Editing Imre Lakatos (section)

==Philosophical work==
=== Philosophy of mathematics ===
{{main|Proofs and Refutations}}

Lakatos's philosophy of mathematics was inspired by both [[Georg Hegel|Hegel]]'s and [[Karl Marx|Marx]]'s [[dialectic]], by [[Karl Popper]]'s theory of knowledge, and by the work of mathematician [[George Pólya]].

The 1976 book ''Proofs and Refutations'' is based on the first three chapters of his 1961 four-chapter doctoral thesis ''Essays in the Logic of Mathematical Discovery''. But its first chapter is Lakatos's own revision of its chapter&nbsp;1 that was first published as ''Proofs and Refutations'' in four parts in 1963–64 in the ''British Journal for the Philosophy of Science''. It is largely taken up by a fictional [[dialogue]] set in a mathematics class. The students are attempting to prove the formula for the [[Euler characteristic]] in [[algebraic topology]], which is a [[theorem]] about the properties of [[polyhedra]], namely that for all polyhedra the number of their vertices ''V'' minus the number of their edges ''E'' plus the number of their faces ''F'' is 2 ({{nobr|''V'' − ''E'' + ''F'' {{=}} 2}}). The dialogue is meant to represent the actual series of attempted proofs that mathematicians historically offered for the [[conjecture]], only to be repeatedly refuted by [[counterexample]]s. Often the students paraphrase famous mathematicians such as [[Cauchy]], as noted in Lakatos's extensive footnotes.

Lakatos termed the polyhedral counterexamples to Euler's formula ''monsters'' and distinguished three ways of handling these objects: Firstly, ''monster-barring'', by which means the theorem in question could not be applied to such objects. Secondly, ''monster-adjustment'', whereby by making a re-appraisal of the ''monster'' it could be ''made'' to obey the proposed theorem. Thirdly, ''exception handling'', a further distinct process. These distinct strategies have been taken up in qualitative physics, where the terminology of ''monsters'' has been applied to apparent counterexamples, and the techniques of ''monster-barring'' and ''monster-adjustment'' recognized as approaches to the refinement of the analysis of a physical issue.<ref>{{cite web |title=Lakatosian Monsters |url=http://harveycohen.net/dragons/Lakatosian_Monsters.htm |access-date=18 January 2015}}</ref>

What Lakatos tried to establish was that no theorem of [[informal mathematics]] is final or perfect. This means that we should not think that a theorem is ultimately true, only that no [[counterexample]] has yet been found. Once a counterexample is found, we adjust the theorem, possibly extending the domain of its validity. This is a continuous way our knowledge accumulates, through the logic and process of proofs and refutations. (If axioms are given for a branch of mathematics, however, Lakatos claimed that proofs from those [[axiom]]s were [[Tautology (logic)|tautological]], i.e. [[logical truth|logically true]].)<ref>See, for instance, Lakatos's ''A renaissance of empiricism in the recent philosophy of mathematics'', section 2, in which he defines a Euclidean system to be one consisting of all logical deductions from an initial set of axioms and writes that "a Euclidean system may be claimed to be true".</ref>

Lakatos proposed an account of mathematical knowledge based on the idea of [[heuristic]]s. In ''Proofs and Refutations'' the concept of "heuristic" was not well developed, although Lakatos gave several basic rules for finding proofs and counterexamples to conjectures. He thought that mathematical "[[thought experiment]]s" are a valid way to discover mathematical conjectures and proofs, and sometimes called his philosophy "quasi-[[empiricism]]".

However, he also conceived of the mathematical community as carrying on a kind of dialectic to decide which [[mathematical proof]]s are [[Validity (logic)|valid]] and which are not. Therefore, he fundamentally disagreed with the "[[formalism (mathematics)|formalist]]" conception of proof that prevailed in [[Gottlob Frege|Frege]]'s and [[Bertrand Russell|Russell]]'s [[logicism]], which defines proof simply in terms of ''formal'' validity.

