Some good Copernican mythbusting

For those who haven’t already seen it Tim O’Neill, Renaissance Mathematicus friend and guest blogger, has posted a superb essay on his excellent blog, History for Atheists, on the myths surrounding the dissemination, publication and reception of Copernicus’ heliocentric theory, The Great Myths 6: Copernicus’ Deathbed Publication. Regular readers of the Renaissance Mathematicus won’t learn anything new but it is an excellent summary of the known historical facts and well worth a read. As with this blog the comments are also well worth reading.

The earliest mention of Copernicus’ theory – Matthew of Miechów’s 1514 catalogue

Go on over and give Tim a boost!



Filed under History of Astronomy, Myths of Science, Renaissance Science, Uncategorized

Michael Mästlin: not just Kepler’s teacher

A large number of significant but minor figures in the history of science tend to get lost in the shadows of those, whom we have raised to god like status within that history; whole hordes have been swallowed up by the shadows cast by Galileo and Newton. Lesley Murdin even wrote an excellent book, Under Newton’s Shadow[1], in acknowledgement of the latter. One figure who suffers from this phenomenon is the German astronomer Michael Mästlin (1550–1631), who if he gets mentioned at all, it is only with reference to his most famous student, Johannes Kepler. Although his substantial influence on Kepler is probably his most important role in the history of astronomy, Mästlin (or Maestlin, as he is usually written in English) deserves to be much better known in his own right.


Michael Mästlin portrait 1619 artist unknown

Michael Mästlin was born 30 September 1550 in Göppingen into a strict Lutheran Protestant family. He was schooled in the convent schools of Königsbronn and Herrenalb. He matriculated at the University of Tübingen in 1658 where he graduated BA in 1569. In the same year he entered the Tübingen Stift (the Lutheran Church hall of residence) with a stipend from the Duke of Württemberg. He graduated MA in in 1571 and completed his theology studies in 1573.

As a student he studied astronomy and mathematics under Philipp Apian (1531–1589) the son of the astronomer, mathematician and cartographer Peter Apian.


Philipp Apian, artist unknown Source: Wikimedia Commons

Philipp had already succeeded his father as professor for mathematics and astronomy in Ingolstadt at the age of twenty-one. Like many others he studied medicine alongside his teaching duties finishing his medical studies later in the Northern Italian universities. In 1569 he was forced by the Jesuits to quit his post in Ingolstadt because of his membership of the Lutheran Church. In the same year he received the professorship in Tübingen. Apian was professor in Tübingen for fourteen years until he was, ironically, forced to resign because he refused to sign the Formula of Concord a document setting out the Lutheran statement of faith and condemning Calvinists.

Apian inspired and guided Mästlin’s interest in astronomy and mathematics. Already in 1570 Mästlin acquired a copy of Copernicus’ De revolutionibus and he would go on to become one of the first university teachers to teach Copernican heliocentricity. From 1573 until he first left Tübingen in 1576 he was Repetens mathematicus (teacher for revision in the mathematical sciences) at the Stift. In 1570 he published the second edition of Erasmus Reinhold’s Prutenicae Tabulae. In 1572 he observed the Supernova, publishing his observations in his Demonstratio Astronomica Loci Stellae Novae the following year. In 1575 he represented the absent Apian as mathematics professor.

In 1576 Mästlin was appointed Diaconus (2nd pastor) in the parish of Backnang, a small town near Stuttgart. His clerical appointment didn’t stop his astronomical activities. He observed the comet of 1577 publishing his Observatio et Demostratio Cometae Aetherei, qui anno … 1577 in 1578. Much is made, in the secondary literature, of Tycho Brahe’s observations of the 1572 supernova and the 1577 comet and how the determination of the supralunar occurrence of both phenomena led to the refutation of the Aristotelian principle of an unchanging heavens. However, at the time Tycho’s were not the only observations and Mästlin’s reports had at least as much if not more influence on the debate as those of Tycho. In fact Tycho named Mästlin as his prime witness confirming his own observations.

