## 1. Life


### 1.1 A Tale of Two Lakatoses



Imre Lakatos was a warm and witty friend and a charismatic and
inspiring teacher ( 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 AVH, 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 Eotvos
College, arguing passionately against the depoliticized (but covertly
bourgeois) scholarship that Eotvos 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 (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, naive 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 may not be 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 Dome Sztojay 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
Nagyvarad (now Oradea in Romania) with false papers under the
name of Molnar. Later, a Hungarian friend, Vilma Balazs,
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 Nagyvarad Lakatos restarted his Marxist group. The co-leader
was his then-girlfriend and subsequent wife, Eva
Revesz. In May, the group was joined by Eva
Izsak, 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
expose: "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, Gyorgy 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, Eva
Izsak, 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 Matyas Rakosi. 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 had taken over (Bandy 2009: ch.
8). He was also a student at Eotvos 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,
Jozsef Revai, who was scheduled to appear in a new show
trial. But when Rakosi decided not to prosecute Revai
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
> [1949]: 58)
> 
> 
> 



An instance of Lakatos's Syme-like behaviour is his 1947
denunciation of the literary critic and philosopher Gyorgy
Lukacs, one of the intellectual luminaries of the communist
movement. Lukacs represented the academically respectable face
of communism, and favoured a gradual and democratic transition to the
dictatorship of the proletariat. Lakatos organized an
"anti-Lukacs meeting...held under the aegis of the
Valosag Circle" to critique Lukacs's
foot-dragging and "Weimarism" (Bandy 2009: 110). Once the
regime was firmly in control, Lukacs 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. (Lukacs 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 Polya's *How to Solve It*).
During this time he was informing on friends and colleagues to the
AVH, 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) Eva Pap and lived at her parents' house
(his father-in-law being the distinguished agronomist, Endre Pap). In
1956 he joined the revisionist Petofi 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 Petofi
Circle who had been a big influence on his first PhD, namely
Gyorgy Lukacs. (See Ropolyi 2002 for the early influence.)
For Lukacs'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 Lukacs 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
> 
> 
> 



and that



> 
> 
> 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".
> (Lukacs 2014 [1919]: ch. 3.)
> 
> 
> 



Thus the Stalinist Lakatos of 1947 had explicitly denounced
Lukacs for not being Stalinist enough, but the revisionist
Lakatos of 1956 was implicitly denouncing Lukacs 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
 SS3.2
 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 Lukacs's Bolshevik writings were
a protest. (See Lukacs 1971 [1923] and 2014 [1919].) Though
factual "refutations" of a research programme are not
always decisive, a Lukacs-like indifference to the facts is,
for Lakatos, the mark of a fundamentally unscientific attitude. In our
opinion, this puts paid to Ropolyi's claim that Lukacs
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 Eotvos (though he was 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. The counterexample is a solid bounded by a pair of nested cubes,
one of which is inside, but does not touch the other:



![A line drawing of two cubes one centered inside the other.](fig1.png)


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 [1966])) 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 since 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:



1. If <hard core plus auxiliary hypotheses>, then *O*
(where *O* represents some observation statement);
2. Not-*O* (Nature shouts "no": the predictions
don't pan out);



Therefore


3. 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 comparative 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
 SS3.6--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
dialetheist (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. Works


### 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 Polya 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:



![a line drawing of what appears to be two cubes one centered inside the other but with the inner cube being a tunnel through the outer cube (the border going from the vertices of the outer cube to the vertices of the inner cube on two opposing faces.](fig2.png)


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), *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 - the formula *V* - *E + F =* 2 holds
good. 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?



![A line drawing of two tetrahedra sharing a single edge.](fig3.png)


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,
 SS3.2).



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 [1918]: 129)
> 
> 
> 



As Lakatos amply documents in **Renaissance**, a
surprising number of labourers in the foundationalist
vinyard--Carnap and Quine, Fraenkel and Godel, 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, naive 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--"reconstruction*s*"--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--it 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?



Lakatos himself was a master of philosophically inspired case-studies
of episodes in the history of science--Feyerabend said he had
turned this into an art form. His "Hegelian" idea that the
"rationally reconstructed" history of thought has primacy
is emphasised in two books, Larvor 1998 and Kadvany 2001. After his
death, a Colloquium was held in Nafplion, Greece, where case-studies
applying Lakatos's ideas to episodes from the history of both
the natural and social sciences were presented by his students and
colleagues. The Proceedings of this "Nafplion Colloquium"
were subsequently published in two volumes--Howson (ed.) 1976 and
Latsis (ed.) 1976. Further case-studies include Zahar 1973 and Urbach
1974.



