §3.6—but to begin with, at least, this was Lakatos’s dominant idea.) To begin with, the unit of scientific evaluation is no longer the individual theory (as with Popper), but the sequence of theories, the research One of Lakatos’s key examples is the predicted return of Halley’s comet programme. We don’t ask ourselves whether this or that theory is scientific which was derived by observing part of its trajectory and using Newtonian or not, or whether it constitutes good or bad science. Rather we ask mechanics to calculate the elongated ellipse in which it was moving. The ourselves whether the sequence of theories, the research programme, is comet duly turned up seventy-two years later, exactly where and when scientific or non-scientific or constitutes good or bad science. Lakatos’s Halley had predicted, a novel fact that could not have been arrived at basic idea is that a research programme constitutes good science—the sort without the aid of Newton’s theory (S&P: 5). Before Newton, astronomers of science it is rational to stick with and rational to work on—if it is might have noticed a comet arriving every seventy-two years (though they progressive, and bad science—the kind of science that is, at least, would have been hard put to it to distinguish that particular comet from intellectually suspect—if it is degenerating. What is it for a research any others), but they could not have been as exact about the time and place programme to be progressive? It must meet two conditions. Firstly it must of its reappearance as Halley managed to be. Newton’s theory delivered be theoretically progressive. That is, each new theory in the sequence must far more precise predictions than the rival heliocentric theory developed have excess empirical content over its predecessor; it must predict novel by Descartes, let alone the earth-centered Ptolemaic cosmology that had and hitherto unexpected facts (FMSRP: 33). Secondly it must be ruled the intellectual roost for centuries. That’s the kind of spectacular empirically progressive. Some of that novel content has to be corroboration that propels a research programme into the lead. And it was corroborated, that is, some of the new “facts” that the theory predicts must a similarly novel prediction, spectacularly confirmed, that dethroned turn out to be true. As Lakatos himself put the point, a research Newton’s physics in favour of Einstein’s. Here’s Lakatos again: programme “is progressive if it is both theoretically and empirically
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This programme made the stunning prediction that if one measures Newtonian programme led to novel facts; the Marxian lagged the distance between two stars in the night and if one measures the behind the facts and has been running fast to catch up with them. distance between them during the day (when they are visible (S&P: 4–5) during an eclipse of the sun), the two measurements will be different. Nobody had thought to make such an observation before Thus good science is progressive and bad science is degenerating and a Einstein’s programme. Thus, in progressive research programme, research programme may either begin or end up as such a degenerate theory leads to the discovery of hitherto unknown novel facts. affair that it ceases to count as science at all. If a research programme (S&P: 5) either predicts nothing new or entails novel predictions that never come to pass, then it may have reached such a pitch of degeneration that it has A degenerating research programme, on the other hand (unlike the theories transformed into a pseudoscience. of Newton and Einstein) either fails to predict novel facts at all, or makes novel predictions that are systematically falsified. Marxism, for example, It is sometimes suggested that in Lakatos’s opinion no theory either is or started out as theoretically progressive but empirically degenerate (novel ought to be abandoned, unless there is a better one in existence (Hacking predictions systematically falsified) and ended up as theoretically 1983: 113). Does this mean that no research programme should be given degenerate as well (no more novel predictions but a desperate attempt to up in the absence of a progressive alternative, no matter how degenerate it explain away unpredicted “observations” after the event). may be? If so, this amounts to the radically anti-sceptical thesis that it is better to subscribe to a theory that bears all the hallmarks of falsehood, Has…Marxism ever predicted a stunning novel fact successfully? such as the current representative of a truly degenerate programme, than to Never! It has some famous unsuccessful predictions. It predicted sit down in undeluded ignorance. (The ancient sceptics, by contrast the absolute impoverishment of the working class. It predicted that thought that it is better not to believe anything at all rather than believe the first socialist revolution would take place in the industrially something that might be false.) We are not sure that this was Lakatos’ most developed society. It predicted that socialist societies would opinion, though he clearly thinks it a mistake to give up on a progressive be free of revolutions. It predicted that there will be no conflict of research programme, unless there is a better one to shift to. But consider interests between socialist countries. Thus the early predictions of again the case of Marxism. What Lakatos seems to be suggesting in the Marxism were bold and stunning but they failed. Marxists passage quoted above, is that it is rationally permissible—perhaps even explained all their failures: they explained the rising living obligatory—to give up on Marxism even if it has no progressive rival, that standards of the working class by devising a theory of imperialism; is, if there is currently no alternative research programme with a set of they even explained why the first socialist revolution occurred in hard core theses about the fundamental character of capitalism and its industrially backward Russia. They “explained” Berlin 1953, ultimate fate. (After all, the later Lakatos probably subscribed to the Budapest 1956, Prague 1968. They “explained” the Russian- Popperian thesis that history in the large is systematically unpredictable. Chinese conflict. But their auxiliary hypotheses were all cooked up In which case there could not be a genuinely progressive programme after the event to protect Marxian theory from the facts. The which foretold the fate of capitalism. At best you could have a conditional
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theory, such as Piketty’s, which says that under capitalism, inequality is But though science aims at truth and therefore at consistency, this does not likely to grow—unless something unexpected happens or unless we decide mean that it can’t put up with a little inconsistency along the way. to do something about it. See Piketty 2014: 35.) So although Lakatos thinks that the scientific community seldom gives up on a programme until The discovery of an inconsistency—or of an anomaly—[need not] something better comes along, it is not clear that he thinks that this is what immediately stop the development of a programme: it may be they always ought to do. rational to put the inconsistency into some temporary, ad hoc quarantine, and carry on with the positive heuristic of the There are numerous departures from Popperian orthodoxy in all this. To programme. (FMSRP: 58) begin with, Lakatos effectively abandons falsifiability as the Demarcation Criterion between science and non-science. A research programme can be Thus it was both rational and scientific for Bohr to persist with his falsifiable (in some senses) but unscientific and scientific but unfalsifiable. research programme, even though its hard core theses on the structure of First, the falsifiable non-science. Every successive theory in a the atom were fundamentally inconsistent (FMSRP: 55–58). So although degenerating research programme can be falsifiable but the programme as Lakatos rejects Hegel’s, claim that there are contradictions in reality whole may not be scientific. This might happen if it only predicted (though not, perhaps in Reality), he also rejects Popper’s thesis that familiar facts or if its novel predictions were never verified. A tired because contradictions imply everything, inconsistent theories exclude purveyor of old and boring truths and/or a persistent predictor of novel nothing and must therefore be rejected as unfalsifiable and unscientific. falsehoods might fail to make the scientific grade. Secondly, the non- For Lakatos, Bohr’s theory of the atom is fundamentally inconsistent, but falsifiable science. In Lakatos’s opinion, it need not be a crime to insulate this does not mean that it implies that the moon is made of green cheese. the hard-core of your research programme from empirical refutation. For Thus what Lakatos seems to be suggesting is here (though he is not as Popper, it is a sin against science to defend a refuted theory by explicit as he might be) is that, when it comes to assessing scientific “introducing ad hoc some auxiliary assumption, or by re-interpreting the research programmes, we should sometimes employ a contradiction- theory ad hoc in such a way that it escapes refutation” (C&R, 48). Not so tolerant logic; that is a logic that rejects the principle, explicitly endorsed for Lakatos, though this is not to say that when it comes to ad hocery by Popper, that anything whatever follows from a contradiction (FMSRP: “anything goes”. 58 n. 2). In today’s terminology, Lakatos is a paraconsistentist (since he implicitly denies that from a contradiction anything follows) but not a Thirdly, Lakatos’s Demarcation Criterion is a lot more forgiving than dialethist (since he explicitly denies that there are true contradictions). Popper’s. For a start, an inconsistent research programme need not be Thus he is neither a follower of Popper with respect to theories nor a condemned to the outer darkness as hopelessly unscientific. This is not follower of Hegel with respect to reality (see Priest 2006 and 2002, because any of its constituent theories might be true. Lakatos rejects the especially ch. 7, and Brown and Priest 2015). Hegelian thesis that there are contradictions in reality. “If science aims at truth, it must aim at consistency; if it resigns consistency, it resigns truth.” There is another respect in which Lakatos’s Demarcation Criterion is more forgiving than Popper’s. For Popper, if a theory is not falsifiable, then it’s