In the years between 1578 and 1580 Mästlin constructed his own observing instruments, a quadrant and a Jacob’s staff. In 1580 he published his Ephemerides Novae … for the years 1577 to 1590. His highly visible astronomical activities led to Mästlin being appointed professor for astronomy and mathematics at the University of Heidelberg in 1580, which had become Protestant in 1556. In Heidelberg he published the first edition of his astronomy textbook, Epitome Astronominae, a standard Ptolemaic geocentric work 1582, which over the years would see six further editions.


In 1584 he was called back to Tübingen his alma mater to succeed his own teacher Philipp Apian as professor for the mathematical sciences, a post that he would hold for more than 47 years until his death in 1631.

Whilst still at Heidelberg Mästlin, as a leading Protestant mathematicus was consulted by the rulers of the German Protestant states on whether to adopt the new Gregorian calendar. In his Gründtlicher Bericht von der allgemeinen und nunmehr bei 1600 Jahren von dem ersten Kaiser Julio bis jetzt gebrauchten jarrechnung oder kalender (Rigorous report on the general and up till now for 1600 years used calculation of years or calendar from the first Caesar Julio), published in 1583, he rejected the new calendar on mathematical and astronomical grounds, noting it was not clear how it was calculated, (this information didn’t become available until much later) and also on religious grounds. In his anti-Catholic polemic he referred to the Pope as “seiner Heilosigkeyt”, that is “his Awfulness”, a pun in German on Heiligkeit meaning holiness and Heillos meaning awful. Mästlin played a central role in the rejection of the calendar reform in the Protestant states, who only adopted the Gregorian calendar in 1700.

It is in his role as professor in Tübingen that is best known and in particular his relationship with his most famous student Johannes Kepler. Kepler studied in Tübingen from 1589 till 1594, like Mästlin as a stipendiary in the Tübinger Stift. From Mästlin’s lectures Kepler learnt about the heliocentric system of Copernicus and not through some sort of secret instruction as is often falsely claimed. It was almost certainly Mästlin who recommended Kepler for the post of mathematics teacher in Graz and it was definitely Mästlin who convinced Kepler to accept the post. The two stayed close after Kepler’s move to Graz and exchanged many long letters on a range of subject. In 1596 Mästlin assisted Kepler in getting his Mysterium Cosmographicum published, adding Rheticus’ Narratio Prima, as an appendix to the work thereby demonstrating his strong support for the Copernican hypothesis.


Strangely after 1600 Mästlin began to distance himself from his most famous pupil, no longer answering all of his letters and declining to help when Kepler was desperately looking for a new position. This cooling of their relationship from the side of the mentor has never been satisfactorily explained but two things probably played a role. On the scientific side Mästlin strongly disapproved of Kepler’s attempts to explain the physical cause of planetary motion, admonishing him to stick to the astronomer’s role of providing mathematical models of that motion and to leave the explanations to the philosophers. Also a thorn in Mästlin’s highly devout Lutheran eyes was Kepler’s sympathy for other religious viewpoints, which led to his being excluded from communion.

Kepler’s was by no means Mästlin’s only renowned student. His most notorious student is certainly Johann Valentin Andreae (1586–1654)


Johann Valentin Andreae Source: Wikimedia Commons

author of the Rosicrucian Chymische Hochzeit Christiani Rosencreutz anno 1459 (Chymical Wedding of Christian Rosenkreutz) published in 1616

valentin_hochzeit_1616_0005_800pxand the Christian utopia Reipublicae Christianopolitanae descriptio (Description of the Republic of Christianopolis) published in 1619. Andreae initially studied theology and mathematics in Tübingen from 1602 till 1605. Mästlin as his mathematics teacher had a major influence on him also introducing him to Kepler with whom he corresponded until the latters death. He was also responsible for Andeae coming into contact with Kepler’s close friend Christoph Besold, who introduced Andreae to the esoteric studies that would lead to his Rosicrucian activities. Andeae’s utopia is, like that from Bacon, one that is endowed with natural philosophy and the mathematical science.


