Saturday, 21 September 2013

How economics suffers from de-politicised mathematics

What I aim to do in this post is explain why I think the de-politicisation of mathematics has been a bad thing.  I have written about how before the Second World War prominent mathematicians were active in public life, a phenomena unheard of today. It might be helpful to clarify what I mean by ‘political’, from Greek: πωλιτικωσ politikos, meaning “of, for, or relating to citizens”, it does not imply ideological.  For me, the first indication that the de-politicisation of mathematics is a bad thing comes from where I, instinctively, see the root cause: in the traumas of the collapse of the attempt to establish the logical foundations of mathematics and the First World War.  I think its difficult for good to flow from tragedy. The ultimate indication that the de-politicisation was damaging is that I think that it contributed to the Financial Crisis of 2007-2009 by creating a  myth of the infallibility of mathematics.

I start with DavidHilbert who had been born in Königsburg, the Prussian city of Emmanuel Kant and Euler’s bridges, in 1862 and where  he completed his doctorate in 1885. Ten years later he was appointed the professor of mathematics at the University of Göttingen, the centre of German mathematics, and in 1899 Hilbert published Grundlagen der Geometrie (‘Foundations of Geometry’), which placed non-Euclidean geometry on a basis of 21 axioms. At the time a number of people, including the British mathematician and philosopher Bertrand Russell, were working on establishing the ‘pure truth’ of mathematics by placing it on a firm logical basis and Hilbert’s work on laying the foundations of geometry was part of this broader effort put mathematics into a clear, consistent, framework.

In 1900, at the second World Conference of Mathematics held in Paris, Hilbert presented an optimistic vision of how mathematics would develop and set 10 (later extended to 23) problems to be solved by mathematicians. In particular he made the claim that “In mathematics there is no ignorabimus” [6, p 445] (‘ignorabimus’ means ‘will not know’ and sets a boundary on knowledge).

Problems with the logical foundations of mathematics emerged in the first decade of the twentieth century with Russell’s Paradox and Hilbert’s response was to create the ‘Hilbert program’ in 1920, the search for a finite set of consistent axioms at the root of (existing) mathematics, an Elements or Grundlagen der Geometrie for all of mathematics. In a paper he presented in 1917, Axiomatisches Denken (‘Axiomatic Thinking’), he argued that at the heart of many fields that mathematics was concerned with there were the axioms
If we consider a particular theory more closely, we always see that a few distinguished propositions of the field of knowledge underlie the construction of the framework of concepts, and these propositions then suffice by themselves for the construction, in accordance with logical principles, of the entire framework.…The procedure of the axiomatic method, as it is expressed here, amounts to a deepening of the foundations of the individual domains of knowledge — a deepening that is necessary for every edifice that one wishes to expand and build higher while preserving its stability. ...If the theory of a field of knowledge—that is, the framework of concepts that represents it—is to serve its purpose of orienting and ordering, then it must satisfy two requirements above all: first it should give us an overview of the independence and dependence of the propositions of the theory; second, it should give us a guarantee of the consistency of all the propositions of the theory. In particular, the axioms of each theory are to be examined from these two points of view. [3, pp 1108–1109]
Hilbert was arguing that the axiomatic method was needed at that time for the same reason that Cauchy, who had imposed rigor on mathematics following the French Revolution, had been needed a hundred years or so earlier: mathematics had developed so quickly in the sixty years after Riemann that it needed to stop and take stock. Hilbert’s ‘formalism’ provides a mechanism for generating sound mathematics,
mathematical proofs are thus seen as a vehicle for making truth flow from axioms to theorems via logical deductions as sanctioned by rules of logic [11, p 292]
I get the sense that in years after his nation's defeat Hilbert is struggling to make sense of a rapidly changing world. Although he focuses on the mathematics, the changes in mathematics had, by and large, occurred before 1908 and I think his search for order in mathematics was a projection of his search for order in society.  Isn't cod psychology marvelous.

The process involved in axiomatisation turns mathematics, in Hilbert's own words,  into “a game played according to certain simple rules with meaningless marks on paper.” To more intuitive mathematicians, like Poincaré, it turned mathematics into a machine, sucking the inspiration out of it, “the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages”.

