1.1 A Tale of Two Lakatoses
Imre Lakatos was a warm and witty friend and a charismatic and inspiring teacher (see Feyerabend 1975a). He was also a fallibilist, and a professed foe of elitism and authoritarianism, taking a dim view of what he described as the Wittgensteinian “thought police” (owing to the Orwellian tendency on the part of some Wittgensteinians to suppress dissent by constricting the language, dismissing the stuff that they did not like as 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 prominent part in opposing the socialist student radicals at the LSE in 1968, arguing passionately against the politicization of scholarship (LTD; Congden 2002).
But in the earlier and Hungarian phase of his life, Lakatos was a Stalinist revolutionary, the leader of a communist cell who persuaded a young comrade that it was her duty to the revolution to commit suicide, since otherwise she was likely to be arrested by the Nazis and coerced into betraying the valuable young cadres who constituted the group (Bandy 2009: ch. 5; Long 1998 and 2002; Congden 1997). So far from being a fallibilist, the young Lakatos displayed a cocksure self-confidence in his grasp of the historical situation, enough to exclude any alternative solution to the admittedly appalling problems that this group of young and mostly Jewish communists were facing in Nazi-occupied Hungary. (“Is there no other way?” the young comrade asked. The answer, apparently, was “No”; Long 2002: 267.) After the Soviet victory, during the late 1940s, he was an eager co-conspirator in the creation of a Stalinist state, in which the 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 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 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 and comrades (Bandy 2009: ch. 14; Long 2002). And he took a prominent part, as a Stalinist student radical, in trying to purge the University of Debrecen of “reactionary” professors and students and in undermining the prestigious but unduly independent Eötvös College, arguing passionately against the depoliticized (but covertly bourgeois) scholarship that Eötvös allegedly stood for (Bandy 2009: chs. 4 and 9; Long 1998 and 2002).
1.2 Life and Works: The Second World and the Third
To the many that knew and loved the later Lakatos, some of these facts are 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 question as Lakatos was wont to draw a Popperian distinction between World 3—the world of theories, propositions and arguments—and World 2—the psychological world of beliefs, decisions and desires. And he was sometimes inclined to suggest that in assessing a philosopher’s work we should confine ourselves to World 3 considerations, leaving the subjectivities of World 2 to one side (see, for instance, F&AM: 140).
So does a philosopher’s life have any bearing on his works? We take our cue from the writings of Lakatos himself. Of course, there were facts about his early career that Lakatos would not have wanted to be widely known, and which he managed to keep concealed from his Western friends and colleagues during his lifetime. But what does his official philosophy have to say about the relevance of biographical data to intellectual history?
In “The History of Science and its Rational Reconstructions” (HS&IRR) Lakatos develops a theory of how to do the history of science, which, with some adjustments, can be blown up into an account of how to do intellectual history in general. For Lakatos, the default assumption in the 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 as “How to explain the apparent motions of the heavenly bodies consistently with a plausible mechanics?”). A “rational reconstruction” in the history of science, employs a theory of (scientific) rationality in conjunction with an account of the problems as they appeared to the scientists in question to display some intellectual episode as a series of rational responses to the problem-situation. On the whole, it is a plus for a theory of [scientific] rationality if it can display the history of science as a relatively rational affair and a strike against it if it cannot. Thus in Lakatos’s opinion, naïve versions of Popper’s falsificationism are in a sense falsified by the history of science, since they represent too much of it as an irrational affair with too many scientists hanging on to hypotheses that they ought to have recognized as refuted. If the rational reconstruction succeeds—that is if we can display some intellectual development as a rational response to the problem situation—then we have an “internal” history of the developments in question. If not, then the “rational reconstruction of history needs to be supplemented by an empirical (socio-psychological) ‘external history’” (HS&IRR: 102). Non-rational or “external” factors sometimes interfere with the rational development of science. “No rationality theory will ever solve problems like why Mendelian genetics disappeared in Soviet Russia in the 1950s” [the reason being that Lysenko, a Stalin favourite, acquired hegemonic status within the world of Soviet biology and persecuted the Mendelians] (HS&IRR: 114).
(Perhaps this marks an important departure from Hegel. For a true Hegelian, everything can, in the last analysis, be seen as rationally required for the self-realization of the Absolute. Hence all history is “internal” in something like Lakatos’s sense, since the “cunning of reason” ensures that apparently irrational impulses are subordinated to the ultimate goal of history.)
Is there, so to speak, an “internal” history of Lakatos’s intellectual development that can be displayed as rational? Or must it be partly explained in terms of “external” influences? The answer depends on the account of rationality that we adopt and the problem situation that we take him to have been addressing.
Whether or not a particular theoretical (or practical) choice is susceptible to an internal explanation depends, in part, on the actor’s problem. Consider, for example, Descartes’ theory of the vortices, namely that the planets are whirled round the sun by a fluid medium which itself contains little whirlpools in which the individual planets are swimming. Descartes’ theory of the vortices, is fairly rational if we take it as an attempt (in the light of what was then known) to explain the motion of the heavenly bodies in a way that is consistent with Copernican astronomy. But it is a lot more rational if we take to be an attempt to explain the motion of the heavenly bodies in a way that is consistent with Copernican astronomy without formally contradicting the Church’s teaching that the earth does not move. (The earth goes round the sun but it does not move with respect to the fluid medium that whirls it round the sun, and, for Descartes, motion is defined as motion with respect to the contiguous matter.) So do we read Descartes’ theory as a fairly rational attempt to solve one problem which is distorted by an external factor or as a very rational attempt solve a related but more complex problem? Well the answer is not clear, but if we want to understand Descartes intellectual development we need to know that it was an important constraint on his theorizing that his views should be formally consistent with the doctrines of the Church.
Similarly, it is important in understanding Lakatos’s theorizing to realize (for example) that in later life he wanted to develop a demarcation criterion between science and non-science that left Soviet Marxism (though not perhaps all forms of Marxism) on the non-scientific side of the divide. And this holds whether we regard this constraint as a non-rational external factor or as a constituent of his problem situation and hence internal to a rational reconstruction of his intellectual development. Biographical facts can be relevant to understanding a thinker’s ideas since they can help to illuminate the problem situation to which they were addressed.
Furthermore, the big issue with respect to Lakatos’s development is how much of the old Hegelian-Marxist remained in the later post-Popperian 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 earlier self—either the self that secretly persisted or the self that the later Lakatos was reacting against.
1.3 From Stalinist Revolutionary to Methodologist of Science
Imre Lakatos was born Imre Lipsitz in Debrecen, eastern Hungary, on November 9, 1922, the only child of Jewish parents, Jacob Marton Lipsitz and Margit Herczfeld. Lakatos’s parents parted when he was very young and he was largely brought up by his grandmother and his mother who worked as a beautician. The Hungary into which Lakatos was born was a kingdom without a king ruled by an admiral without a navy, the “Regent” Admiral Horthy, who had gained his naval rank in the service of the then-defunct Austro-Hungarian Empire. The regime was authoritarian, a sort of fascism-lite. After a brilliant school career, during which he won mathematics competitions and a multitude of prizes, Lakatos entered Debrecen University in 1940. Lakatos graduated in Physics, Mathematics, and Philosophy in 1944. During his time at Debrecen he became a committed communist, attending illegal underground communist meetings and, in 1943, starting his own illegal study group.
No-one who attended Imre’s groups has forgotten the intensity and brilliance of the atmosphere. “He opened the world to me!” a participant said. Even those who were later disillusioned with communism or ashamed of acts they committed remember the sense of inspiration, clear thinking and hope for a new society they felt in Imre’s secret seminars. (Long 2002: 265)
However, in Lakatos’s group the emphasis was on preparing the young cadres for the coming communist revolution, rather than engaging in public propaganda or antifascist resistance activities (Bandy 2009: ch. 3).
In March 1944 the Germans invaded Hungary to forestall its attempts to negotiate a separate peace. (The Hungarian government had allied with the 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 a mistake.) Admiral Horthy, whose anti-Semitism was a more gentlemanly affair than that of the Nazis (he was fine with systematic discrimination but apparently drew the line at mass-murder), was forced to accept a collaborationist government led by Döme Sztójay as prime minister. The new regime had none of Horthy’s humanitarian scruples and began a policy of enthusiastic and systematic cooperation with the Nazi genocide program. In May, Lakatos’s mother, grandmother and other relatives were forced into the Debrecen ghetto, thence to die in Auschwitz—the fate of about 600,000 Hungarian Jews. Lakatos’s father, a wine merchant, managed to get away and survived the war, eventually ending up in Australia. A little earlier, in March, Lakatos himself had managed to escape from Debrecen to Nagváryad (now Oradea in Romania) with false papers under the name of Molnár. Later, a Hungarian friend, Vilma Balázs, recalled that
Imre [had been] very close to his mother and they were quite poor. He often blamed himself for her death and wondered if he could have saved her. (Bandy 2009: 32)
In Nagváryad Lakatos restarted his Marxist group. The co-leader was his then-girlfriend and subsequent wife, Éva Révész. In May, the group was 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 that she would be captured and forced to betray them, hence her duty both to the group and to the cause was to commit suicide. A member of the group took her across country to Debrecen and gave her cyanide (Congden 1997, Long 2002, Bandy 2009, ch. 5). To lovers of Russian literature, the episode recalls Dostoevsky’s The Possessed/Demons (based in part on the real life Nechaev affair). In Dostoevsky’s novel the anti-Tsarist revolutionary, Pyotr Verkhovensky, posing as the representative of a large revolutionary organization, tries to solidify the provincial cell of which he is the chief by getting the rest of group to share in the murder of a 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 Sergei Nechaev.) Hence the title of Congden’s 1997 exposé “Possessed: Imre Lakatos’s Road to 1956”. But to communists or former communists of Lakatos’s generation, it recalled a different book: Chocolate, by the Bolshevik writer Aleksandr Tarasov-Rodianov. This is a stirring tale of revolutionary self-sacrifice in which the hero is the chief of the local Cheka (the forerunner of the KGB). Popular in Hungary, it encouraged a romantic cult of revolutionary ruthlessness and sacrifice in its (mostly) youthful readers. As one of Lakatos’s contemporaries, György Magosh put it,
How that book inspired us. How we longed to be professional revolutionaries who could be hanged several times a day in the interest of the working class and of the great Soviet Union. (Bandy 2009: 31)
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 for Lakatos himself, a chance remark in his most famous paper suggests something about his attitude.
One has to appreciate the dare-devil attitude of our methodological falsificationist [or perhaps as he would have said in an earlier phase of his career, the conscientious Leninist]. He feels himself to be a hero who, faced with two catastrophic alternatives, dares to reflect coolly on their relative merits and [to] choose the lesser evil. (FMSRP: 28)
If you admire the hero who has the courage to make the tough choice between two catastrophic alternatives, isn’t there a temptation to manufacture catastrophic alternatives so that you can heroically choose between them?
Late in 1944, following a Soviet victory, Lakatos returned to Debrecen, and changed his name from the Germanic Jewish Lipsitz to the Hungarian proletarian Lakatos (meaning “locksmith”). He became active in the now legal Communist Party and in two leftist youth and student organizations, the Hungarian Democratic Youth Federation (MADISZ) and the Debrecen University Circle (DEK). As one of the leaders of the DEK, Lakatos agitated for the dismissal of reactionary professors from Debrecen and the exclusion of reactionary students.
