Imre Lakatos
Encyclopedia
Imre Lakatos 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.
, Hungary
in 1922. He received a degree in mathematics, physics
, and philosophy
from the University of Debrecen
in 1944. He avoided Nazi persecution of Jews by changing his name to Imre Molnár. His mother and grandmother died in Auschwitz. He became an active communist
during the Second World War
. He changed his last name 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.
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 doctorate in philosophy in 1961 from the University of Cambridge
. 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 John Watkins. 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 highly-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
".
Lakatos remained at the London School of Economics until his sudden death in 1974 of a brain haemorrhage, aged just 51. The Lakatos Award
was set up by the school in his memory.
In January 1971 he became editor of the internationally prestigious British Journal for the Philosophy of Science until his death in 1974, after which it was then edited jointly for many years by his LSE colleagues John 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
organised 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.
' dialectic
, by Karl Popper
's theory of knowledge, and by the work of mathematician George Polya
.
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 (V)ertices minus the number of their (E)dges plus the number of their (F)aces 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 counterexample
s. Often the students paraphrase famous mathematicians such as Cauchy, as noted in Lakatos's extensive footnotes.
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 axiom
s were tautological
, i.e. logically true.)
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 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.
. 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.
falsificationism
and the revolutionary structure of science described by Kuhn. Popper's theory as often (inaccurately) reported
implied that scientists should give up a theory as soon as they encounter any falsifying evidence, immediately replacing it with increasingly 'bold and powerful' new hypotheses. However, Kuhn described science as consisting of periods of normal science in which scientists continue to hold their theories in the face of anomalies, interspersed with periods of great conceptual change. Popper acknowledged that excellent new theories may be inconsistent with apparently empirically well supported older theories. For example, he pointed out in Objective Knowledge (at page 200) that "in Newton's theory Kepler's laws are only approximately valid – that is, strictly invalid – if we take into account the mutual attraction between the planets", so that (in precise terms) Newton's theories were inconsistent with Kepler's third law
. However, whereas Kuhn implied that good scientists ignored or discounted evidence against their theories Popper regarded counter evidence as something to be dealt with, either by explaining it, or eventually modifying the theory. Popper was not describing actual behaviour of scientists, but what a scientist should do. Kuhn was mostly describing actual behaviour.
Lakatos sought a methodology that would harmonize these apparently contradictory points of view, a methodology that could provide a rational account of scientific progress, consistent with the historical record.
For Lakatos, what we think of as a 'theory' may actually be a succession of slightly different theories and experimental techniques developed over time, that share some common idea, or what Lakatos called their ‘hard core’. Lakatos called such changing collections 'Research Programmes'. The scientists involved in a programme will attempt to shield the theoretical core from falsification attempts behind a protective belt of auxiliary hypotheses. Whereas Popper was generally regarded as disparaging such measures as 'ad hoc', Lakatos wanted to show that adjusting and developing a protective belt is not necessarily a bad thing for a research programme. Instead of asking whether a hypothesis is true or false, Lakatos wanted us to ask whether one research programme is better than another, so that there is a rational basis for preferring it. He showed that in some cases one research programme can be described as progressive while its rivals are degenerating. A progressive research programme is marked by its growth, along with the discovery of stunning novel facts, development of new experimental techniques, more precise predictions, etc. A degenerating research program is marked by lack of growth, or growth of the protective belt that does not lead to novel facts.
Lakatos claimed that he was extending Popper's ideas, which had themselves developed over time. He contrasted Popper, the crude falsificationist, who existed only in the minds of critics and followers who had not understood Popper's writings, Popper1, the author of what Popper actually wrote, and Popper2, who was supposed to be Popper as reinterpreted by his pupil Lakatos, though many commentators believe that Popper2 just is Lakatos. The idea that it is often not possible to show decisively which of two theories or research programmes is better at a particular point in time whereas subsequent developments may show that one is 'progressive' while the other is 'degenerating', and therefore less acceptable, was a major contribution both to philosophy of science and to history of science. Whether it was Popper's idea or Lakatos' idea, or, most likely, a combination, is of less importance.