What is more, the language of the formalists, ‘symbolic logic’, becomes so rarefied that only mathematicians can understand it and this raises a philosophical question as to whether there exists a better language to express mathematics in, and a psychological one, why do we believe in the language. Poincaré’s intuitive approach to mathematics, that it is exact and true as a consequence of the human intellect, was taken up by a Dutch mathematician Bertus Brouwer. Brouwer sees the roots of intuitionism as in the rational response to the collapse of Kant’s ‘neo-Platonic’ idea that some concepts, such as the axioms of mathematics, exist independent of experience that seemed to fall apart when it became apparent that Euclidean geometry was not the be-all and end-all of geometry in the 1850s [3, pp 1171–1172]. The ‘neo-intuitionists’
can never feel assured of the exactness of a mathematical theory by such guarantees as the proof of its being non-contradictory, the possibility of defining its concepts by a finite number of words, or the practical certainty that it will never lead to a misunderstanding in human relations. [1, p 86]
A consequence of this was a rejection of the Law of Excluded Middle. Laplace had said that a mathematical statement must be written in a way that meant it could be proved to be true or false, the ‘Law of the Excluded Middle’, where the middle ground, ambiguity, was out of bounds. Technically, a statement is either true or its negation is true. Either ‘this cat is red’ or ‘this cat is not red’ is true. The foundations of mathematics had been rocked by Cantor late in the nineteenth century with the introduction of infinite sets, which are critical because a continuum is an infinite set. Brouwer argued that statements like “there is a sequence of 100 9’s in the decimal expansion of pi” (which as an irrational number has an infinite number of digits) can not be proved to be false (it can be shown to be true if the sequence is found, but it will take an infinitely long time to search the full sequence). In mathematics if we cannot rely on the Law of Excluded Middle and  we cannot rely on truth/falsity of mathematical statements that apply to continuous phenomena, then how can we rely on any scientific statements to be true or false? Hilbert’s reaction was “Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists”.  And Hilbert is right, we can admire the honesty of the Intuitionists in the same way we can admire the simplicity of the Amish; but we wouldn't like to live like them.

In the end, it was a Platonist, Kurt Gödel, who believed in God and that mathematics exists independently of human thought, who showed that Hilbert’s Program could not be completed, first in a lecture in Hilbert’s home-town of Königsburg and then in a formal paper published in 1931, ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems’.

France’s most prolific mathematician of the twentieth century was Nicolas Bourbaki. France did not protect its intellectuals during the First World War, on the principle that all citizens are equal, and as a consequence it lost a generation of mathematicians and by the mid-1930s most of the living lecturers were about to retire. In 1934, dissatisfied with the quality of the text books being used by this ‘old–guard’ [14, p 104–105] Bourbaki decided to take matters in hand and do for post-Cantor, post-Riemann mathematics what Euclid’s The Elements or Fibonacci’s Liber Abaci had done centuries earlier. He would produce a definitive series of text books for modern mathematics, starting with Set Theory, followed by Algebra, Topology, Functions, Vector Spaces and finishing with Integration. This had to be done, because the innovations of the late nineteenth century had been so profound and, just as Hilbert had realised, mathematics needed to be placed on a stable and coherent framework, if it was not it would lose its status as ‘the art of certain knowledge’.

But Bourbaki worked in the aftermath of Gödel and so the axiomatic approach of Hilbert seemed vain. Bourbaki’s approach was to start from extremely abstract generalisations and only when these had been discussed in detail would special cases, real-world applications of mathematics, be introduced. As  Roy Weintraub has written
Bourbaki came to uphold the primacy of the pure over the applied, the rigorous over the intuitive, the essential over the frivolous [14, p 102]
Bourbaki was the twentieth-century successor to Plato and Kant in that the focus was on the generalisations, the Idealised Forms of mathematics.  The  influence of Bourbaki would reach its peak in the fifties and sixties, not just in his native France but significantly in  the United States.