We are aware that this move on our part is incompatible with the traditional and often voiced “autonomy” of the university [Lakatos stated], but respect for autonomy, in our view, cannot mean that we have to tolerate the strengthening of fascism and reaction. (Bandy 2009: 59 and 61)
Lakatos moved to Budapest in 1946. He became a graduate student at Budapest University, but spent much of his time working towards the communist takeover of Hungary. This was a slow-motion affair, characterized by the infamous “salami tactics” of the Communist leader Mátyás Rákosi. Lakatos worked chiefly in the Ministry of Education, evaluating the credentials of university teachers and making lists of those who should be dismissed as untrustworthy once the communists took over (Bandy 2009: ch. 8). He was also a student at Eötvös College, but attacked it publicly as an elitist and bourgeois institution. The College, and others like it, was closed in 1950 after the communist takeover. In 1947 Lakatos gained his doctorate from Debrecen University for a thesis entitled “On the Sociology of Concept Formation in the Natural Sciences”. In 1948, after the communist takeover was substantially complete, he gained a scholarship to undertake further study in Moscow.
Lakatos flew to Moscow in January 1949, only to be recalled for “un-Party-like” behaviour in July. What these “un-party-like” activities were is something of a mystery but even more of a mystery is why, having returned from Moscow under a cloud, he seemed so cool, calm and collected. Lakatos’s biographers, Long and Bandy, speculate that he was being held in reserve to prepare a case against the communist education chief, József Révai, who was scheduled to appear in a new show trial. But when Rákosi decided not to prosecute Révai after all, Lakatos was thrown to the wolves (Bandy 2009: ch. 12; Long 2002). He was arrested in April 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 prison camp at Recsk.
However Lakatos was probably doomed anyway. In later life Lakatos was big admirer of Orwell’s Nineteen Eighty-Four. Perhaps he recognized himself in Orwell’s description of the Party intellectual (and expert on Newspeak) Syme:
Unquestionably Syme will be vaporized, Winston thought again. He thought it with a kind of sadness, although well knowing that Syme…was fully capable of denouncing him as a thought-criminal if he saw any reason for doing so. There was something subtly wrong with Syme. There was something that he lacked: discretion, aloofness, a sort of saving stupidity. You could not say that he was unorthodox. He believed in the principles of Ingsoc, he venerated Big Brother, he rejoiced over victories, he hated heretics…. Yet a faint air of disreputability always clung to him. He said things that would have been better unsaid, he had read too many books…. (Orwell 2008 : 58)
An instance of Lakatos’s Syme-like behaviour is his 1947 denunciation of the literary critic and philosopher György Lukács, one of the intellectual luminaries of the communist movement. Lukács represented the academically respectable face of communism, and favoured a gradual and democratic transition to the dictatorship of the proletariat. Lakatos organized an “anti-Lukács meeting…held under the aegis of the Valóság Circle” to critique Lukács’s foot-dragging and “Weimarism” (Bandy 2009: 110). Once the regime was firmly in control, Lukács was indeed censured for his undue concessions to bourgeois democracy, and he spent the early fifties under a cloud. But in 1947, Lakatos’s criticisms were deemed premature and he got into trouble because of his un-Party-like activities. (Lukács himself referred to the episode as a “cliquish kaffe klatsch”.) In Communist Hungary it was important not to be “one pamphlet behind” the Party line (Bandy 2009: 92). Lakatos was the sort of over-zealous communist who was sometimes a couple of pamphlets ahead.
After his release from Recsk in September 1953 (minus several teeth), Lakatos remained for a while, a loyal Stalinist. He eked out a living in the Mathematics Institute of the Hungarian Academy of Science, reading, researching and translating (including a translation into Hungarian of George Pólya’s How to Solve It). During this time he was informing on friends and colleagues to the ÁVH., the Hungarian secret police, though he subsequently claimed that he did not pass on anything incriminating (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 the Stalinist Marxism that had been his creed. He married (as his second wife) Éva Pap and lived at her parents’ house (his father-in-law being the distinguished agronomist, Endre Pap). In 1956 he joined the revisionist Petőfi Circle and delivered a stirring speech on “On Rearing Scholars” which at least burnt his bridges with Stalinism:
The very foundation of scholarly education is to foster in students and postgrads a respect for facts, for the necessity of thinking precisely, and to demand proof. Stalinism, however, branded this as bourgeois objectivism. Under the banner of partinost [Party-like] science and scholarship, we saw a vast experiment to create a science without facts, without proofs.
… a basic aspect of the rearing of scholars must be an endeavour to promote independent thought, individual judgment, and to develop conscience and a sense of justice. Recent years have seen an entire ideological campaign against independent thinking and against believing one’s own senses. This was the struggle against empiricism [Laughter and applause]. (Bandy 2009: 221. Bandy quotes the transcripts which seem to differ slightly from the prepared text in the Lakatos archives, reprinted in F&AM)
But Lakatos was not just explicitly repudiating Stalinism. He was also implicitly criticizing another prominent member of the Petőfi Circle who had been a big influence on his first PhD, namely György Lukács (see Ropolyi 2002 for the early influence). For Lukács’s work is pervaded by just the kind of hostility towards empiricism and disdain for facts that Lakatos is denouncing in his speech, as well as an arts-sider’s contempt for the natural sciences, all of which would have been anathema to the later Lakatos. Indeed Lukács was notorious for the view that that
even if the development of science had proved all Marx’s assertions to be false…we could accept this scientific criticism without demur and still remain Marxists—as long as we adhered to the Marxist method
the orthodox Marxist who realizes that…the time has come for the expropriation of the exploiters, will respond to the vulgar-Marxist litany of “facts” which contradict this process with the words of Fichte, one of the greatest of classical German philosophers: “So much the worse for the facts”. (Lukács 2014 : ch. 3)
Thus the Stalinist Lakatos of 1947 had explicitly denounced Lukács for not being Stalinist enough, but the revisionist Lakatos of 1956 was implicitly denouncing Lukács for being methodologically too much of a Stalinist. For the later Lakatos, what was wrong with “orthodox Marxism” was chiefly that its novel factual predictions had been systematically falsified (see §3.2 below). But that was pretty much the complaint of early revisionists such as Bernstein (see Kolakowski 1978: ch. 4, and it was against that kind of revisionism that Lukács’s Bolshevik writings were a protest (see Lukács 1971  and 2014 ). Though factual “refutations” of a research programme are not always decisive, a Lukács-like indifference to the facts is, for Lakatos, the mark of a fundamentally unscientific attitude. In our opinion, this puts paid to Ropolyi’s opinion that Lukács continued to be a major influence on the later Lakatos.
Lakatos left Hungary in November 1956 after the Soviet Union crushed the short-lived Hungarian revolution. He walked across the border into Austria with his wife and her parents. Within two months he was at King’s College Cambridge, with a Rockefeller Fellowship to write a PhD under the supervision of R.B. Braithwaite, which he completed in 1959 under the title “Essays in the Logic of Mathematical Discovery”. If we set aside his romantic adventures, the story of Lakatos’s life thereafter is largely the story of his work, though we should not forget his activities as an academic politician. Even his friendship with Feyerabend and his friendship and subsequent bust-up with Popper were very much work-related. In Britain his academic career was meteoric. In 1960 he was appointed Assistant Lecturer in Karl Popper’s department at the London School of Economics. By 1969 he was Professor of Logic, with a worldwide reputation as a philosopher of science. During the student revolts of the 1960s, which in Britain were centred on the LSE, Lakatos became an establishment figure. He wrote a “Letter to the Director of the London School of Economics” defending academic freedom and academic autonomy, which was widely circulated. It denounces the student radicals for allegedly trying to do what he himself had done at Debrecen and Eötvös (though he is careful to conceal the parallel, citing Nazi and Muscovite precedents instead) (LTD: 247).
Lakatos died suddenly in 1974 of a heart attack at the height of his powers. He was 51.
2. Lakatos’s Big Ideas
Imre Lakatos has two chief claims to fame.
2.1 Against Formalism in Mathematics
The first is his Philosophy of Mathematics, especially as set forth in “Proofs and Refutations” (1963–64) a series of four articles, based on his PhD thesis, and written in the form of a many-sided dialogue. These were subsequently combined in a posthumous book and published, with additions, in 1976. The title is an allusion to a famous paper of Popper’s, “Conjectures and Refutations” (the signature essay of his best-known collection), in which Popper outlines his philosophy of science. Lakatos’s point is that the development of mathematics is much more like the development of science as portrayed by Popper than is commonly supposed, and indeed much more like the development of science as portrayed by Popper than Popper himself supposed.
What Lakatos does not make so much of (though he does not conceal it either) is that in his view the development of mathematics is also much more like the development of thought in general as analysed by Hegel than Hegel himself supposed. There is thesis, antithesis and synthesis, “Hegelian language, which [Lakatos thinks would], generally be capable of describing the various developments in mathematics” (P&R: 146). Thus there is a certain sense in which Lakatos out-Hegels Hegel, giving a dialectical analysis of a discipline (mathematics) that Hegel himself despised as insufficiently dialectical (see Larvor 1998, 1999, 2001). Hence Feyerabend’s gibe (which Lakatos took in good part) that Lakatos was a Pop-Hegelian, the bastard child of Popperian father and a Hegelian mother (F&AM: 184–185).
Proofs and Refutations is a critique of “formalist” philosophies of mathematics (including formalism proper, logicism and intuitionism), which, in Lakatos’s view, radically misrepresent the nature of mathematics as an intellectual enterprise. For Lakatos, the development of mathematics should not be construed as series of Euclidean deductions where the contents of the relevant concepts has been carefully specified in advance so as to preclude equivocation. Rather, these water-tight deductions from well-defined premises are the (perhaps temporary) end-points of an evolutionary, and indeed a dialectical, process in which the constituent concepts are initially ill-defined, open-ended or ambiguous but become sharper and more precise in the context of a protracted debate. The proofs 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).]
Lakatos is also keen to display the development of mathematics as a rational affair even though the proofs (to begin with) are often lacking in logical rigour and the key concepts are often open-ended and unclear
The idea—expressed so clearly by Seidel [and clearly endorsed by Lakatos himself]—that a proof can be respectable without being flawless, was a revolutionary one in 1847, and, unfortunately, still sounds revolutionary today. (P&R: 139)
A corollary of this is that in mathematics many of the “proofs” are not really proofs in the full sense of the word (that is, demonstrations that proceed deductively from apodictic premises via unquestionable rules of inference to certain conclusions) and that many of the “refutations” are not really refutations either, since something rather like the “refuted” thesis often survives the refutation and arises refreshed and invigorated from the dialectical process.
This becomes apparent early on in the dialogue, when the Popperian Gamma protests at the Teacher’s insouciance with respect to refutation, a counterexample to Euler’s thesis (and therefore to Cauchy’s proof) that, for all regular polyhedra, the number of vertices, minus the number of edges, plus the number of faces equals two (\(V-E + F = 2\)). The counterexample is a solid bounded by a pair of nested cubes, one of which is inside, but does not touch the other:
For this hollow cube, \(V - E + F\) (including both the inner and the outer ones) \(= 4\). According to Gamma, this simply refutes Euler’s conjecture and disproves Cauchy’s proof:
GAMMA: Sir, your composure baffles me. A single counterexample refutes a conjecture as effectively as ten. The conjecture and its proof have completely misfired. Hands up! You have to surrender. Scrap the false conjecture, forget about it and try a radically new approach.