On its first publication as an article in the ''British Journal for the Philosophy of Science'' in 1963–64, ''Proofs and Refutations'' became highly influential on new work in the philosophy of mathematics, although few agreed with Lakatos's strong disapproval of formal proof. Before his death he had been planning to return to the philosophy of mathematics and apply his theory of research programmes to it. Lakatos, Worrall and Zahar use [[Henri Poincaré|Poincaré]] (1893)<ref>Poincaré, H. (1893). "Sur la Généralisation d'un Théorème d'Euler relatif aux Polyèdres", ''Comptes Redus des Séances de l'Académie des Sciences'', '''117''' p. 144, as cited in Lakatos, Worrall and Zahar, p. 162.</ref> to answer one of the major problems perceived by critics, namely that the pattern of mathematical research depicted in ''Proofs and Refutations'' does not faithfully represent most of the actual activity of contemporary mathematicians.<ref>Lakatos, Worrall and Zahar (1976), ''Proofs and Refutations'' {{ISBN|0-521-21078-X}}, pp. 106–126, note that Poincaré's formal proof (1899) "Complèment à l'Analysis Situs", ''Rediconti del Circolo Matematico di Palermo'', '''13''', pp. 285–343, [[Rewriting|rewrite]]s Euler's conjecture into a [[tautology (logic)|tautology]] of vector algebra.</ref>

==== Cauchy and uniform convergence ====
In a 1966 text ''Cauchy and the continuum'', Lakatos re-examines the history of the calculus, with special regard to [[Augustin-Louis Cauchy]] and the concept of uniform convergence, in the light of [[non-standard analysis]]. Lakatos is concerned that historians of mathematics should not judge the evolution of mathematics in terms of currently fashionable theories. As an illustration, he examines Cauchy's proof that the sum of a series of continuous functions is itself continuous. Lakatos is critical of those who would see Cauchy's proof, with its failure to make explicit a suitable convergence hypothesis, merely as an inadequate approach to Weierstrassian analysis. Lakatos sees in such an approach a failure to realize that Cauchy's concept of the continuum differed from currently dominant views.

===Research programmes===<!-- This section is linked from [[Quasi-empiricism in mathematics]] -->
Lakatos's second major contribution to the philosophy of science was his model of the "research programme",<ref>Lakatos, Imre. (1970). "Falsification and the methodology of scientific research programmes". In: Lakatos, Musgrave eds. (1970), pp. 91–195.</ref> which he formulated in an attempt to resolve the perceived conflict between [[Karl Popper|Popper's]] [[falsifiability|falsificationism]] and the revolutionary structure of science described by [[Thomas Samuel Kuhn|Kuhn]]. Popper's standard of falsificationism was widely taken to imply that a theory should be abandoned as soon as any evidence appears to challenge it, while Kuhn's descriptions of scientific activity were taken to imply that science is most fruitful during periods in which popular, or "normal", theories are supported despite known anomalies. Lakatos's model of the research programme aims to combine Popper's adherence to empirical validity with Kuhn's appreciation for conventional consistency.

