Imre Lakatos (UK: /ˈlækətɒs/, US: /-toʊs/; Hungarian: Lakatos Imre [ˈlɒkɒtoʃ ˈimrɛ]; November 9, 1922 – February 2, 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes.
Imre Lakatos, c. 1960s
|Born||November 9, 1922|
|Died||February 2, 1974 (aged 51)|
|Alma mater||University of Debrecen|
Moscow State University
University of Cambridge
|Philosophy of mathematics, philosophy of science, history of science, epistemology, politics|
|Method of proofs and refutations, methodology of scientific research programmes, methodology of historiographical research programmes, positive vs. negative heuristics, progressive vs. degenerative research programmes, rational reconstruction, mathematical quasi-empiricism, criticism of logical positivism and formalism|
Lakatos was born Imre (Avrum) Lipschitz to a Jewish family in Debrecen, Hungary in 1922. He received a degree in mathematics, physics, and philosophy from the University of Debrecen in 1944. In March 1944 the Germans invaded Hungary and Lakatos along with Éva Révész, his then-girlfriend and subsequent wife, formed soon after that event a Marxist resistance group. In May of that year, the group was joined by Éva Izsák, a 19-year-old Jewish antifascist activist. Lakatos, considering that there was a risk that she would be captured and forced to betray them, decided that her duty to the group was to commit suicide. Subsequently, a member of the group took her to Debrecen and gave her cyanide.
During the occupation, Lakatos avoided Nazi persecution of Jews by changing his name to Imre Molnár. His mother and grandmother died in Auschwitz. He changed his surname once again to Lakatos (Locksmith) in honor of Géza Lakatos.
After the war, from 1947, he worked as a senior official in the Hungarian ministry of education. He also continued his education with a PhD at Debrecen University awarded in 1948, and also attended György Lukács's weekly Wednesday afternoon private seminars. He also studied at the Moscow State University under the supervision of Sofya Yanovskaya in 1949. When he returned, however, he found himself on the losing side of internal arguments within the Hungarian communist party and was imprisoned on charges of revisionism from 1950 to 1953. More of Lakatos' activities in Hungary after World War II have recently become known. In fact, Lakatos was a hardline Stalinist and, despite his young age, had an important role between 1945 and 1950 (his own arrest and jailing) in building up the Communist rule, especially in cultural life and the academia, in Hungary. Preceding his fleeing to Vienna he confessed he has worked as an informer of State Protection Authority.
After his release, Lakatos returned to academic life, doing mathematical research and translating George Pólya's How to Solve It into Hungarian. Still nominally a communist, his political views had shifted markedly and he was involved with at least one dissident student group in the lead-up to the 1956 Hungarian Revolution.
After the Soviet Union invaded Hungary in November 1956, Lakatos fled to Vienna, and later reached England. He received a PhD in philosophy in 1961 from the University of Cambridge; his thesis advisor was R. B. Braithwaite. The book Proofs and Refutations: The Logic of Mathematical Discovery, published after his death, is based on this work.
Lakatos never obtained British citizenship. In 1960, he was appointed to a position in the London School of Economics, where he wrote on the philosophy of mathematics and the philosophy of science. The LSE philosophy of science department at that time included Karl Popper, Joseph Agassi and J. O. Wisdom. It was Agassi who first introduced Lakatos to Popper under the rubric of his applying a fallibilist methodology of conjectures and refutations to mathematics in his Cambridge PhD thesis.
With co-editor Alan Musgrave, he edited the often cited Criticism and the Growth of Knowledge, the Proceedings of the International Colloquium in the Philosophy of Science, London, 1965. Published in 1970, the 1965 Colloquium included well-known speakers delivering papers in response to Thomas Kuhn's The Structure of Scientific Revolutions.
In January 1971, he became editor of the British Journal for the Philosophy of Science, which J. O. Wisdom had built up before departing in 1965, and he continued as editor until his death in 1974, after which it was then edited jointly for many years by his LSE colleagues John W. N. Watkins and John Worrall, Lakatos's ex-research assistant.
His last LSE lectures in scientific method in Lent Term 1973 along with parts of his correspondence with his friend and critic Paul Feyerabend have been published in For and Against Method (ISBN 0-226-46774-0).