However, Urbach's paper, which was written with Lakatos's
active collaboration and encouragement (F&AM: 348-34),
represents something of an "own goal" for the MSRP. Urbach
argued that the environmentalist programme in IQ Studies, which tries
to explain intergroup differences in tested intelligence as due to
environmental causes, was a degenerating research programme. At least
it was degenerating when compared to its hereditarian rival which puts
these differences down to differences in hereditary endowments. The
tables were dramatically turned just thirteen years later with the
discovery of the Flynn effect (1987) which showed massive differences
in intergroup IQs which simply *could not* be explained by
hereditary differences. (The groups in question were genetically
identical, the higher scoring groups being the children or the
grandchildren of the lower scoring groups. See Flynn 1987 and 2009.)
Thus the supposedly "degenerate" programme was propelled
into the lead. Of course the MSRP *allows* for such dramatic
reversals of fortune, but it is at least a bit embarrassing if a
programme damned as degenerate by both the Master and one of his chief
disciples is spectacularly vindicated just thirteen years later.


### 3.6 "Popper on Demarcation and Induction" (1974)



"Popper on demarcation and induction" (PDI) was written in
1970 for the Popper volume in the *Library of Living
Philosophers* series (Schilpp (ed.) 1974). Sadly, it caused a
major falling out with Popper despite the generous praise in its
opening sections:



> 
> 
> Popper's ideas represent the most important development in the
> philosophy of the twentieth century; an achievement in the
> tradition--and on the level--of Hume, Kant, or Whewell.
> ... More than anyone else, he changed my life. I was nearly forty
> when I got into the magnetic field of his intellect. His philosophy
> helped me to make a final break with the Hegelian outlook which I had
> held for nearly twenty years. (PDI: 139.)
> 
> 
> 



Much of the paper is devoted to criticizing Popper's demarcation
criterion and arguing for his own. Most of these criticisms have been
canvased already. Lakatos argues, for instance, that Popper's
falsificationism can be falsified



> 
> 
> by showing that the best scientific achievements were unscientific [by
> Popper's standards] and that the best scientists, in their
> greatest moments, broke the rules of Popper's game of science
> (PDI: 146).
> 
> 
> 



But Lakatos also develops a criticism that has nothing much to do with
the differences between his demarcation criterion and Popper's,
indeed a criticism that seems equally telling against Popper's
philosophy and his own.



Lakatos points out that when Popper first wrote his classic *Logik
der Forschung* (LSD) in the early 1930s, the correspondence theory
of truth was regarded with deep suspicion by the empiricist
philosophers that he was trying to convince. Accordingly Popper was
careful to state that



> 
> 
> in the logic of science here outlined it is possible to avoid using
> the concepts "true" and "false" ... We
> need not say that the theory is "false" [or
> "falsified"], but we may say instead that it is
> contradicted by a certain set of accepted basic statements. Nor need
> we say of basic statements that they are "true" or
> "false", for we may interpret their acceptance as the
> result of a conventional decision, and the accepted statements as
> results of this decision. (LSD: 273-274.)
> 
> 
> 



But shortly thereafter Popper met Tarski who convinced him that the
correspondence theory of truth was philosophically respectable, and
this liberated him to declare that truth, or truth-likeness was the
object of the scientific enterprise (LSD: 273n). Lakatos apparently
endorses this development.



> 
> 
> Tarski's rehabilitation of the correspondence theory of
> truth...stimulated Popper to complement his logic of discovery
> with his own theory of verisimilitude and of approximation to the
> Truth, an achievement marvellous both in its simplicity and in its
> problem-solving power. (PDI: 154.)
> 
> 
> 



But Lakatos points out a problem. There is now a disconnect between
the *game* of science and the *aim* of science. The
*game* of science consists in putting forward falsifiable,
risky and problem-solving conjectures and sticking with the unrefuted
and the well-corroborated ones. But the *aim* of science
consists in developing true or truth-like theories about a largely
mind-independent world. And Popper has given us no reason to suppose
that by playing the game we are likely to achieve the aim. After all,
a theory can be falsifiable, unfalsified, problem-solving and
well-corroborated without being true.



To restore the connection between the game and its aim Lakatos makes a
plea with Popper for a "whiff of
'inductivism'" (PDI: 159). What is this whiff?