Less notorious but more scientific than Andeae was the polymath Wilhelm Schickard (1592–1635), who as well as being one of Mästlin’s students was also part of the circle of scholars around Besold and Andeae.


Wilhelm Schickard, artist unknown Source: Wikimedia Commons

Like Mästlin and Kepler, a student on the Tübinger Stift he graduated MA in 1611 and went on to study theology. 1613 he received the first of various clerical posts. In 1617 he meet and got to know Kepler, in Württemberg for his mother’s witch trial. He provided engravings and woodcuts for Kepler’s magnum opus the Harmonice Mundi. In 1619 he was appointed professor for Hebrew in Tübingen and here first displayed his talent for logical analysis and leaning. He invented the Rota Hebræa two rotating discs to help his students learn Hebraic conjugations. He also wrote a Horologium Hebræum a Hebrew textbook in 24 capitals, each of which was learnable in one hour. During his time as professor for Hebrew he was also an active astronomer amongst other things producing highly accurate ephemerides. Schickard was a skilled instrument maker and in 1623 he designed and built the earliest known calculating machine, his Rechenuhr (calculating clock) with the intension of helping Kepler with his astronomical calculations. His calculating machine could only add and subtract but included a set of Napier’s Bones in the form of cylinders to aid multiplication and division. He started to build one for Kepler but it got destroyed in a fire. Knowledge of Schickard’s calculating machine got lost in the seventeenth century but was rediscovered in the twentieth century amongst Kepler’s letters by Max Casper. Bruno von Freytag-Löringhoff reconstructed the machine in the 1960s.


In 1631 Schickard succeeded Mästlin as professor of the mathematical sciences at Tübingen. Like Mästlin, Mästlin’s teacher Apian as well as Kepler, Schickard also worked as a surveyor and cartographer.

Schickards Rechenmaschine

Schickard’s Rechenuhr. Reconstruction by Bruno Baron von Freytag-Löringhoff

Throughout his career as an astronomer Mästlin stood in contact and corresponded with nearly all the leading astronomers in Europe. During his later years as professor Mästlin continued working as an active astronomer. Like Kepler he observed the nova of 1604 and the comet of 1618. In 1628 he is known to have observed a lunar eclipse and a second one together with Schickard in 1630. Mästlin was the first person to publish an account of earthshine, the illumination of the moon by sunlight reflected from the earth.


Earthshine reflected from the Moon during conjunction with Venus (left) Source: Wikimedia Commons

Largely forgotten today, except in his role as Kepler’s teacher and an early Copernican, Mästlin was viewed in his own lifetime as one of Europe’s leading astronomers and that with justification.




[1] Lesley Murdin, Under Newton’s Shadow, Astronomical Practices in the Seventeenth Century, Adam Hilger Ltd., Bristol and Boston, 1985


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California, preaching on the burning shore

California, I’ll be knocking on the golden door

Like an angel, standing in a shaft of light

Rising up to paradise, I know I’m gonna shine.

The Grateful Dead – Estimated Prophet


So, as regular readers are well aware I have just spent ten days in the Golden State and as is my wont this is a brief blog post about that journey. I went to California principally to attend scifoo18 in Mountain View. Scifoo is an exclusive, invitation only unconference held once a year at Google in Mountain View by Digital Science, Google, Nature and O’Reilly; foo is short for Friends of O’Reilly. For reasons that I don’t understand I got invited to scifoo15 and had an amazing, once in a lifetime or so I thought, experience. You can read about it here. For reasons that I understand even less I got invited back this year and after some hesitation decided to go, supported by my very generous readers.

Once again the experience of almost nonstop conversations for two days with three hundred people, all of them experts in their diverse scientific fields, proved to be an exhilarating, mind warping experience that defies description. I will however just mention two things. Firstly, I got to meet and briefly to talk with Stewart Brand, Merry Prankster and man behind both The Whole Earth Catalog and The WELL. I owned a copy of the 1971 Last Whole Earth Catalog and meeting Stewart Brand was for me like meeting one of your favourite rock stars.