TEACHER: I agree with you that the conjecture has received a severe criticism by Alpha’s counterexample. But it is untrue that the proof has “completely misfired”. If, for the time being, you agree to my earlier proposal to use the word “proof” for a “thought-experiment which leads to decomposition of the original conjecture into subconjectures”, instead of using it in the sense of a “ guarantee of certain truth”, you need not draw this conclusion. My proof certainly proved Euler’s conjecture in the first sense, but not necessarily in the second. You are interested only in proofs which “prove” what they have set out to prove. I am interested in proofs even if they do not accomplish their intended task. Columbus did not reach India but he discovered something quite interesting.
Thus even in his earlier work, when he is still a professed disciple of Popper, Lakatos is already a rather dissident Popperian. Firstly, there are the hat-tips to Hegel as well as to Popper that crop up from time to time in Proofs and Refutations including the passage where he praises (and condemns) them both in the same breath. (“Hegel and Popper represent the only fallibilist traditions in modem philosophy, but even they both made the mistake of reserving a privileged infallible status for mathematics”. P&R: 139n.1.) Given that Hegel was anathema to Popper (witness his famous or notorious anti-Hegel “scherzo” in The Open Society and Its Enemies, (1945 )) this strongly suggests that Lakatos took his Popper with a large pinch of salt. Secondly, for Popper himself a proof is a proof and a refutation is supposed to kill a scientific conjecture stone-dead. Thus non-demonstrative proofs and non-refuting refutations mark a major departure from Popperian orthodoxy.
2.2 Improving on Popper in the Philosophy of Science
The dissidence continues with Lakatos’s second major contribution to philosophy, his “Methodology of Scientific Research Programmes” or MSRP (developed in detail in in his FMSRP), a radical revision of Popper’s Demarcation Criterion between science and non-science, leading to a novel theory of scientific rationality. This is arguably a lot more realistic than the Popperian theory it was designed to supplant (or, in earlier formulations, the Popperian theory that it was designed to amend). For Popper, a theory is only scientific if is empirically falsifiable, that is if it is possible to specify observation statements which would prove it wrong. A theory is good science, the sort of theory you should stick with (though not the sort of thing you should believe as Popper did not believe in belief), if it is refutable, risky, and problem-solving and has stood up to successive attempts at refutation. It must be highly falsifiable, well-tested but (thus far) unfalsified.
Lakatos objects that although there is something to be said for Popper’s criterion, it is far too restrictive, since it would rule out too much of everyday scientific practice (not to mention the value-judgments of the scientific elite) as unscientific and irrational. For scientists often persist—and, it seems, rationally persist—with theories, such as Newtonian celestial mechanics that by Popper’s standards they ought to have rejected as “refuted”, that is theories that (in conjunction with other assumptions) have led to falsified predictions. A key example for Lakatos is the “Precession of Mercury” that is, the anomalous behaviour of the perihelion of Mercury, which shifts around the Sun in a way that it ought not to do if Newton’s mechanics were correct and there were no other sizable body influencing its orbit. The problem is that there seems to be no such body. The difficulty was well known for decades but it did not cause astronomers to collectively give up on Newton until Einstein’s theory came along. Lakatos thought that the astronomers were right not to abandon Newton even though Newton eventually turned out to be wrong and Einstein turned out to be right.
Again, Copernican heliocentric astronomy was born “refuted” because of the apparent non-existence of stellar parallax. If the earth goes round the sun then the apparent position of at least some of the fixed stars (namely the closest ones) ought to vary with respect to the more distant ones as the earth is moving with respect to them. Some parts of the night sky should 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 the sun, and hence at the other end of its orbit). But for nearly three centuries after the publication of Copernicus’ De Revolutionibus 1543, no such differences were observed. In fact, there is a very slight difference in the apparent positions of the nearest stars depending on the earth’s position in its orbit, but the difference is so very slight as to be almost undetectable. Indeed it was completely undetectable until 1838 when sufficiently powerful telescopes and measuring techniques were able to detect it, by which time the heliocentric view had long been regarded as an established fact. Thus astronomers had not given up on either Copernicus or his successors despite this apparent falsification.
But if scientists often persist with “refuted” theories, either the scientists are being unscientific or Popper is wrong about what constitutes good science, and hence about what scientists ought to do. Lakatos’s idea is to construct a methodology of science, and with it a demarcation criterion, whose precepts are more in accordance with scientific practice.
How does it work? Well, falsifiability continues to play a part in Lakatos’s conception of science but its importance is somewhat diminished. Instead of an individual falsifiable theory which ought to be rejected as soon as it is refuted, we have a sequence of falsifiable theories characterized by shared a hard core of central theses that are deemed irrefutable—or, at least, refutation-resistant—by methodological fiat. This sequence of theories constitutes a research programme.
The shared hard core of this sequence of theories is often unfalsifiable in two senses of the term.
Firstly scientists working within the programme are typically (and rightly) reluctant to give up on the claims that constitute the hard core.
Secondly the hard core theses by themselves are often devoid of empirical consequences. For example, Newtonian mechanics by itself—the three laws of mechanics and the law of gravitation—won’t tell you what you will see in the night sky. To derive empirical predictions from Newtonian mechanics you need a whole host of auxiliary hypotheses about the positions, masses and relative velocities of the heavenly bodies, including the earth. (This is related to Duhem’s thesis that, generally speaking, theoretical propositions—and indeed sets of theoretical propositions—cannot be conclusively falsified by experimental observations, since they only entail observation-statements in conjunction with auxiliary hypotheses. So when something goes wrong, and the observation statements that they entail turn out to be false, we have two intellectual options: modify the theoretical propositions or modify the auxiliary hypotheses. See Ariew 2014.) For Lakatos an individual theory within a research programme typically consists of two components: the (more or less) irrefutable hard core plus a set of auxiliary hypotheses. Together with the hard core these auxiliary hypotheses entail empirical predictions, thus making the theory as a whole—hard core plus auxiliary hypotheses—a falsifiable affair.
What happens when refutation strikes, that is when the hard core in conjunction with the auxiliary hypotheses entails empirical predictions which turn out to be false? What we have essentially is a modus tollens argument in which science supplies one of the premises and nature (plus experiment and observation) supplies the other:
- If <hard core plus auxiliary hypotheses>, then O (where O represents some observation statement);
- Not-O (Nature shouts “no”: the predictions don’t pan out);
- Not <hard core plus auxiliary hypotheses>.
But logic leaves us with a choice. The conjunction of the hard core plus the auxiliary hypotheses has to go, but we can retain either the hard core or the auxiliary hypotheses. What Lakatos calls the negative heuristic of the research programme, bids us retain the hard core but modify the auxiliary hypotheses:
The negative heuristic of the programme forbids us to direct the modus tollens at this “hard core”. Instead, we must use our ingenuity to articulate or even invent “auxiliary hypotheses”, which form a protective belt around this core, and we must redirect the modus tollens to these. It is this protective belt of auxiliary hypotheses which has to bear the brunt of tests and gets adjusted and re-adjusted, or even completely replaced, to defend the thus-hardened core. (FMSRP: 48)
Thus when refutation strikes, the scientist constructs a new theory, the next in the sequence, with the same hard core but a modified set of auxiliary hypotheses. How is she supposed to do this? Well, associated with the hard core, there is what Lakatos calls the positive heuristic of the programme.
The positive heuristic consists of a partially articulated set of suggestions or hints on how to change, develop the “refutable variants” of the research programme, how to modify, sophisticate, the “refutable” protective belt. (FMSRP: 50)
For example, if a planet is not moving in quite the smooth ellipse that it ought to follow a) if Newtonian mechanics were correct and b) if there were nothing but the sun and the planet itself to worry about, then the positive heuristic of the Newtonian programme bids us look for another heavenly body whose gravitational force might be distorting the first planet’s orbit. Alternatively, if stellar parallax is not observed, we can try to refute this apparent refutation by refining our instruments and making more careful measurements and observations.
Lakatos evidently thinks that when one theory in the sequence has been refuted, scientists can legitimately persist with the hard core without being in too much of a hurry to construct the next refutable theory in the sequence. The fact that some planetary orbits are not quite what they ought to be should not lead us to abandon Newtonian celestial mechanics, even if we don’t yet have a testable theory about what exactly is distorting them. It is worth remarking too that the auxiliary hypotheses play a rather paradoxical part in Lakatos’s methodology. On the one hand, they connect the central theses of the hard core with experience, allowing to them to figure in testable, and hence, refutable theories. On the other hand, they insulate the theses of the hard core from refutation, since when the arrow of modus tollens strikes, we direct it at the auxiliary hypotheses rather than the hard core.
So far we have had an account of what scientists typically do do and what Lakatos thinks that they ought to do. But what about the Demarcation Criterion between science and non-science or between good science and bad? Even if it is sometimes rational to persist with the hard core of a theory when the hard core plus some set of auxiliary hypotheses has been refuted, there must surely be some circumstances in which is it rational to give it up! The Methodology of Scientific Research Programme has got to be something more than a defence of scientific pig-headedness! As Lakatos himself puts the point:
Now, Newton’s theory of gravitation, Einstein’s relativity theory, quantum mechanics, Marxism, Freudianism [the last two stock examples of bad science or pseudo-science for Popperians], are all research programmes, each with a characteristic hard core stubbornly defended, each with its more flexible protective belt and each with its elaborate problem-solving machinery. Each of them, at any stage of its development, has unsolved problems and undigested anomalies. All theories, in this sense, are born refuted and die refuted. But are they [all] equally good? (S&P: 4–5)
Lakatos, of course, thinks not. Some science is objectively better than other science and some science is so unscientific as to hardly qualify as science at all. So how does he distinguish between “a scientific or progressive programme” and a “pseudoscientific or degenerating one”? (S&P: 4–5)
To begin with, the unit of scientific evaluation is no longer the individual theory (as with Popper), but the sequence of theories, the research programme. We don’t ask ourselves whether this or that theory is scientific or not, or whether it constitutes good or bad science. Rather we ask ourselves whether the sequence of theories, the research programme, is scientific or non-scientific or constitutes good or bad science. Lakatos’s basic idea is that a research programme constitutes good science—the sort of science it is rational to stick with and rational to work on—if it is progressive, and bad science—the kind of science that is, at least, intellectually suspect—if it is degenerating. What is it for a research programme to be progressive? It must meet two conditions. Firstly it must be theoretically progressive. That is, each new theory in the sequence must have excess empirical content over its predecessor; it must predict novel and hitherto unexpected facts (FMSRP: 33). Secondly it must be empirically progressive. Some of that novel content has to be corroborated, that is, some of the new “facts” that the theory predicts must turn out to be true. As Lakatos himself put the point, a research programme “is progressive if it is both theoretically and empirically progressive, and degenerating if it is not” (FMSRP: 34). Thus a research programme is degenerating if the successive theories do not deliver novel predictions or if the novel predictions that they deliver turn out to be false.
Novelty is, in part, a competitive notion. The novelty of a research programme’s predictions is defined with respect to its rivals. A prediction is novel if the theory not only predicts something not predicted by the previous theories in the sequence, but if the predicted observation is predicted neither by any rival programme that might be in the offing nor by the conventional wisdom. A programme gets no brownie points by predicting what everyone knows to be the case but only by predicting observations which come as some sort of a surprise. (There is some ambiguity here and some softening later on—see below §3.6—but to begin with, at least, this was Lakatos’s dominant idea.)