Lakatos and his colleague Spiro Latsis organized an international conference devoted entirely to historical case studies in Lakatos's methodology of research programmes in physical sciences and economics, to be held in Greece in 1974, and which still went ahead following Lakatos's death in February 1974. These case studies in such as Einstein's relativity programme, Fresnel's wave theory of light and neoclassical economics, were published by Cambridge University Press in two separate volumes in 1976, one devoted to physical sciences and Lakatos's general programme for rewriting the history of science, with a concluding critique by his great friend Paul Feyerabend, and the other devoted to economics.
The 1976 book Proofs and Refutations is based on the first three chapters of his four chapter 1961 doctoral thesis Essays in the logic of mathematical discovery. But its first chapter is Lakatos's own revision of its chapter 1 that was first published as Proofs and Refutations in four parts in 1963–4 in The British Journal for the Philosophy of Science. It is largely taken up by a fictional dialogue set in a mathematics class. The students are attempting to prove the formula for the Euler characteristic in algebraic topology, which is a theorem about the properties of polyhedra, namely that for all polyhedra the number of their Vertices minus the number of their Edges plus the number of their Faces is 2: (V – E + F = 2). The dialogue is meant to represent the actual series of attempted proofs which mathematicians historically offered for the conjecture, only to be repeatedly refuted by counterexamples. Often the students paraphrase famous mathematicians such as Cauchy, as noted in Lakatos's extensive footnotes.
Lakatos termed the polyhedral counter examples to Euler's formula monsters and distinguished three ways of handling these objects: Firstly, monster-barring, by which means the theorem in question could not be applied to such objects. Secondly, monster-adjustment whereby by making a re-appraisal of the monster it could be made to obey the proposed theorem. Thirdly, exception handling, a further distinct process. These distinct strategies have been taken up in qualitative physics, where the terminology of monsters has been applied to apparent counter-examples, and the techniques of monster-barring and monster-adjustment recognized as approaches to the refinement of the analysis of a physical issue.
What Lakatos tried to establish was that no theorem of informal mathematics is final or perfect. This means that we should not think that a theorem is ultimately true, only that no counterexample has yet been found. Once a counterexample, i.e. an entity contradicting/not explained by the theorem is found, we adjust the theorem, possibly extending the domain of its validity. This is a continuous way our knowledge accumulates, through the logic and process of proofs and refutations. (If axioms are given for a branch of mathematics, however, Lakatos claimed that proofs from those axioms were tautological, i.e. logically true.)
Lakatos proposed an account of mathematical knowledge based on the idea of heuristics. In Proofs and Refutations the concept of 'heuristic' was not well developed, although Lakatos gave several basic rules for finding proofs and counterexamples to conjectures. He thought that mathematical 'thought experiments' are a valid way to discover mathematical conjectures and proofs, and sometimes called his philosophy 'quasi-empiricism'.
However, he also conceived of the mathematical community as carrying on a kind of dialectic to decide which mathematical proofs are valid and which are not. Therefore, he fundamentally disagreed with the 'formalist' conception of proof which prevailed in Frege's and Russell's logicism, which defines proof simply in terms of formal validity.
On its first publication as a paper in The British Journal for the Philosophy of Science in 1963–4, Proofs and Refutations became highly influential on new work in the philosophy of mathematics, although few agreed with Lakatos' strong disapproval of formal proof. Before his death he had been planning to return to the philosophy of mathematics and apply his theory of research programmes to it. Lakatos, Worrall and Zahar use Poincaré (1893) to answer one of the major problems perceived by critics, namely that the pattern of mathematical research depicted in Proofs and Refutations does not faithfully represent most of the actual activity of contemporary mathematicians.
In a 1966 text published as (Lakatos 1978), Lakatos re-examines the history of the calculus, with special regard to Augustin-Louis Cauchy and the concept of uniform convergence, in the light of non-standard analysis. Lakatos is concerned that historians of mathematics should not judge the evolution of mathematics in terms of currently fashionable theories. As an illustration, he examines Cauchy's proof that the sum of a series of continuous functions is itself continuous. Lakatos is critical of those who would see Cauchy's proof, with its failure to make explicit a suitable convergence hypothesis, merely as an inadequate approach to Weierstrassian analysis. Lakatos sees in such an approach a failure to realize that Cauchy's concept of the continuum differed from currently dominant views.
