Today is once again the anniversary of the day I started this blog nine years ago. Nine years‽ I have difficulty believing that I have really churned out blog posts on a regular basis, with only minor breaks, for nine years now. It has become something of a tradition that on my blog anniversary I post something autobiographical and I have decided this year to maintain that tradition and explain why, when asked, I always name Imre Lakatos’ Proof and Refutations not just as my favourite book but as the most important/influential book in my life.
As regular readers might have gathered my life has been anything but the normal career path one might expect from a white, middle class, British man born and raised in Northeast Essex. It has taken many twists and turns, detoured down one or other dark alleyway, gone off the rails once or twice and generally not taken the trajectory that my parents and school teachers might have hoped or expected it to take.
In 1970 I went to university in Cardiff to study archaeology but after one year I decided that archaeology was not what I wanted to do and dropped out. I however continued to live in Cardiff apart from some time I spent living in Belgium and but that’s another story. During this period of my life I earned my living doing a myriad of different things whilst I was supposedly trying to work out what it was that I actually wanted to do. As I’ve said on several occasions I became addicted to the history of mathematics at the age of sixteen and during this phase of my life I continued to teach myself both the history of maths and more generally the history of science.
In 1976 my life took another left turn when I moved to Malmö in Sweden.
This was not my first attempt to move to Sweden there had been another abortive attempt a couple of years earlier but that is also another story. This time the move was not instituted by me but by my then partner K. K was a qualified nursery nurse and had applied for a job looking after the children of a pair of doctors in Malmö and her application had been successful. The couple agreed to my accompanying K on the condition that to pay my part of the rent of the flat (that went with the job) I would look after their garden until such time as I found work.
So after witnessing the rained out but brilliant Bob Marley open air in Cardiff football stadium in the summer of 76, we set of for a new life in Malmö. Not having employment my role was to do the cleaning, shopping, cooking and looking after the garden, all things I had been doing for years so no sweat. This left me with a lot of spare time and it wasn’t long before I discovered the Malmö public library. The Swedes are very pragmatic about languages; it is a country with a comparatively small population that lives from international trade so they start learning English in kindergarten. The result in that the public library has lots and lots of English books including a good section on the history and philosophy of mathematics and science, which soon became my happy hunting ground. Card catalogues sorted by subject are a great invention for finding new reading matter on the topic of your choice.
At that point in life I was purely a historian of mathematics with a bit of history of science on the side but in Malmö public library I discovered two books that would change that dramatically. The first was Stephan Körner’s The Philosophy of Mathematics–mathematics has a philosophy I didn’t know that–and the second was Karl Popper’s collection of papers, Conjectures and Refutations: The Growth of Scientific Knowledge. Both found their way back to our flat and were consumed with growing enthusiasm. From that point in my life I was no longer a historian of mathematics and science but had become that strange two-headed beast a historian and philosopher of mathematics and science.
Given the fundamental difference between empirical science and logically formal mathematics my next move might seem to some to be somewhat strange. However, I began to consider the question whether it would be possible to construct a Popperian philosophy of mathematics based on falsification. I gave this question much thought but made little progress. In 1977, for reasons I won’t expand upon here, we returned to the UK and Cardiff.
In Cardiff I continued to pursue my interest in both the histories and philosophies of mathematics and science. In those days I bought my books in a little bookshop in the Morgan Arcade in Cardiff.
One day the owner, whose name I can’t remember but who knew my taste in books said, “I’ve got something here that should interest you” and handed me a copy of Imre Lakatos’ Proofs and Refutations: The Logic of Mathematical Discovery. I now for the first time held in my hands a Popperian philosophy of mathematics or as Lakatos puts it a philosophy of mathematics based on the theories of George Pólya, Karl Popper and Georg Hegel, a strange combination.