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not scientific and that’s that. It’s an either/or affair. For Lakatos being Apart from the fact that, for Lakatos, a) it can be rational to persist with a scientific is a matter of more or less, and the more the less can vary over “falsified” theory, and indeed with theory that is actually inconsistent— time. A research programme can be scientific at one stage, less scientific both anathema to Popper—and that b) that for Lakatos “all theories are (or non-scientific) at another (if it ceases to generate novel predictions and born refuted and die refuted” (S&P: 5) so that there are no unrefuted cannot digest its anomalies) but can subsequently stage a comeback, conjectures for the virtuous scientist to stick with (thus making what recovering its scientific status. Thus the deliverances of the Criterion are Popper would regard as good science practically impossible), Lakatos’s matters of degree, and they are matters of degree that can vary from one methodology of scientific research programmes replaces two of Popper’s time to another. We can seldom say absolutely that a research programme criteria with one. For Popper has one criterion to distinguish science from is not scientific. We can only say that it is not looking very scientifically non-science (or science from pseudoscience if it is a theory with scientific healthy right now, and that the prospects for a recovery do not look good. pretensions) and another to distinguish good science from bad science. In Thus Lakatos is much more of a fallibilist than Popper. For Popper, we can Popper’s view, a theory is scientific if it is empirically falsifiable and non- tell whether a theory is scientific or not by investigating its logical scientific if it is not. Being scientific or not is an absolute affair, a matter of implications. For Lakatos our best guesses might turn out to be mistaken, either/or, since a theory is scientific so long as there are some observations since the scientific status of a research programme is determined, in part, that would falsify it. Being good science is a matter of degree, since a by its history, not just by its logical character, and history, as Popper theory may give more or less hostages to empirical fortune, depending on himself proclaimed, is essentially unpredictable. the boldness of its empirical predictions. For Lakatos on the other hand, non-science or pseudo-science is at one end of a continuum with the best There is another divergence from Popper which helps to explain the science at the other end of the scale. Thus a theory—or better, a research above. Lakatos collapses two of Popper’s distinctions into one; the programme—can start out as genuinely scientific, gradually becoming less distinction between science and non-science and the distinction between so over the course of time (which was Lakatos’s view of Marxism) good science and bad. As Lakatos himself put the point in his lectures at without altogether giving up the scientific ghost. Was the Marxism of the LSE: Lakatos’s day bad science or pseudo-science? From Lakatos’ point of view, the question does not have a determinate answer, the point being that The demarcation problem may be formulated in the following it isn’t good science since it represents a degenerating research terms: what distinguishes science from pseudoscience? This is an programme. But although Lakatos evidently considered Marxism to be in extreme way of putting it, since the more general problem, called bad way, he could not consign it to the dustbin of history as definitively the Generalized Demarcation Problem, is really the problem of the finished, since (as he often insisted) degenerating research programmes appraisal of scientific theories, and attempts to answer the can sometimes stage a comeback. question: when is one theory better than another? We are, naturally, assuming a continuous scale whereby the value zero corresponds to a pseudo scientific theory and positive values to theories considered scientific in a higher or lesser degree. (F&AM: 20)
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3. Works distinguished the “context of discovery” from the “context of justification”, consigned the former to the realm of empirical psychology, 3.1 Proofs and Refutations (1963–4, 1976) and thought it a matter of “unregimented insight and good fortune”, hardly a fit subject for philosophical analysis. Philosophy of mathematics consists As we have seen, Lakatos’s first major publication in Britain was the of the logical analysis of completed theories. Formalism manifests this dialogue “Proofs and Refutations” which originally appeared as a series of orthodoxy and “disconnects the history of mathematics from the four journal articles. The dialogue is dedicated to George Pólya for his philosophy of mathematics” (P&R: 1). Against the orthodoxy, Lakatos “revival of mathematical heuristic” and to Karl Popper for his critical paraphrased Kant (the paraphrase has become almost as famous as the philosophy. original):
Proofs and Refutations is a highly original production. The issues it the history of mathematics…has become blind, while the discusses are far removed from what was then standard fare in the philosophy of mathematics… has become empty. (P&R: 2) philosophy of mathematics, dominated by logicism, formalism and intuitionism, all attempting to find secure foundations for mathematics. Its [Lakatos had stated this Kantian aphorism more generally at a conference theses are radical. And its dialogue form makes it a literary as well as a in Oxford in 1961: “History of science without philosophy of science is philosophical tour de force. blind. Philosophy of science without history of science is empty”. See Hanson 1963: 458.] Its official target is “formalism” or “metamathematics”. But (as we have noted) “formalism” doesn’t just mean “formalism” proper, as this term is Suppose we agree with Lakatos that there is room for heuristics or a logic usually understood in the Philosophy of Mathematics. For Lakatos or discovery. Still, orthodoxy could insist that discovery is one thing, “formalism” includes not just Hilbert’s programme but also logicism and justification another, and that the genesis of ideas has nothing to do with even intuitionism. Formalism sees mathematics as the derivation of their justification. Lakatos, more radically, disputed this. First, he rejected theorems from axioms in formalised mathematical theories. The the foundationalist or justificationist project altogether: mathematics has philosophical project is to show that the axioms are true and the proofs no foundation in logic, or set theory, or anything else. Second, he insisted valid, so that mathematics can be seen as the accumulation of eternal that the way in which a theory grows plays an essential role in its truths. An additional philosophical question is what these truths are about, methodological appraisal. This is as much a central theme of his the question of mathematical ontology. philosophy of empirical science as it is of his philosophy of mathematics.
Lakatos, by contrast, was interested in the growth of mathematical As noted above, Proofs and Refutations takes the form of an imaginary knowledge. How were the axioms and the proofs discovered? How does dialogue between a teacher and a group of students. It reconstructs the mathematics grow from informal conjectures and proofs into more formal history of attempts to prove the Descartes-Euler conjecture about proofs from axioms? Logical empiricist (and Popperian) orthodoxy polyhedra, namely, that for all polyhedra, the number of vertices minus the
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number of edges plus the number of faces is two (V − E + F = 2). The with by incorporating the refuted premise or lemma into the original teacher presents an informal proof of this conjecture, due to Cauchy. This conclusion, as a condition of its correctness. For example, a picture- is a “thought experiment which suggest a decomposition of the original frame is a polyhedron with a hole or tunnel in it, for which conjecture into subconjectures or lemmas” from which the original V − E + F = 0). conjecture is supposed to follow. We now have, as well as the original conjecture or conclusion, the subconjectures or premises, and the meta- conjecture that the latter entail the former. Clearly, this kind of “informal proof” is quite different from the “formalist” idea that an informal proof is a formal proof with gaps (PP2: 63). Equally clearly, any of these conjectures might be refuted by counterexamples.