Lakatos's second major contribution to the philosophy of science was his model of the 'research programme', which he formulated in an attempt to resolve the perceived conflict between Popper's falsificationism and the revolutionary structure of science described by Kuhn. Popper's standard of falsificationism was widely taken to imply that a theory should be abandoned as soon as any evidence appears to challenge it, while Kuhn's descriptions of scientific activity were taken to imply that science was most constructive when it upheld a system of popular, or 'normal', theories, despite anomalies. Lakatos' model of the research programme aims to combine Popper's adherence to empirical validity with Kuhn's appreciation for conventional consistency.
A Lakatosian research programme is based on a hard core of theoretical assumptions that cannot be abandoned or altered without abandoning the programme altogether. More modest and specific theories that are formulated in order to explain evidence that threatens the 'hard core' are termed auxiliary hypotheses. Auxiliary hypotheses are considered expendable by the adherents of the research programme—they may be altered or abandoned as empirical discoveries require in order to 'protect' the 'hard core'. Whereas Popper was generally read as hostile toward such ad hoc theoretical amendments, Lakatos argued that they can be progressive, i.e. productive, when they enhance the programme's explanatory and/or predictive power, and that they are at least permissible until some better system of theories is devised and the research programme is replaced entirely. The difference between a progressive and a degenerative research programme lies, for Lakatos, in whether the recent changes to its auxiliary hypotheses have achieved this greater explanatory/predictive power or whether they have been made simply out of the necessity of offering some response in the face of new and troublesome evidence. A degenerative research programme indicates that a new and more progressive system of theories should be sought to replace the currently prevailing one, but until such a system of theories can be conceived of and agreed upon, abandonment of the current one would only further weaken our explanatory power and was therefore unacceptable for Lakatos. Lakatos's primary example of a research programme that had been successful in its time and then progressively replaced is that founded by Isaac Newton, with his three laws of motion forming the 'hard core'.
The Lakatosian research programme deliberately provides a framework within which research can be conducted on the basis of 'first principles' (the 'hard core') which are shared by those involved in the research programme and accepted for the purpose of that research without further proof or debate. In this regard, it is similar to Kuhn's notion of a paradigm. Lakatos sought to replace Kuhn's paradigm, guided by an irrational 'psychology of discovery', with a research programme no less coherent or consistent yet guided by Popper's objectively valid logic of discovery.
Lakatos was following Pierre Duhem's idea that one can always protect a cherished theory (or part of one) from hostile evidence by redirecting the criticism toward other theories or parts thereof. (See Confirmation holism and Duhem–Quine thesis). This aspect of falsification had been acknowledged by Popper.
Popper's theory, falsificationism, proposed that scientists put forward theories and that nature 'shouts NO' in the form of an inconsistent observation. According to Popper, it is irrational for scientists to maintain their theories in the face of Nature's rejection, as Kuhn had described them doing. For Lakatos, however, "It is not that we propose a theory and Nature may shout NO; rather, we propose a maze of theories, and nature may shout INCONSISTENT". The continued adherence to a programme's 'hard core', augmented with adaptable auxiliary hypotheses, reflects Lakatos's less strict standard of falsificationism.
Lakatos saw himself as merely extending Popper's ideas, which changed over time and were interpreted by many in conflicting ways. In his 1968 paper "Criticism and the Methodology of Scientific Research Programmes", Lakatos contrasted Popper0, the "naive falsificationist" who demanded unconditional rejection of any theory in the face of any anomaly (an interpretation Lakatos saw as erroneous but that he nevertheless referred to often); Popper1, the more nuanced and conservatively interpreted philosopher; and Popper2, the "sophisticated methodological falsificationist" that Lakatos claims is the logical extension of the correctly interpreted ideas of Popper1 (and who is therefore essentially Lakatos himself). It is, therefore, very difficult to determine which ideas and arguments concerning the research programme should be credited to whom.
While Lakatos dubbed his theory "sophisticated methodological falsificationism", it is not "methodological" in the strict sense of asserting universal methodological rules by which all scientific research must abide. Rather, it is methodological only in that theories are only abandoned according to a methodical progression from worse theories to better theories—a stipulation overlooked by what Lakatos terms "dogmatic falsificationism". Methodological assertions in the strict sense, pertaining to which methods are valid and which are invalid, are, themselves, contained within the research programmes that choose to adhere to them, and should be judged according to whether the research programmes that adhere to them prove progressive or degenerative. Lakatos divided these 'methodological rules' within a research programme into its 'negative heuristics', i.e., what research methods and approaches to avoid, and its 'positive heuristics', i.e., what research methods and approaches to prefer. While the 'negative heuristic' protects the hard core, the 'positive heuristic' directs the modification of the hard core and auxiliary hypotheses in a general direction.
