Proofs and Refutations - The Method of Proofs and Refutations

The Method of Proofs and Refutations

Though the book is written as a narrative, an actual method of investigation, that of "proofs and refutations", is developed. In Appendix I, Lakatos summarizes this method by the following list of stages:

1. Primitive conjecture.
2. Proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures).
3. "Global" counterexamples (counterexamples to the primitive conjecture) emerge.
4. Proof re-examined: the "guilty lemma" to which the global counter-example is a "local" counterexample is spotted. This guilty lemma may have previously remained "hidden" or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem - the improved conjecture - supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.

He goes on and gives further stages that might sometimes take place:

1. Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at cross-roads of different proofs, and thus emerge as of basic importance.
2. The hitherto accepted consequences of the original and now refuted conjecture are checked.
3. Counterexamples are turned into new examples - new fields of inquiry open up.