In the dialogue, the students, who are rather advanced, demonstrate the point—they demolish the Teacher’s “proof” by producing So if we define a polyhedron as “normal” if it has no holes or tunnels counterexamples. The counterexamples are of three kinds: in it, we can restrict the original conjecture to “normal” polyhedra and avoid this refutation. The trouble with this method is that it reduces (1) Counterexamples to the conclusion that are not also the content of the original conjecture, and an empty tautology counterexamples to any of the premises (“global but not local threatens—“For all Eulerian polyhedra (polyhedra for which counterexamples”): These establish that the conclusion does not really V − E + F = 2), V − E + F = 2”. More particularly, a blanket follow from the stated premises. They require us to improve the proof, exclusion of polyhedra with holes or tunnels rules out some polyhedra to unearth the “hidden lemma” which the counterexample also refutes, for which V − E + F = 2, despite the presence of a hole—a cube with so that it becomes a “local as well as global” counterexample—see a square hole drilled through it and two ring-shaped faces being an (3), below. example. This suggests a deeper problem than finding the domain of validity of the original conjecture—finding a general relationship (2) Counterexamples to one of the premises that are not also between V, E and F for all polyhedra whatsoever. counterexamples to the conclusion (“local but not global counterexamples”): These require us to improve the proof by We see from this analysis what Lakatos calls the “dialectical unity of replacing the refuted premise with a new premise which is not subject proofs and refutations”. Counterexamples help us to improve our proof by to the counterexample and which (we hope) will do as much to finding hidden lemmas. And proofs help us improve our conjecture by establish the conclusion as the original refuted premise did. finding conditions on its validity. Either way, or both ways, mathematical knowledge grows. And as it grows, its concepts are refined. We begin with (3) Counterexamples both to the conclusion and to (at least one of) the a vague, unarticulated notion of what a polyhedron is. We have a premises (“global and local counterexamples”): These can be dealt conjecture about polyhedra and an informal proof of it. Counterexamples
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or refutations “stretch” our original concept: is a picture frame a genuine history. The real history will chime in in the footnotes, most of polyhedron, or a cylinder, or two polyhedra joined along a single edge? which are to be taken, therefore, as an organic part of the essay. (P&R: 5)
This device, first necessitated by the dialogue form, became a pervasive theme of Lakatos’s writings. It was to attract much criticism, most of it centred around the question whether rationally reconstructed history was real history at all. The trouble is that the rational and the real can come Attempts to rescue our conjecture from refutation yield “proof-generated apart quite radically. At one point in Proofs and Refutations a character in definitions” like that of a “normal polyhedron”. the dialogue makes a historical claim which, according to the relevant Is there any limit to this process of “concept-stretching”, or any distinction footnote, is false. Lakatos says that the statement to be drawn between interesting and frivolous concept-stretching? Can this although heuristically correct (i.e. true in a rational history of process yield, not fallible conjectures and proofs, but certainty? Lakatos’s mathematics) is historically false. This should not worry us: actual editors distinguish the certainty of proofs from the certainty of the axioms history is frequently a caricature of its rational reconstructions. from which all proofs must proceed. They claim that rigorous proof- (P&R: 21) procedures have been attained, and that “There is no serious sense in which such proofs are fallible” (P&R: 57). Quite so. But only because we On occasions, Lakatos’s sense of humour ran away with him, as when the have decided not to “stretch” the logical concepts that lie behind those text contains a made-up quotation from Galileo, and the footnote says that rigorous and formalizable proof-procedures. A rigorous proof in classical he “was unable to trace this quotation” (P&R: 62). (Though this does logic may not be valid in intuitionistic or paraconsistent logics. And the rather smack of his youthful habit of winning arguments with “bourgeois” key point is that a proof, however rigorous, only establishes that if the students by fabricating on-the-spot quotations from the authorities they axioms are true, then so is the theorem. If the axioms themselves remain respected. See Bandy 2009: 122.) Horrified critics protested that rationally fallible, then so do the theorems rigorously derived from them. Providing reconstructed history is a caricature of real history, not in fact real history foundations for mathematics requires the axioms to be made certain, by at all but rather “philosophy fabricating examples”. One critic said that deriving them from logic or set theory or something else. Lakatos claimed philosophers of science should not be allowed to write history of science. that this foundational project had collapsed (see below, §3.2). This academic trade unionism is misguided. You do not falsify history by pointing out that what ought to have happened did not, in fact, happen. To what extent is this imaginary dialogue a contribution to the history of mathematics? Lakatos explained that There is an important pedagogic point to all this, too. The dialectic of proofs and refutations can generate, in the ways explained above, quite The dialogue form should reflect the dialectic of the story: it is complicated definitions of mathematical concepts, definitions that can only meant to contain a sort of rationally reconstructed or “distilled”
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really be understood by considering the process that gave rise to them. But posits an infinity of objects in order to ensure that every natural number mathematics teaching is not historical, or even quasi-historical. (One sense has a successor; and the Axiom of Choice (which Russell refers to as the in which Lakatos’s theory is dialectical: it represents a process as rational multiplicative axiom)—were either not self-evident, not logical or both. even though the terms of the debate are not clearly defined.) But students Russell’s fall-back position was to argue that mathematics was not nowadays are presented with the latest definitions at the outset, and justified by being derivable from his axioms but that his axioms were required to learn them and apply them, without ever really understanding justified because the truths of mathematics could be derived from them them. whilst avoiding contradictions:
One question about Proofs and Refutations is whether the heuristic When pure mathematics is organized as a deductive system…it patterns depicted in it apply to the whole of mathematics. While some becomes obvious that, if we are to believe in the truth of pure aspects clearly are peculiar to the particular case-study of polyhedra, the mathematics, it cannot be solely because we believe in the truth of general patterns are not. Lakatos himself applied them in a second case- the set of premises. Some of the premises are much less obvious study, taken from the history of analysis in the nineteenth century than some of their consequences, and are believed chiefly because (“Cauchy and the Continuum”, 1978c). of their consequences. (Russell 2010 [1918]: 129)
3.2 “Regress” and “Renaissance” As Lakatos amply documents in Renaissance, a surprising number of labourers in the foundationalist vinyard—Carnap and Quine, Fraenkel and The onslaught on formalism continues in a pair of papers “Infinite Regress Gödel, Mostowski and von Neumann—were prepared to make similar and the Foundations of Mathematics” (1962) and “A Renaissance of noises. Lakatos dubs this development “empiricism” (or “quasi- Empiricism in the Recent Philosophy of Mathematics?” (1967a). Here empiricism”) and hails it on the one hand whilst condemning it on the Popper predominates and Hegel recedes. Regress is a critique of both other. logicism and formalism proper (that is, Hilbert’s programme), concentrating primarily on Russell. Russell sought to rescue mathematics Why “empiricism”? Not because it revives Mill’s idea that the truths of from doubt and uncertainty by deriving the totality of mathematics from arithmetic are empirical generalizations, but because it ascribes to self-evident logical axioms via stipulative definitions and water-tight rules mathematics the same kind of hypothetico-deductive structure that the of inference. But the discovery of Russell’s Paradox and the felt need to empirical sciences supposedly display, with axioms playing the part of deal with the Liar and related paradoxes blew this ambition sky-high. For theories and their mathematical consequences playing the part of some of the axioms that Russell was forced to posit—the Theory of Types observation-statements (or in Lakatos’s terminology, “potential falsifiers”). which Lakatos sees, in effect, as a monster-barring definition (elevated Why does Lakatos hail the “empiricism” that he also condemns? Because into an axiom) that avoids the paradoxes by excluding self-referential it means that mathematics has the same kind epistemic structure that propositions as meaningless; the Axiom of Reducibility which is needed to science has according to Popper. It’s a matter of axiomatic conjectures that relax the unduly restrictive Theory of Types; the Axiom of Infinity which