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 difficulty with falsificationism had been acknowledged by Popper.
Falsificationism, (Popper
's theory), 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, yet this is what Kuhn had described them as doing. But for Lakatos, "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". This inconsistency can be resolved without abandoning our Research Programme by leaving the hard core alone and altering the auxiliary hypotheses. One example given is Newton
's three laws of motion
. Within the Newtonian system (research programme) these are not open to falsification as they form the programme's hard core. This research programme provides a framework within which research can be undertaken with constant reference to presumed first principles which are shared by those involved in the research programme, and without continually defending these first principles. In this regard it is similar to Kuhn's notion of a paradigm.
Lakatos also took the view that a research programme contained 'methodological rules', some that instruct on what paths of research to avoid (he called this the 'negative heuristic') and some that instruct on what paths to pursue (he called this the 'positive heuristic').
Lakatos claimed that not all changes of the auxiliary hypotheses within research programmes (Lakatos calls them 'problem shifts') are equally as acceptable. He took the view that these 'problem shifts' can be evaluated both by their ability to explain apparent refutations and by their ability to produce new facts. If it can do this then Lakatos claims they are progressive. However if they do not, if they are just 'ad-hoc' changes that do not lead to the prediction of new facts, then he labels them as degenerate.
Lakatos took the view that if a research programme is progressive, then it is rational for scientists to keep changing the auxiliary hypotheses in order to hold on to it in the face of anomalies. However, if a research programme is degenerate, then it faces danger from its competitors: it can be 'falsified' by being 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 (as he considered that Kuhn took them to be).
"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, Velikovsky's planetary cosmogony, Freudian psychoanalysis, 20th century Soviet Marxism, Lysenko's biology
, Bohr
's Quantum Mechanics post-1924, astrology, psychiatry, sociology, neo-classical economics, and Darwin's theory.
Almost 20 years after Lakatos's 1973 'challenge' on the scientificity of Darwin, in her 1991 The Ant and the Peacock (pp31–2), 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, allegedly explained by descent with modification. She concluded that "our usual idea of corroboration as requiring the successful prediction of novel facts...Darwinian theory was not strong on temporally novel predictions". She was equivocal about whether it did or did not make any novel predictions, only saying " For the most part this evidence was already well known, thoroughly documented by pre-Darwinian natural history.[Italics added]" Cronin did not state what other part of the evidence was not already well known, but did then assert that it was scientific on the weaker Zahar criterion of providing independent novel explanation of old already well known facts. However, she failed to demonstrate that it provided any confirmed nomological-deductive explanation of any old facts of likeness within evolutionary diversity, making an assertion that it did so, without proof.
published in The British Journal for the Philosophy of Science found Milton Friedman
's methodology to be 'pseudo-scientific' in terms of Lakatos's evaluative philosophy of science, according to which the demarcation between scientific and pseudo-scientific theories consists of their at least predicting testable empirical novel facts or not. Latsis claimed that Friedman's instrumentalist methodology of neoclassical economics had never predicted any novel facts. In its defense in a three-page letter to Latsis in December 1972, Friedman counter-claimed that the neoclassical monopoly competition model had in fact shown empirical progress by predicting phenomena not previously observed that were also subsequently confirmed by empirical evidence. But he did not identify any specific economic phenomenon as an example of any such successfully predicted positive novel fact.
In early 1973, as Editor of the Journal, Lakatos invited Friedman to submit a discussion note based on his December 1972 letter to Latsis for publication in a symposium on the issue of the scientific status or not of neoclassical economics. Lakatos even assured Friedman he would have the last word. But Friedman never took up Lakatos's invitation. Three years later, in 1976, Friedman was awarded the Nobel Prize for Economics "for his achievements in the fields of consumption analysis, monetary history and theory and for his demonstration of the complexity of stabilization policy".
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 1970 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.'
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:
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
and science
Philosophy of science
The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
, 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.
Life
Lakatos was born Imre (Avrum) Lipsitz to a Jewish family in DebrecenDebrecen
Debrecen , is the second largest city in Hungary after Budapest. Debrecen is the regional centre of the Northern Great Plain region and the seat of Hajdú-Bihar county.- Name :...