Lakatos was born Imre Lipschitz in Debrecen, Hungary in 1922. He studied mathematics, physics and philosophy graduating from the University of Debrecen in 1944. Following the German invasion in 1944 he formed a Marxist resistance group with his girlfriend and later wife. During the occupation he changed his Jewish name to Molnár to avoid persecution. After the War he changed it again to Lakatos in honour of his grandmother, who had died in Auschwitz. After the War he became a civil servant in the ministry of education and took a PhD from the University of Debrecen in 1948. He also studied as a post doc at the University of Moscow. Involved in political infighting he was imprisoned for revisionism from 1950 to 1953. One should point out that in the post War period Lakatos was a hard-line Stalinist and strong supporter of the communist government. His imprisonment however changed his political views and he began to oppose the government. Out of prison he returned to academic life and translated Georg Pólya’s How to Solve It into Hungarian. When the Russians invaded in 1956, Lakatos fled to the UK via Vienna. He now took a second PhD at the University of Cambridge in 1961 under R.B. Braithwaite. In 1960 he was appointed to a position at the LSE where he remained until his comparatively early death at the age of 51 in 1974.
The book that I had acquired is a large part of Lakatos’ 1961 PhD thesis, published in book form posthumously, and extends Popper’s philosophy of logical discovery into the realm of mathematics. In his seminal work The Logic of Scientific Discovery, (which I had read shortly after discovering his Conjectures and Refutations) Karl Popper moved the discussion in the philosophy of science from justification to discovery. Most previous work in the philosophy of science had been devoted to attempting to justify the truth of accepted scientific theories; Popper’s work was concerned on a formal level at how we arrive at those theories. The same situation existed in the philosophy of mathematics. Philosophers of mathematics were concerned with the logical justification of proven mathematical theorems. Lakatos turned his attention instead to the historical evolution of mathematical theorem.
Proofs and Refutations is written in the form of a Socratic dialogue, although the discussion has more than two participants. A teacher and his class, the students all have Greek letters for names, who are trying to determine the relationship between the number of vertices, edges and faces in polyhedra, V-E+F = 2; a formula now known as the Euler characteristic or Euler’s Gem. The discussion in the class follows and mirrors the evolution in spacial geometry that led to the discovery of this formula. Lakatos giving references to the historical origins of each step in the footnotes. The discussion takes the reader down many byways and cul de sacs and on many detours and around many corners where strange things are waiting to surprise the unwary reader.
The book is thoroughly researched and brilliantly written: erudite and witty, informative on a very high level but a delight to read. I don’t think I can express in words the effect that reading this book had on me. It inspired me to reach out to new heights in my intellectual endeavours, although I knew from the very beginning that I could never possibly reach the level on which Lakatos resided. Before reading Proofs and Refutations, history of mathematics had been a passionate hobby for me; afterwards it became the central aim in my life. I applied to go back to university in Cardiff to study philosophy, having already matriculated six years earlier to study archaeology this meant a one to one interview with a head of department. I completely blew the interview; I always do!
In 1980 I moved to Germany and in 1981 I applied to go to university in Erlangen to study mathematics, which I was able to do after having spent a year learning German. I wanted to choose philosophy as my subsidiary, which meant an interview with a professor. The man I met was Christian Thiel, a historian of logic and mathematics although I didn’t know that at the time, who was just starting his first year as professor, although he had earlier studied in Erlangen. We clicked immediately and although he no longer remembers on that day we discussed the theories of Imre Lakatos. As I documented here Christian Thiel became my mentor and is indirectly more than somewhat responsible for this blog
I have read a lot of books in my life and I continue to do so, although now much more slowly than in the past, but no book has ever had the same impact on me as Proofs and Refutations did the first time I read it. This is why I always name it when asked questions like, what was the most important book you have read or what is your all time favourite book.
 Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, eds. John Worrall and Elie Zahar, CUP, Cambridge etc., 1976
 How to Solve It is a wonderful little volume describing methods for solving mathematical problems; its methodology can also be used for a much wider range of problems and not just mathematical ones.
 Part of the thesis had been published as a series of four papers paper under the title Proofs and Refutations in The British Journal for the Philosophy of Science, 14 1963-64. The main part of the book is an expanded version of those original papers.
 I recommend David S. Richeson, Euler’s Gem, University Press Group Ltd., Reprint 2012