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can be mathematically refuted. (The difference between science and bunk. The prospects for an inductive logic that allows you to derive mathematics consists in the differences between the potential falsifiers.) scientific theories from sets of observation statements, thus providing them with a weak or probabilistic justification, are dim indeed. There is no Why does Lakatos condemn the “empiricism” that he also commends? inductive logic according to which real-life scientific theories can be Because Russell, like most of his supporters, succumbs to the “inductivist” inferred, “partially proved” or “confirmed (by facts) to a certain degree”’ illusion that the axioms can be confirmed by the truth of their (Changes: 133). But Lakatos sought to prove his point by analysing the consequences. In Lakatos’s opinion this is simply a mistake. Truth can Popper/Carnap debate and reversing the common verdict that Carnap had trickle down from the axioms to their consequences and falsity can flow won and that Popper had lost. And here he faced a problem. As Fox upwards from the consequences to the axioms (or at least to the axiom (1981) explains: set). But neither truth nor probability nor justified belief can flow up from the consequences to the axioms from which they follow. Here Lakatos out- The facts on which the verdict was based were that Popper’s Poppers Popper, portraying not just science but even mathematics as a claimed refutations of Carnap all failed, through either fallacy or collection of unsupported conjectures that can be refuted but not misrepresentation, and that Carnap was a careful, precise, irenic confirmed, anything else being condemned as to “inductivism”. However thinker, in the habit of stating as his conclusions exactly what his the inductivism that Lakatos scornfully rejects in Renaissance is just the premises warranted. The standards on which the verdict was based kind of inductivism that he would be recommending to Popper just a few were the respectable professional ones by which we mark third- years later. year essays. The verdict was: Carnap gets an A+, and Popper’s refusal to wither away is a moral and intellectual embarrassment. 3.3 “Changes in the Problem of Inductive Logic” (1968) (Fox 1981: 94)
In 1964 Lakatos turned from the history and philosophy of mathematics to Lakatos’s strategy was to accept the facts but reverse the value-judgment the history and philosophy of the empirical sciences. He organised a by developing the twin concepts of a degenerating research programme famous International Colloquium in the Philosophy of Science, held in and a degenerating problem-shift and applying them to Carnap’s London in 1965. Participants included Tarski, Quine, Carnap, Kuhn, and successive endeavours. But Carnap’s programme was philosophico- Popper. The Proceedings ran to four volumes (Lakatos (ed.) 1967 & 1968, mathematical rather than scientific. So what was wrong with it could not and Lakatos and Musgrave (eds.) 1968 & 1970). Lakatos himself be that it failed to predict novel facts or that its predictions were mostly contributed three major papers to these proceedings. The first of these falsified. For it was not in the business of predicting empirical (Renaissance) has been dealt with already. The second, “Changes in the observations whether novel or otherwise. (Indeed Lakatos’s concept of a Problem of Inductive Logic” (Changes), analyses the debate between degenerating philosophical programme seems to have preceded his Carnap and Popper regarding the relations between theory and evidence in concept of a degenerating scientific programme.) So what was wrong with science. It is remarkable both for its conclusions and for its methodology. Carnap’s enterprise? In an effort to solve his original problem, Carnap had The conclusion, to put it bluntly, is that a certain brand of inductivism is to solve a series of sub-problems. Some were solved, others were not,
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generating sub-sub-problems of their own. Some of these were solved, sophisticated falsificationist positions, attributing them to “Popper0, others were not, generating sub-sub-sub-problems and sub-sub-sub-sub- Popper1 and Popper2”—or as he otherwise put it, “proto-Popper, pseudo- problems etc. Since some of these sub-problems (or sub-sub-problems) Popper and proper-Popper”. (Popper did not appreciate being were solved, the programme appeared to its proponents be busy and disassembled into temporal or ideological parts and protested “I am not a progressive. But it was drifting further and further away from achieving its Trinity”.) original objectives. Lakatos’s methodology has been seen, rightly, as an attempt to reconcile Now for Lakatos, such problem-shifts are not necessarily degenerating. If Popper’s falsificationism with the views of Thomas Kuhn. Popper saw a programme ends up solving a problem that it did not set out to solve, that science as consisting of bold explanatory conjectures, and dramatic is all fine and dandy so long as the problem that it succeeds in solving is refutations that led to new conjectures. Kuhn (and Polanyi before him) more interesting and important than the problem that it did set out to solve. objected that
But one may solve problems less interesting than the original one; No process yet disclosed by the historical study of scientific indeed, in extreme cases, one may end up solving (or trying to development at all resembles the methodological stereotype of solve) no other problems but those which one has oneself created falsification by direct comparison with nature. (Kuhn 1962: 77) while trying to solve the original problem. In such cases we may talk about a degenerating problem-shift. (Changes: 128–9) Instead, science consists of long periods of “normal science”, paradigm- based research, where the task is to force nature to fit the paradigm. When Thus Carnap starts off with the exciting problem of showing how nature refuses to comply, this is not seen as a refutation, but rather as an scientific theories can be partially confirmed by empirical facts and ends anomaly. It casts doubt, not on the ruling paradigm, but on the ingenuity of up with technical papers about drawing different coloured balls out of an the scientists—“only the practitioner is blamed, not his tools”. It is only in urn. In Lakatos’s opinion this does not constitute intellectual progress. extraordinary periods of “revolutionary science” that anything like Carnap had lost the plot. Popperian refutations occur.
3.4 “Falsification and the Methodology of Scientific Research Lakatos proposed a middle-way, in which Kuhn’s socio-psychological Programmes” (1970) tools were replaced by logico-methodological ones. The basic unit of appraisal is not the isolated testable theory, but rather the “research The best-known of Lakatos’s “Conference Proceedings” is Criticism and programme” within which a series of testable theories is generated. Each the Growth of Knowledge, which became an international best-seller. It theory produced within a research programme contains the same common contains Lakatos’s important paper “Falsification and the Methodology of or “hard core” assumptions, surrounded by a “protective belt” of auxiliary Scientific Research Programmes” (FMSRP) which we have discussed hypotheses. When a particular theory is refuted, adherents of a programme already. A briefer account of this methodology had already appeared do not pin the blame on their hard-core assumptions, which they render (Lakatos 1968a), in which Lakatos distinguished dogmatic, naïve and
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“irrefutable by fiat”. Instead, criticism is directed at the hypotheses in the The use of the plural—“reconstructions”—is important. There is more “protective belt” and they are modified to deal with the problem. than one way of rationally reconstructing history, and how you do it Importantly, these modifications are not random—they are in the best depends upon what you count as rational and what not—depends, in short, cases guided by the heuristic principles implicit in the “hard core” of the in your theory of rationality. There is not one “rational history”—as Hegel programme. A programme progresses theoretically if the new theory may have thought—but several competing ones. And, in a remarkable solves the anomaly faced by the old and is independently testable, making dialectical turn, Lakatos proposed that one can evaluate competing new predictions. A programme progresses empirically if at least one of theories of rationality by asking how well they enable one to reconstruct these new predictions is confirmed. the history of science (whether it be mathematics or empirical science). The thought is that if your philosophy of science, or theory of scientific Notice that a programme can make progress, both theoretically and rationality, deems most of “great science” irrational, then something is empirically, even though every theory produced within it is refuted. A wrong with it. Contrariwise, the more of the history of “great science” programme degenerates if its successive theories are not theoretically your theory of rationality deems rational, the better that theory is. progressive (because it predicts no novel facts), or not empirically progressive (because novel predictions get refuted). Furthermore, and The obvious worry is that this meta-criterion for theories of scientific contrary to Kuhn’s idea that normally science is dominated by a single rationality threatens to deprive the philosophy of science of any critical paradigm, Lakatos claimed that the history of science typically consists of bite. Will not the best philosophy of science simply say that whatever competing research programmes. A scientific revolution occurs when a scientists do is rational, that scientific might is right, that the best degenerating programme is superseded by a progressive one. It acquires methodology is Feyerabend’s “Anything goes”? Lakatos’s Kantian hegemonic status though its rivals may persist as minority reports. epigram “Philosophy of science without history of science is empty; history of science without philosophy of science is blind” threatens to Kuhn saw all this as vindicating his own view, albeit with different eliminate the philosophy of science altogether, in favour of historical- terminology (Kuhn 1970: 256, 1977: 1). But this missed the significance sociological studies of the decisions of scientific communities. (One of us of replacing Kuhn’s socio-psychological descriptions with logico- discusses this problem, and attempts to disarm the worry, in Musgrave methodological ones. It also missed Lakatos’s claim that there are always 1983.) competing programmes or paradigms. Hegemony is seldom as total as Kuhn seems to suggest. Another worry, which is perhaps less obvious, is that Lakatos seems to be implicitly appealing to the kind of inductive principle that he scorns 3.5 “The History of Science and Its Rational Reconstructions” elsewhere. Isn’t he saying that a sequence of successes in the history of (1971) science displaying key episodes as rational tends to confirm a theory of scientific rationality? As we have seen, in Proofs and Refutations Lakatos had already joked that “actual history is frequently a caricature of its rational reconstructions”.