, Hungary
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
in 1922. He received a degree in mathematics, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, and philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
from the University of Debrecen
University of Debrecen
The University of Debrecen is a university located in Debrecen, Hungary. It is the oldest continuously operating institution of higher education in Hungary .-History:...
in 1944. He avoided Nazi persecution of Jews by changing his name to Imre Molnár. His mother and grandmother died in Auschwitz. He became an active communist
Communism
Communism is a social, political and economic ideology that aims at the establishment of a classless, moneyless, revolutionary and stateless socialist society structured upon common ownership of the means of production...
during the Second World War
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
. He changed his last name once again to Lakatos (Locksmith) in honor of Géza Lakatos
Géza Lakatos
Knight Géza Lakatos de Csíkszentsimon was a general in Hungary during World War II who served briefly as Prime Minister of Hungary, under governor Miklós Horthy from August 29, 1944, until October 15,...
.
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
Georg Lukács
György Lukács was a Hungarian Marxist philosopher and literary critic. He is a founder of the tradition of Western Marxism. He contributed the concept of reification to Marxist philosophy and theory and expanded Karl Marx's theory of class consciousness. Lukács' was also an influential literary...
weekly Wednesday afternoon private seminars. He also studied at the Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
under the supervision of Sofya Yanovskaya
Sofya Yanovskaya
Sofya Aleksandrovna Yanovskaya was a mathematician and historian, specializing in the history of mathematics, mathematical logic, and philosophy of mathematics...
in 1949. When he returned, however, he found himself on the losing side of internal arguments within the Hungarian communist party
Hungarian Workers' Party
The Hungarian Working People's Party was the ruling communist party of Hungary from 1948 to 1956. It was formed by a merger of the Hungarian Communist Party and the Social Democratic Party. Its leader was Mátyás Rákosi until 1956, then Ernő Gerő in the same year for three months, and eventually...
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.
After his release, Lakatos returned to academic life, doing mathematical research and translating George Pólya
George Pólya
George Pólya was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory...
's How to Solve It
How to Solve It
How to Solve It is a small volume by mathematician George Pólya describing methods of problem solving.- Four principles :How to Solve It suggests the following steps when solving a mathematical problem:...
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
1956 Hungarian Revolution
The Hungarian Revolution or Uprising of 1956 was a spontaneous nationwide revolt against the government of the People's Republic of Hungary and its Soviet-imposed policies, lasting from 23 October until 10 November 1956....
.
After the Soviet Union
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
invaded Hungary in November 1956, Lakatos fled to Vienna
Vienna
Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...
, and later reached England. He received a doctorate in philosophy in 1961 from the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
. The book Proofs and Refutations: The Logic of Mathematical Discovery
Proofs and Refutations
Proofs and Refutations is a book by the philosopher Imre Lakatos expounding his view ofthe 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...
, published after his death, is based on this work.
Lakatos never obtained British Citizenship
British nationality law
British nationality law is the law of the United Kingdom that concerns citizenship and other categories of British nationality. The law is complex because of the United Kingdom's former status as an imperial power.-History:...
.
In 1960 he was appointed to a position in the London School of Economics
London School of Economics
The London School of Economics and Political Science is a public research university specialised in the social sciences located in London, United Kingdom, and a constituent college of the federal University of London...
, where he wrote on the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
and the philosophy of science
Philosophy of science
The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
. The LSE philosophy of science department at that time included Karl Popper
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
, Joseph Agassi
Joseph Agassi
Joseph Agassi is an Israeli academic with contributions in logic, scientific method, and philosophy. He studied under Karl Popper and taught at the London School of Economics. He later taught at the University of Hong Kong, the University of Illinois, Boston University, and York University in...
and John Watkins. 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
Alan Musgrave
Alan Musgrave is an English born New Zealand philosopher. Musgrave was educated at the London School of Economics with a BA Honours Philosophy and Economics 1961. Sir Karl Popper supervised Musgrave's PhD which was completed in 1969. Musgrave worked as Popper's Research Assistant initially then...
