There seem to be three separate worries raised to Lakatos's thought that
'history of mathematics without philosophy of mathematics is blind':
1) philosophy may hold up the progress of mathematics.
2) adopting a philosophy of mathematics relevant to contemporary mathematics would make us misunderstand the mathematics of the past.
3) concentrating on the official philosophy of mathematics of an era would make us misunderstand the mathematics that time.
Worry (1): This is an issue about philosophy and so not Lakatos's concern here, but his general attitude that one should always engage in criticism (in the positive sense) is. One can point to examples where a principle is made explicit, e.g., 'one is not a number', and then made a piece of dogma. For Lakatos, the first act is beneficial but not the second. Philosophers of mathematics today seem so concerned about the dogma worry (and perhaps also the fear of appearing foolish in daring to make any comment that would have any bearing on what mathematicians do) that they pride themselves in 'leaving mathematics as it is'. This was not Lakatos's way: uncritically generated mathematics was to be thrown out. Personally, I believe that today we're too far off the pace to say even as much as this, but I do think we can contribute to debates that flare up (or smoulder on) in math departments. I hope my efforts to elucidate the senses of the term 'natural' as it used by mathematicians (chap 9 of my book) will prove interesting to them. Resources from historians (use of 'natural' in earlier times) and from practitioners of science studies in general are relevant to this project.
As for worry (2): of course! Although, with sensitivity it might throw up a novel way of reading the past.
Worry(3): Milo Gardner's qualms may be allayed by adopting a Collingwoodian approach, where the philosopher would interest herself in the constellation of presuppositions underlying a practice in some period. Then one needn't think through the mathematics of an age merely in light of the writings of the 'official' philosophers of the day. E.g., for the early 19th century make sure you've read the views of Gauss, Galois, Dirichlet, etc. on mathematics.
Replying to Elizabeth Hind:
"The past needs to be studied for its own sake and on its own terms. That is history, and to try and deal with mathematical texts on their own terms we need to understand the philosophy of the time in which they were written. This is not what Lakatos had in mind."
Lakatos certainly forces matters in Proofs and Refutations, and so misses out all the delicious part of the story of what fed into Poincare's Analysis Situs, e.g., Riemann's work. But compared to what was going on at the time, 1963, P&R is a lone oasis in a very wide desert.
"In the introduction to your book you choose to quote from Hardy: "These
parts of mathematics [parts which have a utility] are, on the whole, rather dull; they are the parts which have the least aesthetic value." "It is not possible to justify the life of any genuine professional mathematician on the ground of the 'utility' of his work" I defend anyone's right to do research which seems to have no utility. However, I find this attitude is prevalent among people engaged in history of maths. Because of this I feel that history of mathematics concentrates on those bits which are aesthetic, what you would consider to be 'real mathematics'. Yet, surely by doing that we are missing out on some very important episodes in human history?"
I refuse to admit this charge. If you read on, I wasn't quoting Hardy to agree with his notions of uselessness. In fact, if you follow up the caveat I mention, you'll find Hardy wasn't dogmatic on this issue. I can't comment on the charges you lay against historians, but as for myself the task of
understanding the relationships between what gets called pure mathematics, applied mathematics, mathematical physics and theoretical physics is an essential task. And if you get to chapter 10 of my book, you'll see I call for work on the use of mathematics in theoretical computer science. I'd be interested to hear what historians think about Sarah Cuomo's allocation of space in her Ancient Mathematics between the mathematics of the 'person-on-the-street' and that of the 'professional' mathematician.
There's a certain relevance here to current debates in Britain as to whether to stop having mathematics as a compulsory subject to 16, 'numeracy' being deemed what is necessary to survive in a modern society.