Lakatos claimed that not all changes of the auxiliary hypotheses of a research programme (which he calls 'problem shifts') are equally productive or acceptable. He took the view that these 'problem shifts' should be evaluated not just by their ability to defend the 'hard core' by explaining apparent anomalies, but also by their ability to produce new facts, in the form of predictions or additional explanations. Adjustments that accomplish nothing more than the maintenance of the 'hard core' mark the research programme as degenerative.
Lakatos' model provides for the possibility of a research programme that is not only continued in the presence of troublesome anomalies but that remains progressive despite them. For Lakatos, it is essentially necessary to continue on with a theory that we basically know cannot be completely true, and it is even possible to make scientific progress in doing so, as long as we remain receptive to a better research programme that may eventually be conceived of. In this sense, it is, for Lakatos, an acknowledged misnomer to refer to 'falsification' or 'refutation', when it is not the truth or falsity of a theory that is solely determining whether we consider it 'falsified', but also the availability of a less false theory. A theory cannot be rightfully 'falsified', according to Lakatos, until it is superseded by a better (i.e. more progressive) research programme. This is what he says is happening in the historical periods Kuhn describes as revolutions and what makes them rational as opposed to mere leaps of faith or periods of deranged social psychology, as Kuhn argued.
According to the demarcation criterion of pseudoscience 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). Progressive scientific theories are those which have their novel facts confirmed and degenerate scientific theories, which can degenerate so much 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....The idea of growth and the concept of empirical character are soldered into one." See pages 34–5 of The Methodology of Scientific Research Programmes, 1978.
Lakatos's own key examples of pseudoscience were Ptolemaic astronomy, Immanuel Velikovsky's planetary cosmogony, Freudian psychoanalysis, 20th century Soviet Marxism, Lysenko's biology, Niels Bohr's Quantum Mechanics post-1924, astrology, psychiatry, sociology, neoclassical economics, and Darwin's theory.
In his 1973 Scientific Method Lecture 1 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 Darwin, in her 1991 The Ant and the Peacock, LSE lecturer and ex-colleague of Lakatos, Helena Cronin, attempted to establish that Darwinian theory was empirically scientific in respect of at least being supported by evidence of likeness in the diversity of life forms in the world, explained by descent with modification. She wrote that
our usual idea of corroboration as requiring the successful prediction of novel facts...Darwinian theory was not strong on temporally novel predictions. ... however familiar the evidence and whatever role it played in the construction of the theory, it still confirms the theory.
In his 1970 paper "History of Science and Its Rational Reconstructions" Lakatos proposed a dialectical historiographical meta-method for evaluating different theories of scientific method, namely by means of their comparative success in explaining the actual history of science and scientific revolutions on the one hand, whilst on the other providing a historiographical framework for rationally reconstructing the history of science as anything more than merely inconsequential rambling. The paper started with his now renowned dictum "Philosophy of science without history of science is empty; history of science without philosophy of science is blind."
However, neither Lakatos himself nor his collaborators ever completed the first part of this dictum by showing that in any scientific revolution the great majority of the relevant scientific community converted just when Lakatos's criterion – one programme successfully predicting some novel facts whilst its competitor degenerated – was satisfied. Indeed, for the historical case studies in his 1968 paper "Criticism and the Methodology of Scientific Research Programmes" he had openly admitted as much, commenting 'In this paper it is not my purpose to go on seriously to the second stage of comparing rational reconstructions with actual history for any lack of historicity.'
Paul Feyerabend argued that Lakatos's methodology was not a methodology at all, but merely "words that sound like the elements of a methodology." He argued that Lakatos's methodology was no different in practice from epistemological anarchism, Feyerabend's own position. He wrote in Science in a Free Society (after Lakatos's death) that:
Lakatos realized and admitted that the existing standards of rationality, standards of logic included, were too restrictive and would have hindered science had they been applied with determination. He therefore permitted the scientist to violate them (he admits that science is not "rational" in the sense of these standards). However, he demanded that research programmes show certain features in the long run — they must be progressive.... I have argued that this demand no longer restricts scientific practice. Any development agrees with it.