, he edited the highly-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
The Structure of Scientific Revolutions
The Structure of Scientific Revolutions , by Thomas Kuhn, is an analysis of the history of science. Its publication was a landmark event in the history, philosophy, and sociology of scientific knowledge and it triggered an ongoing worldwide assessment and reaction in — and beyond — those scholarly...
".
Lakatos remained at the London School of Economics until his sudden death in 1974 of a brain haemorrhage, aged just 51. The Lakatos Award
Lakatos Award
The Lakatos Award is given annually for a contribution to the philosophy of science which is widely interpreted as outstanding. The contribution must be in the form of a book published in English during the previous six years....
was set up by the school in his memory.
In January 1971 he became editor of the internationally prestigious British Journal for the Philosophy of Science until his death in 1974, after which it was then edited jointly for many years by his LSE colleagues John 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
Paul Feyerabend
Paul Karl Feyerabend was an Austrian-born philosopher of science best known for his work as a professor of philosophy at the University of California, Berkeley, where he worked for three decades . He lived a peripatetic life, living at various times in England, the United States, New Zealand,...
have been published in For and Against Method (ISBN 0-226-46774-0).
Lakatos and his colleague Spiro Latsis
Spiro Latsis
Dr. Spiro J. Latsis is a Greek businessman with a fortune of US$9.1 billion . He ranked 51st on Forbes's 2006 World's Billionaires list. He is the son of the Greek shipping magnate, John Latsis, who died in 2003....
organised 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
Neoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...
, were published by Cambridge University Press in two separate volumes in 1976, one devoted to physical sciences and Lakatos's general programme for rewriting the history of science, with a concluding critique by his great friend Paul Feyerabend, and the other devoted to economics.
Proofs and refutations, mathematics
Lakatos' philosophy of mathematics was inspired by both Hegel's and MarxKarl Marx
Karl Heinrich Marx was a German philosopher, economist, sociologist, historian, journalist, and revolutionary socialist. His ideas played a significant role in the development of social science and the socialist political movement...
' dialectic
Dialectic
Dialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...
, by Karl Popper
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
's theory of knowledge, and by the work of mathematician George Polya
George Pólya
George Pólya was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory...
.
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
Dialogue
Dialogue is a literary and theatrical form consisting of a written or spoken conversational exchange between two or more people....
set in a mathematics class. The students are attempting to prove the formula for the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, which is a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
about the properties of polyhedra, namely that for all polyhedra the number of their (V)ertices minus the number of their (E)dges plus the number of their (F)aces 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
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
, only to be repeatedly refuted by counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
s. Often the students paraphrase famous mathematicians such as Cauchy, as noted in Lakatos's extensive footnotes.
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
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
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 axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s were tautological
Tautology (logic)
In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
, i.e. logically true.)
Lakatos proposed an account of mathematical knowledge based on the idea of heuristic
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...
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
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
s' are a valid way to discover mathematical conjectures and proofs, and sometimes called his philosophy 'quasi-empiricism
Empiricism
Empiricism is a theory of knowledge that asserts that knowledge comes only or primarily via sensory experience. One of several views of epistemology, the study of human knowledge, along with rationalism, idealism and historicism, empiricism emphasizes the role of experience and evidence,...
'.
However, he also conceived of the mathematical community as carrying on a kind of dialectic to decide which mathematical proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
s are valid
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
and which are not. Therefore he fundamentally disagreed with the 'formalist
Formalism (mathematics)
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....
' conception of proof which prevailed in Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
's and Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
's logicism
Logicism
Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Richard Dedekind...
, 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)
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
to answer one of the major problems perceived by critics, namely that the pattern of mathematical research depicted in Proofs and Refutations does not faithfully represent most of the actual activity of contemporary mathematicians.