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Lakatos himself was a master of philosophically inspired case-studies of 3.6 “Popper on Demarcation and Induction” (1974) episodes in the history of science—Feyerabend said he had turned this into an art form. His “Hegelian” idea that the “rationally reconstructed” history “Popper on demarcation and induction” (PDI) was written in 1970 for the of thought has primacy is emphasised in two books, Larvor 1998 and Popper volume in the Library of Living Philosophers series (Schilpp (ed.) Kadvany 2001. After his death, a Colloquium was held in Nafplion, 1974). Sadly, it caused a major falling out with Popper despite the Greece, where case-studies applying Lakatos’s ideas to episodes from the generous praise in its opening sections: history of both the natural and social sciences were presented by his Popper’s ideas represent the most important development in the students and colleagues. The Proceedings of this “Nafplion Colloquium” philosophy of the twentieth century; an achievement in the were subsequently published in two volumes—Howson (ed.) 1976 and tradition—and on the level—of Hume, Kant, or Whewell. … More Latsis (ed.) 1976. Further case-studies include Zahar 1973 and Urbach than anyone else, he changed my life. I was nearly forty when I got 1974. into the magnetic field of his intellect. His philosophy helped me to However, Urbach’s paper, which was written with Lakatos’s active make a final break with the Hegelian outlook which I had held for collaboration and encouragement (F&AM: 348–34), represents something nearly twenty years. (PDI: 139) of an “own goal” for the MSRP. Urbach argued that the environmentalist Much of the paper is devoted to criticizing Popper’s demarcation criterion programme in IQ Studies, which tries to explain intergroup differences in and arguing for his own. Most of these criticisms have been canvased tested intelligence as due to environmental causes, was a degenerating already. Lakatos argues, for instance, that Popper’s falsificationism can be research programme. At least it was degenerating when compared to its falsified hereditarian rival which puts these differences down to differences in hereditary endowments. The tables were dramatically turned just thirteen by showing that the best scientific achievements were unscientific years later with the discovery of the Flynn effect (1987) which showed [by Popper’s standards] and that the best scientists, in their greatest massive differences in intergroup IQs which simply could not be moments, broke the rules of Popper’s game of science. (PDI: 146) explained by hereditary differences. (The groups in question were genetically identical, the higher scoring groups being the children or the But Lakatos also develops a criticism that has nothing much to do with the grandchildren of the lower scoring groups. See Flynn 1987 and 2009.) differences between his demarcation criterion and Popper’s, indeed a Thus the supposedly “degenerate” programme was propelled into the lead. criticism that seems equally telling against Popper’s philosophy and his Of course the MSRP allows for such dramatic reversals of fortune, but it is own. at least a bit embarrassing if a programme damned as degenerate by both Lakatos points out that when Popper first wrote his classic Logik der the Master and one of his chief disciples is spectacularly vindicated just Forschung (LSD) in the early 1930s, the correspondence theory of truth thirteen years later. was regarded with deep suspicion by the empiricist philosophers that he was trying to convince. Accordingly Popper was careful to state that
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in the logic of science here outlined it is possible to avoid using the To restore the connection between the game and its aim Lakatos makes a concepts “true” and “false” … We need not say that the theory is plea with Popper for a “whiff of “inductivism”” (PDI: 159). What is this “false” [or “falsified”], but we may say instead that it is whiff? contradicted by a certain set of accepted basic statements. Nor need we say of basic statements that they are “true” or “false”, for An inductive principle which connects realist metaphysics with we may interpret their acceptance as the result of a conventional methodological appraisals, verisimilitude with corroboration, decision, and the accepted statements as results of this decision. which reinterprets the rules of the “scientific game” as a— (LSD: 273–274) conjectural—theory about the signs of the growth of knowledge, that is, about the signs of growing verisimilitude of our scientific But shortly thereafter Popper met Tarski who convinced him that the theories. (PDI: 156) correspondence theory of truth was philosophically respectable, and this liberated him to declare that truth, or truth-likeness was the object of the In other words, it is a metaphysical principle which states that highly scientific enterprise (LSD: 273n). Lakatos apparently endorses this falsifiable but well-corroborated theories are (in some sense) more likely development. to be true (or truth-like) than their low-risk counterparts. Corroborations tend to confirm. Thus by playing the game we approximate the aim. Tarski’s rehabilitation of the correspondence theory of truth… Lakatos goes on to urge that this whiff of inductivism is not much of an stimulated Popper to complement his logic of discovery with his ask, since Popper sometimes seems to presuppose it without fully realizing own theory of verisimilitude and of approximation to the Truth, an that he is doing so. achievement marvellous both in its simplicity and in its problem- solving power. (PDI: 154) There are three points to note.
But Lakatos points out a problem. There is now a disconnect between the (1) If this criticism holds good against Popper it is equally good against game of science and the aim of science. The game of science consists in Lakatos himself. He too has a disconnect between the game of science— putting forward falsifiable, risky and problem-solving conjectures and which, when it is played well, consists in developing progressive research sticking with the unrefuted and the well-corroborated ones. But the aim of programmes—and the aim of science—which, like Popper, he takes to be science consists in developing true or truth-like theories about a largely truth (FMSRP: 58). To solve this problem, we need a metaphysical mind-independent world. And Popper has given us no reason to suppose principle which states that highly progressive research programmes are (in that by playing the game we are likely to achieve the aim. After all, a some sense) more likely to be true (or truth-like) than their degenerating theory can be falsifiable, unfalsified, problem-solving and well- rivals. Thus if Popper could do with a whiff of inductivism, the same goes corroborated without being true. for Lakatos. (2) The inductivism that Lakatos recommends to Popper looks remarkably like the inductivism that he condemned in Russell. (“I do not see any way
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out of a dogmatic assertion that we know the inductive principle, or some Lakatos then defines objectivity and rationality in terms of equivalent; the only alternative is to throw over almost everything that is progressive research programmes, and allows an incident in the regarded as knowledge by science and common sense.” Russell 1944: 683, history of science to be objective and rational if its internal history quoted disdainfully by Lakatos at Regress: 18.) But if inductivism is can be written as a sequence of progressive problem shifts. permissible (or even de rigueur) in the Philosophy of Science, perhaps it is (Hacking 1983: 126) permissible (or even de rigueur) in the Philosophy of Mathematics! In which case, the Renaissance of Empiricism in the Philosophy of Progress becomes a surrogate for truth. We don’t ask whether a theory is Mathematics may count as a genuine renaissance after all, since the logical true or not but only whether it is part of a progressive programme. To or set-theoretic axioms may (as Russell supposed) be confirmed (and paraphrase the young Karl Popper, hence rationally believed) because of their mathematical consequences. If in the logic of science [that Lakatos has] outlined it is possible to epistemic support can flow upwards from evidence to theory (where the avoid using the concepts “true” and “false” [which, in Lakatos’s evidence consists of a sequence of novel and successful predictions), opinion, is a jolly good thing!]. (LSD: 273) perhaps it can flow upwards from consequences to axioms. But if Lakatos had really been such an anti-truth-freak, he would not have (3) This episode undermines an influential “Hegelian” reading of Lakatos congratulated Popper on his Tarskian turn. Rather he would have due to Ian Hacking. According to Hacking, condemned him for taking the vacuous concept of truth to be the aim of Lakatos, educated in Hungary in an Hegelian and Marxist science. As for the disconnect between the aim of science and the game of tradition, took for granted the post-Kantian, Hegelian, demolition science, he would have recommended that Popper resolve it by dropping of correspondence theories. (Hacking 1983: 118) the aim and substituting the game (which, according to Hacking, was what Lakatos himself was trying to do). If truth were not the object of the This is an odd assertion as Lakatos explicitly endorses the correspondence exercise, there would be no need for a whiff of inductivism to connect theory on a number of occasions and even declares truth to be the aim of Popper’s method with science’s ultimate objective. But Lakatos did think science, which is why contradictions are intolerable in the long term that a whiff of inductivism was needed to connect Popper’s method with (FMSRP: 58). But in Hacking’s view, Lakatos was science’s objective. Hence Lakatos believed that truth was the object of the scientific enterprise. Whatever the remnants of Hegelianism that down on truth, not just a particular theory of truth. He [did] not Lakatos retained in later life, an aversion to truth (or to the correspondence want a replacement for the correspondence theory, but a theory of truth ) was not one of them. replacement for truth itself. (Hacking 1983: 119)
He found his replacement in the concept of progress.