It is difficult to give any biographical details of Bourbaki for the simple reason that he did not exist. Nicolas Bourbaki is the collective pseudonym for a group of French mathematicians, including Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonne, SzolemMandelbrot (uncle of Benoît Mandelbrot), René de Poussel and AndreWeil (brother of the Socialist-Christian philosopher Simone), who were associated with the École Normale Supérieure.  Generally coming from the educated upper-middle class, fathers were university lecturers rather than teachers, they were almost caricatures of French intellectuals (apart from Poussel who left the group early).  They came up with their plan to rejuvenate mathematics at the Café Capoulade, on the corner of the Boulevard Saint-Michel and Rue Soufflot in Paris’ Latin Quarter. The plan was to operate as a closed ‘secret society’ and produce textbooks employing very precise language and strict formats [14, p 105]. The process of producing the texts was collaborative, and therefore slow and cumbersome. Individuals would write a chapter which was ‘read’ to the whole group, usually at a summer ‘congress’. The group would then tear apart the first draft, and the process repeated until the chapter was unanimously approved. The first book appeared in 1939, with twenty one volumes of part I, “The Fundamental Structures of Analysis” being completed in the late 1950s. By this time mathematics was growing faster than the 8-12 years it took Bourbaki to write a book and through the 1960s the group imploded.

What I find interesting is that the people who made up the core of Bourbaki had come from families who owed their position to the French state, while the movement grew and thrived during a time of incredible political turmoil in France. Between 1920 and the German occupation in 1940, France had over 30 different governments, the longest was Daladier’s 1938 regime that lasted almost two years. Then there was the traumatic occupation followed by another twenty seven governments between 1945 and the establishment of the Fifth Republic in 1958 in the aftermath of a ‘coup’ that recalled de Gaulle from political exile. The First World War was cataclysmic for France and the country took forty years to start a full recovery. As a perfidious Englishman I must admit to admiration for what the country has achieved in recovering from the mess it was in in 1962 (when it withdrew from Algeria and focused on Metropolitan reconstruction).

Ian Stewart, whose first book describing mathematics to non-mathematicians, Concepts of Modern Mathematics, was an exposition of Bourbaki mathematics (as explained in the Preface to the Dover Publications edition) notes that the Bourbaki approach was doomed to failure.
It was a great technique, but it had its limitations—the main one being that it tended to ignore unusual special cases, odd little results about just one example. It was a bit like a general theory of curves that, because it considered circles to be just another special case of much more complicated things, hadn’t appreciated the importance of π. [13, p 497]
and, because of this obsession with the abstract over the practical,
by the end of the sixties, mathematics and physics departments were no longer on speaking terms. [13, p 496]
In 1992 the Nobel Prize winning theoretical physicist, Murray Gell-Mann explained what had happened.
[Bourbaki teaches] a kind of neo-Kantian philosophy in which the laws of nature are nothing but Kantian “categories” used by the human mind to grasp reality …that the structures and objects of mathematics have a reality, that they exist in a sense, somewhere beyond space and time. [5, p 7]
This said, Gell–Mann paints a more optimistic picture
abstract mathematics reached out in so many directions and became so seemingly abstruse that it appeared to have left physics far behind, so that among all the new structures being explored by mathematicians, the fraction that would even be of any interest to science would be so small as not to make it worth the time of a scientist to study them.
But all that has changed in the last decade or two. It has turned out that the apparent divergence of pure mathematics from science was partly an illusion produced by obscurantist, ultra-rigorous language used by mathematicians, especially those of a Bourbaki persuasion, and their reluctance to write up non–trivial examples in explicit detail. When demystified, large chunks of modern mathematics turn out to be connected with physics and other sciences, and these chunks are mostly in or near the most prestigious parts of mathematics, such as differential topology, where geometry, algebra and analysis come together. Pure mathematics and science are finally being reunited and mercifully, the Bourbaki plague is dying out. [5, p 7]
What I see in both the Hilbert and the Bourbaki approaches to mathematics, as well as in the attitudes of mathematicians who emerged in post-Stalinist Soviet science, is a desire to escape the turbulent political realities that surrounded mathematicians.  In response to the  turmoil around them mathematicians seem to wish to create their own Castalia, a place free of politics or economics where the cerebral mathematicians could focus on playing their 'game', as described in Hermann Hesse’s The Glass Bead Game.

Why this is significant is encapsulated in parts of both the US Financial Crisis Inquiry Commission Report and the report of the British Parliamentary Report on Banking Standards ([4, p 44], [10, para 60, vol 2]): today economic authority is based on mathematics. Financial economics produced sophisticated mathematical theorems related to pricing and risk management in the derivative markets and  simply by existing as mathematics they were legitimate. There was no room for debate or discussion because mathematics, based on Hilbert’s formal deduction and Bourbaki’s idealised abstractions, and written in obscure notation, was infallible. It doesn't seem to matter that there were discussion and concerns within mathematics, economics accepted the authority of the theorems and their models simply because they were mathematical.

