how auxiliary hypothesis

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• v.326(7404); 2003 Jun 28

Effect of interpretive bias on research evidence

Ted j kaptchuk.

1 Harvard Medical School, Osher Institute, 401 Park Drive, Boston, MA 02215, USA ude.dravrah.smh@kuhctpak_det

Associated Data

Short abstract.

Doctors are being encouraged to improve their critical appraisal skills to make better use of medical research. But when using these skills, it is important to remember that interpretation of data is inevitably subjective and can itself result in bias.

Facts do not accumulate on the blank slates of researchers' minds and data simply do not speak for themselves. 1 Good science inevitably embodies a tension between the empiricism of concrete data and the rationalism of deeply held convictions. Unbiased interpretation of data is as important as performing rigorous experiments. This evaluative process is never totally objective or completely independent of scientists' convictions or theoretical apparatus. This article elaborates on an insight of Vandenbroucke, who noted that “facts and theories remain inextricably linked... At the cutting edge of scientific progress, where new ideas develop, we will never escape subjectivity.” 2 Interpretation can produce sound judgments or systematic error. Only hindsight will enable us to tell which has occurred. Nevertheless, awareness of the systematic errors that can occur in evaluative processes may facilitate the self regulating forces of science and help produce reliable knowledge sooner rather than later. ​ later.

Interpretative processes and biases in medical science

Science demands a critical attitude, but it is difficult to know whether you have allowed for too much or too little scepticism. Also, where is the demarcation between the background necessary for making judgments (such as theoretical commitments and previous knowledge) and the scientific goal of being objective and free of preconceptions? The interaction between data and judgment is often ignored because there is no objective measure for the subjective components of interpretation. Taxonomies of bias usually emphasise technical problems that can be fixed. 3 The biases discussed below, however, may be present in the most rigorous science and are obvious only in retrospect.

Quality assessment and confirmation bias

The quality of any experimental findings must be appraised. Was the experiment well performed and are the outcomes reliable enough for acceptance? This scrutiny, however, may cause a confirmation bias: researchers may evaluate evidence that supports their prior belief differently from that apparently challenging these convictions. Despite the best intentions, everyday experience and social science research indicates that higher standards may be expected of evidence contradicting initial expectations.

Two examples might be helpful. Koehler asked 297 advanced university science graduate students to evaluate two supposedly genuine experiments after being induced with different “doses” of positive and negative beliefs through false background papers. 4 Questionnaires showed that their beliefs were successfully manipulated. The students gave significantly higher rating to reports that agreed with their manipulated beliefs, and the effect was greater among those induced to hold stronger beliefs. In another experiment, 398 researchers who had previously reviewed experiments for a respected journal were unknowingly randomly assigned to assess fictitious reports of treatment for obesity. The reports were identical except for the description of the intervention being tested. One intervention was an unproved but credible treatment (hydroxycitrate); the other was an implausible treatment (homoeopathic sulphur). Quality assessments were significantly higher for the more plausible version. 5 Such confirmation bias may be common. w1 w2

Definitions of interpretation biases

Confirmation bias —evaluating evidence that supports one's preconceptions differently from evidence that challenges these convictions

Rescue bias —discounting data by finding selective faults in the experiment

Auxiliary hypothesis bias —introducing ad hoc modifications to imply that an unanticipated finding would have been otherwise had the experimental conditions been different

Mechanism bias —being less sceptical when underlying science furnishes credibility for the data

“Time will tell” bias —the phenomenon that different scientists need different amounts of confirmatory evidence

Orientation bias —the possibility that the hypothesis itself introduces prejudices and errors and becomes a determinate of experimental outcomes

Expectation and rescue and auxiliary hypothesis biases

Experimental findings are inevitably judged by expectations, and it is reasonable to be suspicious of evidence that is inconsistent with apparently well confirmed principles. Thus an unexpected result is initially apt to be considered an indication that the experiment was poorly designed or executed. 6 w3 This process of interpretation, so necessary in science, can give rise to rescue bias, which discounts data by selectively finding faults in the experiment. Although confirmation bias is usually unintended, rescue bias is a deliberate attempt to evade evidence that contradicts expectation.

Instances of rescue bias are almost as numerous as letters to the editors in journals. The avalanche of letters in response to the Veterans Administration Cooperative randomised controlled trial examining the efficacy of coronary artery bypass grafting published in 1977 is a well documented example. 7 The trial found no significant difference in mortality between 310 patients treated medically and 286 treated surgically. A subgroup of 113 patients with obstruction of the left main coronary artery, however, clearly benefited from surgery. 8 Instead of settling the clinical question, the trial spurred fierce debate in which supporters and detractors of the surgery perceived flaws that, they claimed, would skew the evidence away from their preconceived position. Each stakeholder found selective faults to justify preexisting positions that reflected their disciplinary affiliations (cardiology v cardiac surgeon), traditions of research (clinical v physiological), and personal experience. 9

Auxiliary hypothesis bias is a form of rescue bias. Instead of discarding contradictory evidence by seeing fault in the experiment, the auxiliary hypothesis introduces ad hoc modifications to imply that an unexpected finding would have been otherwise had the experimental conditions been different. Because experimental conditions can easily be altered in so many ways, adjusting a hypothesis is a versatile tool for saving a cherished theory. w4 Evidence pointing to an unwelcome finding in a randomised controlled trial, for example, can easily be dismissed by arguments against the therapeutic dose, its timing, or how patients were selected. Lakatos termed such reluctance to accept an experimental verdict a scientist's “thick skin.” 10 Thus, when early randomised controlled trials showed that hormone replacement therapy did not reduce the risk of coronary heart disease, 11 advocates of hormone replacement therapy argued that it was still valuable for primary prevention because the study group was women with established coronary heart disease, making the disease too far advanced to benefit from the treatment.

Plausibility and mechanism bias

Evidence is more easily accepted when supported by accepted scientific mechanisms. This understandable tendency to be less sceptical when underlying science furnishes credibility can give rise to mechanism bias. Often, such scientific plausibility underlies and overlaps the other biases I've described. Many examples exist where with hindsight it is clear that plausibility caused systematic misinterpretation of evidence. For example, the early negative evidence for hormone replacement therapy would have undoubtedly been judged less cautiously if a biological rationale had not already created a strong expectation that oestrogens would benefit the cardiovascular system. 12 w5 Similarly, the rationale for antiarrhythmic drugs for myocardial infarction was so imbedded that each of three antiarrhythmic drugs had to be proved harmful individually before each trial could be terminated. 13 w6And the link between Helicobacter pylori and peptic ulcer was rejected initially because the stomach was considered to be too acidic to support bacterial growth. 14

Waiting for more evidence and “time will tell” bias

The position that more evidence is necessary before making a judgment indicates a judicious attitude that is central to a scientific scepticism. None the less, different scientists seem to need different amounts of confirmatory evidence to feel satisfied. This discrepancy in duration conceals a subjective process that easily can become a “time will tell” bias. The evangelist, at one extreme, is quick to accept the data as good evidence (or even proof). Evangelists often have a vested intellectual, professional, or personal commitment and may have taken part in the experiment being assessed. At the other extreme are the snails, who invariably find the data unconvincing, perhaps because of their personal and intellectual investment in old “facts.” At the two extremes, as well as at all points in between, there is no objective way to tell whether good judgment or systematic error is operating. Max Planck described the “time will tell” bias cynically: “a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” 15

Hypothesis and orientation bias

The above categories of potential biases all occur after data are collected. Sometimes, however, conviction may affect the collection of data, creating orientation bias. Psychologists call this the “experimenter's hypothesis as an unintended determinant of experimental results.” 16 Thus, psychology graduate students, when informed that rats were specially bred for maze brightness, found that these rats outperformed those bred for maze dullness, despite both groups really being standard laboratory rats assigned at random. 17 Somehow, experimental and recording errors tend to be larger and more in the direction supporting the hypothesis. w7 w8