A Lakatosian research programme<ref>Bruce J. Caldwell (1991) "[http://public.econ.duke.edu/~bjc18/docs/MSRP%20in%20Economics.pdf The Methodology of Scientific Research Programmes: Criticisms and Conjectures]" in G. K. Shaw ed. (1991) ''Economics, Culture, and Education: Essays in Honor of Mark Blaug''. Aldershot: Elgar, 1991 pp. 95–107.</ref> is based on a ''hard core'' of theoretical assumptions that cannot be abandoned or altered without abandoning the programme altogether. More modest and specific theories that are formulated in order to explain evidence that threatens the "hard core" are termed ''auxiliary hypotheses''. Auxiliary hypotheses are considered expendable by the adherents of the research programme—they may be altered or abandoned as empirical discoveries require in order to "protect" the "hard core". Whereas Popper was generally read as hostile toward such {{lang|la|ad hoc}} theoretical amendments, Lakatos argued that they can be ''progressive'', i.e. productive, when they enhance the programme's explanatory and/or predictive power, and that they are at least permissible until some better system of theories is devised and the research programme is replaced entirely. The difference between a ''progressive'' and a ''degenerative'' research programme lies, for Lakatos, in whether the recent changes to its auxiliary hypotheses have achieved this greater explanatory/predictive power or whether they have been made simply out of the necessity of offering some response in the face of new and troublesome evidence. A degenerative research programme indicates that a new and more progressive system of theories should be sought to replace the currently prevailing one, but until such a system of theories can be conceived of and agreed upon, abandonment of the current one would only further weaken our explanatory power and was therefore unacceptable for Lakatos. Lakatos's primary example of a research programme that had been successful in its time and then progressively replaced is that founded by [[Isaac Newton]], with his three [[Newton's laws of motion|laws of motion]] forming the "hard core".

The Lakatosian research programme deliberately provides a framework within which research can be conducted on the basis of "first principles" (the "hard core"), which are shared by those involved in the research programme and accepted for the purpose of that research without further proof or debate. In this regard, it is similar to Kuhn's notion of a paradigm. Lakatos sought to replace Kuhn's paradigm, guided by an irrational "psychology of discovery", with a research programme no less coherent or consistent, yet guided by Popper's objectively valid [[The Logic of Scientific Discovery|logic of discovery]].

Lakatos was following [[Pierre Duhem]]'s idea that one can always protect a cherished theory (or part of one) from hostile evidence by redirecting the criticism toward other theories or parts thereof. (See ''[[Confirmation holism]]'' and [[Duhem–Quine thesis]]). This aspect of falsification had been acknowledged by Popper.

[[Karl Popper|Popper]]'s theory, falsificationism, proposed that scientists put forward theories and that nature "shouts NO" in the form of an inconsistent observation. According to Popper, it is irrational for scientists to maintain their theories in the face of nature's rejection, as Kuhn had described them doing. For Lakatos, however, "It is not that we propose a theory and Nature may shout NO; rather, we propose a maze of theories, and nature may shout INCONSISTENT".<ref>Lakatos, Musgrave eds. (1970), p. 130.</ref> The continued adherence to a programme's "hard core", augmented with adaptable auxiliary hypotheses, reflects Lakatos's less strict standard of falsificationism.

Lakatos saw himself as merely extending Popper's ideas, which changed over time and were interpreted by many in conflicting ways. In his 1968 article "Criticism and the Methodology of Scientific Research Programmes",<ref name=La68>Lakatos, Imre. (1968). "Criticism and the Methodology of Scientific Research Programmes". ''Proceedings of the Aristotelian Society'' '''69'''(1):149–186 (1968).</ref> Lakatos contrasted ''Popper0'', the "naive falsificationist" who demanded unconditional rejection of any theory in the face of any anomaly (an interpretation Lakatos saw as erroneous but that he nevertheless referred to often); ''Popper1'', the more nuanced and conservatively interpreted philosopher; and ''Popper2'', the "sophisticated methodological falsificationist" that Lakatos claims is the logical extension of the correctly interpreted ideas of ''Popper1'' (and who is therefore essentially Lakatos himself). It is, therefore, very difficult to determine which ideas and arguments concerning the research programme should be credited to whom.