The second thing concerns my very generous sponsoring readers. At the end of scifoo there is a general session and during this people were asked to relate any special experience that they had had during the weekend. After some hesitation I got up and took a microphone and said the following, “ I attended scifoo15 and had a truly mind blowing experience and never expected to get invited back. However, I did get invited back and initially thought I would have to reject the invitation because I’m a freelance historian of science and a retiree and don’t have very much money. Then I set up a GoFundMe and asked my readers to help me come here. I got €1700 in a few days and so I’m here. I want to publicly thank my unbelievably generous readers, who made it possible for me to come here and to meet all of you.” When I finished I got a thunderous round of applause and for the next half an hour people kept coming up to me and saying that was a wonderful story. So I say one last time thank you to all the wonderful people who helped make my scifoo visit possible.

On the Saturday evening I met up with Travis, who used to live in Prague and is responsible for the History of Alchemy podcast. We had met previously when he and his friends visited me in Nürnberg. He now lives in Santa Clara just down the road from my hotel in Sunnyvale. It was a very pleasant evening.

In 2015 having flown all the way to San Francisco I spent several days playing the tourist in San Francisco and Oakland. This time I decided to travel down to Los Angeles. For some time I have been acting as an advisor to an interdisciplinary group of art historians in LA called project AWE. When I said that I would come and visit them in LA they immediately said I could hold a lecture, so I did.

I flew to LA on the Monday morning and spent the day doing very little, recovering from the weekend. On Tuesday I visited Gypsy, a master’s student from the Western Esoteric Institute of Amsterdam University, who was the first person from project AWE to contact me and who had also visited me in Nürnberg. Gypsy cooked me a great meal and then drove me to Venice Beach where I spent a nice afternoon playing at being a tourist including seeing the Pacific ocean for the first time in my life and consuming my first milkshake in about ten thousand year.

On Wednesday I went to The Huntington, a museum, library and botanical garden in San Marino, where Internet friend Joel A Klein is Molina Curator of the History of Medicine and Allied Sciences. Met up with Joel and had a great chat before spending the day seeing as much of the Huntington as is possible in one day, highly recommended.

Thursday was another lazy day before I held my lecture in the evening on Albrecht Dürer as a Renaissance Mathematicus to an audience of art historians and artists, who received my offering with much enthusiasm.


Lecturer plus audience

You can view your friendly neighbourhood blogger stumbling and stuttering his way through that lecture here.

On Friday Zhenya Gershman, art historian and founder of project AWE, took me round the art galleries of the Getty Centre, where she had worked for eleven years. I can tell you that to go round an art gallery with an art historian, who is intimately acquainted with that gallery, is the only way to do it. You can find photos and brief biographies of both Gypsy and Zhenya here.

I would like to thank the people at scifoo, especially the truly amazing Cat Allman, Travis, Gypsy, Joel and Zhenya for making my visit to California a very memorable experience.

And so ends the 2018 adventures of the Renaissance Mathematicus in the Golden State.




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By the time most of you read this I shall be flying over the Atlantic in the direction of California. There will be no new blog posts until I return at the beginning of July but until then I will leave you with the SciFoo18 Wall of Fame, those unbelievably generous people who helped make this trip possible. I say once again from the depths of my heart how grateful I am to all of them; they are all wonderful people. I also want to take the opportunity to thank all of my readers who have helped to sustain this blog over the last nine years, without it there would never have been an invitation to SciFoo in the first place.