Summary points

Evidence does not speak for itself and must be interpreted for quality and likelihood of error

Interpretation is never completely independent of a scientist's beliefs, preconceptions, or theoretical commitments

On the cutting edge of science, scientific interpretation can lead to sound judgment or interpretative biases; the distinction can often be made only in retrospect

Common interpretative biases include confirmation bias, rescue bias, auxiliary hypothesis bias, mechanism bias, “time will tell” bias, and orientation bias

The interpretative process is a necessary aspect of science and represents an ignored subjective and human component of rigorous medical inquiry

Numerous studies have noted that randomised controlled trials sponsored by the pharmaceutical industry consistently favour new therapies. 18 Research outcomes seem to be affected by what the researcher is looking for. It is unclear to what extent these apparent successes are the result of publication bias or matters of study design. Nonetheless, such results are consistent with an orientation bias and explain the fact that some early double blind randomised controlled trials performed by enthusiasts show efficacy—like hyperbaric oxygen for multiple sclerosis 19 w9 or endotoxin antibodies for Gram negative septic shock 20 —whereas subsequent trials cannot replicate the outcome. 19

This article is written from the perspective of philosophy of science. From a statistical point of view, the arguments presented are obviously compatible with a subjectivist or bayesian framework that formally incorporates previous beliefs in calculations of probability. But even if we accept that probabilities measure objective frequencies of events, the arguments still apply. After all, the overall experiment still has to be assessed.

I have argued that research data must necessarily undergo a tacit quality control system of scientific scepticism and judgment that is prone to bias. Nonetheless, I do not mean to reduce science to a naive relativism or argue that all claims to knowledge are to be judged equally valid because of potential subjectivity in science. Recognition of an interpretative process does not contradict the fact that the pressure of additional unambiguous evidence acts as a self regulating mechanism that eventually corrects systematic error. Ultimately, brute data are coercive. However, a view that science is totally objective is mythical, and ignores the human element of medical inquiry. Awareness of subjectivity will make assessment of evidence more honest, rational, and reasonable. 21

Supplementary Material

This article is a shortened version of a paper written for a seminar on bias led by Fredrick Mosteller at Harvard University and reflects his helpful feedback. Peter Goldman criticised earlier versions of the article and helped make it understandable. The comments of Iain Chalmers and Al Fishman have been helpful, as was the dedicated research of Cleo Youtz. All errors and shortcomings of the paper belong solely to the author.

Funding: In part from grants 1R01 AT00402-01 and 1R01 AT001414 from the National Institutes of Health, Bethesda, MD.

Competing interests: None declared.

Method of the Auxiliary Hypothesis

• 1.1 Example
• 2 Similarity to a proof method from Propositional Logic
• 3 References

The method rests on the Criterion of Deduction:

Let $A$ be a relation in $I$, and $I*$ be the the theory obtained by adjoining $A$ to the axioms of $I$. If $B$ is a theorem in $I*$, then $A \implies B$ is a theorem in $I$.

The proof is on the page 30 in the book Theory of Sets by Bourbaki.

Suppose a field, where the inverse of the zero-element and its existence are not defined in axioms.

The auxiliary hypothesis: adjoin the inconsistent assumption (not yet proved) to the axioms that the zero-element has an inverse:

According to the axioms of the field:

Contradiction, since an axiom of the field states the uniqueness of the product. So the zero-element has no inverse by the Method of the auxiliary hypothesis. Otherwise, the set of axioms would be inconsistent.

A common mistake is to assume the non-existence of the inverse for the zero-element from the axioms of the field. It is a consequence, not a definition per se.

$\blacksquare$

Similarity to a proof method from Propositional Logic

If $a \land \neg b$ leads to contradiction, then the proposition $(a \implies b)$ is true .

Theory of Sets by Bourbaki

• Proven Results

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Imre Lakatos

Imre Lakatos (1922–1974) was a Hungarian-born philosopher of mathematics and science who rose to prominence in Britain, having fled his native land in 1956 when the Hungarian Uprising was suppressed by Soviet tanks. He was notable for his anti-formalist philosophy of mathematics (where “formalism” is not just the philosophy of Hilbert and his followers but also comprises logicism and intuitionism) and for his “Methodology of Scientific Research Programmes” or MSRP, a radical revision of Popper’s Demarcation Criterion between science and non-science which gave rise to a novel theory of scientific rationality.

Although he lived and worked in London, rising to the post of Professor of Logic at the London School of Economics (LSE), Lakatos never became a British citizen, but died a stateless person. Despite the star-studded array of academic lords and knights who were willing to testify on his behalf, neither MI5 nor the Special Branch seem to have trusted him, and no less a person than Roy Jenkins, the then Home Secretary, signed off on the refusal to naturalize him. (See Bandy 2009: ch. 16, which includes the transcripts of successive interrogations.)

Nonetheless, Lakatos’s influence, particularly in the philosophy of science, has been immense. According to Google Scholar, by the 25 th of January 2015, that is, just twenty-five days into the new year, thirty-three papers had been published citing Lakatos in that year alone , a citation rate of over one paper per day. Introductory texts on the Philosophy of Science typically include substantial sections on Lakatos, some admiring, some critical, and many an admixture of the two (see for example Chalmers 2013 and Godfrey-Smith 2003). The premier prize for the best book in the Philosophy of Science (funded by the foundation of a wealthy and academically distinguished disciple, Spiro Latsis) is named in his honour. Moreover, Lakatos is one of those philosophers whose influence extends well beyond the confines of academic philosophy. Of the thirty-three papers citing Lakatos published in the first twenty-five days of 2015, at most ten qualify as straight philosophy. The rest are devoted to such topics as educational theory, international relations, public policy research (with special reference to the development of technology), informatics, design science, religious studies, clinical psychology, social economics, political economy, mathematics, the history of physics and the sociology of the family. Thus Imre Lakatos was very much more than a philosophers’ philosopher.

First, we discuss Lakatos’s life in relation to his works. Lakatos’s Hungarian career has now become a big issue in the critical literature. This is partly because of disturbing facts about Lakatos’s early life that have only come to light in the West since his death, and partly because of a dispute between the “Hungarian” and the “English” interpreters of Lakatos’s thought, between those writers (not all of them Magyars) who take the later Lakatos to be much more of a Hegelian (and perhaps much more of a disciple of György Lukács) than he liked to let on, and those who take his Hegelianism to be an increasingly residual affair, not much more, in the end, than a habit of “coquetting” with Hegelian expressions (Marx, Capital : 103). Just as there are analytic Marxists who think that Marx’s thought can be rationally reconstructed without the Hegelian coquetry and dialectical Marxists who think that it cannot, so also there are analytic Lakatosians who think that Lakatos’s thought can be largely reconstructed without the Hegelian coquetry and dialectical Lakatosians who think that it cannot (see for instance Kadvany 2001 and Larvor 1998). Obviously, we cannot settle the matter in an Encyclopedia entry but we hope to say enough to illuminate the issue. (Spoiler alert: so far as the Philosophy of Science is concerned, we tend to favor the English interpretation. We are more ambivalent with respect to the Philosophy of Mathematics.)

Secondly we discuss Lakatos’s big ideas, the two contributions that constitute his chief claims to fame as a philosopher, before moving on (thirdly) to a more detailed discussion of some of his principal papers. We conclude with a section on the Feyerabend/Lakatos Debate. Lakatos was a provocative and combative thinker, and it falsifies his thought to present it as less controversial (and perhaps less outrageous) than it actually was.