> 
> 
> An inductive principle which connects realist metaphysics with
> methodological appraisals, verisimilitude with corroboration, which
> reinterprets the rules of the "scientific game" as
> a--conjectural--theory about the *signs* of the
> *growth of knowledge*, that is, about the signs of *growing
> verisimilitude of our scientific theories* (PDI: 156).
> 
> 
> 



In other words, it is a metaphysical principle which states that
highly falsifiable but well-corroborated theories are (in some sense)
more likely to be true (or truth-like) than their low-risk
counterparts. Corroborations tend to confirm. Thus by playing the game
we approximate the aim. Lakatos goes on to urge that this whiff of
inductivism is not much of an ask, since Popper sometimes seems to
presuppose it without fully realizing that he is doing so.



There are three points to note.



(1) If this criticism holds good against Popper it is equally good
against Lakatos himself. He too has a disconnect between the
*game* of science--which, when it is played well, consists
in developing progressive research programmes--and the
*aim* of science--which, like Popper, he takes to be truth
(FMSRP: 58). To solve this problem, we need a metaphysical principle
which states that highly progressive research programmes are (in some
sense) more likely to be true (or truth-like) than their degenerating
rivals. Thus if Popper could do with a whiff of inductivism, the same
goes for Lakatos.



(2) The inductivism that Lakatos recommends to Popper looks remarkably
like the inductivism that he condemned in Russell. ("I do not
see any way out of a dogmatic assertion that we *know* the
inductive principle, or some equivalent; the only alternative is to
throw over almost everything that is regarded as knowledge by science
and common sense." Russell 1944: 683, quoted disdainfully by
Lakatos at **Regress**: 18.) But if inductivism is
permissible (or even *de rigueur*) in the Philosophy of
Science, perhaps it is permissible (or even *de rigueur*) in
the Philosophy of Mathematics! In which case, the Renaissance of
Empiricism in the Philosophy of Mathematics may count as a genuine
renaissance after all, since the logical or set-theoretic axioms may
(as Russell supposed) be confirmed (and hence rationally believed)
because of their mathematical consequences. If epistemic support can
flow upwards from evidence to theory (where the evidence consists of a
sequence of novel and successful predictions), perhaps it can flow
upwards from consequences to axioms.



(3) This episode undermines an influential "Hegelian"
reading of Lakatos due to Ian Hacking. According to Hacking,



> 
> 
> Lakatos, educated in Hungary in an Hegelian and Marxist tradition,
> took for granted the post-Kantian, Hegelian, demolition of
> correspondence theories (Hacking 1983: 118).
> 
> 
> 



This is an odd assertion as Lakatos explicitly endorses the
correspondence theory on a number of occasions and even declares truth
to be the aim of science, which is why contradictions are intolerable
in the long term (FMSRP: 58). But in Hacking's view, Lakatos
was



> 
> 
> down on truth, not just a particular theory of truth. He [did] not
> want a replacement for the correspondence theory, but a replacement
> for truth itself (Hacking 1983: 119).
> 
> 
> 



He found his replacement in the concept of progress.



> 
> 
> Lakatos then defines objectivity and rationality in terms of
> progressive research programmes, and allows an incident in the history
> of science to be objective and rational if its internal history can be
> written as a sequence of progressive problem shifts (Hacking 1983:
> 126).
> 
> 
> 



Progress becomes a surrogate for truth. We don't ask whether a
theory is true or not but only whether it is part of a progressive
programme. To paraphrase the young Karl Popper,



> 
> 
> in the logic of science [that Lakatos has] outlined it is possible to
> avoid using the concepts "true" and "false"
> [which, in Lakatos's opinion, is a jolly good thing!] (LSD:
> 273).
> 
> 
> 



But if Lakatos had really been such an anti-truth-freak, he would not
have congratulated Popper on his Tarskian turn. Rather he would have
condemned him for taking the vacuous concept of truth to be the aim of
science. As for the disconnect between the *aim* of science and
the *game* of science, he would have recommended that Popper
resolve it by dropping the aim and substituting the game (which,
according to Hacking, was what Lakatos himself was trying to do). If
truth were not the object of the exercise, there would be no need for
a whiff of inductivism to connect Popper's method with
science's ultimate objective. But Lakatos *did* think
that a whiff of inductivism was needed to connect Popper's
method with science's objective. Hence Lakatos believed that
truth was the object of the scientific enterprise. Whatever the
remnants of Hegelianism that Lakatos retained in later life, an
aversion to truth (or to the correspondence theory of truth ) was not
one of them.


### 3.7 "Why Did Copernicus's Research Programme Supersede Ptolemy's?" (1976)



Lakatos's last publication was an historical a case-study,
co-authored with Elie Zahar and published after his death. It argues
that the methodology of scientific research programmes can explain the
Copernican Revolution as a rational process by which an earlier theory
(Ptolemy's geocentric theory of the Cosmos) was dethroned in
favour another objectively better one (Copernicus's heliocentric
theory). It thus demonstrates the rationality of the Copernican
Revolution (one of the most dramatic episodes in the history of
thought) and confirms the MSRP as a theory of scientific rationality
(so long as we accept the inductive principle that the more
"great science" that a demarcation criterion can represent
as rational, the more likely it is to be correct).