The Renaissance Mathematicus SciFoo18 Wall of Fame

 1) Jonathan Dresner 2) John Kane 3) Anonymous

4) Luke Dury 5) Charles Hueneman 6) Andreas Sommer

7) Cornelis J. Schilt 8) Kevin Quiggle 9) William Connolley

10) Gypsy Van Melle Seaton 11) Anonymous

12) Fawn Nguyen 13) Anonymous 14) Anonymous

15) Gavin Moodie 16) Seb Falk 17) Anonymous

18) Ash Jogalekar 19) Noah Greenstein 20) Anonymous

21) Anonymous 22) Gene Dannen 23) Anonymous

24) Anonymous 25) Anonymous 26) Anonymous

27) Anonymous 28) Michael Traynor 29) Anonymous

30) Gerald Cummins 31) Heribert Watzke

32) Anders Ehrnberg 33) Friends of Charles Darwin

34) Anonymous 35) Anonymous 36) Anonymous

37) Sally Osborn 38) Paul Coxon 39) Matthew Cobb

40) Tim O’Neill 41) Tony Mann 42) Meg Rosenburg

43) Isaac Cowan 44) Timoer Frelink

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The comet huntress

If people ask what sort of historian I am, if I am being somewhat formal I answer a narrative historian of the contextual history of science. That’s quite a mouthful and if I have to explain it I say, I’m a storyteller. I tell stories from the history of science, not anecdotes but factually based stories. For me the most important aspect of those stories is that the scientific elements are embedded in the social, political, cultural, religious, intellectual and economic contexts of their time. Science for all of its supposed objectivity does not live outside of human culture but is an integral part of it. If I was asked to give an example of how my approach to the history of science works in practice I might well point people towards Emily Winterburn’s The Quiet Revolution of Caroline Herschel.[1]

Caroline Herschel001

In this book Emily Winterburn has delivered up a perfect example of how to write a narrative history of the contextual history of science.

Emily Winterburn, who is an Internet friend, wrote an excellent doctoral thesis, which I have as a pdf, on The Herschels: A scientific family in training as a part time student at Imperial College. I assumed that she would turn this work into a book but she chose instead to concentrate her efforts on just one member of this extraordinary family of astronomers, Caroline.

Caroline Herschel002

This is not a full biography of Caroline Herschel but is an in depth look at just one decade in her life, the ten years in which she ceased to be merely Williams sister and assistant and became a fully fledged observational astronomer in her own right. What also makes this decade in her life so interesting for the historian is that Caroline kept journals all her life and also wrote autobiographies for her relatives but for some reason she destroyed her personal records for just these ten years, her, in scientific terms, most productive ones.

Winterburn’s book does not just concentrate on Caroline’s astronomical discoveries, comets and nebulae, but embeds these in the full social, political, economic and cultural context in which Caroline lived and worked during this period. In particular Winterburn illuminates the strategies and tactics that Caroline and also William used to help her gain access to the exclusively male world of late eighteenth century science. Alone this aspect of the book makes it a valuable piece of eighteenth century British social and cultural history and well worth the purchase price.

Winterburn deals in detail with the shifting relationships that Caroline had with the various members of her family, especially both her domestic and her working relationships with her elder brother William. She illuminates, in particular, very nicely how that working relationship evolved as Caroline became not just William’s assistant but an astronomer in her own right also how her role as assistant also evolved as William’s research interests changed over the years. On the domestic side both William’s marriage and the advent of his son John Herschel also caused significant changes in Caroline’s life.

With increasing fame Caroline a shy and retiring person also had to deal with more and more contacts with people outside of her close family circle, something that was in many senses a cause of stress for her. However she still managed to develop personal relationships with leading astronomers such as Lalande and Maskelyne, a slow process that Winterburn illustrates very skilfully.

There is much in depth background material on eighteenth century astronomy, the role of women at this time, particularly in science, and the social structures of Georgian England. This is not just a portrait of an important pioneering female astronomer but a full contextual description of what it meant to be a pioneering female astronomer in the late eighteenth century. Winterburn has written a masterpiece of contextual, narrative history of science, which is also a prime example of first class feminist historiography. She writes superbly with a light touch and her book is a delight to read. I started it in my hospital bed and when I came home it became my bedtime reading. The short but information packed chapters are just the right length for reading before turning off the light and I was quite sad when I finished the book and had to find something else to finish the day.

