Mathematics and first principles
Lakatos's contribution to the philosophy of mathematics was, to put it simply, definitive: the subject will never be the same again. For decades the philosophy of mathematics was about foundations, set theory, paradoxes, axioms, formal logic and infinity - an agenda set by Bertrand Russell, among others, beefed up by the truly wonderful discoveries of Kurt Gödel. Lakatos made us think instead about what most research mathematicians do. He wrote an amazing philosophical dialogue around the proof of a seemingly elementary but astonishingly deep geometrical idea pioneered by Euler. It is a work of art - I rank it right up there with the dialogues composed by Hume or Berkeley or Plato.
Praise indeed, and yet elsewhere, as I commented at pp. 7-8 of a paper discussing Michael Friedman's Dynamics of Reason, Hacking describes Lakatos as a 'deflator' when it comes to mathematics. By this he means that Lakatos is showing that as mathematics proceeds, if it is carried out properly with plenty of critical discussion, a point will be arrived at where the definitions are such that results will follow easily from them. A theory which was initially driven by (quasi)-empirical facts has become merely a collection of analytic statements, true by virtue of meaning. Now, I think this is to get Lakatos very wrong, as I suggest on p.8 of the Friedman review. Yes, it's all about having good definitions, but they're good for Lakatos to the extent that they're right, or at least more right than their predecessors.
Urs Schreiber and I have been discussing related ideas about when one feels one has understood a construction properly. The strange thing is that there's almost a disincentive to reformulate a field to make it as well-organised as possible so as to allow a principled understanding. Some of this may be down to the temporary advantage you'll gain if you alone thoroughly grasp a field and can produce a string of new results which appear to your rivals to be arrived at rather mysteriously. But there's also this other issue that you will be thought to have made the results of the field trivial, or true simply in virtue of meaning.
I don't recall anywhere Lakatos providing us with the philosophical resources to help us ward off the charges of deflation or trivialisation. Instead, I think we should look to Alasdair MacIntyre's revival of Aristotelianism, in particular pages 184-5 of MacIntyre A. (1998) 'First Principles, Final Ends and Contemporary Philosophical Issues' in K. Knight (ed.) The MacIntyre Reader, Polity Press, pp. 171-201. This paper will appear in the first of two volumes of collected papers with Cambridge University Press this April. It's no easy matter to read an extract from one of his papers from a standing start, but here is a taste of what he says:
That first principles expressed as judgments are analytic does not, of course, entail that they are or could be known to be true a priori. Their analyticity, the way in which subject-expressions include within their meaning predicates ascribing essential properties to the subject and certain predicates have a meaning such that they necessarily can only belong to that particular type of subject, is characteristically discovered as the outcome of some prolonged process of empirical enquiry. That type of enquiry is one in which, according to Aristotle, there is a transition from attempted specification of essences by means of prescientific definitions, specifications which require acquaintances with particular instances of the relevant kind (Posterior Analytics, 93a21-9), even although a definition by itself will not entail the occurrence of such instances, to the achievement of genuinely scientific definitions in and through which essences are to be comprehended. (184)
...the analyticity of the first principles is not Kantian analyticity, let alone positivist analyticity. (185)
MacIntyre goes on discuss truth in terms of adequacy of the intellect to its subject matter. Clearly relating these ideas to mathematics needs an extended treatment, beyond what I have given in my How Mathematicians May Fail to be Fully Rational, e.g., pp. 12-13.