Apart from the intrinsic interest of the subject, the paper marks a
modification of Lakatos's conception of factual novelty and
hence a modification to the MSRP. For the earlier Lakatos, a fact
counts as novel with respect to a research programme if it is not
predicted by any of its rivals and if it is not already known. In
WDCRPSP Lakatos accepts an amendment due to his co-author Elie Zahar.
Zahar's original problem was our old friend the Precession of
Mercury. This was explained by Einstein's
programme--specifically the General Theory of
Relativity--but not by Newton's, and this was generally
thought to count in Einstein's favour. The difficulty is that in
Lakatos's lexicon the Precession of Mercury did not count as a
novel fact. After all it had been known to astronomers for nearly a
century. Thus, given the original version of the MSRP, the discovery
that that the General Theory could explain the Precession of Mercury
(whilst Newton's theory could not) did not mean that
Einstein's programme was any more progressive than
Newton's. (*That* had to be argued on other grounds.) But
this is such a counterintuitive result that it suggests a defect in
the MSRP. Zahar's modification is that a fact counts as a novel
prediction with respect to a research programme if a) it is not
predicted by any of the programme's rivals and b) either it is
*not* already known or if it *is* already known, the
hard core of the programme was not devised to explain it.



By this modified criterion the Precession of Mercury counts as a novel
fact with respect to Einstein's programme. For the General
Theory was designed to solve a *different* set of problems. The
prediction that if the General Theory were correct, the perihelion of
Mercury would shift as it does without the influence of any other
heavenly body came as an "unexpected present from
Schwarzschild" (the man who did the sums). It was therefore
"an unintended by-product of Einstein"s programme'
(WDCRPSP: 185). So despite its antiquity, the Precession of Mercury
counts as a novel fact or a novel prediction with respect to
Einstein's programme, thus making the programme a lot more
progressive. Some might regard Zahar's amendment as a
suspiciously *ad hoc* move, but *ad hoc* or not, it
looks like an improvement on the original MSRP. Lakatos and Zahar go
on to use this idea to explain why Copernicus's programme very
properly superseded Ptolemy's.


## 4. Mincemeat Unmade: Lakatos versus Feyerabend



According to his friend Paul Feyerabend, Lakatos was "was a
fascinating person, an outstanding thinker and the best philosopher of
science of our strange and uncomfortable century" (Feyerabend
1975a: 1). Writing in 1981, John Fox raised a cynical eyebrow:



> 
> 
> As when Lakatos similarly praises Popper, it is easy to suspect
> indirect self-advertisement: building up one's opponent so that
> the announced victory is taken as winning a world title (Fox 1981:
> 92).
> 
> 
> 



With Motterlini's publication of the Feyerabend/Lakatos
correspondence (F&AM), Fox's suspicions have been amply
confirmed. It is quite clear that Lakatos and Feyerabend were engaged
in a self-conscious campaign of mutual boosterism, leading up to a
planned epic encounter between a fallibilistic rationalism, as
represented by Lakatos, and epistemological anarchism, as represented
by Feyerabend. As Feyerabend put it "I was to attack the
rationalist position, Imre was to restate and defend it, making
mincemeat of me in the process" (Feyerabend 1975b: preface).
This Battle of the Titans was to consist of Feyerabend's
*Against Method* and Lakatos's projected reply, which is
referred to, in their correspondence, by the mysterious acronym
"MAM".



Sometimes the mutual boosterism went a bit too far, causing pain and
distress to serious-minded philosophers who regarded Popperian
critical rationalism as a bulwark against a resurgent Nazism:



> 
> 
> Hans Albert is on the verge of suicide [writes Lakatos to Feyerabend].
> Allegedly somebody told him that in Kiel you will describe critical
> rationalism as a "mental disease", and he thinks that will
> be the end of Reason in Germany. I told him that though you are AN
> EXTREMELY GREAT MAN, that you will not bring Nazism back
> single-handedly.... (F&AM: 291).
> 
> 
> 



But although they had interested motives for talking each other up, it
is clear that the mutual admiration between Feyerabend and Lakatos was
quite sincere. Each genuinely regarded the other as the man to
beat.