Lakatos and Feyerabend planned to produce a joint work in which Lakatos would develop a rationalist description of science and Feyerabend would attack it. The correspondence between Lakatos and Feyerabend, where the two discussed the project, has since been reproduced, with commentary, by Matteo Motterlini.
1974 in philosophyAcceptability
Acceptability is the characteristic of a thing being subject to acceptance for some purpose. A thing is acceptable if it is sufficient to serve the purpose for which it is provided, even if it is far less usable for this purpose than the ideal example. A thing is unacceptable (or has the characteristic of unacceptability) if it deviates so far from the ideal that it is no longer sufficient to serve the desired purpose, or if it goes against that purpose. From a logical perspective, a thing can be said to be acceptable if it has no characteristics that make it unacceptable:
We say that a theory Δ is acceptable if for any wff α, Δ does not prove both α and ¬α.
Hungarian mathematician Imre Lakatos developed a concept of acceptability "taken as a measure of the approximation to the truth". This concept was criticized in its applicability to philosophy as requiring that better theories first be eliminated. Acceptability is also a key premise of negotiation, wherein opposing sides each begin from a point of seeking their ideal solution, and compromise until they reach a solution that both sides find acceptable:
When a proposal or counter-proposal is received by an agent, it has to decide whether it is acceptable. If it is, the agent can agree to it; if not, and alternative that is acceptable to the receiving agent needs to be generated. Acceptability is determined by searching the hierarchy. If the proposal is a specification of at least one acceptable goal, the proposal is acceptable. If it is the specification of at least one unacceptable goal, the proposal is clearly unacceptable.
Where an unacceptable proposal has been made, "a counterproposal is generated if there are any acceptable ones that have had already been explored". Since the acceptability of proposition to a participant in a negotiation is only known to that participant, the participant may act as though a proposal that is actually acceptable to them is not, in order to obtain a more favorable proposal.Bold hypothesis
Bold hypothesis (or "bold conjecture") is a concept in the philosophy of science of Karl Popper, first explained in his debut The Logic of Scientific Discovery (1935) and subsequently elaborated in writings such as Conjectures and Refutations: The Growth of Scientific Knowledge (1963). The concept is nowadays widely used in the philosophy of science and in the philosophy of knowledge. It is also used in the social and behavioural sciences.Colin Howson
Colin Howson (born 1945) is a British philosopher. He is Professor of Philosophy at the University of Toronto, where he joined the faculty on July 1, 2008. Previously, he was Professor of Logic at the London School of Economics. He completed a PhD on the philosophy of probability in 1981. In the late 1960s he had been a research assistant of Imre Lakatos at LSE.Imre
Imre is a Hungarian masculine first name, which is also in Estonian use. The origin of the name is not clear. Some argue that it derived from the Gothic Amalareiks, or from the High German Emmerich whose Latinized version is Emericus. Its English equivalent is Henry.