I have not yet come across what I feel is a credible reason why economics has become so enamored with formalist mathematics. Lawson [7, Ch 10] argues it is because mathematics confers authority, but gives no explanation as to why mathematics should have this power. Lawson challenges the power but one senses that he feels mathematics exists independently of human thought, and this mathematics is irrelevant to social phenomena. He does not seem to think that mathematics could be a product of economic intuition, not just physical intuition. Weitntraub [14] offers a narrative of how mathematical ideas crossed over into economics, without giving what I think is a compelling argument as to why mathematical formalism became so significant. Mirowski argues that ‘Cyborg science’, did not spontaneously emerge but was “constructed by a new breed of science managers” [9, p 15] and it was these managers that promoted the mathematisation of economics. While the emergence of ‘Cyborg science’ as a dominant theme of post-war science may well have been constructed, there is something spontaneous in Wiener  Turing and Kolmogorov, the leading twentieth century mathematicians of the US, UK and USSR, all independently having a youthful interest in biology, becoming mathematicians and making contributions in probability and going on to work in computation.

My own belief is that the critical process was the interaction between (particularly American) economists and mathematicians in the Second World War working on problems of Operations Research. At the outbreak of the war in 1939 the vast majority of soldiers and politicians would not have thought mathematicians had much to offer the war effort, the attitude among the military is still often that “war is a human activity that cannot be reduced to mathematical formulae” [12, p 3]. However, operational researchers had laid the foundations for Britain’s survival in the dark days of 1940-1941, Turing and his code-breakers had enabled the allies to keep one step ahead of the Nazis and Allied scientists had ensured that the scarce resources of men and arms were effectively allocated to achieving different objectives. Alan Bullock argues that the blitzkrieg was the only military tactic available to the Nazis, since they had neither the capability nor the capacity to manage more complex operations [2, pp 588–594] . By the end of the war, it could be argued that the war had been won as much through the efforts of awkward engineers as square-jawed commandos and the Supreme Commander of Allied Forces in Europe and Chief of the U.S. Army, General Eisenhower was calling for more scientists to support the military [12, p 64].

It is hardly surprising that in the post-war years economists embraced mathematics. Pre-war generals would have made the same sort of objections to mathematics that economists had. However, after the war the success of Operations Research could be compared to the failure of economists in the lead up to, and in the aftermath of, the Great Depression that had dominated the decade before the war. But possibly more significant than this theory is the fact that many post-war economists had worked alongside mathematicians on military and government policy problems during the war. Samuelson who was instrumental in introducing stochastic calculus had worked in Wiener’s lab at MIT addressing gun-control problems during the war [8, p 63–64].

Personally I feel prominent economists became over awed by the successes of mathematics, through, for example, observing mathematicians’ abilities to transform apparently random sequences of letters into meaningful messages, something that must have seemed magical and resonant to the economic problem of interpreting data. The problem is codes are generated deterministically  but the same cannot be said for economic data. I believe it was a synthesis of the post First World War traumas of mathematics and the post-Second World War optimism and confidence of economics that created the explosion of mathematical economics in the 1950s-1960s.

Today mathematical finance is possibly the most abstract branch of applied mathematics, while mathematical physics is complex it is still connected to sensible phenomena and amenable to intuition, and this state seems to be atypical of the relationship between mathematics and economics which is concerned with more abstract phenomena.  The situation is not irrecoverable, but, as I have said before, it requires a much tighter integration of non-mathematical economists and un-economic mathematicians.  I look with envy at my colleagues carrying out research  in biology using the same mathematical technology I use but, as one said recently, their papers do not need to prove a theorem and clear results are admired, not technical brilliance.


[1] L. E. J. Brouwer. Intuitionism and formalism. Bulletin of the American Mathematical Society, 20(2):81–96, 1913.

[2] A. Bullock. Hitler and Stalin: Parallel lives. Fontana, 1993.

[3] W. B. Ewald. From Kant to Hilbert: A source book in the foundations of mathematics, volume II. Oxford University Press, 1996.

[4] FCIC. The Financial Crisis Inquiry Report. Technical report, The National Commission on the Causes of the Financial and Economic Crisis in the United States, 2011.

[5] M. Gell-Mann. Nature conformable to herself. Bulletin of the Santa Fe Institute, 7(1):7–8, 1992.