Note: In referring to Lakatos’s chief works (and to a couple of Popper’s) we have employed a set of acronyms rather than the name/date system, hoping that this will be more perspicuous to readers. The acronyms are explained in the Bibliography.

1.1 A Tale of Two Lakatoses

• 1.2 Life and Works: The Third World and the Second

1.3 From Stalinist Revolutionary to Methodologist of Science

2.1 against formalism in mathematics, 2.2 improving on popper in the philosophy of science, 3.1 proofs and refutations (1963–4, 1976), 3.2 “regress” and “renaissance”, 3.3 “changes in the problem of inductive logic” (1968).

• 3.4 “Falsification and the Methodology of Scientific Research Programmes” (1970)
• 3.5 “The History of Science and Its Rational Reconstructions” (1971)
• 3.6 “Popper on Demarcation and Induction” (1974)
• 3.7 “Why Did Copernicus’s Research Programme Supersede Ptolemy’s?” (1976)
• 4. Mincemeat Unmade: Lakatos versus Feyerabend
• Works by Lakatos
• Secondary Literature
• Other Internet Resources
• Related Entries

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 ÁVH, the Hungarian secret police, as an informant by keeping tabs on his friends and comrades (Bandy 2009: ch. 14; Long 2002). And he took a prominent part, as a Stalinist student radical, in trying to purge the University of Debrecen of “reactionary” professors and students and in undermining the prestigious but unduly independent Eötvös College, arguing passionately against the depoliticized (but covertly bourgeois) scholarship that Eötvös allegedly stood for (Bandy 2009: chs. 4 and 9; Long 1998 and 2002).

1.2 Life and Works: The Second World and the Third

To the many that knew and loved the later Lakatos, some of these facts are difficult to digest. But how relevant are they to assessing his philosophy, which was largely the product of his British years? This is an important question as Lakatos was wont to draw a Popperian distinction between World 3—the world of theories, propositions and arguments—and World 2—the psychological world of beliefs, decisions and desires. And he was sometimes inclined to suggest that in assessing a philosopher’s work we should confine ourselves to World 3 considerations, leaving the subjectivities of World 2 to one side (F&AM: 140).

So does a philosopher’s life have any bearing on his works? We take our cue from the writings of Lakatos himself. Of course, there were facts about his early career that Lakatos would not have wanted to be widely known, and which he managed to keep concealed from his Western friends and colleagues during his lifetime. But what does his official philosophy have to say about the relevance of biographical data to intellectual history?

In “The History of Science and its Rational Reconstructions” (HS&IRR) Lakatos develops a theory of how to do the history of science, which, with some adjustments, can be blown up into an account of how to do intellectual history in general. For Lakatos, the default assumption in the history of science is that the scientists in question are engaged in a more-or-less rational effort to solve a set of (relatively) “pure” problems (such as “How to explain the apparent motions of the heavenly bodies consistently with a plausible mechanics?”). A “rational reconstruction” in the history of science, employs a theory of (scientific) rationality in conjunction with an account of the problems as they appeared to the scientists in question to display some intellectual episode as a series of rational responses to the problem-situation. On the whole, it is a plus for a theory of [scientific] rationality if it can display the history of science as a relatively rational affair and a strike against it if it cannot. Thus in Lakatos’s opinion, naïve versions of Popper’s falsificationism are in a sense falsified by the history of science, since they represent too much of it as an irrational affair with too many scientists hanging on to hypotheses that they ought to have recognized as refuted. If the rational reconstruction succeeds—that is if we can display some intellectual development as a rational response to the problem situation—then we have an “internal” history of the developments in question. If not, then the “rational reconstruction of history needs to be supplemented by an empirical (socio-psychological) ‘external history’” (HS&IRR: 102). Non-rational or “external” factors sometimes interfere with the rational development of science. “No rationality theory will ever solve problems like why Mendelian genetics disappeared in Soviet Russia in the 1950s” [the reason being that Lysenko, a Stalin favourite, acquired hegemonic status within the world of Soviet biology and persecuted the Mendelians] (HS&IRR: 114).

(Perhaps this marks an important departure from Hegel. For a true Hegelian, everything can, in the last analysis, be seen as rationally required for the self-realization of the Absolute. Hence all history is “internal” in something like Lakatos’s sense, since the “cunning of reason” ensures that apparently irrational impulses are subordinated to the ultimate goal of history.)

Is there, so to speak, an “internal” history of Lakatos’s intellectual development that can be displayed as rational? Or must it be partly explained in terms of “external” influences? The answer depends on the account of rationality that we adopt and the problem situation that we take him to have been addressing.

Whether or not a particular theoretical (or practical) choice is susceptible to an internal explanation depends, in part, on the actor’s problem. Consider, for example, Descartes’ theory of the vortices, namely that the planets are whirled round the sun by a fluid medium which itself contains little whirlpools in which the individual planets are swimming. Descartes’ theory of the vortices, is fairly rational if we take it as an attempt (in the light of what was then known) to explain the motion of the heavenly bodies in a way that is consistent with Copernican astronomy. But it is a lot more rational if we take to be an attempt to explain the motion of the heavenly bodies in a way that is consistent with Copernican astronomy without formally contradicting the Church’s teaching that the earth does not move . (The earth goes round the sun but it does not move with respect to the fluid medium that whirls it round the sun, and, for Descartes, motion is defined as motion with respect to the contiguous matter.) So do we read Descartes’ theory as a fairly rational attempt to solve one problem which is distorted by an external factor or as a very rational attempt solve a related but more complex problem? Well the answer 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.

Imre Lakatos was born Imre Lipsitz in Debrecen, eastern Hungary, on November 9, 1922, the only child of Jewish parents, Jacob Marton Lipsitz and Margit Herczfeld. Lakatos’s parents parted when he was very young and he was largely brought up by his grandmother and his mother who worked as a beautician. The Hungary into which Lakatos was born was a kingdom without a king ruled by an admiral without a navy, the “Regent” Admiral Horthy, who had gained his naval rank in the service of the then-defunct Austro-Hungarian Empire. The regime was authoritarian, a sort of fascism-lite. After a brilliant school career, during which he won mathematics competitions and a multitude of prizes, Lakatos entered Debrecen University in 1940. Lakatos graduated in Physics, Mathematics, and Philosophy in 1944. During his time at Debrecen he became a committed communist, attending illegal underground communist meetings and, in 1943, starting his own illegal study group.

No-one who attended Imre’s groups has forgotten the intensity and brilliance of the atmosphere. “He opened the world to me!” a participant said. Even those who were later disillusioned with communism or ashamed of acts they committed, remember the sense of inspiration, clear thinking and hope for a new society they felt in Imre’s secret seminars. (Long 2002: 265)

However, in Lakatos’s group the emphasis was on preparing the young cadres for the coming communist revolution, rather than engaging in public propaganda or antifascist resistance activities (Bandy 2009: ch. 3).

In March 1944 the Germans invaded Hungary to forestall its attempts to negotiate a separate peace. (The Hungarian government had allied with the Axis powers, in the hopes of recovering some of the territories lost at the Treaty of Trianon in 1920. By 1944 they had begun to realize that this was a mistake.) Admiral Horthy, whose anti-Semitism was a more gentlemanly affair than that of the Nazis (he was fine with systematic discrimination but apparently drew the line at mass-murder), was forced to accept a collaborationist government led by Döme Sztójay as prime minister. The new regime had none of Horthy’s humanitarian scruples and began a policy of enthusiastic and systematic cooperation with the Nazi genocide program. In May, Lakatos’s mother, grandmother and other relatives were forced into the Debrecen ghetto, thence to die in Auschwitz—the fate of about 600,000 Hungarian Jews. Lakatos’s father, a wine merchant, managed to get away and survived the war, eventually ending up in Australia. A little earlier, in March, Lakatos himself had managed to escape from Debrecen to Nagyvárad (now Oradea in Romania) with false papers under the name of Molnár. Later, a Hungarian friend, Vilma Balázs, recalled that

Imre [had been] very close to his mother and they were quite poor. He often blamed himself for her death and wondered if he could have saved her (Bandy 2009: 32).