Cauchy and uniform convergence
In a 1966 text published as (Lakatos 1978), Lakatos re-examines the history of the calculus, with special regard to Augustin-Louis Cauchy and the concept of uniform convergence, in the light of non-standard analysisNon-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
. 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
Lakatos' contribution to the philosophy of science was an attempt to resolve the perceived conflict between Popper'sKarl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
falsificationism
Falsifiability
Falsifiability or refutability of an assertion, hypothesis or theory is the logical possibility that it can be contradicted by an observation or the outcome of a physical experiment...
and the revolutionary structure of science described by Kuhn. Popper's theory as often (inaccurately) reported
implied that scientists should give up a theory as soon as they encounter any falsifying evidence, immediately replacing it with increasingly 'bold and powerful' new hypotheses. However, Kuhn described science as consisting of periods of normal science in which scientists continue to hold their theories in the face of anomalies, interspersed with periods of great conceptual change. Popper acknowledged that excellent new theories may be inconsistent with apparently empirically well supported older theories. For example, he pointed out in Objective Knowledge (at page 200) that "in Newton's theory Kepler's laws are only approximately valid – that is, strictly invalid – if we take into account the mutual attraction between the planets", so that (in precise terms) Newton's theories were inconsistent with Kepler's third law
Kepler's laws of planetary motion
In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....
. However, whereas Kuhn implied that good scientists ignored or discounted evidence against their theories Popper regarded counter evidence as something to be dealt with, either by explaining it, or eventually modifying the theory. Popper was not describing actual behaviour of scientists, but what a scientist should do. Kuhn was mostly describing actual behaviour.
Lakatos sought a methodology that would harmonize these apparently contradictory points of view, a methodology that could provide a rational account of scientific progress, consistent with the historical record.
For Lakatos, what we think of as a 'theory' may actually be a succession of slightly different theories and experimental techniques developed over time, that share some common idea, or what Lakatos called their ‘hard core’. Lakatos called such changing collections 'Research Programmes'. The scientists involved in a programme will attempt to shield the theoretical core from falsification attempts behind a protective belt of auxiliary hypotheses. Whereas Popper was generally regarded as disparaging such measures as 'ad hoc', Lakatos wanted to show that adjusting and developing a protective belt is not necessarily a bad thing for a research programme. Instead of asking whether a hypothesis is true or false, Lakatos wanted us to ask whether one research programme is better than another, so that there is a rational basis for preferring it. He showed that in some cases one research programme can be described as progressive while its rivals are degenerating. A progressive research programme is marked by its growth, along with the discovery of stunning novel facts, development of new experimental techniques, more precise predictions, etc. A degenerating research program is marked by lack of growth, or growth of the protective belt that does not lead to novel facts.
Lakatos claimed that he was extending Popper's ideas, which had themselves developed over time. He contrasted Popper, the crude falsificationist, who existed only in the minds of critics and followers who had not understood Popper's writings, Popper1, the author of what Popper actually wrote, and Popper2, who was supposed to be Popper as reinterpreted by his pupil Lakatos, though many commentators believe that Popper2 just is Lakatos. The idea that it is often not possible to show decisively which of two theories or research programmes is better at a particular point in time whereas subsequent developments may show that one is 'progressive' while the other is 'degenerating', and therefore less acceptable, was a major contribution both to philosophy of science and to history of science. Whether it was Popper's idea or Lakatos' idea, or, most likely, a combination, is of less importance.
Lakatos was following Pierre Duhem's
Pierre Duhem
Pierre Maurice Marie Duhem was a French physicist, mathematician and philosopher of science, best known for his writings on the indeterminacy of experimental criteria and on scientific development in the Middle Ages...
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
Confirmation holism
Confirmation holism, also called epistemological holism is the claim that a single scientific theory cannot be tested in isolation; a test of one theory always depends on other theories and hypotheses....
and Duhem-Quine thesis). This difficulty with falsificationism had been acknowledged by Popper.
Falsificationism, (Popper
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
's theory), 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, yet this is what Kuhn had described them as doing. But for Lakatos, "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". This inconsistency can be resolved without abandoning our Research Programme by leaving the hard core alone and altering the auxiliary hypotheses. One example given is Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
's three laws of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
. Within the Newtonian system (research programme) these are not open to falsification as they form the programme's hard core. This research programme provides a framework within which research can be undertaken with constant reference to presumed first principles which are shared by those involved in the research programme, and without continually defending these first principles. In this regard it is similar to Kuhn's notion of a paradigm.