[6] D. Hilbert. Mathematical problems. Bulletin of the American Mathematical Society, 8(10):437–479, 1902.

[7] T. Lawson. Reorienting Economics. Taylor & Francis, 2012.

[8] D. MacKenzie. An Engine, Not a Camera: How Financial Models Shape Markets. The MIT Press, 2008.

[9] P. Mirowski. Machine dreams: Economic agents as cyborgs. History of Political Economy, 29(1):13–40, 1998.

[10] PCBS. Changing Banking for Good. Technical report, The Parliamentary Commission on Banking Standards, 2013.

[11] Y. Rav. A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica, 15(3):291–320, 2007.

[12] C. R. Schrader. History of Operations Research in the United States Army, Volume I: 1942–1962. U. S. Government Printing Office, 2006.

[13] I. Stewart. Bye–Bye Bourbaki: Paradigm shifts in mathematics. The Mathematical Gazette, 79(486):496–498, 1995.

[14] E. R. Weintraub. How Economics Became a Mathematical Science. Duke University Press, 2002. 

Wednesday, 18 September 2013

Political Science: The lost culture of mathematics?

I have been thinking a lot about how political mathematicians should be. This had been at the back of my mind for a while but it came to the fore when I read Pindyck’s paper that motivated Insincerity in Climate Science. Today, mathematicians are supposed to be ‘pure scientists’ standing aloof of political discussion, but in the history of my own domain of mathematical probability there is a rich tradition of mathematicians being involved in political acts. This is intended to be the first of two posts, the second will consider the (negative) impact of the de-politicisation of mathematics.
Poincaré’s impact on mathematics was profound, and as influential as any of his contemporaries. But more than being an ‘ivory tower’ researcher, he was a teacher, a conscientious administrator, to the detriment of his own research, and took an active part in human affairs. For example, Mawhin reports an episode when the French papers linked the wet weather to the passage of a comet, which Poincaré debunked with “humour”.
However not all his activities were motivated by a sense of fun. In 1894, Alfred Dreyfus, a French army officer from a Jewish family in the Alsace, on the border with Germany, was arrested and then court-martialled for spying for the Germans. There was a public outcry that the trial had been a fraud and in 1899 there was a retrial. Part of the evidence against Dreyfus was supplied by Alphonse Bertillon, a biometrician who worked as a police officer and in the 1880s developed anthropometry, a technique for uniquely identifying people on the basis of eleven physical measurements, such as height, length of foot and length or ear and was the forerunner of fingerprinting. Sherlock Holmes is thought to have been born within a year of Bertillion and started his career as a consulting detective in 1881. Holmes refers to Bertillion in The Hound of the Baskervilles and in The Naval Treaty. Bertillion’s evidence at the second trial was emphatic
In the collection of observations and agreements that constitute my demonstration, there is no place for doubt; and this is with not only theoretical but material certitude, that with the feeling of responsibility following from such an absolute certitude, in all honesty, I affirm, today as in 1894, under oath, that the memorandum is the work of the defendant. [9]
Mawhin describes Poincaré’s involvement
Such a statement was philosophically unbearable to Poincaré. In a letter written at the request of Painlevé [another prominent French mathematician who was active in politics] and read to the court, he strongly reacted against the use of probability theory in Bertillon’s conclusions:

Nothing in it has any scientific character. I do not know if the defendant will be sentenced, but if he is, it must be on other evidence. It is impossible that such an argument makes any impression on free-minded people who have received a solid scientific education. [9]
Poincaré was first elected to the Académie des Sciences in 1887 becoming President in 1906 and on the basis of the breadth of his research he became the only person elected to every one of the five sections of the Académie, geometry, mechanics, physics, geography and navigation. In 1908 he was elected to the pantheon of French intellectuals, the Académie Française, following the footsteps of d’Alembert, Condorcet, Laplace and Fourier, being elected director just before his sudden death in 1912. Mawhin, a mathematician himself, notes that there has been no mathematician in the Académie Française since 1941, reflecting the fact that recently, mathematicians have not added any ‘style’ (perhaps élan is a better term) to their work [9].
Émile Borel was born in Montauban in 1871, his father a Protestant minister and his mother coming from a family of wealthy wool merchants. He was a brilliant student and came top of both the entrance examinations to the École Polytechnique and the École Normale Supérieure. His family were keen for Émile to go to the École Polytechnique as it was a better route into business, but Emile chose the École Normale, since it would lead to a career in science.
By the time Borel had completed his thesis in 1894 he had already published six papers on other topics, and was sent to the University of Lille for three years ‘apprenticeship’, returning to the École Normale in 1897 with a further twenty-two papers to his name. His rise was meteoric, becoming Deputy-Director of the École Normale in 1910. He has named after him at east ten mathematical notions, including the fundamental Borel sigma-algebra, and a crater on the moon.
Borel had become interested in probability around 1905, and in 1921 he was appointed to be the professor of Probability and Mathematical Physics at the University of Paris. Borel’s attitude to probability was noteable for such an outstanding mathematician. As Eberhard. Knobloch puts it
[Borel] took for the most part an opposed view [to other mathematicians at the time] because of his realistic attitude toward mathematics. He stressed the important and practical value of probability theory. He emphasised the applications to the different sociological, biological, physical, and mathematical sciences. He preferred to elucidate these applications instead of looking for an axiomatisation of probability theory. Its essential peculiarities were for him unpredictability, indeterminism, and discontinuity. [5]
Around 1921 he began to consider situations where winning a game did not depend solely on chance, but also on the skill of the players [6, p 33], such as the game of baccarrat which had been studied by Joseph Bertrand in 1899, and, unknown to Borel, Her that was solved in 1713 by the first Earl Waldegrave, an illegitimate grandson of James II/VII, while in exile in France using ‘mixed’ strategies over two hundred years before von Neumann-Morgenstern. Borel published a series of papers on the general theme of ‘Games that Involve Chance and the Skill of the Players’ in 1924.
Borel, driven by the ‘hunch’ that in exploring these problems, new mathematics would emerge [3, p 84], approached these problems first by considering a game of two players, both of whom could adopt three similar strategies, and working out whether the ‘best’ strategy existed, or if it did not exist, what was the best set of ‘mixed’ strategies. The main difference between Borel’s work and that of Waldegrave is described by the joint biographer of Norbert Weiner and John von Neumann, Steve Heims
The first step in a proper mathematical theory of games is to provide a suitable description of games in mathematical language. Such a description must contain all necessary information concerning any game but should contain no irrelevant information. Irrelevant information would impede insight into the mathematical problem. But one can devise such a suitable description only after one is very clear about the mathematical problems that one wishes to pose in relation to games. This was first done by ...Borel in 1921.[3, p 83]
Borel showed how the simple three strategy game could be extended to the case where there was a continuum, an infinite number, of strategies and commented that
The problems of probability and analysis that one might raise concerning the art of war or of economic and financial speculation are not without analogy to the problems concerning games, but they generally have a much higher degree of complexity [2, p 20]
Despite this clear precedent, von Neumann was clear in stating that when he was working on games, he was unaware of Borel’s efforts [6, p 45].
Borel had volunteered for military service in 1914 and initially commanded an artillery battery, but then worked in research for most of the war and was awarded the Croix de Guerre. The war had had a profound effect on Borel, half his students and his adopted son had been killed, and in 1920 he resigned from the Deputy Directorship of the École Normale and he became active in leftist politics, being elected to the Chamber of Deputies in 1924 and becoming the Minister of the Navy in 1925 under the mathematician Prime Minister Paul Painlevé’s government which lasted six months. (As a side note one of Borel’s political opponents was Raymond Poincaré, Henri’s cousin). During the Second World War, in his seventies, he was an active member of the French Resistance and wrote a number of books on the practical usefulness of probability. He died at the ripe old age of eighty-five.