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3.7 “Why Did Copernicus’s Research Programme Supersede a novel prediction with respect to a research programme if a) it is not Ptolemy’s?” (1976) predicted by any of the programme’s rivals and b) either it is not already known or if it is already known, the hard core of the programme was not Lakatos’s last publication was an historical a case-study, co-authored with devised to explain it. Elie Zahar and published after his death. It argues that the methodology of scientific research programmes can explain the Copernican Revolution as By this modified criterion the Precession of Mercury counts as a novel fact a rational process by which an earlier theory (Ptolemy’s geocentric theory with respect to Einstein’s programme. For the General Theory was of the Cosmos) was dethroned in favour another objectively better one designed to solve a different set of problems. The prediction that if the (Copernicus’s heliocentric theory). It thus demonstrates the rationality of General Theory were correct, the perihelion of Mercury would shift as it the Copernican Revolution (one of the most dramatic episodes in the does without the influence of any other heavenly body came as an history of thought) and confirms the MSRP as a theory of scientific “unexpected present from Schwarzschild” (the man who did the sums). It rationality (so long as we accept the inductive principle that the more was therefore “an unintended by-product of Einstein”s programme’ “great science” that a demarcation criterion can represent as rational, the (WDCRPSP: 185). So despite its antiquity, the Precession of Mercury more likely it is to be correct). counts as a novel fact or a novel prediction with respect to Einstein’s programme, thus making the programme a lot more progressive. Some Apart from the intrinsic interest of the subject, the paper marks a might regard Zahar’s amendment as a suspiciously ad hoc move, but ad modification of Lakatos’s conception of factual novelty and hence a hoc or not, it looks like an improvement on the original MSRP. Lakatos modification to the MSRP. For the earlier Lakatos, a fact counts as novel and Zahar go on to use this idea to explain why Copernicus’s programme with respect to a research programme if it is not predicted by any of its very properly superseded Ptolemy’s. rivals and if it is not already known. In WDCRPSP Lakatos accepts an amendment due to his co-author Elie Zahar. Zahar’s original problem was 4. Mincemeat Unmade: Lakatos versus Feyerabend our old friend the Precession of Mercury. This was explained by Einstein’s programme—specifically the General Theory of Relativity—but not by According to his friend Paul Feyerabend, Lakatos was “was a fascinating Newton’s, and this was generally thought to count in Einstein’s favour. person, an outstanding thinker and the best philosopher of science of our The difficulty is that in Lakatos’s lexicon the Precession of Mercury did strange and uncomfortable century” (Feyerabend 1975a: 1). Writing in not count as a novel fact. After all it had been known to astronomers for 1981, John Fox raised a cynical eyebrow: nearly a century. Thus, given the original version of the MSRP, the discovery that that the General Theory could explain the Precession of As when Lakatos similarly praises Popper, it is easy to suspect Mercury (whilst Newton’s theory could not) did not mean that Einstein’s indirect self-advertisement: building up one’s opponent so that the programme was any more progressive than Newton’s. (That had to be announced victory is taken as winning a world title. (Fox 1981: 92) argued on other grounds.) But this is such a counterintuitive result that it suggests a defect in the MSRP. Zahar’s modification is that a fact counts as
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With Motterlini’s publication of the Feyerabend/Lakatos correspondence Feyerabend’s epistemological anarchism is sometimes summed up by the (F&AM), Fox’s suspicions have been amply confirmed. It is quite clear slogan “Anything goes” but that is a little misleading. His point is rather that Lakatos and Feyerabend were engaged in a self-conscious campaign this: If you want a set of methodological rules distinguishing between of mutual boosterism, leading up to a planned epic encounter between a good science and bad science, the only thing that won’t exclude some of fallibilistic rationalism, as represented by Lakatos, and epistemological what you (Dear Reader) regard as the best science is the principle anarchism, as represented by Feyerabend. As Feyerabend put it “I was to “Anything goes”. Anything else would rule out what is widely regarded as attack the rationalist position, Imre was to restate and defend it, making some of the best science as unscientific. Thus a large proportion of mincemeat of me in the process” (Feyerabend 1975b: preface). This Battle Feyerabend’s Against Method is devoted to “praising” Galileo for his of the Titans was to consist of Feyerabend’s Against Method and Lakatos’s allegedly anti-Popperian practices and his dodgy (but progressive) projected reply, which is referred to, in their correspondence, by the rhetorical tricks. Everyone agrees that Galileo was a great scientist. But if mysterious acronym “MAM”. Galileo was great, then the rules that supposedly constitute great science are defective since they would exclude some of the greatest of Galileo’s Sometimes the mutual boosterism went a bit too far, causing pain and great deeds. distress to serious-minded philosophers who regarded Popperian critical rationalism as a bulwark against a resurgent Nazism: But what about Lakatos? Feyerabend poses a dilemma. Suppose we apply the Lakatos’s methodology of scientific research programmes in a Hans Albert is on the verge of suicide [writes Lakatos to conservative or rigouristic spirit. Scientists are urged to abandon Feyerabend]. Allegedly somebody told him that in Kiel you will degenerating research programmes in favour of the progressive, and grant- describe critical rationalism as a “mental disease”, and he thinks giving agencies are urged to defund them. After all, such programmes are that will be the end of Reason in Germany. I told him that though condemned by the Demarcation Criterion as bad science or even non- you are AN EXTREMELY GREAT MAN, that you will not bring science! At the very least, the adherents of degenerating research Nazism back single-handedly…. (F&AM: 291) programmes must bear the stigma of irrationality, owning up to their scientific sins. But in that case Lakatos’s MSRP would be condemning But although they had interested motives for talking each other up, it is some research programmes to death as bad science or even non-science clear that the mutual admiration between Feyerabend’s and Lakatos was that might otherwise recover their progressive (and hence their scientific) quite sincere. Each genuinely regarded the other as the man to beat. status. Thus Lakatos would be vulnerable to the same criticism that he Feyerabend’s criticism of Lakatos is summed up in his joking dedication himself applies to Popper—he would be excluding some of the best to Against Method: To IMRE LAKATOS Friend and Fellow-Anarchist. In science as unscientific (that is, research programmes that have suffered a other words Feyerabend’s charge is that for all his law-and-order degenerating phase only to stage a magnificent comeback). In response to pretensions as a defender of the rationality of science and a critic of this, Lakatos distinguished appraisal from advice, and said that the task of pseudoscience, Lakatos is really an epistemic anarchist malgré lui. the philosopher of science is to issue rules of appraisal, not to advise scientists (or grant-giving agencies) about what they ought to do. The