In Nagyvárad Lakatos restarted his Marxist group. The co-leader was his then-girlfriend and subsequent wife, Éva Révész. In May, the group was joined by Éva Izsák, a 19-year-old Jewish antifascist activist who needed lodgings with a non-Jewish family. Lakatos decided that there was a risk that she would be captured and forced to betray them, hence her duty, both to the group and to the cause, was to commit suicide. A member of the group took her across country to Debrecen and gave her cyanide (Congden 1997, Long 2002, Bandy 2009, ch. 5). To lovers of Russian literature, the episode recalls Dostoevsky’s The Possessed/Demons (based in part on the real-life Nechaev affair). In Dostoevsky’s novel the anti-Tsarist revolutionary, Pyotr Verkhovensky, posing as the representative of a large revolutionary organization, tries to solidify the provincial cell of which he is the chief by getting the rest of group to share in the murder of a dissident member who supposedly poses a threat to the group. (It does not work for the fictional Pytor Verkhovensky and it did work for the real-life Sergei Nechaev.) Hence the title of Congden’s 1997 exposé: “Possessed: Imre Lakatos’s Road to 1956”. But to communists or former communists of Lakatos’s generation, it recalled a different book: Chocolate , by the Bolshevik writer Aleksandr Tarasov-Rodianov. This is a stirring tale of revolutionary self-sacrifice in which the hero is the chief of the local Cheka (the forerunner of the KGB). Popular in Hungary, it encouraged a romantic cult of revolutionary ruthlessness and sacrifice in its (mostly) youthful readers. As one of Lakatos’s contemporaries, György Magosh put it,

How that book inspired us. How we longed to be professional revolutionaries who could be hanged several times a day in the interest of the working class and of the great Soviet Union (Bandy 2009: 31).

It was in that spirit, that the ardent young Marxist, Éva Izsák, could be persuaded that it was her duty to kill herself for the sake of the cause. As for Lakatos himself, a chance remark in his most famous paper suggests something about his attitude.

One has to appreciate the dare-devil attitude of our methodological falsificationist [or perhaps as he would have said in an earlier phase of his career, the conscientious Leninist]. He feels himself to be a hero who, faced with two catastrophic alternatives, dares to reflect coolly on their relative merits and [to] choose the lesser evil. (FMSRP: 28)

If you admire the hero who has the courage to make the tough choice between two catastrophic alternatives, isn’t there a temptation to manufacture catastrophic alternatives so that you can heroically choose between them?

Late in 1944, following a Soviet victory, Lakatos returned to Debrecen, and changed his name from the Germanic Jewish Lipsitz to the Hungarian proletarian Lakatos (meaning “locksmith”). He became active in the now legal Communist Party and in two leftist youth and student organizations, the Hungarian Democratic Youth Federation (MADISZ) and the Debrecen University Circle (DEK). As one of the leaders of the DEK, Lakatos agitated for the dismissal of reactionary professors from Debrecen and the exclusion of reactionary students.

We are aware that this move on our part is incompatible with the traditional and often voiced “autonomy” of the university [Lakatos stated], but respect for autonomy, in our view, cannot mean that we have to tolerate the strengthening of fascism and reaction (Bandy 2009: 59 and 61).

Lakatos moved to Budapest in 1946. He became a graduate student at Budapest University, but spent much of his time working towards the communist takeover of Hungary. This was a slow-motion affair, characterized by the infamous “salami tactics” of the Communist leader Mátyás Rákosi. Lakatos worked chiefly in the Ministry of Education, evaluating the credentials of university teachers and making lists of those who should be dismissed as untrustworthy once the communists had taken over (Bandy 2009: ch. 8). He was also a student at Eötvös College, but attacked it publicly as an elitist and bourgeois institution. The College, and others like it, was closed in 1950 after the communist takeover. In 1947 Lakatos gained his doctorate from Debrecen University for a thesis entitled “On the Sociology of Concept Formation in the Natural Sciences”. In 1948, after the communist takeover was substantially complete, he gained a scholarship to undertake further study in Moscow.

Lakatos flew to Moscow in January 1949, only to be recalled for “un-Party-like” behaviour in July. What these “un-party-like” activities were is something of a mystery but even more of a mystery is why, having returned from Moscow under a cloud, he seemed so cool, calm and collected. Lakatos’s biographers, Long and Bandy, speculate that he was being held in reserve to prepare a case against the communist education chief, József Révai, who was scheduled to appear in a new show trial. But when Rákosi decided not to prosecute Révai after all, Lakatos was thrown to the wolves (Bandy 2009: ch. 12; Long 2002). He was arrested in April 1950 on charges of revisionism and, after a period in the cellars of the secret police (including, of course, torture), he was condemned to the prison camp at Recsk.

However Lakatos was probably doomed anyway. In later life Lakatos was big admirer of Orwell’s Nineteen Eighty-Four . Perhaps he recognized himself in Orwell’s description of the Party intellectual (and expert on Newspeak) Syme:

Unquestionably Syme will be vaporized, Winston thought again. He thought it with a kind of sadness, although well knowing that Syme…was fully capable of denouncing him as a thought-criminal if he saw any reason for doing so. There was something subtly wrong with Syme. There was something that he lacked: discretion, aloofness, a sort of saving stupidity. You could not say that he was unorthodox. He believed in the principles of Ingsoc, he venerated Big Brother, he rejoiced over victories, he hated heretics…. Yet a faint air of disreputability always clung to him. He said things that would have been better unsaid, he had read too many books…. (Orwell 2008 [1949]: 58)

An instance of Lakatos’s Syme-like behaviour is his 1947 denunciation of the literary critic and philosopher György Lukács, one of the intellectual luminaries of the communist movement. Lukács represented the academically respectable face of communism, and favoured a gradual and democratic transition to the dictatorship of the proletariat. Lakatos organized an “anti-Lukács meeting…held under the aegis of the Valóság Circle” to critique Lukács’s foot-dragging and “Weimarism” (Bandy 2009: 110). Once the regime was firmly in control, Lukács was indeed censured for his undue concessions to bourgeois democracy, and he spent the early fifties under a cloud. But in 1947, Lakatos’s criticisms were deemed premature and he got into trouble because of his un-Party-like activities. (Lukács himself referred to the episode as a “cliquish kaffe klatsch ”.) In Communist Hungary it was important not to be “one pamphlet behind” the Party line (Bandy 2009: 92). Lakatos was the sort of over-zealous communist who was sometimes a couple of pamphlets ahead.

