Proofs and Refutations: The Logic of Mathematical Discovery

"I first came across this book at university in a course on the philosophy of mathematics. Looking back, it was one of my first experiences of how maths could be different to how I was taught it. In the book, Lakatos takes a particular area of mathematics to do with shape and recreates an imaginary dialogue where he and the characters in the book go through this extraordinary process of developing what mathematicians would call conjectures. So they are developing predictions about how a particular property of a shape might operate, or which types of shape it might be true for. There’s this amazing description of a process which, for me, is at the heart of doing mathematics, which is coming up with conjectures, coming up with predictions and then the search for counter-examples that don’t fit that conjecture, and, as a result of that, the refinement of the conjecture itself. It’s quite a scientific process: you come up with an idea and you try and refute it. There’s a phrase, “mathematizing”, which is about trying to capture the process of doing mathematics as a mathematician and this book is just a beautiful description of that. It was very important in the development of my own thinking, just about how mathematics can then become alive. OK, other people may have gone down these steps of logic before, but that doesn’t make the process any less valid. Those ended up being guiding ideas for me as a teacher in my classroom, trying to set up situations where the children can come up with conjectures, can come up with predictions and they can work on testing those ideas, and trying to find examples that don’t work and so on. That’s absolutely right. Again, it puts across this more human side of the subject, where this is about mathematicians, as a community, deciding what the standards are for rigour and proof. One of the misleading things about the subject is that because a lot of this work happened a long time ago — and those standards have been agreed on for hundreds of years — it can appear there was never any choice about them. Actually, there was a lot of choice and a lot of debate about how things should be defined. The connection with how misleading the presentation in textbooks is, is important as well. Generally, in a textbook, a mathematical theorem will be presented at the beginning of a chapter and then there will be some very neat proof offered next. You get no sense at all, really, of the toing and froing that took place to arrive at this, or how this theorem has come about, or even, in some cases, what problem it was there to solve. It’s as if, in the textbook, you get this very, very condensed nugget of a theorem and its proof and the whole messy genesis of it is somehow airbrushed out."