Lakatos in a nutshell

Scientists join research programmes. Research programmes are characterized by a small hard core of 2-5 postulates that guide development of theories and experiments. The hard core is not questioned from within the research programme.

To be progressive, a research program must produce a series of dramatic ("novel") predictions that are confirmed by experiment. This is in contrast to the mainstream account of science, which emphasizes that it's rational to believe in a theory only if its predictions are not (yet) refuted. Lakatos's argument is that real scientists don't abandon beliefs because they're refuted. Indeed, "theories grow in a sea of anomalies, and counterexamples are merrily ignored."

While anomalies or counterexamples are generally shelved to deal with later, some are too telling to ignore. Scientists react by producing an protective belt of auxiliary hypotheses. Those are of two sorts:

1. The good kind are theories in their own right that also lead to novel predictions and confirmations.
2. "ad hoc" hypotheses are those purely created to defend the research programme, to explain away counterexamples. They don't lead to useful predictions.

Note that you can't tell from the outside which category a protective theory falls into. That's discovered over time.

Unlike the hard core, parts of the protective belt can be dropped or replaced.

A research programme is degenerating if:

1. it does not lead to stunning new predictions (at least occasionally...);
2. all its bold predictions are falsified; and
3. it does not grow in steps which "follow the spirit of the programme". That most likely means that it's no longer building by finding implications of its hard core. Instead, the researchers spend their time constructing ad hoc protective theories.

A research programme can recover from degenerating and become progressive again.

The wikipedia article has more detail. It's pretty good as of this episode's publication date.

References

The standard reference is Lakatos's Philosophical Papers, Volume 1: The Methodology of Scientific Research Programmes, 1978, Currie & Worrell (editors). I personally found a series of Lakatos's transcribed lectures more useful for this episode. They're in For and Against Method: Including Lakatos's Lectures on Scientific Method and the Lakatos-Feyerabend Correspondence, 1995, Motterlini (editor). Lakatos and Feyerabend were both friends and sparring partners with very different views about science. Unfortunately, Feyerabend didn't save most of Lakatos's letters, and Feyerabend's letters tend more toward gossip than debate about issues. It's quite a loss, given that Lakatos died young (age 51).

I don't mention it in the podcast, but Lakatos's Proofs and Refutations: The Logic of Mathematical Discovery, 1976, Worrall and Zaher (editors) is a wonderful book. It's a series of fictional conversations between a teacher and his students that recapitulates the history of Euler's polyhedron formula, V-E+F=2.  As with his later Methodology did for science, Lakatos demonstrates that mathematics isn't just a steady accumulation of knowledge. Mathematicians don't just play the definition-theorem-proof game; they also use techniques like "monster barring". You'll be surprised by how entertaining it is.

Immanuel Velikovsky, Worlds in Collision, (first edition, 1950). Wikipedia article.
The Millikin oil drop experiment.
The manifesto for Agile software development.
Kent Beck and Cynthia Andres, Extreme Programming Explained: Embrace Change (second edition, 2004).
Edward Yourdon, Death March (first edition 1997).
My Bothered Bolsheviks are described in Stephen Kotkin, Stalin: Volume I: Paradoxes of Power (1878-1928), 2014.

Credits

Imre Lakatos courtesy Library of the London School of Economics and Political Science, No restrictions, via Wikimedia Commons