After his release from Recsk in September 1953 (minus several teeth), Lakatos remained for a while, a loyal Stalinist. He eked out a living in the Mathematics Institute of the Hungarian Academy of Science, reading, researching and translating (including a translation into Hungarian of George Pólya’s How to Solve It ). During this time he was informing on friends and colleagues to the ÁVH, the Hungarian secret police, though he subsequently claimed that he did not pass on anything incriminating (Long, 2002: 290 ). It was whilst working at the Mathematics Institute that he first gained access to the works of Popper. Gradually he turned against the Stalinist Marxism that had been his creed. He married (as his second wife) Éva Pap and lived at her parents’ house (his father-in-law being the distinguished agronomist, Endre Pap). In 1956 he joined the revisionist Petőfi Circle and delivered a stirring speech on “On Rearing Scholars” which at least burnt his bridges with Stalinism:

The very foundation of scholarly education is to foster in students and postgrads a respect for facts, for the necessity of thinking precisely, and to demand proof. Stalinism, however, branded this as bourgeois objectivism. Under the banner of partinost [Party-like] science and scholarship, we saw a vast experiment to create a science without facts, without proofs. … a basic aspect of the rearing of scholars must be an endeavour to promote independent thought, individual judgment, and to develop conscience and a sense of justice. Recent years have seen an entire ideological campaign against independent thinking and against believing one’s own senses. This was the struggle against empiricism [Laughter and applause] (Bandy 2009: 221. Bandy quotes the transcripts which seem to differ slightly from the prepared text in the Lakatos archives, reprinted in F&AM)

But Lakatos was not just explicitly repudiating Stalinism. He was also implicitly criticizing another prominent member of the Petőfi Circle who had been a big influence on his first PhD, namely György Lukács. (See Ropolyi 2002 for the early influence.) For Lukács’s work is pervaded by just the kind of hostility towards empiricism and disdain for facts that Lakatos is denouncing in his speech, as well as an arts-sider’s contempt for the natural sciences, all of which would have been anathema to the later Lakatos. Indeed Lukács was notorious for the view that that

even if the development of science had proved all Marx’s assertions to be false…we could accept this scientific criticism without demur and still remain Marxists—as long as we adhered to the Marxist method

the orthodox Marxist who realizes that…the time has come for the expropriation of the exploiters, will respond to the vulgar-Marxist litany of “facts” which contradict this process with the words of Fichte, one of the greatest of classical German philosophers: “So much the worse for the facts”. (Lukács 2014 [1919]: ch. 3.)

Thus the Stalinist Lakatos of 1947 had explicitly denounced Lukács for not being Stalinist enough, but the revisionist Lakatos of 1956 was implicitly denouncing Lukács for being methodologically too much of a Stalinist. For the later Lakatos, what was wrong with “orthodox Marxism” was chiefly that its novel factual predictions had been systematically falsified (see §3.2 below). But that was pretty much the complaint of early revisionists such as Bernstein (see Kolakowski 1978: ch. 4) and it was against that kind of revisionism that Lukács’s Bolshevik writings were a protest. (See Lukács 1971 [1923] and 2014 [1919].) Though factual “refutations” of a research programme are not always decisive, a Lukács-like indifference to the facts is, for Lakatos, the mark of a fundamentally unscientific attitude. In our opinion, this puts paid to Ropolyi’s claim that Lukács continued to be a major influence on the later Lakatos.

Lakatos left Hungary in November 1956 after the Soviet Union crushed the short-lived Hungarian revolution. He walked across the border into Austria with his wife and her parents. Within two months he was at King’s College Cambridge, with a Rockefeller Fellowship to write a PhD under the supervision of R.B. Braithwaite, which he completed in 1959 under the title “Essays in the Logic of Mathematical Discovery”. If we set aside his romantic adventures, the story of Lakatos’s life thereafter is largely the story of his work, though we should not forget his activities as an academic politician. Even his friendship with Feyerabend and his friendship and subsequent bust-up with Popper were very much work-related. In Britain his academic career was meteoric. In 1960 he was appointed Assistant Lecturer in Karl Popper’s department at the London School of Economics. By 1969 he was Professor of Logic, with a worldwide reputation as a philosopher of science. During the student revolts of the 1960s, which in Britain were centred on the LSE, Lakatos became an establishment figure. He wrote a “Letter to the Director of the London School of Economics” defending academic freedom and academic autonomy, which was widely circulated. It denounces the student radicals for allegedly trying to do what he himself had done at Debrecen and Eötvös (though he 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.

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:

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.

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:

• If <hard core plus auxiliary hypotheses>, then O (where O represents some observation statement);
• Not- O (Nature shouts “no”: the predictions don’t pan out);
• Not <hard core plus auxiliary hypotheses>.

But logic leaves us with a choice. The conjunction of the hard core plus the auxiliary hypotheses has to go, but we can retain either the hard core or the auxiliary hypotheses. What Lakatos calls the negative heuristic of the research programme, bids us retain the hard core but modify the auxiliary hypotheses:

The negative heuristic of the programme forbids us to direct the modus tollens at this “hard core”. Instead, we must use our ingenuity to articulate or even invent “auxiliary hypotheses”, which form a protective belt around this core, and we must redirect the modus tollens to these. It is this protective belt of auxiliary hypotheses which has to bear the brunt of tests and gets adjusted and re-adjusted, or even completely replaced, to defend the thus-hardened core. (FMSRP: 48)

Thus when refutation strikes, the scientist constructs a new theory, the next in the sequence, with the same hard core but a modified set of auxiliary hypotheses. How is she supposed to do this? Well, associated with the hard core, there is what Lakatos calls the positive heuristic of the programme.

The positive heuristic consists of a partially articulated set of suggestions or hints on how to change, develop the “refutable variants” of the research programme, how to modify, sophisticate, the “refutable” protective belt. (FMSRP: 50)

For example, if a planet is not moving in quite the smooth ellipse that it ought to follow a) if Newtonian mechanics were correct and b) if there were nothing but the sun and the planet itself to worry about, then the positive heuristic of the Newtonian programme bids us look for another heavenly body whose gravitational force might be distorting the first planet’s orbit. Alternatively, if stellar parallax is not observed, we can try to refute this apparent refutation by refining our instruments and making more careful measurements and observations.

Lakatos evidently thinks that when one theory in the sequence has been refuted, scientists can legitimately persist with the hard core without being in too much of a hurry to construct the next refutable theory in the sequence. The fact that some planetary orbits are not quite what they ought to be should not lead us to abandon Newtonian celestial mechanics, even if we don’t yet have a testable theory about what exactly is distorting them. It is worth remarking too that the auxiliary hypotheses play a rather paradoxical part in Lakatos’s methodology. On the one hand, they connect the central theses of the hard core with experience, allowing to them to figure in testable, and hence, refutable theories. On the other hand, they insulate the theses of the hard core from refutation, since when the arrow of modus tollens strikes, we direct it at the auxiliary hypotheses rather than the hard core.

So far we have had an account of what scientists typically do do and what Lakatos thinks that they ought to do. But what about the Demarcation Criterion between science and non-science or between good science and bad? Even if it is sometimes rational to persist with the hard core of a theory when the hard core plus some set of auxiliary hypotheses has been refuted, there must surely be some circumstances in which is it rational to give it up! The Methodology of Scientific Research Programme has got to be something more than a defence of scientific pig-headedness! As Lakatos himself puts the point:

Now, Newton’s theory of gravitation, Einstein’s relativity theory, quantum mechanics, Marxism, Freudianism [the last two stock examples of bad science or pseudo-science for Popperians], are all research programmes, each with a characteristic hard core stubbornly defended, each with its more flexible protective belt and each with its elaborate problem-solving machinery. Each of them, at any stage of its development, has unsolved problems and undigested anomalies. All theories, in this sense, are born refuted and die refuted. But are they [all] equally good? (S&P: 4–5.)

Lakatos, of course, thinks not. Some science is objectively better than other science and some science is so unscientific as to hardly qualify as science at all. So how does he distinguish between “a scientific or progressive programme” and a “pseudoscientific or degenerating one”? (S&P: 4–5.)

To begin with, the unit of scientific evaluation is no longer the individual theory (as with Popper), but the sequence of theories, the research programme . We don’t ask ourselves whether this or that theory is scientific or not, or whether it constitutes good or bad science. Rather we ask ourselves whether the sequence of theories, the research programme, is scientific or non-scientific or constitutes good or bad science. Lakatos’s basic idea is that a research programme constitutes good science—the sort of science it is rational to stick with and rational to work on—if it is progressive , and bad science—the kind of science that is, at least, intellectually suspect —if it is degenerating . What is it for a research programme to be progressive? It must meet two conditions. Firstly it must be theoretically progressive. That is, each new theory in the sequence must have excess empirical content over its predecessor; it must predict novel and hitherto unexpected facts (FMSRP: 33). Secondly it must be empirically progressive. Some of that novel content has to be corroborated, that is, some of the new “facts” that the theory predicts must turn out to be true. As Lakatos himself put the point, a research programme “is progressive if it is both theoretically and empirically progressive, and degenerating if it is not” (FMSRP: 34). Thus a research programme is degenerating if the successive theories do not deliver novel predictions or if the novel predictions that they deliver turn out to be false.