One of Lakatos’s key examples is the predicted return of Halley’s comet which was derived by observing part of its trajectory and using Newtonian mechanics to calculate the elongated ellipse in which it was moving. The comet duly turned up seventy-two years later, exactly where and when Halley had predicted, a novel fact that could not have been arrived at without the aid of Newton’s theory (S&P: 5). Before Newton, astronomers might have noticed a comet arriving every seventy-two years (though they would have been hard put to it to distinguish that particular comet from any others), but they could not have been as exact about the time and place of its reappearance as Halley managed to be. Newton’s theory delivered far more precise predictions than the rival heliocentric theory developed by Descartes, let alone the earth-centered Ptolemaic cosmology that had ruled the intellectual roost for centuries. That’s the kind of spectacular corroboration that propels a research programme into the lead. And it was a similarly novel prediction, spectacularly confirmed, that dethroned Newton’s physics in favour of Einstein’s. Here’s Lakatos again:
This programme made the stunning prediction that if one measures the distance between two stars in the night and if one measures the distance between them during the day (when they are visible during an eclipse of the sun), the two measurements will be different. Nobody had thought to make such an observation before Einstein’s programme. Thus, in progressive research programme, theory leads to the discovery of hitherto unknown novel facts. (S&P: 5)
A degenerating research programme, on the other hand (unlike the theories of Newton and Einstein) either fails to predict novel facts at all, or makes novel predictions that are systematically falsified. Marxism, for example, started out as theoretically progressive but empirically degenerate (novel predictions systematically falsified) and ended up as theoretically degenerate as well (no more novel predictions but a desperate attempt to explain away unpredicted “observations” after the event).
Has…Marxism ever predicted a stunning novel fact successfully? Never! It has some famous unsuccessful predictions. It predicted the absolute impoverishment of the working class. It predicted that the first socialist revolution would take place in the industrially most developed society. It predicted that socialist societies would be free of revolutions. It predicted that there will be no conflict of interests between socialist countries. Thus the early predictions of Marxism were bold and stunning but they failed. Marxists explained all their failures: they explained the rising living standards of the working class by devising a theory of imperialism; they even explained why the first socialist revolution occurred in industrially backward Russia. They “explained” Berlin 1953, Budapest 1956, Prague 1968. They “explained” the Russian-Chinese conflict. But their auxiliary hypotheses were all cooked up after the event to protect Marxian theory from the facts. The Newtonian programme led to novel facts; the Marxian lagged behind the facts and has been running fast to catch up with them. (S&P: 4–5)
Thus good science is progressive and bad science is degenerating and a research programme may either begin or end up as such a degenerate affair that it ceases to count as science at all. If a research programme 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 transformed into a pseudoscience.
It is sometimes suggested that in Lakatos’s opinion no theory either is or ought to be abandoned, unless there is a better one in existence (Hacking 1983: 113). Does this mean that no research programme should be given up in the absence of a progressive alternative, no matter how degenerate it 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, such as the current representative of a truly degenerate programme, than to sit down in undeluded ignorance. (The ancient sceptics, by contrast thought that it is better not to believe anything at all rather than believe something that might be false.) We are not sure that this was Lakatos’ opinion, though he clearly thinks it a mistake to give up on a progressive research programme, unless there is a better one to shift to. But consider again the case of Marxism. What Lakatos seems to be suggesting in the passage quoted above, is that it is rationally permissible—perhaps even obligatory—to give up on Marxism even if it has no progressive rival, that is, if there is currently no alternative research programme with a set of hard core theses about the fundamental character of capitalism and its ultimate fate. (After all, the later Lakatos probably subscribed to the Popperian thesis that history in the large is systematically unpredictable. In which case there could not be a genuinely progressive programme which foretold the fate of capitalism. At best you could have a conditional theory, such as Piketty’s, which says that under capitalism, inequality is likely to grow—unless something unexpected happens or unless we decide to do something about it. See Piketty 2014: 35.) So although Lakatos thinks that the scientific community seldom gives up on a programme until something better comes along, it is not clear that he thinks that this is what they always ought to do.
There are numerous departures from Popperian orthodoxy in all this. To begin with, Lakatos effectively abandons falsifiability as the Demarcation Criterion between science and non-science. A research programme can be falsifiable (in some senses) but unscientific and scientific but unfalsifiable. First, the falsifiable non-science. Every successive theory in a degenerating research programme can be falsifiable but the programme as whole may not be scientific. This might happen if it only predicted familiar facts or if its novel predictions were never verified. A tired purveyor of old and boring truths and/or a persistent predictor of novel falsehoods might fail to make the scientific grade. Secondly, the non-falsifiable science. In Lakatos’s opinion, it need not be a crime to insulate the hard-core of your research programme from empirical refutation. For Popper, it is a sin against science to defend a refuted theory by “introducing ad hoc some auxiliary assumption, or by re-interpreting the theory ad hoc in such a way that it escapes refutation” (C&R, 48). Not so for Lakatos, though this is not to say that when it comes to ad hocery “anything goes”.
Thirdly, Lakatos’s Demarcation Criterion is a lot more forgiving than Popper’s. For a start, an inconsistent research programme need not be condemned to the outer darkness as hopelessly unscientific. This is not because any of its constituent theories might be true. Lakatos rejects the 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.” But though science aims at truth and therefore at consistency, this does not mean that it can’t put up with a little inconsistency along the way.
The discovery of an inconsistency—or of an anomaly—[need not] immediately stop the development of a programme: it may be rational to put the inconsistency into some temporary, ad hoc quarantine, and carry on with the positive heuristic of the programme. (FMSRP: 58)
Thus it was both rational and scientific for Bohr to persist with his research programme, even though its hard core theses on the structure of the atom were fundamentally inconsistent (FMSRP: 55–58). So although Lakatos rejects Hegel’s, claim that there are contradictions in reality (though not, perhaps in Reality), he also rejects Popper’s thesis that because contradictions imply everything, inconsistent theories exclude nothing and must therefore be rejected as unfalsifiable and unscientific. For Lakatos, Bohr’s theory of the atom is fundamentally inconsistent, but this does not mean that it implies that the moon is made of green cheese. Thus what Lakatos seems to be suggesting is here (though he is not as explicit as he might be) is that, when it comes to assessing scientific research programmes, we should sometimes employ a contradiction-tolerant logic; that is a logic that rejects the principle, explicitly endorsed by Popper, that anything whatever follows from a contradiction (FMSRP: 58 n. 2). In today’s terminology, Lakatos is a paraconsistentist (since he implicitly denies that from a contradiction anything follows) but not a dialethist (since he explicitly denies that there are true contradictions). Thus he is neither a follower of Popper with respect to theories nor a follower of Hegel with respect to reality (see Priest 2006 and 2002, especially ch. 7, and Brown and Priest 2015).
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 not scientific and that’s that. It’s an either/or affair. For Lakatos being scientific is a matter of more or less, and the more the less can vary over time. A research programme can be scientific at one stage, less scientific (or non-scientific) at another (if it ceases to generate novel predictions and cannot digest its anomalies) but can subsequently stage a comeback, recovering its scientific status. Thus the deliverances of the Criterion are matters of degree, and they are matters of degree that can vary from one time to another. We can seldom say absolutely that a research programme is not scientific. We can only say that it is not looking very scientifically healthy right now, and that the prospects for a recovery do not look good. Thus Lakatos is much more of a fallibilist than Popper. For Popper, we can tell whether a theory is scientific or not by investigating its logical implications. For Lakatos our best guesses might turn out to be mistaken, since the scientific status of a research programme is determined, in part, by its history, not just by its logical character, and history, as Popper himself proclaimed, is essentially unpredictable.
There is another divergence from Popper which helps to explain the above. Lakatos collapses two of Popper’s distinctions into one; the distinction between science and non-science and the distinction between good science and bad. As Lakatos himself put the point in his lectures at the LSE:
The demarcation problem may be formulated in the following terms: what distinguishes science from pseudoscience? This is an extreme way of putting it, since the more general problem, called the Generalized Demarcation Problem, is really the problem of the appraisal of scientific theories, and attempts to answer the 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)
Apart from the fact that, for Lakatos, a) it can be rational to persist with a “falsified” theory, and indeed with theory that is actually inconsistent—both anathema to Popper—and that b) that for Lakatos “all theories are born refuted and die refuted” (S&P: 5) so that there are no unrefuted conjectures for the virtuous scientist to stick with (thus making what Popper would regard as good science practically impossible), Lakatos’s methodology of scientific research programmes replaces two of Popper’s criteria with one. For Popper has one criterion to distinguish science from non-science (or science from pseudoscience if it is a theory with scientific pretensions) and another to distinguish good science from bad science. In Popper’s view, a theory is scientific if it is empirically falsifiable and non-scientific if it is not. Being scientific or not is an absolute affair, a matter of either/or, since a theory is scientific so long as there are some observations that would falsify it. Being good science is a matter of degree, since a theory may give more or less hostages to empirical fortune, depending on 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 science at the other end of the scale. Thus a theory—or better, a research programme—can start out as genuinely scientific, gradually becoming less so over the course of time (which was Lakatos’s view of Marxism) without altogether giving up the scientific ghost. Was the Marxism of 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 it isn’t good science since it represents a degenerating research programme. But although Lakatos evidently considered Marxism to be in bad way, he could not consign it to the dustbin of history as definitively finished, since (as he often insisted) degenerating research programmes can sometimes stage a comeback.
3.1 Proofs and Refutations (1963–4, 1976)
As we have seen, Lakatos’s first major publication in Britain was the dialogue “Proofs and Refutations” which originally appeared as a series of four journal articles. The dialogue is dedicated to George Pólya for his “revival of mathematical heuristic” and to Karl Popper for his critical philosophy.
Proofs and Refutations is a highly original production. The issues it discusses are far removed from what was then standard fare in the philosophy of mathematics, dominated by logicism, formalism and intuitionism, all attempting to find secure foundations for mathematics. Its theses are radical. And its dialogue form makes it a literary as well as a philosophical tour de force.
Its official target is “formalism” or “metamathematics”. But (as we have noted) “formalism” doesn’t just mean “formalism” proper, as this term is usually understood in the Philosophy of Mathematics. For Lakatos “formalism” includes not just Hilbert’s programme but also logicism and even intuitionism. Formalism sees mathematics as the derivation of theorems from axioms in formalised mathematical theories. The philosophical project is to show that the axioms are true and the proofs valid, so that mathematics can be seen as the accumulation of eternal truths. An additional philosophical question is what these truths are about, the question of mathematical ontology.
Lakatos, by contrast, was interested in the growth of mathematical knowledge. How were the axioms and the proofs discovered? How does mathematics grow from informal conjectures and proofs into more formal proofs from axioms? Logical empiricist (and Popperian) orthodoxy distinguished the “context of discovery” from the “context of justification”, consigned the former to the realm of empirical psychology, and thought it a matter of “unregimented insight and good fortune”, hardly a fit subject for philosophical analysis. Philosophy of mathematics consists of the logical analysis of completed theories. Formalism manifests this orthodoxy and “disconnects the history of mathematics from the philosophy of mathematics” (P&R: 1). Against the orthodoxy, Lakatos paraphrased Kant (the paraphrase has become almost as famous as the original):
the history of mathematics…has become blind, while the philosophy of mathematics… has become empty. (P&R: 2)
[Lakatos had stated this Kantian aphorism more generally at a conference in Oxford in 1961: “History of science without philosophy of science is blind. Philosophy of science without history of science is empty”. See Hanson 1963: 458.]