Because the book is based largely on primary sources the bibliography is very short but Winterburn also gives a list of books for potential further reading. There are endnotes (he said through gritted teeth), which are mostly just references and a comprehensive index. In the middle of the book there is a collection of black and white plates.

If you are interested in the history of astronomy, feminist history, Georgian history or just like an excellent read then buy this book, I promise you won’t regret it.






[1] Emily Winterburn, The Quiet Revolution of Caroline Herschel: The Lost Heroine of Astronomy, The History Press, Stroud, 2017


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A multi-functional book for a multi-functional instrument

Probably the most talked about astronomical instrument in recent years is the so-called Antikythera Mechanism, several corroded chunks of bronze gear work found in the sea of the coast of the Greek island of Antikythera at the end of the nineteenth century.


The Antikythera mechanism (Fragment A – front); visible is the largest gear in the mechanism, approximately 140 millimetres (5.5 in) in diameter Source: Wikimedia Commons

Historian of ancient astronomy, Alexander Jones, who was a member of one of the teams investigating and interpreting the mechanism, has now written a book about it, A Portable Cosmos.[1]


I say that he has written a book but in fact it is really several books in one. The first two chapters deal with the story of the original discovery and recovery of the mechanism. They also sketch the history of the succession of investigations and interpretations of the mechanism that have taken place between its discovery and the present. The longest section of the book deals with a detailed description of the external aspects of the mechanism, its dials, scales and pointers. The penultimate chapter is an examination of the physical aspects of the mechanism, its gears and gear shafts. The final chapter, an afterword, is titled The Meaning of the Mechanism. For me, the most fascinating element of the book is that Jones in his explanations of the functions of the dials and pointers delivers up a comprehensive introduction to the histories of astronomy, astrology and cosmology of ancient Babylon and Greece, in fact I would rate it as the best such introduction that I have ever read.

Despite his very obviously high level command of the material Jones does not baffle with science but writes in a light and very accessible style and I for one found the book highly readable. Of interest is the fact that because large parts of the mechanism are missing and what is there is highly damaged there is not a general agreement under the experts, who have worked on the mechanism, about how to interpret the function or purpose of numerous aspects of it. Jones doesn’t just express his own well-informed and well-reasoned explanations but draws his readers’ attention to alternative suggestions and interpretations, explaining why he prefers his own chosen one. Having said this archaeoastronomer Doris Vickers, who recommended the book to me suggested also consulting the official Greek Antikythera Mechanism Research Project website, which has more information and other viewpoints to those of Jones.

The book has a very useful glossary of technical terms, endnotes (regular readers already know my views on endnotes contra footnotes), a comprehensive bibliography so you can read up on those interpretations that deviate from Jones’ and a good index.

To quote a cliché, if you only read one book on the Antikythera Mechanism, then it really should be this one. It kept me occupied and entertained during my recent four days in hospital and proved to be an excellent companion for that period and I would whole heartedly recommended for happier circumstances as well.

[1] Alexander Jones, A Portable Cosmos: Revealing the Antikythera Mechanism, Scientific Wonder of the Ancient World, OUP, Oxford, 2007

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Imre and me – a turning point

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.


Image of Malmö (Elbogen) in Scania, Southern Sweden from a German book (Civitates orbis terrarum, Vol. IV, by G. Braun & F. Hogenberg) .1580 Source: Wikimedia Commons

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.


Morgan Arcade Source: Wikimedia Commons

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[1]. 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.


This is still the copy that I bought on that fateful day in the small bookshop in the Morgan Arcade forty plus years ago

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[2] 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.


Library of the London School of Economics and Political Science – Professor Imre Lakatos, c1960s Source: Wikimedia Commons

The book that I had acquired is a large part of Lakatos’ 1961 PhD thesis, published in book form posthumously[3], 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[4]. 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.


[1] Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, eds. John Worrall and Elie Zahar, CUP, Cambridge etc., 1976

[2] 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.

[3] 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.

[4] I recommend David S. Richeson, Euler’s Gem, University Press Group Ltd., Reprint 2012

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