Bearers of the name include the following (who generally held Hungarian nationality, unless otherwise noted):
Imre Ámos (1907–1944/45), painter
Imre Antal (1935–2008), pianist
Imre Bajor (1957—2014), actor
Imre Bródy (1891–1944), physicist
Imre Bujdosó (b. 1959), Olympic fencer
Imre Csáky (cardinal) (1672–1732), Roman Catholic cardinal
Imre Csermelyi (b. 1988), football player
Imre Cseszneky (1804–1874), agriculturist and patriot
Imre Csiszár (b. 1938), mathematician
Imre Csösz (b. 1969), Olympic judoka
Imre Czomba (b. 1972), Composer and musician
Imre Deme (b. 1983), football player
Imre Erdődy (1889–1973), Olympic gymnast
Imre Farkas (1879–1976), musician
Imre Farkas (b. 1935), Olympic canoeist
Imre Finta (1911–2003), indicted war criminal
Imre Földi, Olympic weightlifter
Imre Friedmann (1921–2007), biologist
Imre Frivaldszky (1799–1870), botanist and entomologist
Imre Garaba (b. 1958), football player
Imre Gedővári (b. 1951), Olympic fencer
Imre Gellért (1888–1981), Olympic gymnast
Imre Gyöngyössy (1930–1994), film director and screenwriter
Imre Harangi (1913–1979), Olympic boxer
Imre Hódos (1928–1989), Olympic wrestler
Imre Hollai (b. 1925), diplomat, President of the United Nations General Assembly
Imre Jenei (b. 1937), Romanian (Hungarian ethnic) football player and coach
Imre Kálmán (1882–1953), operetta composer
Imre Kertész (1929–2016), author and winner of the 2002 Nobel Prize in Literature
Imre König (1899–1992), chess master
Imre Komora (b. 1940), football player
Imre Lakatos (1922–1974), philosopher of mathematics and science
Imre Leader (b. 1963), British mathematician
Imre Madách (1823–1864), writer, poet, lawyer and politician
Imre Makovecz (b. 1935), architect
Imre Mándi (1916–1945), Olympic boxer
Imre Mudin (1887–1918), Olympic track and field athlete
Imre Nagy (1896–1958), politician, twice Prime Minister of Hungary, key figure of the Hungarian Revolution of 1956
Imre Nagy (b. 1933), Olympic pentathlete
Imre Németh (1917–1989), Olympic hammer thrower
Imre of Hungary (ca. 1000-1007–1031), prince and Roman Catholic saint
Imre of Hungary (1174–1204), King of Hungary
Imre Páli (1909–?), Olympic handballer
Imre Polyák (b. 1932), Olympic wrestler
Imre Pozsgay (b. 1933), reform Communist politician
Imre Pulai (b. 1967), Olympic canoer
Imre Rapp (b. 1937), football player
Imre Salusinszky (b. 1955), Australian newspaper columnist
Imre Schlosser (1889–1959), football player
Imre Senkey (1898–?), football player and manager
Imre Steindl (1839–1902), architect
Imre Szabics (b. 1981), football player
Imre Szekeres (b. 1950), politician and Minister of Defence
Imre Szellő (b. 1983), Olympic boxer
Imre Szentpály (1904–1987), Olympic polo player
Imre Taveter (born 1967), Estonian sport sailor
Imre Thököly (1657–1705), statesman, leader of an anti-Habsburg uprising, Prince of Transylvania
Imre Tiidemann (born 1970), Estonian modern pentathlete
Imre Tiitsu (born 1980), Estonian ice sledge hockey player
Imre Tóth (b. 1985), Grand Prix motorcycle racer
Imre Varadi (b. 1959), English football player
Imre Weisshaus (1905–1987), Hungarian-French pianist
Imre Zachár (1890–1954), Olympic water polo player and swimmer
Imre Zámbó (1958–2001), pop singer by the name of Jimmy ZámbóImre Leader
Imre Bennett Leader is a British mathematician and Othello player. He is Professor of Pure Mathematics, specifically combinatorics, at the University of Cambridge.
He was educated at St Paul's School and at Trinity College, Cambridge, and in 1981 he was a member of the United Kingdom team at the International Mathematical Olympiad, where he won a silver medal. In 1999-2001, he led the UK IMO team as its chief trainer.
He has been the most consistently successful Othello player in Britain, winning the national championship 12 times between 1983 and 2016. In 1983, he was runner-up in the World Othello Championship, and in 1988, he was on the UK team that won the World Team Championship.In mathematics, his work has concentrated on combinatorics. He completed his PhD, entitled Discrete Isoperimetric Inequalities and Other Combinatorial Results, in 1989, supervised by Béla Bollobás. Godson of mathematical philosopher Imre Lakatos, he is currently a fellow of Trinity College, University of Cambridge.Informal mathematics
Informal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics. The philosopher Imre Lakatos in his Proofs and Refutations aimed to sharpen the formulation of informal mathematics, by reconstructing its role in nineteenth century mathematical debates and concept formation, opposing the predominant assumptions of mathematical formalism. Informality may not discern between statements given by inductive reasoning (as in approximations which are deemed "correct" merely because they are useful), and statements derived by deductive reasoning.Lakatos
Lakatos is a Hungarian surname (meaning locksmith), and may refer to:
Brent Lakatos, (born 1980), Canadian athlete
Géza Lakatos, a Hungarian general during World War II; briefly served as Prime Minister of Hungary
Imre Lakatos, a philosopher of mathematics and science; the Lakatos Award is named after him
Imre Schlosser-Lakatos, Hungarian footballer
Menyhért Lakatos Hungarian Romani writer
Pál Lakatos, Hungarian boxer
Roby Lakatos, a Romani (Gypsy) violinist from HungaryLakatos Award
The Lakatos Award is given annually for an outstanding contribution to the philosophy of science, widely interpreted. The contribution must be in the form of a book published in English during the previous six years.