Suppose we agree with Lakatos that there is room for heuristics or a logic or discovery. Still, orthodoxy could insist that discovery is one thing, justification another, and that the genesis of ideas has nothing to do with their justification. Lakatos, more radically, disputed this. First, he rejected the foundationalist or justificationist project altogether: mathematics has no foundation in logic, or set theory, or anything else. Second, he insisted that the way in which a theory grows plays an essential role in its methodological appraisal. This is as much a central theme of his philosophy of empirical science as it is of his philosophy of mathematics.
As noted above, Proofs and Refutations takes the form of an imaginary dialogue between a teacher and a group of students. It reconstructs the history of attempts to prove the Descartes-Euler conjecture about polyhedra, namely, that for all polyhedra, the number of vertices minus the number of edges plus the number of faces is two (\(V - E + F = 2\)). The teacher presents an informal proof of this conjecture, due to Cauchy. This is a “thought experiment which suggest a decomposition of the original conjecture into subconjectures or lemmas” from which the original 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 counterexamples. The counterexamples are of three kinds:
(1) Counterexamples to the conclusion that are not also counterexamples to any of the premises (“global but not local counterexamples”): These establish that the conclusion does not really follow from the stated premises. They require us to improve the proof, to unearth the “hidden lemma” which the counterexample also refutes, so that it becomes a “local as well as global” counterexample—see (3), below.
(2) Counterexamples to one of the premises that are not also counterexamples to the conclusion (“local but not global counterexamples”): These require us to improve the proof by replacing the refuted premise with a new premise which is not subject to the counterexample and which (we hope) will do as much to establish the conclusion as the original refuted premise did.
(3) Counterexamples both to the conclusion and to (at least one of) the premises (“global and local counterexamples”): These can be dealt with by incorporating the refuted premise or lemma into the original conclusion, as a condition of its correctness. For example, a picture-frame is a polyhedron with a hole or tunnel in it, for which \(V - E + F = 0\)).
So if we define a polyhedron as “normal” if it has no holes or tunnels in it, we can restrict the original conjecture to “normal” polyhedra and avoid this refutation. The trouble with this method is that it reduces the content of the original conjecture, and an empty tautology threatens—“For all Eulerian polyhedra (polyhedra for which \(V - E + F = 2\)), \(V - E + F = 2\)”. More particularly, a blanket exclusion of polyhedra with holes or tunnels rules out some polyhedra for which \(V - E + F = 2\), despite the presence of a hole—a cube with a square hole drilled through it and two ring-shaped faces being an example. This suggests a deeper problem than finding the domain of validity of the original conjecture—finding a general relationship between V, E and F for all polyhedra whatsoever.
We see from this analysis what Lakatos calls the “dialectical unity of proofs and refutations”. Counterexamples help us to improve our proof by finding hidden lemmas. And proofs help us improve our conjecture by finding conditions on its validity. Either way, or both ways, mathematical knowledge grows. And as it grows, its concepts are refined. We begin with a vague, unarticulated notion of what a polyhedron is. We have a conjecture about polyhedra and an informal proof of it. Counterexamples or refutations “stretch” our original concept: is a picture frame a genuine polyhedron, or a cylinder, or two polyhedra joined along a single edge?
Attempts to rescue our conjecture from refutation yield “proof-generated definitions” like that of a “normal polyhedron”.
Is there any limit to this process of “concept-stretching”, or any distinction to be drawn between interesting and frivolous concept-stretching? Can this process yield, not fallible conjectures and proofs, but certainty? Lakatos’s editors distinguish the certainty of proofs from the certainty of the axioms from which all proofs must proceed. They claim that rigorous proof-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 have decided not to “stretch” the logical concepts that lie behind those rigorous and formalizable proof-procedures. A rigorous proof in classical logic may not be valid in intuitionistic or paraconsistent logics. And the key point is that a proof, however rigorous, only establishes that if the axioms are true, then so is the theorem. If the axioms themselves remain fallible, then so do the theorems rigorously derived from them. Providing foundations for mathematics requires the axioms to be made certain, by deriving them from logic or set theory or something else. Lakatos claimed that this foundational project had collapsed (see below, §3.2).
To what extent is this imaginary dialogue a contribution to the history of mathematics? Lakatos explained that
The dialogue form should reflect the dialectic of the story: it is meant to contain a sort of rationally reconstructed or “distilled” history. The real history will chime in in the footnotes, most of 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 apart quite radically. At one point in Proofs and Refutations a character in the dialogue makes a historical claim which, according to the relevant footnote, is false. Lakatos says that the statement
although heuristically correct (i.e. true in a rational history of mathematics) is historically false. This should not worry us: actual history is frequently a caricature of its rational reconstructions. (P&R: 21)
On occasions, Lakatos’s sense of humour ran away with him, as when the text contains a made-up quotation from Galileo, and the footnote says that he “was unable to trace this quotation” (P&R: 62). (Though this does rather smack of his youthful habit of winning arguments with “bourgeois” students by fabricating on-the-spot quotations from the authorities they respected. See Bandy 2009: 122.) Horrified critics protested that rationally reconstructed history is a caricature of real history, not in fact real history at all but rather “philosophy fabricating examples”. One critic said that philosophers of science should not be allowed to write history of science. 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.
There is an important pedagogic point to all this, too. The dialectic of proofs and refutations can generate, in the ways explained above, quite complicated definitions of mathematical concepts, definitions that can only really be understood by considering the process that gave rise to them. But mathematics teaching is not historical, or even quasi-historical. (One sense in which Lakatos’s theory is dialectical: it represents a process as rational even though the terms of the debate are not clearly defined.) But students nowadays are presented with the latest definitions at the outset, and required to learn them and apply them, without ever really understanding them.
One question about Proofs and Refutations is whether the heuristic patterns depicted in it apply to the whole of mathematics. While some aspects clearly are peculiar to the particular case-study of polyhedra, the general patterns are not. Lakatos himself applied them in a second case-study, taken from the history of analysis in the nineteenth century (“Cauchy and the Continuum”, 1978c).
3.2 “Regress” and “Renaissance”
The onslaught on formalism continues in a pair of papers “Infinite Regress and the Foundations of Mathematics” (1962) and “A Renaissance of Empiricism in the Recent Philosophy of Mathematics?” (1967a). Here Popper predominates and Hegel recedes. Regress is a critique of both logicism and formalism proper (that is, Hilbert’s programme), concentrating primarily on Russell. Russell sought to rescue mathematics from doubt and uncertainty by deriving the totality of mathematics from self-evident logical axioms via stipulative definitions and water-tight rules of inference. But the discovery of Russell’s Paradox and the felt need to deal with the Liar and related paradoxes blew this ambition sky-high. For some of the axioms that Russell was forced to posit—the Theory of Types which Lakatos sees, in effect, as a monster-barring definition (elevated into an axiom) that avoids the paradoxes by excluding self-referential propositions as meaningless; the Axiom of Reducibility which is needed to relax the unduly restrictive Theory of Types; the Axiom of Infinity which posits an infinity of objects in order to ensure that every natural number has a successor; and the Axiom of Choice (which Russell refers to as the multiplicative axiom)—were either not self-evident, not logical or both. Russell’s fall-back position was to argue that mathematics was not justified by being derivable from his axioms but that his axioms were justified because the truths of mathematics could be derived from them whilst avoiding contradictions:
When pure mathematics is organized as a deductive system…it becomes obvious that, if we are to believe in the truth of pure mathematics, it cannot be solely because we believe in the truth of the set of premises. Some of the premises are much less obvious than some of their consequences, and are believed chiefly because of their consequences. (Russell 2010 : 129)
As Lakatos amply documents in Renaissance, a surprising number of labourers in the foundationalist vinyard—Carnap and Quine, Fraenkel and Gödel, Mostowski and von Neumann—were prepared to make similar noises. Lakatos dubs this development “empiricism” (or “quasi-empiricism”) and hails it on the one hand whilst condemning it on the other.
Why “empiricism”? Not because it revives Mill’s idea that the truths of arithmetic are empirical generalizations, but because it ascribes to mathematics the same kind of hypothetico-deductive structure that the empirical sciences supposedly display, with axioms playing the part of theories and their mathematical consequences playing the part of observation-statements (or in Lakatos’s terminology, “potential falsifiers”).
Why does Lakatos hail the “empiricism” that he also condemns? Because it means that mathematics has the same kind epistemic structure that science has according to Popper. It’s a matter of axiomatic conjectures that can be mathematically refuted. (The difference between science and mathematics consists in the differences between the potential falsifiers.)
Why does Lakatos condemn the “empiricism” that he also commends? Because Russell, like most of his supporters, succumbs to the “inductivist” illusion that the axioms can be confirmed by the truth of their consequences. In Lakatos’s opinion this is simply a mistake. Truth can trickle down from the axioms to their consequences and falsity can flow upwards from the consequences to the axioms (or at least to the axiom 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-Poppers Popper, portraying not just science but even mathematics as a collection of unsupported conjectures that can be refuted but not confirmed, anything else being condemned as to “inductivism”. However the inductivism that Lakatos scornfully rejects in Renaissance is just the kind of inductivism that he would be recommending to Popper just a few years later.
3.3 “Changes in the Problem of Inductive Logic” (1968)
In 1964 Lakatos turned from the history and philosophy of mathematics to the history and philosophy of the empirical sciences. He organised a famous International Colloquium in the Philosophy of Science, held in London in 1965. Participants included Tarski, Quine, Carnap, Kuhn, and Popper. The Proceedings ran to four volumes (Lakatos (ed.) 1967 & 1968, and Lakatos and Musgrave (eds.) 1968 & 1970). Lakatos himself contributed three major papers to these proceedings. The first of these (Renaissance) has been dealt with already. The second, “Changes in the Problem of Inductive Logic” (Changes), analyses the debate between Carnap and Popper regarding the relations between theory and evidence in science. It is remarkable both for its conclusions and for its methodology. The conclusion, to put it bluntly, is that a certain brand of inductivism is bunk. The prospects for an inductive logic that allows you to derive scientific theories from sets of observation statements, thus providing them with a weak or probabilistic justification, are dim indeed. There is no inductive logic according to which real-life scientific theories can be inferred, “partially proved” or “confirmed (by facts) to a certain degree”’ (Changes: 133). But Lakatos sought to prove his point by analysing the Popper/Carnap debate and reversing the common verdict that Carnap had won and that Popper had lost. And here he faced a problem. As Fox (1981) explains:
The facts on which the verdict was based were that Popper’s claimed refutations of Carnap all failed, through either fallacy or misrepresentation, and that Carnap was a careful, precise, irenic thinker, in the habit of stating as his conclusions exactly what his premises warranted. The standards on which the verdict was based were the respectable professional ones by which we mark third-year essays. The verdict was: Carnap gets an A+, and Popper’s refusal to wither away is a moral and intellectual embarrassment. (Fox 1981: 94)
Lakatos’s strategy was to accept the facts but reverse the value-judgment by developing the twin concepts of a degenerating research programme and a degenerating problem-shift and applying them to Carnap’s successive endeavours. But Carnap’s programme was philosophico-mathematical rather than scientific. So what was wrong with it could not be that it failed to predict novel facts or that its predictions were mostly falsified. For it was not in the business of predicting empirical observations whether novel or otherwise. (Indeed Lakatos’s concept of a degenerating philosophical programme seems to have preceded his concept of a degenerating scientific programme.) So what was wrong with Carnap’s enterprise? In an effort to solve his original problem, Carnap had to solve a series of sub-problems. Some were solved, others were not, generating sub-sub-problems of their own. Some of these were solved, others were not, generating sub-sub-sub-problems and sub-sub-sub-sub-problems etc. Since some of these sub-problems (or sub-sub-problems) were solved, the programme appeared to its proponents be busy and progressive. But it was drifting further and further away from achieving its original objectives.