Feyerabend's criticism of Lakatos is summed up in his joking
dedication to *Against Method*: To IMRE LAKATOS *Friend and
Fellow-Anarchist*. In other words Feyerabend's charge is
that for all his law-and-order pretensions as a defender of the
rationality of science and a critic of pseudoscience, Lakatos is
really an epistemic anarchist *malgre lui*.
Feyerabend's epistemological anarchism is sometimes summed up by
the slogan "Anything goes" but that is a little
misleading. His point is rather this: *If you want a set of
methodological rules distinguishing between good science and bad
science*, the only thing that *won't* exclude some of
what *you* (Dear Reader) regard as the best science is the
principle "Anything goes". Anything else would rule out
what is widely regarded as some of the best science as unscientific.
Thus a large proportion of Feyerabend's *Against Method*
is devoted to "praising" Galileo for his allegedly
anti-Popperian practices and his dodgy (but progressive) rhetorical
tricks. Everyone agrees that Galileo was a great scientist. But if
Galileo was great, then the rules that supposedly constitute great
science are defective since they would exclude some of the greatest of
Galileo's great deeds.



But what about Lakatos? Feyerabend poses a dilemma. Suppose we apply
the Lakatos's methodology of scientific research programmes in a
conservative or rigouristic spirit. Scientists are urged to abandon
degenerating research programmes in favour of the progressive, and
grant-giving agencies are urged to defund them. After all, such
programmes are condemned by the Demarcation Criterion as *bad*
science or even *non*-science! At the very least, the adherents
of degenerating research programmes must bear the stigma of
irrationality, owning up to their scientific sins. But in that case
Lakatos's MSRP would be condemning some research programmes to
death as bad science or even non-science that might otherwise recover
their progressive (and hence their scientific) status. Thus Lakatos
would be vulnerable to the same criticism that he himself applies to
Popper--he would be excluding some of the best science as
unscientific (that is, research programmes that have suffered a
degenerating phase only to stage a magnificent comeback). In response
to this, Lakatos distinguished appraisal from advice, and said that
the task of the philosopher of science is to issue rules of appraisal,
not to advise scientists (or grant-giving agencies) about what they
ought to do. The Demarcation Criterion can evaluate the current state
of play but it does not tell anyone what to do about it. (To
paraphrase Marx's Thesis XI, "Methodologists hitherto have
attempted to *change* the world of scientific research in
various ways; the point, however is to *appraise* it".)
The MSRP does enjoin a principle of scientific honesty, namely that
the adherents of degenerating research programmes should own up to
their methodological shortcomings, such as the lack of novel
predictions or the falsification of the predictions that they have
made. However, so long as they admit to these failures they can
(rationally?) persist in their degenerate ways.



But in that case Lakatos is gored by the other horn of
Feyerabend's dilemma. For Feyerabend argues that a Demarcation
Criterion that cannot tell anyone what to do or not to do is scarcely
distinguishable from "Anything goes". To revert to
Feyerabend's political analogy, what is the difference between
an anarchist society and a "state" where the
"police" can *appraise* people for their
"criminal" or "law-abiding" behaviour but can
never make an arrest or send anyone to jail? That's a
"state" which isn't a state and a "police
force" which isn't a police force! We have not scientific
law-and-order but anarchy, accompanied by uplifting sermons and
benedictions posthumously bestowed on the mighty scientific dead.



What was Lakatos's response to this dilemma? It is sometimes
suggested, not least by Feyerabend himself, that Lakatos did have, or
would have had, an answer but that he did not live to write it up.
Their correspondence suggests otherwise. Although the *locus
classicus* of Feyerabend's argument is chapter 16 of
*Against Method* (1975b) he had already developed his dilemma
in "Consolations for the Specialist" (1968) and Lakatos
had access to successive versions of the argument in the successive
drafts that Feyerebend sent him in the last is six years of his life.
Yet there is no trace of a counterargument in Lakatos's
surviving letters to Feyerabend. Instead there are a series of
fearsome threats.



> 
> 
> I am now greatly grateful for your depicting me as *God* and
> yourself as the *Devil*. I also return the compliment: for me
> you are the only philosopher worth demolishing. But there is
> *one* trouble: I *can* take you to such little pieces
> that only an electromicroscope can discover you again. Will you be
> very hurt? (F&AM: 268-9.)
> 
> 
> 



However, aside from these threats, a developed answer to
Feyerabend's dilemma is conspicuous by its absence. One is
reminded of King Lear:



> 
> 
> I will do such things,--
>   
> 
> What they are, yet I know not: but they shall be
>   
> 
> The terrors of the earth.
> 
> 
> 



The upshot is that if there is a Lakatosian answer to
Feyerabend's dilemma, it is an answer that has to be concocted
on his behalf. One of us has a go in Musgrave 1976, but for the
Methodology of Scientific Research Programmes, it is still, very much,
an open problem.