**Imre Lakatos** (1922-1974) was a philosopher of mathematics and of science.

## Life

Lakatos was born Imre Lipschitz in Debrecen, Hungary in 1922. He received a degree in mathematics, physics, and philosophy from the University of Debrecen in 1944. He became an active communist during the Second World War.

After the war, he continued his education in Budapest (under György Lukács, among others) and worked as a senior official in the Hungarian ministry of education. 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.

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*, published after his death, is based on this work.

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 and John Watkins.

He remained at the London School of Economics until his death in 1974. The Lakatos Award was set up by the school in his memory.

Parts of his correspondence with his friend and critic Paul Feyerabend have been published in *For and Against Method* (ISBN 0226467740).

## Proofs and refutations

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

The book *Proofs and Refutations* is based on his doctoral thesis. 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. 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 'quote' famous mathematicians such as Cauchy.

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 publication in 1976, *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. One of the major problems perceived by critics is that the pattern of mathematical research depicted in *Proofs and Refutations* does not faithfully represent most of the actual activity of contemporary mathematicians.

## Research programmes

Lakatos' contribution to the philosophy of science was an attempt to resolve the perceived conflict between Popper's Falsificationism and the revolutionary structure of science described by Kuhn. Popper's theory 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.

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 'theories' are actually groups of slightly different theories that share some common idea, or what Lakatos called their 'hard core'. Lakatos called these groups 'Research Programs'. Those scientists involved in the program will shield the theoretical core from falsification attempts behind a protective belt of *auxiliary hypotheses*. Whereas Popper generally disparaged 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 program. Instead of asking whether a hypothesis is true or false, Lakatos wanted us to ask whether a research program is progressive or degenerative. A progressive research program is marked by its growth, along with the discovery of stunning novel facts. A degenerative research program is marked by lack of growth, or growth of the protective belt that does not lead to novel facts.

Lakatos was following the Quinian idea that one can always protect a cherished belief from hostile evidence by redirecting the criticism toward other things that are believed. (See Quine-Duhem 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 Natures 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*"^{1}. This inconstancy can be resolved without abandoning our Research Program by leaving the hard core alone and altering the auxiliary hypotheses.

One example given is Newton's three laws of motion, which define quantities such as force. Within the Newtonian system (research program) these are not open to falsification as they form the programs 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 believed 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 hypothesis within research programmes (Lakatos calls them 'problem shifts') are equally as acceptable. He believe 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^{2}. 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 believed that if a Research program is progressive then it is rational for Scientists to keep changing the auxiliary hypothesis in order to hold on to it in the face of anomalies. However, if a research program is degenerating then it faces danger from its competitors, it can be 'falsified' by being superseded by a better (i.e. more progressive) Research Program. This is what he believes is happening in the historical periods Kuhn describes as revolutions and what makes them rational as opposed to mere leaps of faith (as he believed Kuhn took them to be).

## Notes

**1.** Lakatos ed. (1970), Pg. 130

**2.** As an added complication he further differentiates between empirical and theoretical progressiveness. Theoretical progressiveness is if the new 'theory has more empirical content then the old. Empirically progressiveness is if some of this content is corroborated. (Lakatos ed., 1970, P.118)

## Selected works

- Lakatos ed. (1970).
*Criticism and the Growth of Knowledge*. Cambridge: Cambridge University Press. ISBN 0521078261
- Lakatos (1976).
*Proofs and Refutations*. Cambridge: Cambridge University Press. ISBN 0521290384
- Lakatos (1977).
*The Methodology of Scientific Research Programmes: Philosophical Papers Volume 1*. Cambridge: Cambridge University Press
- Lakatos (1978).
*Mathematics, Science and Epistemology: Philosophical Papers Volume 2*. Cambridge: Cambridge University Press. ISBN 0521217695

## Further Information

- Brendan Larvor (1998).
*Lakatos: An Introduction*. London: Routledge. ISBN 0415142768

Last updated: 08-01-2005 10:03:34