Lakatos also took the view that a research programme contained 'methodological rules', some that instruct on what paths of research to avoid (he called this the 'negative heuristic') and some that instruct on what paths to pursue (he called this the 'positive heuristic').
Lakatos claimed that not all changes of the auxiliary hypotheses within research programmes (Lakatos calls them 'problem shifts') are equally as acceptable. He took the view that these 'problem shifts' can be evaluated both by their ability to explain apparent refutations and by their ability to produce new facts. If it can do this then Lakatos claims they are progressive. However if they do not, if they are just 'ad-hoc' changes that do not lead to the prediction of new facts, then he labels them as degenerate.
Lakatos took the view that if a research programme is progressive, then it is rational for scientists to keep changing the auxiliary hypotheses in order to hold on to it in the face of anomalies. However, if a research programme is degenerate, then it faces danger from its competitors: it can be 'falsified' by being 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 (as he considered that Kuhn took them to be).
Pseudoscience
According to the demarcation criterion of pseudoscience originally proposed by Lakatos, a theory is pseudoscientific if it fails to make any novel predictions of previously unknown phenomena, in contrast with scientific theories, which predict novel fact(s). Progressive scientific theories are those which have their novel facts confirmed and degenerate scientific theories are those whose predictions of novel facts are refuted. As he put it:"A given fact is explained scientifically only if a new fact is predicted with it....The idea of growth and the concept of empirical character are soldered into one." See pages 34–5 of The Methodology of Scientific Research Programmes, 1978.
Lakatos's own key examples of pseudoscience were Ptolemaic astronomy, Velikovsky's planetary cosmogony, Freudian psychoanalysis, 20th century Soviet Marxism, Lysenko's biology
Lysenkoism
Lysenkoism, or Lysenko-Michurinism, also denotes the biological inheritance principle which Trofim Lysenko subscribed to and which derive from theories of the heritability of acquired characteristics, a body of biological inheritance theory which departs from Mendelism and that Lysenko named...
, Bohr
Bohr
Bohr may refer to:People:* Niels Bohr , Danish atomic physicist, Nobel Prize in physics 1922* Aage Bohr , Danish nuclear physicist, Nobel Prize in physics 1975, son of Niels Bohr...
's Quantum Mechanics post-1924, astrology, psychiatry, sociology, neo-classical economics, and Darwin's theory.
Darwin's theory
In his 1973 LSE Scientific Method Lecture 1 he also claimed that "nobody to date has yet found a demarcation criterion according to which Darwin can be described as scientific".Almost 20 years after Lakatos's 1973 'challenge' on the scientificity of Darwin, in her 1991 The Ant and the Peacock (pp31–2), 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, allegedly explained by descent with modification. She concluded that "our usual idea of corroboration as requiring the successful prediction of novel facts...Darwinian theory was not strong on temporally novel predictions". She was equivocal about whether it did or did not make any novel predictions, only saying " For the most part this evidence was already well known, thoroughly documented by pre-Darwinian natural history.[Italics added]" Cronin did not state what other part of the evidence was not already well known, but did then assert that it was scientific on the weaker Zahar criterion of providing independent novel explanation of old already well known facts. However, she failed to demonstrate that it provided any confirmed nomological-deductive explanation of any old facts of likeness within evolutionary diversity, making an assertion that it did so, without proof.
The Milton Friedman neoclassical economics case study
In August 1972, a case study of the methodology of neoclassical economics by Lakatos's London School of Economics colleague Spiro LatsisSpiro Latsis
Dr. Spiro J. Latsis is a Greek businessman with a fortune of US$9.1 billion . He ranked 51st on Forbes's 2006 World's Billionaires list. He is the son of the Greek shipping magnate, John Latsis, who died in 2003....
published in The British Journal for the Philosophy of Science found Milton Friedman
Milton Friedman
Milton Friedman was an American economist, statistician, academic, and author who taught at the University of Chicago for more than three decades...