The Award is in memory of Imre Lakatos and has been endowed by the Latsis Foundation. It is administered by the following committee:
The Director of the London School of Economics (Chairman)
Professor John Worrall (Convenor)
Professor Hans Albert
Professor Nancy Cartwright
Professor Adolf Grünbaum
Professor Philip Kitcher
Professor Alan Musgrave
Professor Michael RedheadThe Committee makes the Award on the advice of an independent and anonymous panel of selectors. The value of the Award is £10,000.
To take up an Award a successful candidate must visit the LSE and deliver a public lecture.List of philosophers of science
This is a chronological list of philosophers of science. For an alphabetical name-list, see Category:Philosophers of science.Odium theologicum
The Latin phrase odium theologicum (literally "theological hatred") is the name originally given to the often intense anger and hatred generated by disputes over theology. It has also been adopted to describe non-theological disputes of a rancorous nature.
John Stuart Mill, discussing the fallibility of the moral consensus in his essay "On Liberty" (1859) refers scornfully to the odium theologicum, saying that, in a sincere bigot, it is one of the most unequivocal cases of moral feeling. In this essay, he takes issue with those who rely on moral feeling rather than reasoned argument to justify their beliefs.Paradigm
In science and philosophy, a paradigm () is a distinct set of concepts or thought patterns, including theories, research methods, postulates, and standards for what constitutes legitimate contributions to a field.Postmodern mathematics
Postmodern mathematics is a thought developed as a result of postmodernism. The theory asserts that there is no such thing as ‘absolutism’ or ultimate truth in mathematics. It also declares that the term ‘mathematics’ can't be used to define a specific object. This thought emerged from the post modernistic idea of the relativity of truth.. The thought also maintains that the ideas in mathematics are subjective and that there is no ‘right’ answer to a mathematical question.Proofs and Refutations
Proofs and Refutations is a 1976 book by philosopher Imre Lakatos expounding his view of the progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron. A central theme is that definitions are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proofs. This gives mathematics a somewhat experimental flavour. At the end of the Introduction, Lakatos explains that his purpose is to challenge formalism in mathematics, and to show that informal mathematics grows by a logic of "proofs and refutations".Rational reconstruction
Rational reconstruction is a philosophical term with several distinct meanings. It is found in the work of Jürgen Habermas and Imre Lakatos.Research program
A research program (UK: research programme) is a professional network of scientists conducting basic research. The term was used by philosopher of science Imre Lakatos to blend and revise the normative model of science offered by Karl Popper's falsificationism and the descriptive model of science offered by Thomas Kuhn's normal science. Lakatos found falsificationism impractical and often not practiced, and found normal science—where a paradigm of science, mimicking an exemplar, extinguishes differing perspectives—more monopolistic than actual.
Lakatos found that many research programmes coexisted. Each had a hard core of theories immune to revision, surrounded by a protective belt of malleable theories. A research programme vies against others to be most progressive. Extending the research programme's theories into new domains is theoretical progress, and experimentally corroborating such is empirical progress, always refusing falsification of the research programme's hard core. A research programme might degenerate—lose progressiveness—but later return to progressiveness.Reuben Hersh
Reuben Hersh (born 1927) is an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. This work challenges and complements mainstream philosophy of mathematics. ("Hersh" is his professional or pen name. His family name is Reuben Laznovsky.)
After receiving a B.A. in English literature from Harvard University in 1946, Hersh spent a decade writing for Scientific American and working as a machinist. After losing his right thumb when working with a band saw, he decided to study mathematics at the Courant Institute of Mathematical Sciences. In 1962, he was awarded a Ph.D. in mathematics from New York University; his advisor was P.D. Lax. He has been affiliated with the University of New Mexico since 1964, where he is now professor emeritus.
Hersh has written a number of technical articles on partial differential equations, probability, random evolutions (example), and linear operator equations. He is the (co)author of four articles in Scientific American, and 12 articles in the Mathematical Intelligencer.
