methodology walkabout

Methodology Walkabout

Every subject has its own methodology. Obviously, some are more interesting than others. Methodologies change over time. As Darwinian creatures under the rubric of evolutionary epistemology we track the changes and adapt.


After I have given a sketch of the Lakatos model below I shall show how it may be applied to Andrew Wiles' proof of Fermat's Last Theorem.


The unit of change for Lakatos is not a paradigm (Kuhn), not a theory or hypothesis or basic statement (Popper), not theories and observation sentences (Quine & the justificationist fallibilists) but a Scientific Research Programme (SRP).

A Scientific Research Programme is a sequence of scientific theories such that each theory in the sequence is a modification of its predecessor.

A Scientific Research Programme has one central sub-component, two accompanying structures and a protective auxiliary belt.

The central sub-component is the hard core, which is a set of theoretical assertions or statements. Any theory which belongs to a Scientific Research Programme must subscribe to the truth of the statements in the hard core. The hard core is "conventionally accepted" and is provisionally not refutable.
The first companion structure is the negative heuristic which is a methodological principle which implies that the theoretical assertions in the hard core are not to be abandoned in the face of anomalies and counter-examples. You may abandon auxiliary observational hypotheses, or basic statements, or statements of initial conditions.

The second companion structure is the positive heuristic which is a partially articulated set of suggestions or hints on how to develop, expand or change the potentially refutable variants of the research programme, and a set of suggestions on how to make the refutable protective belt more sophisticated.

The current theory in a SRP always has a protective belt of auxiliary observational hypotheses, and is able to specify initial conditions. Testable predictions are derived from initial conditions and auxiliary observational hypotheses. The theory's auxiliary belt (TAB) is the set of all observational hypotheses which are testable.

A SRP is said to be progressive if its theoretical growth anticipates its empirical growth. Progressive SRP's have increasing empirical content and predict new facts. A SRP is said to be degenerating if its theoretical growth has slowed down and lags behind its empirical growth, and when it gives post hoc explanations of facts predicted by or discovered by a competitor SRP.


Lakatos says we should regard theory T as falsified if and only if (iff):

1) another theory R has excess empirical content of T; that is it predicts novel facts; i.e. facts improbable in the light of, or even forbidden by T.

2) R explains the previous success of T, i.e., all the unrefuted content of T is included (within the limits of observational error) in the content of R; AND

3) some of the excess content of R is corroborated.

Lakatos believes we should not model the growth of scientific knowledge as a two cornered fight between theory and experiment. We should view it as a three cornered fight between 2 rival theories and an experiment. In the fight the world acts as referee. Lakatos wants to defend a realist view of science.

An example of FALSIFICATION in MATHEMATICS if we use the falsificationist methodology of Imre Lakatos

The monograph PROOFS AND REFUTATIONS by Imre Lakatos, C.U.P. 1976, was based on the first 3 chapters of his 1961 Ph.D. thesis for the London School of Economics.
Reuben Hersh says that Proofs and Refutations is "... an overwhelming work. The effect of its polemical brilliance, its complexity of argument and self-conscious sophistication, its sheer weight of historical learning, is to dazzle the reader."

In PROOFS Lakatos tries to show that the best understanding of what mathematicians do when they produce proofs for theorems is a variation of Popperian falsificationism. That is, mathematicians advance their subject much like natural scientists do by conjecturing the truth of a hypothesis, lemma or theorem and then showing that the theorem is not false or cannot be falsified, by providing a logic chain, from axioms or pre-established accepted mathematical truths to the desired conclusion, the truth of the theorem in question.
In the case of natural science according to Lakatos, you conjecture a whole set of alleged scientific truths, which are embedded in a research programme, and it is in a Darwinian knowledge world that the research programme competes against other research programmes for survival.

One of the most important mathematical proofs of the 20th Century was that of Fermat's Last Theorem by Andrew Wiles. I think that if you break up the proof, which is more than 200 pages long, you get logical chunks which are best understood in the way Lakatos is suggesting we understand them. I shall try to sketch how this works below.