's methodology to be 'pseudo-scientific' in terms of Lakatos's evaluative philosophy of science, according to which the demarcation between scientific and pseudo-scientific theories consists of their at least predicting testable empirical novel facts or not. Latsis claimed that Friedman's instrumentalist methodology of neoclassical economics had never predicted any novel facts. In its defense in a three-page letter to Latsis in December 1972, Friedman counter-claimed that the neoclassical monopoly competition model had in fact shown empirical progress by predicting phenomena not previously observed that were also subsequently confirmed by empirical evidence. But he did not identify any specific economic phenomenon as an example of any such successfully predicted positive novel fact.
In early 1973, as Editor of the Journal, Lakatos invited Friedman to submit a discussion note based on his December 1972 letter to Latsis for publication in a symposium on the issue of the scientific status or not of neoclassical economics. Lakatos even assured Friedman he would have the last word. But Friedman never took up Lakatos's invitation. Three years later, in 1976, Friedman was awarded the Nobel Prize for Economics "for his achievements in the fields of consumption analysis, monetary history and theory and for his demonstration of the complexity of stabilization policy".
Historiographical research programmes
In his 1973 monograph 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 1970 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.'
Feyerabend
Paul FeyerabendPaul Feyerabend
Paul Karl Feyerabend was an Austrian-born philosopher of science best known for his work as a professor of philosophy at the University of California, Berkeley, where he worked for three decades . He lived a peripatetic life, living at various times in England, the United States, New Zealand,...
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
Epistemological anarchism
Epistemological anarchism is an epistemological theory advanced by Austrian philosopher of science Paul Feyerabend which holds that there are no useful and exception-free methodological rules governing the progress of science or the growth of knowledge...
, 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.
See also
- Scientific Community MetaphorScientific community metaphorIn computer science, the Scientific Community Metaphor is a metaphor used to aid understanding scientific communities. The first publications on the Scientific Community Metaphor in 1981 and 1982 involved the development of a programming language named Ether that invoked procedural plans to...
, an approach to programming influenced by Lakatos's work on research programmes. - List of Eastern Bloc defectors
- Lakatos AwardLakatos AwardThe Lakatos Award is given annually for a contribution to the philosophy of science which is widely interpreted as outstanding. The contribution must be in the form of a book published in English during the previous six years....
set up in memory of him
Archives
Imre Lakatos's papers are held at the London School of Economics. His personal library is also held at the School.Further information
- Alex Bandy (2010). Chocolate and Chess. Unlocking Lakatos. Budapest: Akadémiai Kiadó. ISBN 978 963 05 8819 5
- Brendan Larvor (1998). Lakatos: An Introduction. London: Routledge. ISBN 0-415-14276-8
- Jancis Long (1998). "Lakatos in Hungary", Philosophy of the Social Sciences 28, pp. 244–311.
- John Kadvany (2001). Imre Lakatos and the Guises of Reason. Durham and London: Duke University Press. ISBN 0-8223-2659-0; author's Web site: http://www.johnkadvany.com.
- Teun Koetsier (1991). Lakatos' Philosophy of Mathematics: A Historical Approach. Amsterdam etc.: North Holland. ISBN 0-444-88944-2
- Szabo, Arpad The Beginnings of Greek Mathematics (Tr Ungar) Reidel & Akademiai Kiado, Budapest 1978 ISBN 963 05 1416 8
External links
- Science and Pseudoscience (including an MP3 audio file) – Lakatos' 1973 Open UniversityOpen UniversityThe Open University is a distance learning and research university founded by Royal Charter in the United Kingdom...
BBC RadioBBC RadioBBC Radio is a service of the British Broadcasting Corporation which has operated in the United Kingdom under the terms of a Royal Charter since 1927. For a history of BBC radio prior to 1927 see British Broadcasting Company...
talk on the subject - Lakatos’s Hungarian intellectual background The Autumn 2006 MIT Press journal Perspectives on Science devoted to articles on this topic, with article abstracts.
- Economic research programme
- Official russian page