Now for Lakatos, such problem-shifts are not necessarily degenerating. If a programme ends up solving a problem that it did not set out to solve, that is all fine and dandy so long as the problem that it succeeds in solving is more interesting and important than the problem that it did set out to solve.
But one may solve problems less interesting than the original one; indeed, in extreme cases, one may end up solving (or trying to solve) no other problems but those which one has oneself created while trying to solve the original problem. In such cases we may talk about a degenerating problem-shift. (Changes: 128–9)
Thus Carnap starts off with the exciting problem of showing how scientific theories can be partially confirmed by empirical facts and ends up with technical papers about drawing different coloured balls out of an urn. In Lakatos’s opinion this does not constitute intellectual progress. Carnap had lost the plot.
3.4 “Falsification and the Methodology of Scientific Research Programmes” (1970)
The best-known of Lakatos’s “Conference Proceedings” is Criticism and the Growth of Knowledge, which became an international best-seller. It contains Lakatos’s important paper “Falsification and the Methodology of Scientific Research Programmes” (FMSRP) which we have discussed already. A briefer account of this methodology had already appeared (Lakatos 1968a), in which Lakatos distinguished dogmatic, naïve and sophisticated falsificationist positions, attributing them to “Popper0, Popper1 and Popper2”—or as he otherwise put it, “proto-Popper, pseudo-Popper and proper-Popper”. (Popper did not appreciate being disassembled into temporal or ideological parts and protested “I am not a Trinity”.)
Lakatos’s methodology has been seen, rightly, as an attempt to reconcile Popper’s falsificationism with the views of Thomas Kuhn. Popper saw science as consisting of bold explanatory conjectures, and dramatic refutations that led to new conjectures. Kuhn (and Polanyi before him) objected that
No process yet disclosed by the historical study of scientific development at all resembles the methodological stereotype of falsification by direct comparison with nature. (Kuhn 1962: 77)
Instead, science consists of long periods of “normal science”, paradigm-based research, where the task is to force nature to fit the paradigm. When nature refuses to comply, this is not seen as a refutation, but rather as an anomaly. It casts doubt, not on the ruling paradigm, but on the ingenuity of the scientists—“only the practitioner is blamed, not his tools”. It is only in extraordinary periods of “revolutionary science” that anything like Popperian refutations occur.
Lakatos proposed a middle-way, in which Kuhn’s socio-psychological tools were replaced by logico-methodological ones. The basic unit of appraisal is not the isolated testable theory, but rather the “research programme” within which a series of testable theories is generated. Each theory produced within a research programme contains the same common or “hard core” assumptions, surrounded by a “protective belt” of auxiliary hypotheses. When a particular theory is refuted, adherents of a programme do not pin the blame on their hard-core assumptions, which they render “irrefutable by fiat”. Instead, criticism is directed at the hypotheses in the “protective belt” and they are modified to deal with the problem. Importantly, these modifications are not random—they are in the best cases guided by the heuristic principles implicit in the “hard core” of the programme. A programme progresses theoretically if the new theory solves the anomaly faced by the old and is independently testable, making new predictions. A programme progresses empirically if at least one of these new predictions is confirmed.
Notice that a programme can make progress, both theoretically and empirically, even though every theory produced within it is refuted. A programme degenerates if its successive theories are not theoretically progressive (because it predicts no novel facts), or not empirically progressive (because novel predictions get refuted). Furthermore, and contrary to Kuhn’s idea that normally science is dominated by a single paradigm, Lakatos claimed that the history of science typically consists of competing research programmes. A scientific revolution occurs when a degenerating programme is superseded by a progressive one. It acquires hegemonic status though its rivals may persist as minority reports.
Kuhn saw all this as vindicating his own view, albeit with different terminology (Kuhn 1970: 256, 1977: 1). But this missed the significance of replacing Kuhn’s socio-psychological descriptions with logico-methodological ones. It also missed Lakatos’s claim that there are always competing programmes or paradigms. Hegemony is seldom as total as Kuhn seems to suggest.
3.5 “The History of Science and Its Rational Reconstructions” (1971)
As we have seen, in Proofs and Refutations Lakatos had already joked that “actual history is frequently a caricature of its rational reconstructions”. The use of the plural—“reconstructions”—is important. There is more than one way of rationally reconstructing history, and how you do it depends upon what you count as rational and what not—depends, in short, in your theory of rationality. There is not one “rational history”—as Hegel may have thought—but several competing ones. And, in a remarkable dialectical turn, Lakatos proposed that one can evaluate competing theories of rationality by asking how well they enable one to reconstruct the history of science (whether it be mathematics or empirical science). The thought is that if your philosophy of science, or theory of scientific rationality, deems most of “great science” irrational, then something is wrong with it. Contrariwise, the more of the history of “great science” your theory of rationality deems rational, the better that theory is.
The obvious worry is that this meta-criterion for theories of scientific rationality threatens to deprive the philosophy of science of any critical bite. Will not the best philosophy of science simply say that whatever scientists do is rational, that scientific might is right, that the best methodology is Feyerabend’s “Anything goes”? Lakatos’s Kantian epigram “Philosophy of science without history of science is empty; history of science without philosophy of science is blind” threatens to eliminate the philosophy of science altogether, in favour of historical-sociological studies of the decisions of scientific communities. (One of us discusses this problem, and attempts to disarm the worry, in Musgrave 1983.)
Another worry, which is perhaps less obvious, is that Lakatos seems to be implicitly appealing to the kind of inductive principle that he scorns elsewhere. Isn’t he saying that a sequence of successes in the history of science displaying key episodes as rational tends to confirm a theory of scientific rationality?
For other people with the same name, see Lakatos (disambiguation).
The native form of this personal name is Lakatos Imre. This article uses Western name order when mentioning individuals.
Imre Lakatos, c. 1960s
|Born||(1922-11-09)November 9, 1922|
|Died||February 2, 1974(1974-02-02) (aged 51)|
|Alma mater||University of Debrecen|
Moscow State University
University of Cambridge
|Philosophy of mathematics, philosophy of science, history of science, epistemology, politics|
|Method of proofs and refutations, methodology of scientific research programmes, methodology of historiographical research programmes, positive vs. negative heuristics, progressive vs. degenerative research programmes, rational reconstruction, quasi-empiricism in mathematics, criticism of logical positivism and formalism|
Imre Lakatos (UK:,US:; Hungarian: Lakatos Imre[ˈlɒkɒtoʃ ˈimrɛ]; November 9, 1922 – February 2, 1974) was a Hungarianphilosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes.
Lakatos was born Imre (Avrum) Lipschitz to a Jewish family in Debrecen, Hungary in 1922. He received a degree in mathematics, physics, and philosophy from the University of Debrecen in 1944. In March 1944 the Germans invaded Hungary and Lakatos along with Éva Révész, his then-girlfriend and subsequent wife, formed soon after that event a Marxist resistance group. In May of that year, the group was joined by Éva Izsák, a 19-year-old Jewish antifascist activist. Lakatos, considering that there was a risk that she would be captured and forced to betray them, decided that her duty to the group was to commit suicide. Subsequently, a member of the group took her to Debrecen and gave her cyanide.
During the occupation, Lakatos avoided Nazi persecution of Jews by changing his name to Imre Molnár. His mother and grandmother died in Auschwitz. He changed his surname once again to Lakatos (Locksmith) in honor of Géza Lakatos.
After the war, from 1947 he worked as a senior official in the Hungarian ministry of education. He also continued his education with a PhD at Debrecen University awarded in 1948, and also attended György Lukács's weekly Wednesday afternoon private seminars. He also studied at the Moscow State University under the supervision of Sofya Yanovskaya in 1949. When he returned, however, he found himself on the losing side of internal arguments within the Hungarian communist party and was imprisoned on charges of revisionism from 1950 to 1953. More of Lakatos' activities in Hungary after World War II have recently become known. In fact, Lakatos was a hardline Stalinist and, despite his young age, had an important role between 1945 and 1950 (his own arrest and jailing) in building up the Communist rule, especially in cultural life and the academia, in Hungary.
After his release, Lakatos returned to academic life, doing mathematical research and translating George Pólya's How to Solve It into Hungarian. Still nominally a communist, his political views had shifted markedly and he was involved with at least one dissident student group in the lead-up to the 1956 Hungarian Revolution.
After the Soviet Union invaded Hungary in November 1956, Lakatos fled to Vienna, and later reached England. He received a PhD in philosophy in 1961 from the University of Cambridge; his thesis advisor was R. B. Braithwaite. The book Proofs and Refutations: The Logic of Mathematical Discovery, published after his death, is based on this work.
Lakatos never obtained British citizenship. In 1960 he was appointed to a position in the London School of Economics, where he wrote on the philosophy of mathematics and the philosophy of science. The LSE philosophy of science department at that time included Karl Popper, Joseph Agassi and J. O. Wisdom. It was Agassi who first introduced Lakatos to Popper under the rubric of his applying a fallibilist methodology of conjectures and refutations to mathematics in his Cambridge PhD thesis.
With co-editor Alan Musgrave, he edited the often cited Criticism and the Growth of Knowledge, the Proceedings of the International Colloquium in the Philosophy of Science, London, 1965. Published in 1970, the 1965 Colloquium included well-known speakers delivering papers in response to Thomas Kuhn'sThe Structure of Scientific Revolutions.
Lakatos remained at the London School of Economics until his sudden death in 1974 of a heart attack at the age of just 51. The Lakatos Award was set up by the school in his memory.
In January 1971 he became editor of the British Journal for the Philosophy of Science, which J. O. Wisdom had built up before departing in 1965, and he continued as editor until his death in 1974, after which it was then edited jointly for many years by his LSE colleagues John W. N. Watkins and John Worrall, Lakatos's ex-research assistant.
His last LSE lectures in scientific method in Lent Term 1973 along with parts of his correspondence with his friend and critic Paul Feyerabend have been published in For and Against Method (ISBN 0-226-46774-0).
Lakatos and his colleague Spiro Latsis organized an international conference devoted entirely to historical case studies in Lakatos's methodology of research programmes in physical sciences and economics, to be held in Greece in 1974, and which still went ahead following Lakatos's death in February 1974. These case studies in such as Einstein's relativity programme, Fresnel's wave theory of light and neoclassical economics, were published by Cambridge University Press in two separate volumes in 1976, one devoted to physical sciences and Lakatos's general programme for rewriting the history of science, with a concluding critique by his great friend Paul Feyerabend, and the other devoted to economics.
Proofs and refutations, mathematics
Main article: Proofs and Refutations
Lakatos' philosophy of mathematics was inspired by both Hegel's and Marx's dialectic, by Karl Popper's theory of knowledge, and by the work of mathematician George Pólya.