Andrew Wiles and Fermat's Last Theorem

1. Frey at the Oberwolffach conference showed that; If a certain equation in number theory, which he identified, is possible, then the Taniyama-Shimura (T-S) conjecture is false, and Fermat's Last Theorem (F.L.T.) is false.
So Wiles was able to Conjecture that if Frey is correct, then if the Taniyama-Shimura conjecture is true, then Fermat's Last Theorem must also be true. And this is only possible if Frey's equation is false. But the exact way in which the truth of (T-S) would imply (F.L.T.) was not evident to Wiles in the early stages of constructing the proof of (F.L.T.).
2. Ken Ribet was able to show, and communicate to Wiles, a rough sketch or more or less how, given that Frey is correct, (T-S) conjecture implies the truth of (F.L.T.)
3. Wiles therefore settled down to prove the truth of the (T-S) conjecture. There are other more human conjectures at play here. Wiles had to conjecture that he could prove FLT, provided he isolated himself from the Princeton maths dept., provided he immersed himself in certain methods and techniques he was not at home with, etc.
4. Wiles then conjectured that he could use the methods and work of Iwasawa to establish the link between elliptical curves and modular forms. This way ahead did not work, and Wiles had to admit that his conjecture was false.
5. Wiles then conjectured that he could use the methods and work of Kolyvagin-Flach, instead of Iwasawa, with the use of Hecke algebras, to link the infinite number of elliptical curves to the infinite number of modular forms. This conjecture was falsified by Wiles and his assistant Taylor.
6. Wiles then conjectured that both the above methods taken together, Iwasawa and Kolyvagin-Flach, with the use of Hecke algebras, could jointly be used to link elliptical curves to modular forms. Wiles and Taylor were then able create the maths to to prove this, i.e. the conjecture wass not falsified.
7. The proof was then submitted, refereed and published. This was the second time that Wiles had submitted the proof of (F.L.T.) for publication. The first time round a gap was discovered in the logic chain, thus showing a 'local' counter-example to the proof. According to Lakatos a 'local counter example' is one which falsifies a part of the logic chain, but does not threaten the proof as a whole. A error or mistake that threatens the proof as a whole he calls a 'global counter-example'. The function of the referees in both rounds was to try to falsify the logic by finding local or global counter-examples.
8. So in overview, the global conjectures which formed the backbone of the proof of (F.L.T.) are those identified in pargraphs 1, 2 and 6 above. In each case Wiles or Wiles and his assistant Taylor, had to provide a logic chain to establish the truth of the conjecture, in order to show that the conjecture was not false and given the nature of the logic chain probably not falsifiable.

LAKATOS' MODEL applied to:


I take the following example from "The Invisible Ape", by Jerold Lowenstein and Adrienne Zihlman in New Scientist, 1988, and in some places I have, in order to specify their example accurately, borrowed some of the terminology used by Lowenstein and Zihlman.

The example fits the Lakatos analysis, and shows how accepted evidence, which is apparently "absolutely obvious", for a theory, can be overturned by non-obvious, counter-intuitive evidence, which rests on a new technique. It particularly highlights the idea of a degenerating problemshift, as opposed to a progressive problemshift.

The original conjecture was that:
Chimpanzees and gorillas are more closely related to each other than to humans. This was held to be an obvious fact by the physical anthropologists because of the evidence:

They look more like each other. They are hairy; Men are smooth. Chimpanzees and gorillas walk on all fours;we walk on our hind limbs. Chimpanzees and gorillas have short legs and long arms but with not very flexible hands; we have long legs and short arms with very flexible hands. They walk on their knuckles; we do not, and cannot easily walk on our knuckles, since our hands cannot bear much weight. Their brains are small; ours are large. Their canine teeth are large; ours small. Their molars have thin enamel; ours have thick enamel.

For more than 100 years these differences convinced comparative anatomists and physical anthropologists of the truth of the conjecture. If it were true then our common ancestor the invisible conjectured ape would remain a mystery.
Was it a knuckle walker like Chimps? Or a biped like us? Or something very different looking?

According to Lowenstein and Zihlman the attitudes adopted by the physical anthropologists during the debate correspond to the four stages of human reaction to a bereavement:"first denial, then rage, next grief, and finally sadness and reluctant acceptance".

Denial Stage: In the decade of the 1960's, Morris Goodman and Vincent Sarich (U.C Berkeley) showed by immunological tests of blood proteins that humans, chimps and gorillas are closely related. This implied that the three lineages separated only 5 million years ago.
The physical anthropologists denied this. They already had a 14 million year old human ancestor "Ramapithecus". This implied a divergence of apes and humans between 15-20 million years ago.
The molecular data contradicted the morphological evidence and fossil record.

Rage: The issue during the decade of the 1970's was: What kind of evidence best establishes genetic relationships through time; morphology and the fossil record or molecules?

Mary Clair-King and Allen Wilson (U.C. Berkeley) asked: How could humans look so different from chimps and gorillas when all showed the same degree of difference at the molecular level? (from 1%to 5%)
Answer: Only 1% to 5% of the DNA Genome is expressed as proteins. Between 95 and 99% consists of Introns or pseudogenes, so called "junk" genes. This is DNA that goes along for the ride, without doing any work. The "junk" genes replicate from generation to generation without affecting morphology. All this extra DNA is not much use to the organism but it is useful to researchers.