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Demarcation Criterion can evaluate the current state of play but it does not I am now greatly grateful for your depicting me as God and tell anyone what to do about it. (To paraphrase Marx's Thesis XI, yourself as the Devil. I also return the compliment: for me you are “Methodologists hitherto have attempted to change the world of scientific the only philosopher worth demolishing. But there is one trouble: I research in various ways; the point, however is to appraise it”.) The can take you to such little pieces that only an electromicroscope MSRP does enjoin a principle of scientific honesty, namely that the can discover you again. Will you be very hurt? (F&AM: 268–9) adherents of degenerating research programmes should own up to their methodological shortcomings, such as the lack of novel predictions or the However, aside from these threats, a developed answer to Feyerabend’s falsification of the predictions that they have made. However, so long as dilemma is conspicuous by its absence. One is reminded of King Lear: they admit to these failures they can (rationally?) persist in their I will do such things,— degenerate ways. What they are, yet I know not: but they shall be But in that case Lakatos is gored by the other horn of Feyerabend’s The terrors of the earth. dilemma. For Feyerabend argues that a Demarcation Criterion that cannot The upshot is that if there is a Lakatosian answer to Feyerabend’s tell anyone what to do or not to do is scarcely distinguishable from dilemma, it is an answer that has to be concocted on his behalf. One of us “Anything goes”. To revert to Feyerabend’s political analogy, what is the has a go in Musgrave 1976, but for the Methodology of Scientific difference between an anarchist society and a “state” where the “police” Research Programmes, it is still, very much, an open problem. can appraise people for their “criminal” or “law-abiding” behaviour but can never make an arrest or send anyone to jail? That’s a “state” which Bibliography isn’t a state and a “police force” which isn’t a police force! We have not scientific law-and-order but anarchy, accompanied by uplifting sermons Works by Lakatos and benedictions posthumously bestowed on the mighty scientific dead. 1946a: “Citoyen és Munkásosztály” (Citoyen and the working class), What was Lakatos’s response to this dilemma? It is sometimes suggested, Valóság, 1: 77–88. not least by Feyerabend himself, that Lakatos did have, or would have 1946b: “A fizikai Idealizmus Bírálata” (A critique of idealism in physics); had, an answer but that he did not live to write it up. Their correspondence a review of Susan Stebbing’s Philosophy and the Physicists, suggests otherwise. Although the locus classicus of Feyerabend’s Athenaeum, 1: 28–33. argument is chapter 16 of Against Method (1975b) he had already 1947a: “Huszadik Század”, Forum, 1: 316–20. developed his dilemma in “Consolations for the Specialist” (1968) and 1947b: “Eötvös Collégium—Györffy Kollégium”, Valóság, 2: 107–24. Lakatos had access to successive versions of the argument in the 1947c: “Jeges Károly: Megtanulom a fizikát”, Társadalmi Szemle, 1: 472. successive drafts that Feyerebend sent him in the last is six years of his 1947d: “Természettudományos világnézet és demokratikus nevelés” life. Yet there is no trace of a counterargument in Lakatos’s surviving (Scientific worldview and democratic upbringing), Embernevelés, 2: letters to Feyerabend. Instead there are a series of fearsome threats. 63-66.
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1947e: “Modern fizika—modern társadalom” (Modern physics—modern Programmes”, Proceedings of the Aristotelian Society, 69: 149–186. society), in Kemény Gábor (ed.), Továbbképzés és demokrácia. 1968b [Changes]: “Changes in the Problem of Inductive Logic”, in [There is an English translation of this essay in Kampis et al. 2002: Lakatos 1968c, 315–417 [Reprinted as chapter 8 of PP2, cited pages 356-368.] from this version.] 1947f: “‘Haladó tudós’ a demokráciában” (A “progressive scholar” in a 1968c: (ed.), The Problem of Inductive Logic, Amsterdam: North-Holland. democracy), Tovább, June 13. 1968d: (edited with A. Musgrave) Problems in the Philosophy of Science, 1956: Speech at the Pedagogy Debate of the Petőfi Circle on September Amsterdam: North-Holland. 28, 1956; transcript published in András B. Hegedűs (ed.), A Petőfi 1968e: “A Letter to the Director of the London School of Economics”, in Kör vitái (The debates of the Petőfi Circle), Vol. VI, Budapest: C.B. Cox and A.E. Dyson (eds.), Fight for Education, A Black Paper, Intézet and M&uacte;zsák Kiadó, 1992, 34–38. [English translation London: Critical Quarterly Society, 28–31. [Republished as chapter “On rearing scholars” in Motterlini 1999: 375-381.] 12 of Lakatos 1978b (PP2), referred to, in this reprint, as LTD.] 1961: “Essays in the Logic of Mathematical Discovery”. Unpublished 1970a [FMSRP]: “Falsification and the Methodology of Scientific PhD dissertation, Cambridge University. Research Programmes”, in Lakatos 1970b, 91–196 (Republished as 1962 [Regress]: “Infinite Regress and Foundations of Mathematics”, chapter 1 of Lakatos 1978a, PP1, cited pages from this version.) Aristotelian Society Supplementary Volume, 36: 155–94. 1970b, editor with A. Musgrave: Criticism and the Growth of Knowledge, [Republished as chapter 1 of Lakatos 1978b (PP2), cited pages from Cambridge: Cambridge University Press. this version.] 1970c: Discussion of “Knowledge and Physical Reality” by A. Mercier, in 1963: Discussion of “History of Science as an Academic Discipline” by A.D. Breck and W. Yourgrau (eds.), Physics, Logic and History, New A.C. Crombie and M.A. Hoskin, in A.C. Crombie (ed.), Scientific York: Plenum Press, pp. 53–4. Change, London: Heinemann, pp. 781–5. [Republished as chapter 13 1970d: Discussion of ‘Scepticism and the Study of History’ by Richard H. of Lakatos 1978b (PP2).] Popkin, in A.D. Breck and W. Yourgrau (eds.), Physics, Logic and 1963–4: “Proofs and Refutations”, in the British Journal for the History, New York: Plenum Press, pp. 220–3. Philosophy of Science, 14: 1–25, 120–139, 221–243, 296–342. 1971a [HS&IRR]: “The History of Science and its Rational [Reprinted in Lakatos 1976c (P&R). cited pages from this version.] Reconstructions”, in R.C. Buck and R.S. Cohen (eds.), PSA 1970: 1967a [Renaissance]: “A Renaissance of Empiricism in the Recent Boston Studies in the Philosophy of Science, 8, Dordrecht: Reidel, pp. Philosophy of Mathematics”, in Lakatos 1967b: 199–202. 91–135. [Republished as chapter 2 of Lakatos 1978a (PP1), cited [Republished in an expanded form as 1978b (PP2), cited pages from pages from this version] this version.] 1971b: “Replies to Critics”, in R.C. Buck and R.S. Cohen (eds.): PSA 1967b: (ed.), Problems in the Philosophy of Mathematics, Amsterdam: 1970: Boston Studies in the Philosophy of Science, 8, Dordrecht: North-Holland. Reidel, pp. 174–82. 1968a: “Criticism and the Methodology of Scientific Research 1974a: “Discussion Remarks on Papers by Ne‘eman, Yahil, Beckler,