The 1976 book Proofs and Refutations is based on the first three chapters of his four chapter 1961 doctoral thesis Essays in the logic of mathematical discovery. But its first chapter is Lakatos's own revision of its chapter 1 that was first published as Proofs and Refutations in four parts in 1963–4 in The British Journal for the Philosophy of Science. It is largely taken up by a fictional dialogue set in a mathematics class. The students are attempting to prove the formula for the Euler characteristic in algebraic topology, which is a theorem about the properties of polyhedra, namely that for all polyhedra the number of their Vertices minus the number of their Edges plus the number of their Faces is 2: (V – E + F = 2). The dialogue is meant to represent the actual series of attempted proofs which mathematicians historically offered for the conjecture, only to be repeatedly refuted by counterexamples. Often the students paraphrase famous mathematicians such as Cauchy, as noted in Lakatos's extensive footnotes.
Lakatos termed the polyhedral counter examples to Euler's formula monsters and distinguished three ways of handling these objects: Firstly, monster-barring, by which means the theorem in question could not be applied to such objects. Secondly, monster-adjustment whereby by making a re-appraisal of the monster it could be made to obey the proposed theorem. Thirdly, exception handling, a further distinct process. Interestingly, these distinct strategies have been taken up in qualitative physics, where the terminology of monsters has been applied to apparent counter-examples, and the techniques of monster-barring and monster-adjustment recognized as approaches to the refinement of the analysis of a physical issue.
What Lakatos tried to establish was that no theorem of informal mathematics is final or perfect. This means that we should not think that a theorem is ultimately true, only that no counterexample has yet been found. Once a counterexample, i.e. an entity contradicting/not explained by the theorem is found, we adjust the theorem, possibly extending the domain of its validity. This is a continuous way our knowledge accumulates, through the logic and process of proofs and refutations. (If axioms are given for a branch of mathematics, however, Lakatos claimed that proofs from those axioms were tautological, i.e. logically true.)
Lakatos proposed an account of mathematical knowledge based on the idea of heuristics. In Proofs and Refutations the concept of 'heuristic' was not well developed, although Lakatos gave several basic rules for finding proofs and counterexamples to conjectures. He thought that mathematical 'thought experiments' are a valid way to discover mathematical conjectures and proofs, and sometimes called his philosophy 'quasi-empiricism'.
However, he also conceived of the mathematical community as carrying on a kind of dialectic to decide which mathematical proofs are valid and which are not. Therefore, he fundamentally disagreed with the 'formalist' conception of proof which prevailed in Frege's and Russell's logicism, which defines proof simply in terms of formal validity.
On its first publication as a paper in The British Journal for the Philosophy of Science in 1963–4, Proofs and Refutations became highly influential on new work in the philosophy of mathematics, although few agreed with Lakatos' strong disapproval of formal proof. Before his death he had been planning to return to the philosophy of mathematics and apply his theory of research programmes to it. Lakatos, Worrall and Zahar use Poincaré (1893) to answer one of the major problems perceived by critics, namely that the pattern of mathematical research depicted in Proofs and Refutations does not faithfully represent most of the actual activity of contemporary mathematicians.
Cauchy and uniform convergence
In a 1966 text published as (Lakatos 1978), Lakatos re-examines the history of the calculus, with special regard to Augustin-Louis Cauchy and the concept of uniform convergence, in the light of non-standard analysis. Lakatos is concerned that historians of mathematics should not judge the evolution of mathematics in terms of currently fashionable theories. As an illustration, he examines Cauchy's proof that the sum of a series of continuous functions is itself continuous. Lakatos is critical of those who would see Cauchy's proof, with its failure to make explicit a suitable convergence hypothesis, merely as an inadequate approach to Weierstrassian analysis. Lakatos sees in such an approach a failure to realize that Cauchy's concept of the continuum differed from currently dominant views.
Lakatos's second major contribution to the philosophy of science was his model of the 'research programme', which he formulated in an attempt to resolve the perceived conflict between Popper'sfalsificationism and the revolutionary structure of science described by Kuhn. Popper's standard of falsificationism was widely taken to imply that a theory should be abandoned as soon as any evidence appears to challenge it, while Kuhn's descriptions of scientific activity were taken to imply that science was most constructive when it upheld a system of popular, or 'normal', theories, despite anomalies. Lakatos' model of the research programme aims to combine Popper's adherence to empirical validity with Kuhn's appreciation for conventional consistency.
A Lakatosian research programme is based on a hard core of theoretical assumptions that cannot be abandoned or altered without abandoning the programme altogether. More modest and specific theories that are formulated in order to explain evidence that threatens the 'hard core' are termed auxiliary hypotheses. Auxiliary hypotheses are considered expendable by the adherents of the research programme—they may be altered or abandoned as empirical discoveries require in order to 'protect' the 'hard core'. Whereas Popper was generally read as hostile toward such ad hoc theoretical amendments, Lakatos argued that they can be progressive, i.e. productive, when they enhance the programme's explanatory and/or predictive power, and that they are at least permissible until some better system of theories is devised and the research programme is replaced entirely. The difference between a progressive and a degenerative research programme lies, for Lakatos, in whether the recent changes to its auxiliary hypotheses have achieved this greater explanatory/predictive power or whether they have been made simply out of the necessity of offering some response in the face of new and troublesome evidence. A degenerative research programme indicates that a new and more progressive system of theories should be sought to replace the currently prevailing one, but until such a system of theories can be conceived of and agreed upon, abandonment of the current one would only further weaken our explanatory power and was therefore unacceptable for Lakatos. Lakatos's primary example of a research programme that had been successful in its time and then progressively replaced is that founded by Isaac Newton, with his three laws of motion forming the 'hard core'.
The Lakatosian research programme deliberately provides a framework within which research can be conducted on the basis of 'first principles' (the 'hard core') which are shared by those involved in the research programme and accepted for the purpose of that research without further proof or debate. In this regard, it is similar to Kuhn's notion of a paradigm. Lakatos sought to replace Kuhn's paradigm, guided by an irrational 'psychology of discovery', with a research programme no less coherent or consistent yet guided by Popper's objectively valid logic of discovery.
Lakatos was following Pierre Duhem's idea that one can always protect a cherished theory (or part of one) from hostile evidence by redirecting the criticism toward other theories or parts thereof. (See Confirmation holism and Duhem–Quine thesis). This aspect of falsification had been acknowledged by Popper.
Popper's theory, falsificationism, proposed that scientists put forward theories and that nature 'shouts NO' in the form of an inconsistent observation. According to Popper, it is irrational for scientists to maintain their theories in the face of Nature's rejection, as Kuhn had described them doing. For Lakatos, however, "It is not that we propose a theory and Nature may shout NO; rather, we propose a maze of theories, and nature may shout INCONSISTENT". The continued adherence to a programme's 'hard core', augmented with adaptable auxiliary hypotheses, reflects Lakatos's less strict standard of falsificationism.
Lakatos saw himself as merely extending Popper's ideas, which changed over time and were interpreted by many in conflicting ways. In his 1968 paper "Criticism and the Methodology of Scientific Research Programmes", Lakatos contrasted Popper0, the "naive falsificationist" who demanded unconditional rejection of any theory in the face of any anomaly (an interpretation Lakatos saw as erroneous but that he nevertheless referred to often); Popper1, the more nuanced and conservatively interpreted philosopher; and Popper2, the "sophisticated methodological falsificationist" that Lakatos claims is the logical extension of the correctly interpreted ideas of Popper1 (and who is therefore essentially Lakatos himself). It is, therefore, very difficult to determine which ideas and arguments concerning the research programme should be credited to whom.
While Lakatos dubbed his theory "sophisticated methodological falsificationism", it is not "methodological" in the strict sense of asserting universal methodological rules by which all scientific research must abide. Rather, it is methodological only in that theories are only abandoned according to a methodical progression from worse theories to better theories—a stipulation overlooked by what Lakatos terms "dogmatic falsificationism". Methodological assertions in the strict sense, pertaining to which methods are valid and which are invalid, are, themselves, contained within the research programmes that choose to adhere to them, and should be judged according to whether the research programmes that adhere to them prove progressive or degenerative. Lakatos divided these 'methodological rules' within a research programme into its 'negative heuristics', i.e., what research methods and approaches to avoid, and its 'positive heuristics', i.e., what research methods and approaches to prefer. While the 'negative heuristic' protects the hard core, the 'positive heuristic' directs the modification of the hard core and auxiliary hypotheses in a general direction.
Lakatos claimed that not all changes of the auxiliary hypotheses of a research programme (which he calls 'problem shifts') are equally productive or acceptable. He took the view that these 'problem shifts' should be evaluated not just by their ability to defend the 'hard core' by explaining apparent anomalies, but also by their ability to produce new facts, in the form of predictions or additional explanations. Adjustments that accomplish nothing more than the maintenance of the 'hard core' mark the research programme as degenerative.
Lakatos' model provides for the possibility of a research programme that is not only continued in the presence of troublesome anomalies but that remains progressive despite them. For Lakatos, it is essentially necessary to continue on with a theory that we basically know cannot be completely true, and it is even possible to make scientific progress in doing so, as long as we remain receptive to a better research programme that may eventually be conceived of. In this sense, it is, for Lakatos, an acknowledged misnomer to refer to 'falsification' or 'refutation', when it is not the truth or falsity of a theory that is solely determining whether we consider it 'falsified', but also the availability of a less false theory. A theory cannot be rightfully 'falsified', according to Lakatos, until it is superseded by a better (i.e. more progressive) research programme. This is what he says is happening in the historical periods Kuhn describes as revolutions and what makes them rational as opposed to mere leaps of faith or periods of deranged social psychology, as Kuhn argued.
According to the demarcation criterion of pseudoscience originally proposed by Lakatos, a theory is pseudoscientific if it fails to make any novel predictions of previously unknown phenomena, in contrast with scientific theories, which predict novel fact(s). Progressive scientific theories are those which have their novel facts confirmed and degenerate scientific theories are those whose predictions of novel facts are refuted. As he put it:
"A given fact is explained scientifically only if a new fact is predicted with it....The idea of growth and the concept of empirical character are soldered into one." See pages 34–5 of The Methodology of Scientific Research Programmes, 1978.
Lakatos's own key examples of pseudoscience were Ptolemaic astronomy, Immanuel Velikovsky's planetary cosmogony, Freudianpsychoanalysis, 20th century Soviet Marxism,Lysenko's biology, Niels Bohr's Quantum Mechanics post-1924, astrology, psychiatry, sociology, neoclassical economics, and Darwin's theory.
In his 1973 LSE Scientific Method Lecture 1 he also claimed that "nobody to date has yet found a demarcation criterion according to which Darwin can be described as scientific".
Almost 20 years after Lakatos's 1973 challenge to the scientificity of Darwin, in her 1991 The Ant and the Peacock, LSE lecturer and ex-colleague of Lakatos, Helena Cronin, attempted to establish that Darwinian theory was empirically scientific in respect of at least being supported by evidence of likeness in the diversity of life forms in the world, explained by descent with modification. She wrote that
our usual idea of corroboration as requiring the successful prediction of novel facts...Darwinian theory was not strong on temporally novel predictions. ... however familiar the evidence and whatever role it played in the construction of the theory, it still confirms the theory.