This DNA is not impeded by natural selection. It accumulates mutations at an even faster rate than the coding sequences of DNA. So it provides a fast biological clock for timing evolutionary differences, and a possible way of resolving the problem.
Three techniques called DNA sequencing, Mitochondrial DNA sequencing, and DNA hybridisation, suggest three different lines of evidence which support the chimp/human similarity, and chimp/human versus gorilla difference.
DNA Hybridisation bears same relationship to DNA sequencing as protein immunology does to protein sequencing.
It compares two large molecules by measuring their overall similarities and differences.
The double strands in the human DNA helix can be "melted" by heat into single strands. Then these can be "melted" with strands from chimpanzee molecules into a single strand. The hybrid is a chimp-human strand. The hybrids separate at a lower temperature (like sections of a defective zipper) than the human or chimp originals because of mismatches, in the base pairs. A 1º difference of temp. represents a 1% difference in the sequence.

Grief: Charles Sibley and Jon Ahlquist (Yale) found that chimp-human hybrids are 20% more stable than chimp-gorilla or human-gorilla hybrids. This implies that gorillas split off from chimps/humans approximately 1 million years before the chimps/humans split from each other.

Sadness and reluctant acceptance: In the mid 1980's after 20 years of defence the physical anthropologists and palaeontologists abandoned their original hypothesis that Ramapithecus was a human ancestor, and that chimps and gorillas are more closely related to each other than to humans. On the basis of the bone and teeth fragments that they thought supported their original theory, and which, with benefit of hindsight, led them astray, the palaeontologists now proclaim that Ramapithecus is an ancestor to orang-utans.

+ + +

Its easy to explain the conflict above using Lakatos' model....
The example is a perfect fit for the Lakatos model, becuse it shows how accepted evidence, which is apparently "absolutely obvious", for a theory, can be overturned by non-obvious, counter-intuitive evidence, which rests on a new technique. It particularly highlights the Lakatosian idea of a degenerating problemshift, as opposed to a progressive problemshift.


You start with the data to be explained. This maybe some feature like sex, or aggression or altruism or parental investment or whatever. (or it might be an aspect of leadership; or some aspect of how leaders lead, or of why others follow their leadership etc. How leaders are chosen, or what features they have)

You bring the assumptions and techniques and tools of SOCIO-BIOLOGY (a.k.a. POPULATION BIOLOGY) to bear on the data.
What you are looking for is an applied explanation for some aspect of the chosen feature which relies also on some of the supplementary theories listed below.

So for the chosen aspect of sex, or aggression, or altruism, or parental investment or leadership or whatever) one needs:

1. an explanation of the the chosen feature in primate & hunter gatherer societies & now in modern humans;
followed by,
(2) how it is physically operated in real time in a phenotype through memory lookup and the application of rules or precedents;
followed by (3) what happens at the neuroscience level in the left cerebral hemisphere that co-ordinates the chosen feature;
followed by (4) how this aspect is developed in pre-teenagers & teenagers;
followed by an (5) explication of the modifications of brain circuitry in our ancestors which allowed them to benefit (their inclusive fitness etc. from the chosen feature.

The set of hierarchical explanations used would invoke the supplementary theories listed below.

Supplementary intermediate theories - which are supplementary to - sociobiology are used in (hierarchical reductive) explanations for this data;

* evolutionary psychology (linking biology to culture)
*cognitive science ( linking biology to culture)
*cognitive neuroscience (linking the social or tribal mind through the individual mind to matter)
*behavioural genetics (linking group behaviour to individuals to genes)
*genetics (linking the individual to genes)

If you set the socio-biological explanations, supported by specialised sub-explanations from, cognitive science, cognitive neuroscience, behavioural genetics and genetics for your chosen feature side by side with an explanation from sociology under standard social science model (SSSM) or some standard theory of psychology under (SSSM) there could be very little contest from the old paradigm. And there would be very little common ground. The explanations under the evolutionary rubric, (with specialised sub-explanations from, cognitive science, cognitive neuroscience, behavioural genetics and genetics) would be incomparably richer than those from the old SSSM paradigm. And they would also be scientific, something which the old SSSM is not.

& & &

The major theories: Regular Biology & Sociobiology ( a.k.a. Population Biology)

Data Box A
Data to be explained by the major theory
1. behaviour of human populations & human culture & “social or tribal mind”
Supplementary intermediate theories used in hierarchical reductive explanations
* evolutionary psychology
*cognitive science
*cognitive neuroscience
*behavioural genetics

& & &

On the old (not yet discarded) paradigms the explanatory job above was supposed to be done by the SSSM using a standard theory of sociology (Durkheim / structural functionalism/ etc) or some version of Marxism with a little help from, so called, .

Data Box B
Data to be explained by the major theory
1. behaviour of human individuals & individual human culture & individual mind
Intermediate theories used in hierarchical reductive explanations
* evolutionary psychology
*cognitive science
*cognitive neuroscience
*behavioural genetics
* genetics

& & &

On the old (not yet discarded) paradigms the explanatory job above was supposed to be done by behaviourism or Freudianism or some other chosen theory of psychology using the same misguided assumptions from the SSSM.

+ + +