While Lakatos dubbed his theory "sophisticated methodological falsificationism", it is not "methodological" in the strict sense of asserting universal methodological rules by which all scientific research must abide. Rather, it is methodological only in that theories are only abandoned according to a methodical progression from worse theories to better theories—a stipulation overlooked by what Lakatos terms "dogmatic falsificationism". Methodological assertions in the strict sense, pertaining to which methods are valid and which are invalid, are, themselves, contained within the research programmes that choose to adhere to them, and should be judged according to whether the research programmes that adhere to them prove progressive or degenerative. Lakatos divided these "methodological rules" within a research programme into its "negative heuristics", i.e., what research methods and approaches to avoid, and its "positive heuristics", i.e., what research methods and approaches to prefer. While the "negative heuristic" protects the hard core, the "positive heuristic" directs the modification of the hard core and auxiliary hypotheses in a general direction.<ref>{{Cite book |title=Great readings in clinical science: essential selections for mental health professionals |date=2012 |publisher=Pearson |others=Lilienfeld, Scott O., 1960–, O'Donohue, William T. |isbn=9780205698035 |location=Boston |oclc=720560483}}</ref>

Lakatos claimed that not all changes of the auxiliary hypotheses of a research programme (which he calls "problem shifts") are equally productive or acceptable. He took the view that these "problem shifts" should be evaluated not just by their ability to defend the "hard core" by explaining apparent anomalies, but also by their ability to produce new facts, in the form of predictions or additional explanations.<ref>Theoretical progressiveness is if the new theory has more empirical content than the old. Empirical progressiveness is if some of this content is corroborated. (Lakatos ed., 1970, p. 118).</ref> Adjustments that accomplish nothing more than the maintenance of the "hard core" mark the research programme as degenerative.

Lakatos's model provides for the possibility of a research programme that is not only continued in the presence of troublesome anomalies but that remains progressive despite them. For Lakatos, it is essentially necessary to continue on with a theory that we basically know cannot be completely true, and it is even possible to make scientific progress in doing so, as long as we remain receptive to a better research programme that may eventually be conceived of. In this sense, it is, for Lakatos, an acknowledged misnomer to refer to "falsification" or "refutation", when it is not the truth or falsity of a theory that is solely determining whether we consider it "falsified", but also the availability of a ''less false'' theory. A theory cannot be rightfully "falsified", according to Lakatos, until it is superseded by a better (i.e. more progressive) research programme. This is what he says is happening in the historical periods Kuhn describes as revolutions and what makes them rational as opposed to mere leaps of faith or periods of deranged social psychology, as Kuhn argued.

===Pseudoscience===
According to the [[Demarcation problem|demarcation]] criterion of [[pseudoscience]] proposed by Lakatos, a theory is pseudoscientific if it fails to make any novel predictions of previously unknown phenomena or its predictions were mostly falsified, in contrast with scientific theories, which predict novel fact(s).<ref>See/hear Lakatos's 1973 Open University BBC Radio talk [https://www.lse.ac.uk/philosophy/science-and-pseudoscience-overview-and-transcript ''Science and Pseudoscience'' ].</ref> Progressive scientific theories are those that have their novel facts confirmed, and degenerate scientific theories, which can degenerate so much that they become pseudo-science, are those whose predictions of novel facts are refuted. As he put it:
: "A given fact is explained scientifically only if a new fact is predicted with it&nbsp;... The idea of growth and the concept of empirical character are soldered into one." See pages 34–35 of ''The Methodology of Scientific Research Programmes'', 1978.