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Sambursky, Elkana, Agassi, Mendelsohn”, in Y. Elkana (ed.), The Papers: Volume 2), J. Worrall and G. Currie (eds.), Cambridge: Interaction Between Science and Philosophy, Atlantic Highlands, Cambridge University Press. New Jersey: Humanities Press, pp. 41, 155–6, 163, 165, 167, 280–3, 1978c: “Cauchy and the Continuum:the Significance of Non-Standard 285–6, 288–9, 292, 294–6, 427–8, 430–1, 435. Analysis for the History and Philosophy of Mathematics”. [Published 1974b [PDI]: “Popper on Demarcation and Induction”, in P.A. Schilpp as chapter 5 of PP1] (ed.), The Philosophy of Karl Popper, La Salle: Open Court, 241–73. 1999a: “Lectures on Scientific Method” in Motterlini 1999: 19–109 [Republished as chapter 3 of Lakatos 1978a (PP1), cited pages from 1999b: “Lakatos-Feyerabend Correspondence” in Motterlini 1999: 119– this version.] 374. 1974c: “The Role of Crucial Experiments in Science”, Studies in the 1999c: “On Rearing Scholars” in Motterlini 1999: 375–381. History and Philosophy of Science, 4: 309–25. 1999d: “The Intellectuals’ Betrayal of Reason” in Motterlini 1999: 393– 1974d [S&P]: “Science and Pseudoscience”, in Vesey, G. (ed.), 397. Philosophy in the Open, Open University Press. [Republished as the introduction to Lakatos1978a (PP1), cited pages from this version.] Secondary Literature 1976a: [UT] “Understanding Toulmin”, Minerva, 14: 126–43. Ariew, R., 2014, “Pierre Duhem”, The Stanford Encyclopedia of [Republished as chapter 11 of Lakatos 1978b.] Philosophy (Fall 2014 edition), E.N. Zalta (ed.). URL= 1976b [Renaissance]: “A Renaissance of Empiricism in the Recent
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(trans.), New York: Knopf. Appraising Lakatos, Dordrecht: Kluwer. Feyerabend, P.K., 1968, “Consolation for the Specialist” in Lakatos (and Kuhn, T.S., 1962, The Structure of Scientific Revolutions, Chicago: Musgrave) 1970b: 197–230. University of Chicago Press. –––, 1975a, “Imre Lakatos”, British Journal for the Philosophy of Science, –––, 1970, “Reflections on My Critics”, in Lakatos (and Musgrave) 26: 1–18. 1970b: 231–278. –––, 1975b, Against Method, London: New Left Books. –––, 1977, The Essential Tension, Chicago: University of Chicago Press. Flynn, J.R., 1987, “Massive IQ Gains in Fourteen Nations: What IQ Tests Larvor, B., 1998, Lakatos: An Introduction, London: Routledge Really Measure” Psychological Bulletin, 101: 171–191. –––, 1999, “Lakatos’ Mathematical Hegelianism” The Owl of Minerva, –––, 2009, What Is Intelligence: Beyond the Flynn Effect, expanded 31(1): 23–44. [Larvor 1999 available online] edition, Cambridge: Cambridge University Press. –––, 2001, “What is Dialectical Philosophy of Mathematics?” Philosophia Fox, J., 1981, “Critical notice: Appraising Lakatos”, Australasian Journal Mathematica, 9(1): 212–229. of Philosophy, 59(1): 92–103. doi:10.1080/00048408112340071 Latsis, S.J. (ed.), 1976, Method and Appraisal in Economics, Cambridge: Gavroglu, K., Y. Goudaroulis, and P. Nicolaccopoulos (eds.), 1989, Imre Cambridge University Press. Lakatos and Theories of Scientific Change, Boston Studies in the Long, J., 1998, “Lakatos in Hungary”, Philosophy of the Social Sciences, Philosophy of Science, 111, Dordrecht/Boston/London: Kluwer 28: 244–311. Academic Publishers. –––, 2002, “The Unforgiven: Imre Lakatos’ Life in Hungary”, in G. Godfrey-Smith, P., 2003, Theory and Reality: An Introduction to the Kampis, et al. 2002: 263-302. Philosophy of Science, Chicago: University of Chicago Press. Lukács, G., 1971 [1923], History and Class Consciousness: Studies in Hacking, I., 1983, Representing and Intervening, Cambridge: Cambridge Marxist Dialectics, R. Livingstone (trans.), Cambridge University Press. Massachusetts: MIT Press. Hanson, N.R., 1963, “Commentary”, in A.C. Crombie (ed), Scientific –––, 2014 [1919], Tactics and Ethics 1919–1929, M. McColgan (trans.), Change, London: Heinemann, 458–466. London: Verso. Hegel, W.G.F., 2008 [1820/21], The Philosophy of Right, Knox and Marx, K., 1976, Capital (Volume 1), B. Fowkes (trans.), Harmondsworth, Houlgate (trans.), Oxford: Oxford University Press. Penguin. Howson, C. (ed.), 1976, Method and Appraisal in the Physical Sciences, Motterlini, M. (ed.), 1999 [F&AM], For and Against Method, including Cambridge: Cambridge University Press. Lakatos’s Lectures on Scientific Method and the Lakatos-Feyerabend Kolakowski, L., 1978, Main Currents of Marxism (Volume 2: The Golden Correspondence, Chicago: University of Chicago Press. Age), P. Falla (trans.), Oxford: Oxford University Press. Musgrave, A.E., 1976, “Method or Madness” in R.S. Cohen, et al. 1976: Kadvany, J., 2001, Imre Lakatos and the Guises of Reason, Durham and 457-491. London: Duke University Press. –––, 1983, “Facts and Values in Science Studies”, in R.W. Home (ed), Kampis, George, Ladislav Kvasz, and Michael Stöltzner (eds.), 2002, Science Under Scrutiny, Australasian Studies in History and
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Philosophy of Science, volume 3, Dordrecht: Reidel, pp. 49–79. Journal for the Philosophy of Science, 25: 99–135, 235–259. Orwell, G., 2008 [1949], Nineteen Eighty-Four, Harmondsworth: Penguin, Zahar, E., 1973, “Why did Einstein’s Programme supersede Lorentz’s?”, originally published in 1949 London: Secker & Warburg. British Journal for the Philosophy of Science, 24: 95–123, 223–262. Piketty, T., 2014, Capital in the Twenty-First Century, Cambridge, MA: Harvard University Press. Academic Tools Pólya, G., 1945, How to Solve It, Princeton: Princeton University Press. Popper, K.R., 1963 [C&R], Conjectures and Refutations: The Growth of How to cite this entry. Scientific Knowledge, London: Routledge. Preview the PDF version of this entry at the Friends of the SEP –––, 1966, The Open Society and Its Enemies, 5th edition (2 Volumes), Society. London: Routledge. First edition in 1945. Look up this entry topic at the Indiana Philosophy Ontology –––, 2002 [LSD], The Logic of Scientific Discovery, London: Routledge Project (InPhO). Classics; originally published as Logik der Forschung, Vienna: J. Enhanced bibliography for this entry at PhilPapers, with links Springer, 1935. to its database. Priest, G., 2002, Beyond the Limits of Thought, 2nd edition, Oxford: Oxford University Press. First edition in 1995. Other Internet Resources –––, 2006, In Contradiction, 2nd edition, Oxford: Oxford University Press. First edition in 1987, Leiden: Martinus Nijhoff. Lakatos, webpages on Lakatos at the London School of Economics. Radnitzky, G. and G. Andersson (eds.), 1978, Progress and Rationality in From Budapest: the story of Imre Lakatos, at the Philosopher's Zone. Science (Boston Studies in the Philosophy of Science Volume 58), Dordrecht/Boston/London: Reidel Related Entries Roplyi, L., 2002, “Lakatos and Lukács” in Kampis et al. 2002: 303–338. Russell, B., 2010 [1918], The Philosophy of Logical Atomism, London: Brouwer, Luitzen Egbertus Jan | Copernicus, Nicolaus | Descartes, René | Routledge. Descartes, René: physics | Feyerabend, Paul | Galileo Galilei | Hegel, –––, 1944, “Reply to Criticisms”, in P. Schilpp (ed.), The Philosophy of Georg Wilhelm Friedrich | Hilbert, David: program in the foundations of Bertrand Russell, Evanston and Chicago: Northwestern University mathematics | Kuhn, Thomas | laws of nature: ceteris paribus | logic: Press, pp. 679–741. paraconsistent | logicism and neologicism | Lukács, Georg [György] | Schilpp, P.A. (ed.), 1944, The Philosophy of Bertrand Russell, Evanston: Marx, Karl | mathematics, philosophy of | mathematics, philosophy of: North Western University Press. formalism | mathematics, philosophy of: intuitionism | mathematics: Schilpp, P.A. (ed.), 1974, The Philosophy of Karl Popper, La Salle: Open constructive | mathematics: explanation in | mathematics: inconsistent | Court. mathematics: non-deductive methods in | Newton, Isaac | Popper, Karl | Urbach, P., 1974, “Progress and Degeneration in the “IQ Debate””, British Principia Mathematica | probability, interpretations of | rationality:
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historicist theories of | Russell, Bertrand | science: and pseudo-science | scientific progress | scientific revolutions | set theory: Zermelo's axiomatization of | skepticism: ancient
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