Rational reconstructions of the history of science
In his 1970 paper "History of Science and Its Rational Reconstructions" Lakatos proposed a dialectical historiographical meta-method for evaluating different theories of scientific method, namely by means of their comparative success in explaining the actual history of science and scientific revolutions on the one hand, whilst on the other providing a historiographical framework for rationally reconstructing the history of science as anything more than merely inconsequential rambling. The paper started with his now renowned dictum "Philosophy of science without history of science is empty; history of science without philosophy of science is blind."
However neither Lakatos himself nor his collaborators ever completed the first part of this dictum by showing that in any scientific revolution the great majority of the relevant scientific community converted just when Lakatos's criterion – one programme successfully predicting some novel facts whilst its competitor degenerated – was satisfied. Indeed, for the historical case studies in his 1968 paper "Criticism and the Methodology of Scientific Research Programmes" he had openly admitted as much, commenting 'In this paper it is not my purpose to go on seriously to the second stage of comparing rational reconstructions with actual history for any lack of historicity.'
Paul Feyerabend argued that Lakatos's methodology was not a methodology at all, but merely "words that sound like the elements of a methodology." He argued that Lakatos's methodology was no different in practice from epistemological anarchism, Feyerabend's own position. He wrote in Science in a Free Society (after Lakatos's death) that:
Lakatos realized and admitted that the existing standards of rationality, standards of logic included, were too restrictive and would have hindered science had they been applied with determination. He therefore permitted the scientist to violate them (he admits that science is not "rational" in the sense of these standards). However, he demanded that research programmes show certain features in the long run — they must be progressive.... I have argued that this demand no longer restricts scientific practice. Any development agrees with it.
Lakatos and Feyerabend planned to produce a joint work in which Lakatos would develop a rationalist description of science and Feyerabend would attack it. The correspondence between Lakatos and Feyerabend, where the two discussed the project, has since been reproduced, with commentary, by Matteo Motterlini.
- ^E. Reck (ed.), The Historical Turn in Analytic Philosophy, Springer, 2016: ch. 4.2.
- ^Kostas Gavroglu, Yorgos Goudaroulis, P. Nicolacopoulos (eds.), Imre Lakatos and Theories of Scientific Change, Springer, 2012, p. 211.
- ^András Máté (2006). "Árpád Szabó and Imre Lakatos, Or the relation between history and philosophy of mathematics". Perspectives on Science. 14 (3): 282–301. doi:10.1162/posc.2006.14.3.282.
- ^Philosophy of Science: Popper and Lakatos, lecture on the philosophy of science of Karl Popper and Imre Lakatos, delivered to master's students at the University of Sussex in November 2014.
- ^ abcImre Lakatos (Stanford Encyclopedia of Philosophy)
- ^Bandy 2010.[page needed]
- ^Scheffler, Israel (2007), Gallery of Scholars: A Philosopher's Recollections, Philosophy and education, 13, Springer, p. 42, ISBN 9781402027109 .
- ^Donald A. Gillies. "Review. Matteo Motterlini (ed). Imre Lakatos. Paul K Feyerabend. Sull'orlo della scienza: pro e contro il metodo. (On the threshold of Science: for and against method)." The British Journal of the Philosophy of Science. Vol. 47, No. 3, Sep., 1996. https://www.jstor.org/stable/687992
- ^See Lakatos's 5 Jan 1971 letter to Paul Feyerabend pp 233–4 in Motterlini's 1999 For and Against Method
- ^These were respectively Method and Appraisal in the Physical Sciences: The Critical Background to Modern Science 1800–1905 by Colin Howson (ed.) and Method and Appraisal in Economics by Spiro J. Latsis (ed.)
- ^"Lakatosian Monsters". Retrieved 18 January 2015.
- ^See, for instance, Lakatos' A renaissance of empiricism in the recent philosophy of mathematics, section 2, in which he defines a Euclidean system to be one consisting of all logical deductions from an initial set of axioms and writes that "a Euclidean system may be claimed to be true".
- ^Poincaré, H. (1893). "Sur la Généralisation d'un Théorème d'Euler relatif aux Polyèdres", Comptes Redus des Séances de l'Académie des Sciences, 117 p. 144, as cited in Lakatos, Worrall and Zahar, p. 162
- ^Lakatos, Worrall and Zahar (1976), Proofs and RefutationsISBN 0-521-21078-X, pp. 106–126, note that Poincaré's formal proof (1899) "Complèment à l'Analysis Situs", Rediconti del Circolo Matematico di Palermo, 13, pp. 285–343, rewrites Euler's conjecture into a tautology of vector algebra.
- ^Lakatos, Imre. (1970). "Falsification and the methodology of scientific research programmes." In: Lakatos, Musgrave eds. (1970), pp. 91–195.
- ^Bruce J. Caldwell (1991) "The Methodology of Scientific Research Programmes: Criticisms and Conjectures" in G. K. Shaw ed. (1991) Economics, Culture, and Education: Essays in Honor of Mark Blaug Aldershot: Elgar, 1991 pp. 95–107
- ^Lakatos, Musgrave eds. (1970), p. 130
- ^ abLakatos, Imre. (1968). "Criticism and the Methodology of Scientific Research Programmes." Proceedings of the Aristotelian Society69(1):149–186 (1968).
- ^Great readings in clinical science : essential selections for mental health professionals. Lilienfeld, Scott O., 1960-, O'Donohue, William T. Boston: Pearson. 2012. ISBN 9780205698035. OCLC 720560483.
- ^Theoretical progressiveness is if the new 'theory has more empirical content than the old. Empirical progressiveness is if some of this content is corroborated. (Lakatos ed., 1970, p. 118)
- ^See/hear Lakatos's 1973 Open University BBC Radio talk Science and Pseudoscience at his LSE website at www.lse.ac.uk/lakatos
- ^Lakatos notably only condemned specifically Soviet Marxism as pseudoscientific, as opposed to Marxism in general. In fact at the very end of his very last LSE lectures on Scientific Method in 1973, he finished by posing the question of whether Trotsky's theoretical development of Marxism was scientific, and commented that "Nobody has ever undertaken a critical history of Marxism with the aid of better methodological and historiographical instruments. Nobody has ever tried to find an answer to questions like: were Trotsky's unorthodox predictions simply patching up a badly degenerating programme, or did they represent a creative development of Marx's programme? To answer similar questions, we would really need a detailed analysis which takes years of work. So I simply do not know the answer, even if I am very interested in it."[p109 Motterlini 1999] However, in his 1976 On the Critique of Scientific Reason Feyerabend claimed Vladimir Lenin's development of Marxism in his auxiliary theory of colonial exploitation had been 'Lakatos scientific' because it was "accompanied by a wealth of novel predictions (the arrival and structure of monopolies being one of them)." And he continued by claiming both Rosa Luxemburg's and Trotsky's developments of Marxism were close to what Lakatos regarded as scientific: "And whoever has read Rosa Luxemburg's reply to Bernstein's criticism of Marx or Trotsky's account of why the Russian Revolution took place in a backward country (cf. also Lenin , vol. 19, pp. 99ff.) will see that Marxists are pretty close to what Lakatos would like any upstanding rationalist to do..." [See footnote 9 of p. 315 of Howson (ed.) 1976.]
- ^Published in For and Against Method: Imre Lakatos and Paul Feyerabend by Matteo Motterlini (ed.), University of Chicago Press 1999
- ^Cronin, H., The Ant and the Peacock: Altruism and Sexual Selection from Darwin to Today, Cambridge University Press, 1993. pp. 31–32. 
- ^Lakatos, Imre. (1970). "History of Science and Its Rational Reconstructions." PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association. (JSTOR link).
- ^See How to Defend Society Against Science
- ^Paul Feyerabend (1978). Science in a Free Society. London: NLB. ISBN 0-86091-008-3
- ^Motterlini, M. (1999). For and Against Method. Chicago: UCP. ISBN 9780226467757
- Oxford Dictionary of National Biography
- Cronin, Helena (1991) The Ant and the Peacock Cambridge University Press
- Howson, Colin, Ed. Method and Appraisal in the Physical Sciences: The Critical Background to Modern Science 1800–1905 Cambridge University Press 1976 ISBN 0-521-21110-7
- Kampis, Kvaz & Stoltzner (eds.) Appraising Lakatos: Mathematics, Methodology and the Man, Vienna Circle Institute Library, Kluwer 2002 ISBN 1-4020-0226-2
- Lakatos, Musgrave ed. (1970). Criticism and the Growth of Knowledge. Cambridge: Cambridge University Press. ISBN 0-521-07826-1
- Lakatos (1976). Proofs and Refutations. Cambridge: Cambridge University Press. ISBN 0-521-29038-4
- Lakatos (1978). The Methodology of Scientific Research Programmes: Philosophical Papers Volume 1. Cambridge: Cambridge University Press
- Lakatos (1978). Mathematics, Science and Epistemology: Philosophical Papers Volume 2. Cambridge: Cambridge University Press. ISBN 0521217695
- Lakatos, I.: Cauchy and the continuum: the significance of nonstandard analysis for the history and philosophy of mathematics. Math. Intelligencer 1 (1978), no. 3, 151–161 (paper originally presented in 1966).
- Lakatos, I., and Feyerabend P., For and against Method: including Lakatos's Lectures on Scientific Method and the Lakatos-Feyerabend Correspondence, ed. by Matteo Motterlini, Chicago University Press, (451 pp), 1999, ISBN 0-226-46774-0
- Latsis, Spiro J. Ed. Method and Appraisal in Economics Cambridge University Press 1976 ISBN 0-521-21076-3
- Popper, K R, (1972), Objective knowledge: an evolutionary approach, Oxford (Clarendon Press) 1972 (bibliographic summary, no text).
- Maxwell, Nicholas (2017) Karl Popper, Science and Enlightenment, UCL Press, London. Free online.
- Zahar, Elie (1973) Why Einstein's programme superseded Lorentz'sBritish Journal for the Philosophy of Science
- Zahar, Elie (1988) Einstein's Revolution: A study in heuristic Open Court 1988
- Alex Bandy (2010). Chocolate and Chess. Unlocking Lakatos. Budapest: Akadémiai Kiadó. ISBN 978-963-05-8819-5
- Brendan Larvor (1998). Lakatos: An Introduction. London: Routledge. ISBN 0-415-14276-8
- Jancis Long (1998). "Lakatos in Hungary", Philosophy of the Social Sciences28, pp. 244–311.
- John Kadvany (2001). Imre Lakatos and the Guises of Reason. Durham and London: Duke University Press. ISBN 0-8223-2659-0; author's web site: http://www.johnkadvany.com.
- Teun Koetsier (1991). Lakatos' Philosophy of Mathematics: A Historical Approach. Amsterdam etc.: North Holland. ISBN 0-444-88944-2
- Szabó, Árpád The Beginnings of Greek Mathematics (Tr Ungar) Reidel & Akadémiai Kiadó, Budapest 1978 ISBN 963-05-1416-8
- Musgrave, Alan; Pigden, Charles. "Imre Lakatos". In Zalta, Edward N.Stanford Encyclopedia of Philosophy.
- Science and Pseudoscience (including an MP3 audio file) – Lakatos' 1973 Open UniversityBBC Radio talk on the subject
- O'Connor, John J.; Robertson, Edmund F., "Imre Lakatos", MacTutor History of Mathematics archive, University of St Andrews .
- Lakatos's Hungarian intellectual background The Autumn 2006 MIT Press journal Perspectives on Science devoted to articles on this topic, with article abstracts.
- Official Russian page