Lakatos's own key examples of pseudoscience were [[Ptolemaic system|Ptolemaic]] astronomy, [[Immanuel Velikovsky]]'s planetary cosmogony, [[Freud]]ian [[psychoanalysis]], 20th-century [[Soviet Marxism|''Soviet'' Marxism]],<ref>Lakatos notably only condemned specifically ''Soviet'' Marxism as pseudoscientific, as opposed to Marxism in general. In fact, at the very end of his last LSE lectures on Scientific Method in 1973, he finished by posing the question of whether [[Trotsky]]'s theoretical development of Marxism was scientific, and commented that "Nobody has ever undertaken a critical history of Marxism with the aid of better methodological and historiographical instruments. Nobody has ever tried to find an answer to questions like: were Trotsky's unorthodox predictions simply patching up a badly degenerating programme, or did they represent a creative development of Marx's programme? To answer similar questions, we would really need a detailed analysis which takes years of work. So I simply do not know the answer, even if I am very interested in it." [Motterlini 1999, p. 109] However, in his 1976 ''On the Critique of Scientific Reason'' Feyerabend claimed that [[Vladimir Lenin]]'s development of Marxism in his auxiliary theory of colonial exploitation had been "Lakatos-scientific" because it was "accompanied by a wealth of novel predictions (the arrival and structure of monopolies being one of them)". And he continued by claiming that both Rosa Luxemburg's and Trotsky's developments of Marxism were close to what Lakatos regarded as scientific: "And whoever has read Rosa Luxemburg's reply to Bernstein's criticism of Marx or Trotsky's account of why the Russian Revolution took place in a backward country (cf. also Lenin [1968], vol. 19, pp. 99ff.) will see that Marxists are pretty close to what Lakatos would like any upstanding rationalist to do&nbsp;..." [See footnote 9 of p. 315 of Howson (ed.) 1976].</ref> [[Lysenkoism|Lysenko's biology]], [[Niels Bohr]]'s quantum mechanics post-1924, [[astrology]], [[psychiatry]], and [[neoclassical economics]].

====Darwin's theory====
In his 1973 Scientific Method Lecture 1<ref>Published in ''For and Against Method: Imre Lakatos and Paul Feyerabend'' by Matteo Motterlini (ed.), University of Chicago Press, 1999.</ref> at the London School of Economics, he also claimed that "nobody to date has yet found a demarcation criterion according to which Darwin can be described as scientific".

Almost 20 years after Lakatos's 1973 challenge to the scientificity of [[Charles Darwin|Darwin]], in her 1991 ''The Ant and the Peacock'', LSE lecturer and ex-colleague of Lakatos, [[Helena Cronin]], attempted to establish that Darwinian theory was empirically scientific in respect of at least being supported by evidence of likeness in the diversity of life forms in the world, explained by descent with modification. She wrote that
<blockquote>
our usual idea of corroboration as requiring the successful prediction of novel facts&nbsp;... Darwinian theory was not strong on temporally novel predictions.&nbsp;... however familiar the evidence and whatever role it played in the construction of the theory, it still confirms the theory.<ref>Cronin, H., ''The Ant and the Peacock: Altruism and Sexual Selection from Darwin to Today'', Cambridge University Press, 1993. pp. [https://books.google.com/books?id=y0wTY3z3nggC&q=%22For+darwinian+theory%2C+the+evidence%22 31–32].</ref>
</blockquote>

===Rational reconstructions of the history of science===
In his 1970 article "History of Science and Its Rational Reconstructions"<ref name=":0">{{cite journal |author=Lakatos, Imre |date=1970 |title=History of Science and Its Rational Reconstructions |journal=PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association |volume=1970 |pages=91–136 |doi=10.1086/psaprocbienmeetp.1970.495757 |jstor=495757 |s2cid=145197122 |url=https://www.jstor.org/stable/495757}}</ref> Lakatos proposed a dialectical historiographical meta-method for evaluating different theories of scientific method, namely by means of their comparative success in explaining the actual [[history of science]] and [[scientific revolution]]s on the one hand, whilst on the other providing a historiographical framework for rationally reconstructing the history of science as anything more than merely inconsequential rambling. The article started with his now renowned dictum "Philosophy of science without history of science is empty; history of science without philosophy of science is blind".

However, neither Lakatos himself nor his collaborators ever completed the first part of this dictum by showing that in any scientific revolution the great majority of the relevant scientific community converted just when Lakatos's criterion – one programme successfully predicting some novel facts whilst its competitor degenerated – was satisfied. Indeed, for the historical case studies in his 1968 article "Criticism and the Methodology of Scientific Research Programmes"<ref name=La68/> he had openly admitted as much, commenting: "In this paper it is not my purpose to go on seriously to the second stage of comparing [[rational reconstruction]]s with actual history for any lack of historicity."