Novelty is, in part, a 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 §3.6 —but to begin with, at least, this was Lakatos’s dominant idea.)

One of Lakatos’s key examples is the predicted return of Halley’s comet which was derived by observing part of its trajectory and using Newtonian mechanics to calculate the elongated ellipse in which it was moving. The comet duly turned up seventy-two years later, exactly where and when Halley had predicted, a novel fact that could not have been arrived at without the aid of Newton’s theory (S&P: 5). Before Newton, astronomers might have noticed a comet arriving every seventy-two years (though they would have been hard put to it to distinguish that particular comet from any others), but they could not have been as exact about the time and place of its reappearance as Halley managed to be. Newton’s theory delivered far more precise predictions than the rival heliocentric theory developed by Descartes, let alone the earth-centered Ptolemaic cosmology that had ruled the intellectual roost for centuries. That’s the kind of spectacular corroboration that propels a research programme into the lead. And it was a similarly novel prediction, spectacularly confirmed, that dethroned Newton’s physics in favour of Einstein’s. Here’s Lakatos again:

This programme made the stunning prediction that if one measures the distance between two stars in the night and if one measures the distance between them during the day (when they are visible during an eclipse of the sun), the two measurements will be different. Nobody had thought to make such an observation before Einstein’s programme. Thus, in progressive research programme, theory leads to the discovery of hitherto unknown novel facts. (S&P: 5.)

A degenerating research programme, on the other hand (unlike the theories of Newton and Einstein) either fails to predict novel facts at all, or makes novel predictions that are systematically falsified. Marxism, for example, started out as theoretically progressive but empirically degenerate (novel predictions systematically falsified) and ended up as theoretically degenerate as well (no more novel predictions but a desperate attempt to explain away unpredicted “observations” after the event).

Has…Marxism ever predicted a stunning novel fact successfully? Never! It has some famous unsuccessful predictions. It predicted the absolute impoverishment of the working class. It predicted that the first socialist revolution would take place in the industrially most developed society. It predicted that socialist societies would be free of revolutions. It predicted that there will be no conflict of interests between socialist countries. Thus the early predictions of Marxism were bold and stunning but they failed. Marxists explained all their failures: they explained the rising living standards of the working class by devising a theory of imperialism; they even explained why the first socialist revolution occurred in industrially backward Russia. They “explained” Berlin 1953, Budapest 1956, Prague 1968. They “explained” the Russian-Chinese conflict. But their auxiliary hypotheses were all cooked up after the event to protect Marxian theory from the facts. The Newtonian programme led to novel facts; the Marxian lagged behind the facts and has been running fast to catch up with them. (S&P: 4–5.)

Thus good science is progressive and bad science is degenerating and a research programme may either begin or end up as such a degenerate affair that it ceases to count as science at all. If a research programme either predicts nothing new or entails novel predictions that never come to pass, then it may have reached such a pitch of degeneration that it has transformed into a pseudoscience.

It is sometimes suggested that in Lakatos’s opinion no theory either is or ought to be abandoned, unless there is a better one in existence (Hacking 1983: 113). Does this mean that no research programme should be given up in the absence of a progressive alternative, no matter how degenerate it may be ? If so, this amounts to the radically anti-sceptical thesis that it is better to subscribe to a theory that bears all the hallmarks of falsehood, such as the current representative of a truly degenerate programme, than to sit down in undeluded ignorance. (The ancient sceptics, by contrast thought that it is better not to believe anything at all rather than believe something that might be false.) We are not sure that this was Lakatos’ opinion, though he clearly thinks it a mistake to give up on a progressive research programme, unless there is a better one to shift to. But consider again the case of Marxism. What Lakatos seems to be suggesting in the passage quoted above, is that it is rationally permissible—perhaps even obligatory—to give up on Marxism even if it has no progressive rival, that is, if there is currently no alternative research programme with a set of hard core theses about the fundamental character of capitalism and its ultimate fate. (After all, the later Lakatos probably subscribed to the Popperian thesis that history in the large is systematically unpredictable. In which case there could not be a genuinely progressive programme which foretold the fate of capitalism. At best you could have a conditional theory, such as Piketty’s, which says that under capitalism, inequality is likely to grow— unless something unexpected happens or unless we decide to do something about it. See Piketty 2014: 35.) So although Lakatos thinks that the scientific community seldom gives up on a programme until something better comes along, it is not clear that he thinks that this is what they always ought to do.

There are numerous departures from Popperian orthodoxy in all this. To begin with, Lakatos effectively abandons falsifiability as the Demarcation Criterion between science and non-science. A research programme can be falsifiable (in some senses) but unscientific and scientific but unfalsifiable. First, the falsifiable non-science. Every successive theory in a degenerating research programme can be falsifiable but the programme as whole may not be scientific. This might happen if it only predicted familiar facts or if its novel predictions were never verified. A tired purveyor of old and boring truths and/or a persistent predictor of novel falsehoods might fail to make the scientific grade. Secondly, the non-falsifiable science. In Lakatos’s opinion, it need not be a crime to insulate the hard-core of your research programme from empirical refutation. For Popper, it is a sin against science to defend a refuted theory by “introducing ad hoc some auxiliary assumption, or by re-interpreting the theory ad hoc in such a way that it escapes refutation” (C&R, 48). Not so for Lakatos, though this is not to say that when it comes to ad hocery “anything goes”.

Thirdly, Lakatos’s Demarcation Criterion is a lot more forgiving than Popper’s. For a start, an inconsistent research programme need not be condemned to the outer darkness as hopelessly unscientific. This is not because any of its constituent theories might be true . Lakatos rejects the Hegelian thesis that there are contradictions in reality. “If science aims at truth, it must aim at consistency; if it resigns consistency, it resigns truth.” But though science aims at truth and therefore at consistency, this does not mean that it can’t put up with a little inconsistency along the way.

The discovery of an inconsistency—or of an anomaly—[need not] immediately stop the development of a programme: it may be rational to put the inconsistency into some temporary, ad hoc quarantine, and carry on with the positive heuristic of the programme (FMSRP: 58).

Thus it was both rational and scientific for Bohr to persist with his research programme, even though its hard core theses on the structure of the atom were fundamentally inconsistent (FMSRP: 55–58). So although Lakatos rejects Hegel’s claim that there are contradictions in reality (though not, perhaps in Reality), he also rejects Popper’s thesis that because contradictions imply everything, inconsistent theories exclude nothing and must therefore be rejected as unfalsifiable and unscientific. For Lakatos, Bohr’s theory of the atom is fundamentally inconsistent, but this does not mean that it implies that the moon is made of green cheese. Thus what Lakatos seems to be suggesting is here (though he is not as explicit as he might be) is that, when it comes to assessing scientific research programmes, we should sometimes employ a contradiction-tolerant logic; that is a logic that rejects the principle, explicitly endorsed by Popper, that anything whatever follows from a contradiction (FMSRP: 58 n. 2). In today’s terminology, Lakatos is a paraconsistentist (since he implicitly denies that from a contradiction anything follows) but not a 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.

As we have seen, Lakatos’s first major publication in Britain was the dialogue “Proofs and Refutations” which originally appeared as a series of four journal articles. The dialogue is dedicated to George Pólya for his “revival of mathematical heuristic” and to Karl Popper for his critical philosophy.