Hersh is best known as the coauthor with Philip J. Davis of The Mathematical Experience (1981), which won a National Book Award in Science.Hersh advocates what he calls a "humanist" philosophy of mathematics, opposed to both Platonism (so-called "realism") and its rivals nominalism/fictionalism/formalism. He holds that mathematics is real, and its reality is social-cultural-historical, located in the shared thoughts of those who learn it, teach it, and create it. His article "The Kingdom of Math is Within You" (a chapter in his Experiencing Mathematics, 2014) explains how mathematicians' proofs compel agreement, even when they are inadequate as formal logic. He sympathizes with the perspectives on mathematics of Imre Lakatos and Where Mathematics Comes From, George Lakoff and Rafael Nunez, Basic Books.Theory choice
Theory choice was a main problem in the philosophy of science in the early 20th century, and under the impact of the new and controversial theories of relativity and quantum physics, came to involve how scientists should choose between competing theories.
The classical answer would be to select the theory which was best verified, against which Karl Popper argued that competing theories should be subjected to comparative tests and the one chosen which survived the tests. If two theories could not, for practical reasons, be tested one should prefer the one with the highest degree of empirical content, said Popper in The Logic of Scientific Discovery.
Mathematician and physicist Henri Poincaré instead, like many others, proposed simplicity as a criterion. One should choose the mathematically simplest or most elegant approach. Many have sympathized with this view, but the problem is that the idea of simplicity is highly intuitive and even personal, and that no one has managed to formulate it in precise and acceptable terms.
Popper's solution was subsequently criticized by Thomas S. Kuhn in The Structure of Scientific Revolutions. He denied that competing theories (or paradigms) could be compared in the way that Popper had claimed, and substituted instead what can be briefly described as pragmatic success. This led to an intense discussion with Imre Lakatos and Paul Feyerabend the best known participants.
The discussion has continued, but no general and uncontroversial solution to the problem of formulating objective criteria to decide which is the best theory has so far been formulated. The main criteria usually proposed are to choose the theory which provides the best (and novel) predictions, the one with the highest explanatory potential, the one which offers better problems or the most elegant and simple one. Alternatively a theory may be preferable if it is better integrated into the rest of contemporary knowledge.Truth by consensus
In philosophy, truth by consensus is the process of taking statements to be true simply because people generally agree upon them. Imre Lakatos characterizes it as a "watered down" form of provable truth propounded by some sociologists of knowledge, particularly Thomas Kuhn and Michael Polanyi.Philosopher Nigel Warburton argues that the truth by consensus process is not a reliable way of discovering truth. That there is general agreement upon something does not make it actually true.
There are two main reasons for this:
One reason Warburton discusses is that people are prone to wishful thinking. People can believe an assertion and espouse it as truth in the face of overwhelming evidence and facts to the contrary, simply because they wish that things were so.
The other one is that people are gullible, and easily misled.Another unreliable method of determining truth is by determining the majority opinion of a popular vote. This is unreliable because on many questions the majority of people are ill-informed. Warburton gives astrology as an example of this. He states that while it may be the case that the majority of the people of the world believe that people's destinies are wholly determined by astrological mechanisms, given that most of that majority have only sketchy and superficial knowledge of the stars in the first place, their views cannot be held to be a significant factor in determining the truth of astrology. The fact that something "is generally agreed" or that "most people believe" something should be viewed critically, asking the question why that factor is considered to matter at all in an argument over truth. He states that the simple fact that a majority believes something to be true is unsatisfactory justification for believing it to be true.Warburton makes a distinction between the fallacy of truth by consensus and the process of democracy in decision-making. Democracy is preferable to other processes not because it results in truth, but because it provides for equal participation by multiple special-interest groups, and the avoidance of tyranny.Weinberger characterizes Jürgen Habermas as a proponent of a consensus theory of truth, and criticizes that theory as unacceptable on the following grounds: First, even if everyone's opinion is in agreement, those opinions may all nonetheless be erroneous. Second, truth by consensus is conceived as a limit that is approached via an idealized process of discourse; however, it has not been proven that discourse even tends towards such a limit, or that discourse even tends towards one single limit, and thus it is not proven that truth is the limit that is approached by ideal discourse and consensus.