Proofs and Refutations is a highly original production. The issues it discusses are far removed from what was then standard fare in the philosophy of mathematics, dominated by logicism, formalism and intuitionism, all attempting to find secure foundations for mathematics. Its theses are radical. And its dialogue form makes it a literary as well as a philosophical tour de force.

Its official target is “formalism” or “metamathematics”. But (as we have noted) “formalism” doesn’t just mean “formalism” proper, as this term is usually understood in the Philosophy of Mathematics. For Lakatos “formalism” includes not just Hilbert’s programme but also logicism and even intuitionism. Formalism sees mathematics as the derivation of theorems from axioms in formalised mathematical theories. The philosophical project is to show that the axioms are true and the proofs valid, so that mathematics can be seen as the accumulation of eternal truths. An additional philosophical question is what these truths are about , the question of mathematical ontology.

Lakatos, by contrast, was interested in the growth of mathematical knowledge. How were the axioms and the proofs discovered? How does mathematics grow from informal conjectures and proofs into more formal proofs from axioms? Logical empiricist (and Popperian) orthodoxy distinguished the “context of discovery” from the “context of justification”, consigned the former to the realm of empirical psychology, and thought it a matter of “unregimented insight and good fortune”, hardly a fit subject for philosophical analysis. Philosophy of mathematics consists of the logical analysis of completed theories. Formalism manifests this orthodoxy and “disconnects the history of mathematics from the philosophy of mathematics” (P&R: 1). Against the orthodoxy, Lakatos paraphrased Kant (the paraphrase has become almost as famous as the original):

the history of mathematics…has become blind , while the philosophy of mathematics… has become empty . (P&R: 2)

[Lakatos had stated this Kantian aphorism more generally at a conference in Oxford in 1961: “History of science without philosophy of science is blind. Philosophy of science without history of science is empty”. See Hanson 1963: 458.]

Suppose we agree with Lakatos that there is room for heuristics or a logic or discovery. Still, orthodoxy could insist that discovery is one thing, justification another, and that the genesis of ideas has nothing to do with their justification. Lakatos, more radically, disputed this. First, he rejected the foundationalist or justificationist project altogether: mathematics has no foundation in logic, or set theory, or anything else. Second, he insisted that the way in which a theory grows plays an essential role in its methodological appraisal. This is as much a central theme of his philosophy of empirical science as it is of his philosophy of mathematics.

As noted above, Proofs and Refutations takes the form of an imaginary dialogue between a teacher and a group of students. It reconstructs the history of attempts to prove the Descartes-Euler conjecture about polyhedra, namely, that for all polyhedra, the number of vertices minus the number of edges plus the number of faces is two ( V – E + F = 2). The teacher presents an informal proof of this conjecture, due to Cauchy. This is a “ thought experiment which suggest a decomposition of the original conjecture into subconjectures or lemmas ” from which the original conjecture is supposed to follow. We now have, as well as the original conjecture or conclusion, the subconjectures or premises, and the meta-conjecture that the latter entail the former. Clearly, this kind of “informal proof” is quite different from the “formalist” idea that an informal proof is a formal proof with gaps (PP2: 63). Equally clearly, any of these conjectures might be refuted by counterexamples.

In the dialogue, the students, who are rather advanced, demonstrate the point—they demolish the Teacher’s “proof” by producing counterexamples. The counterexamples are of three kinds:

(1) Counterexamples to the conclusion that are not also counterexamples to any of the premises (“global but not local counterexamples”): These establish that the conclusion does not really follow from the stated premises. They require us to improve the proof, to unearth the “hidden lemma” which the counterexample also refutes, so that it becomes a “local as well as global” counterexample—see (3), below.

(2) Counterexamples to one of the premises that are not also counterexamples to the conclusion (“local but not global counterexamples”): These require us to improve the proof by replacing the refuted premise with a new premise which is not subject to the counterexample and which (we hope) will do as much to establish the conclusion as the original refuted premise did.

(3) Counterexamples both to the conclusion and to (at least one of) the premises (“global and local counterexamples”): These can be dealt with by incorporating the refuted premise or lemma into the original conclusion, as a condition of its correctness. For example, a picture-frame is a polyhedron with a hole or tunnel in it:

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?

Attempts to rescue our conjecture from refutation yield “proof-generated definitions” like that of a “normal polyhedron”.

Is there any limit to this process of “concept-stretching”, or any distinction to be drawn between interesting and frivolous concept-stretching? Can this process yield, not fallible conjectures and proofs, but certainty? Lakatos’s editors distinguish the certainty of proofs from the certainty of the axioms from which all proofs must proceed. They claim that rigorous proof-procedures have been attained, and that “There is no serious sense in which such proofs are fallible” (P&R: 57). Quite so. But only because we have decided not to “stretch” the logical concepts that lie behind those rigorous and formalizable proof-procedures. A rigorous proof in classical logic may not be valid in intuitionistic or paraconsistent logics. And the key point is that a proof, however rigorous, only establishes that if the axioms are true, then so is the theorem. If the axioms themselves remain fallible, then so do the theorems rigorously derived from them. Providing foundations for mathematics requires the axioms to be made certain, by deriving them from logic or set theory or something else. Lakatos claimed that this foundational project had collapsed (see below, §3.2 ).

To what extent is this imaginary dialogue a contribution to the history of mathematics? Lakatos explained that

The dialogue form should reflect the dialectic of the story: it is meant to contain a sort of rationally reconstructed or “distilled” history. The real history will chime in in the footnotes, most of which are to be taken, therefore, as an organic part of the essay (P&R: 5).

This device, first necessitated by the dialogue form, became a pervasive theme of Lakatos’s writings. It was to attract much criticism, most of it centred around the question whether rationally reconstructed history was real history at all. The trouble is that the rational and the real can come apart quite radically. At one point in Proofs and Refutations a character in the dialogue makes a historical claim which, according to the relevant footnote, is false. Lakatos says that the statement

although heuristically correct ( i.e. true in a rational history of mathematics) is historically false. This should not worry us: actual history is frequently a caricature of its rational reconstructions. (P&R: 21)

On occasions, Lakatos’s sense of humour ran away with him, as when the text contains a made-up quotation from Galileo, and the footnote says that he “was unable to trace this quotation” (P&R: 62). (Though this does rather smack of his youthful habit of winning arguments with “bourgeois” students by fabricating on-the-spot quotations from the authorities they respected. See Bandy 2009: 122.) Horrified critics protested that rationally reconstructed history is a caricature of real history, not in fact real history at all but rather “philosophy fabricating examples”. One critic said that philosophers of science should not be allowed to write history of science. This academic trade unionism is misguided. You do not falsify history by pointing out that what ought to have happened did not, in fact, happen.

There is an important pedagogic point to all this, too. The dialectic of proofs and refutations can generate, in the ways explained above, quite complicated definitions of mathematical concepts, definitions that can only really be understood by considering the process that gave rise to them. But mathematics teaching is not historical, or even quasi-historical. (One sense in which Lakatos’s theory is dialectical: it represents a process as rational even though the terms of the debate are not clearly defined.) But students nowadays are presented with the latest definitions at the outset, and required to learn them and apply them, without ever really understanding them.

One question about Proofs and Refutations is whether the heuristic patterns depicted in it apply to the whole of mathematics. While some aspects clearly are peculiar to the particular case-study of polyhedra, the general patterns are not. Lakatos himself applied them in a second case-study, taken from the history of analysis in the nineteenth century (“Cauchy and the Continuum”, 1978c).

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 Gödel, Mostowski and von Neumann—were prepared to make similar noises. Lakatos dubs this development “empiricism” (or “quasi-empiricism”) and hails it on the one hand whilst condemning it on the other.

Why “empiricism”? Not because it revives Mill’s idea that the truths of arithmetic are empirical generalizations, but because it ascribes to mathematics the same kind of hypothetico-deductive structure that the empirical sciences supposedly display, with axioms playing the part of theories and their mathematical consequences playing the part of observation-statements (or in Lakatos’s terminology, “potential falsifiers”).

Why does Lakatos hail the “empiricism” that he also condemns? Because it means that mathematics has the same kind epistemic structure that science has according to Popper. It’s a matter of axiomatic conjectures that can be mathematically refuted. (The difference between science and mathematics consists in the differences between the potential falsifiers.)

Why does Lakatos condemn the “empiricism” that he also commends? Because Russell, like most of his supporters, succumbs to the “inductivist” illusion that the axioms can be confirmed by the truth of their consequences. In Lakatos’s opinion this is simply a mistake. Truth can trickle down from the axioms to their consequences and falsity can flow upwards from the consequences to the axioms (or at least to the axiom set). But neither truth nor probability nor justified belief can flow up from the consequences to the axioms from which they follow. Here Lakatos out-Poppers Popper, portraying not just science but even mathematics as a collection of unsupported conjectures that can be refuted but not confirmed, anything else being condemned as to “inductivism”. However the inductivism that Lakatos scornfully rejects in Renaissance is just the kind of inductivism that he would be recommending to Popper just a few years later.

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.

Research Hypothesis In Psychology: Types, & Examples

Saul McLeod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul McLeod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

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• Rescue Bias
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• ABC Approach
• Abortion Rate
• Abortion Ratio
• Absenteeism
• Absolute Effect
• Absolute Poverty Level
• Absolute Rate
• Absolute Risk (AR)
• Absolute Risk Increase (ARI)
• Absolute Risk Reduction (ARR)
• Accelerated Failure-Time Model
• Acceptable Risk
• Acceptance Sampling
• Accessory Reservoir
• Accumulation of Risk
• Acquaintance Network
• Acquired Immunodeficiency Syndrome (AIDS)
• Active Life Expectancy
• Activities of Daily Living (ADL) Scale
• Activity Setting
• Actuarial Rate
• Actuarial Table
• Additive Model
• Adult Literacy Rate
• Adverse Health Event
• Adverse Reaction
• Adverse Selection
• Aetiology, Aetiological
• Age Dependency Ratio
• Agency Relationship
• Agent (Of Disease)
• Age-Period-Cohort Analysis
• Age-Sex Pyramid
• Age-Sex Register
• Age-Specific Fertility Rate
• Age-Specific Rate
• Age Standardization
• Aggregation Bias
• Aggregate Surveillance
• Aggregative Fallacy
• Aging of The Population
• “Agnostic” Analysis
• Airborne Infection
• Algorithm, Clinical
• Allocation Bias
• Allocation Concealment
• Allostatic Load
• Alma-Ata Declaration
• Alpha Error
• Alpha Level
• Analysis of Variance (ANOVA)
• Analytical Study
• Anamorphic Map
• Ancillary Study
• Anecdotal Evidence
• Animal Model
• Annual Parasite Index (API)
• Anthropometry
• Anthropophilic
• Antigenic Drift
• Antigenicity
• Antigenic Shift
• Apgar Score
• Applied Epidemiology
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A form of Rescue Bias and thus of Interpretive Bias , which occurs in introducing ad hoc modifications to imply that an unanticipated finding would have occurred otherwise had the experimental conditions been different. Because experimental conditions can easily be altered in many ways, adjusting a hypothesis is a versatile tool for saving a cherished theory.... ...

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The logic of scientific methodology

Facts do not a science make

On auxiliary hypotheses

• By PeterM in .Popper, Karl , The Logic of Scientific Discovery

8 October 2013

As regards auxiliary hypotheses we propose to lay down the rule that only those are acceptable whose introduction does not diminish the degree of falsifiability or testability of the system in question, but, on the contrary, increases it. … If the degree of falsifiability is increased, then introducing the hypothesis has actually strengthened the theory: the system now rules out more than it did previously: it prohibits more. We can also put it like this. The introduction of an auxiliary hypothesis should always be regarded as an attempt to construct a new system; and this new system should then always be judged on the issue of whether it would, if adopted, constitute a real advance in our knowledge of the world. An example of an auxiliary hypothesis which is eminently acceptable in this sense is Pauli’s exclusion principle. An example of an unsatisfactory auxiliary hypothesis would be the contraction hypothesis of Fitzgerald and Lorentz which had no falsifiable consequences but merely served to restore the agreement between theory and experiment … .  [62]

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How to Do Things with Theory: The Instrumental Role of Auxiliary Hypotheses in Testing

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• Published: 23 October 2019
• Volume 86 , pages 1453–1468, ( 2021 )

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• Corey Dethier   ORCID: orcid.org/0000-0002-1240-8391 1  

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[I]n saying these words, we are doing something ...rather than reporting something. Austin ( 1962 , p. 13)

Pierre Duhem’s influential argument for holism relies on a view of the role that background theory plays in testing: according to this still common account of “auxiliary hypotheses,” elements of background theory serve as truth-apt premises in arguments for or against a hypothesis. I argue that this view is mistaken. Rather than serving as truth-apt premises in arguments, auxiliary hypotheses are employed as (reliability-apt) “epistemic tools”: instruments that perform specific tasks in connecting our theoretical questions with the world but that are not (or not usually) premises in arguments. On the resulting picture, the acceptability of an auxiliary hypothesis depends not on its truth but on contextual factors such as the task or purpose it is put to and the other tools employed alongside it.

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An A Priori Refutation of the Classical Pessimistic Induction

Grounding and the argument from explanatoriness, duhem’s problem revisited: logical versus epistemic formulations and solutions.

See, for example, Longino ( 1990 ), Mayo ( 1996 ), Azzouni ( 2000 ), Norton ( 2003 ), Woodward ( 2003 ), Wilson ( 2006 ) and Morgan and Morrison ( 1999 ).

I’m not interested in diving into questions of what Duhem really meant, and will generally refer to the schematized “Duhemian” from now on; for a discussion of Duhem’s intended argument, see Ariew ( 2018 ).

This may seem obvious, but the issue of just how heterozygous populations are was a fraught one for mid-twentieth century biology; see, e.g., Lewontin ( 1974 ).

Since I began writing this essay, I’ve become aware that Elgin ( 2017 ) employs HW models to make a similar point to the one made here. For an influential philosophical analysis of HW models, see Sober ( 1984 ).

Of course, this description is idealized. The inclusion of other stochastic processes such as drift or complicating factors such as linkage disequilibrium mean that such inferences will necessarily be statistical. But the simple description suffices for our point.

Bokulich ( 2018 ) refers to a similar function as “subtracting”; see also Norton and Suppe ( 2001 ).

Compare Filion and Moir ( 2018 , p. 739), who employ a related notion of reliability in a more technical context.

I’m implicitly assuming a “materialist” account of induction here (see Norton 2003 ), but I don’t think that this claim relies on any controversial aspect of Norton’s account.

Of course, Edman’s condition is an idealization. We might expect an unreliable thermometer (like an unreliable clock) to be relatively close. How an instrument is broken matters, and we’ll rarely have no information on that front. The same point can be made about theoretical instruments, however, and so the idealization shouldn’t undermine the point.

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Acknowledgements

I would like to thank James Nguyen, Anjan Chakravartty, audiences at Notre Dame and at PSA 2018 in Seattle, and two anonymous reviewers for comments on earlier versions of this essay.

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Dethier, C. How to Do Things with Theory: The Instrumental Role of Auxiliary Hypotheses in Testing. Erkenn 86 , 1453–1468 (2021). https://doi.org/10.1007/s10670-019-00164-9

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