In 1897 a young German mathematician. Felix Hausdorff wrote a paper Das Risico bei Zufallsspielen (‘The risk in random games’) that proposed measuring risk by the expected square of the shortfall, developing the expected shortfall proposed by Teitens. Hausdorff is perhaps the most tragic of the German mathematicians of his age. Born in 1868 in what was then Prussian Breslau, but is now Polish Wroclaw, into a family of wealthy Jewish textile merchants, he graduated from the University of Leipzig in 1891 with a doctorate in mathematics applied to astronomy. However, Hausdorff was more interested in Nietzsche’s modernist philosophy and contemporary literature than mathematics, publishing, the same year as his book on parlour games, Sant’Ilario. Gedanken aus der Landschaft Zarathustras (‘Sant’ Ilario: Thoughts from the Landscape of Zarathustra’), a collection of aphorisms related to the German philosopher, Friedrich Nietzsche’s work, ‘Thus spoke Zarathustra’.
Hausdorff’s involvement with Nietzschean philosophy was not political but cultural and had a profound effect on mathematics. At the time Nietzsche dominated the intellectual circles that Hausdorff was mixing with. At the same time as Poincaré was thinking about mathematical recurrence, Nietzsche discussed the philosophical concept of ‘eternal recurrence’ that featured in both Hindu and Ancient Egyptian religions, in Die fröhliche Wissenschaft (1882) (‘The Gay Science’). According to the historian of mathematics Moritz Epple, when Hausdorff tried to reconcile Nietzsche’s philosophy with Poincaré’s mathematics he was sucked into Cantor’s world of transfinite numbers and point-set topology. By 1904 Hausdorff stopped writing literature and in 1910 became a professor at the University of Bonn, publishing the Grundzüge der Mengenlehre (‘Foundations of Set Theory’) in 1914, which would become the standard text on the subject, introducing new concepts of space and measurement that would go on to twentieth century analysis.
Hausdorff, like all other Jewish mathematicians in Germany at the time, was expelled from the University of Bonn in 1934 and tried, unsuccessfully, to emigrate. The University tried to protect him, but in January 1941 he received orders that he would be deported to a concentration camp, and on 25 January 1942 he wrote to a friend
By the time you receive these lines, we three will have solved the problem in another way - in the way which you have continually attempted to dissuade us. ...What has been done against the Jews in recent months arouses well-founded anxiety that we will no longer be allowed to experience a bearable situation. ...Forgive us, that we still cause you trouble beyond death; I am convinced that you will do what you are able to do (and which perhaps is not very much). Forgive us also our desertion! We wish you and all our friends will experience better times.
and the following day, along with his wife and sister-in-law, committed suicide.[10]
Andrei Andreyevich Markov was born in 1856 in the provincial city of Ryazan, but in the 1860s his father became the estate manager for a Russian princess and the family moved to St. Petersburg. Andrei falls into the class of mathematicians who were sickly as children but showed remarkable aptitude for mathematics (His younger brother Vladimir was similarly gifted in mathematics, but died of tuberculosis when he was 25.), and in 1874 entered St Petersburg University to study mathematics and physics. He submitted his Masters thesis in 1880, which caught the attention of Chebyshev, Russia’s greatest mathematician of the nineteenth century, and then studied for his doctorate while teaching at the university.
Markov was working on problems in probability in the midst of the Russian Revolution of 1905 (Bloody Sunday and the revolt on the Battleship Potemkin) and Markov was an active supporter of the revolutionaries. He had been elected to the Russian Academy of Sciences, but the Tsar forced his removal and when the Romanov’s celebrated the tercentenary of their rule over Russia in 1913, Markov ostentatiously celebrated the bicentenary of the Law of Large Numbers. After the February Revolution of 1917, Markov requested to be sent to the provinces, where he taught for the next four years and when he returned to St Petersburg he was seriously ill, and died in 1922. [10, Markov]
Andrei Nikolaevich Kolmogorov was born at the end of April 1903 in Tambov, about half-way between the Crimea and the Kolmogorov family home near Yaroslav, on the Volga River some 150 miles north-west of Moscow. Andrei’s mother was described as ‘independent’, which might explain why this unmarried woman was giving birth some 300 miles from home as she travelled north from the Crimea. Maryia did not survive her child’s birth and the baby Andrei was raised by one of his mother’s sisters, Vera. Despite the bad start to his life, Andrei was bought up in a loving home, made comfortable by the fact that his grandfather was a local, though minor, noble. Not much is known about Kolmogorov’s father, other than after training as an agriculturalist and becoming involved in revolutionary politics, he had been exiled to Yaroslav were he met the Kologorovs, who were also involved in revolutionary activity. After the Russian Revolution he was appointed tot the Ministry of Agriculture and was killed fighting the White Russian general, Denikin in 1919. ([4], [13])
Vera and Andrei moved to Moscow in 1910 and Andrei went to a progressive private school, where his favourite subjects were biology and history. On account of the difficult situation in Moscow in the aftermath of the October Revolution of 1917, Kolmogorov left the city and worked on the construction of a railway between 1918 and 1920. He returned to Moscow and was admitted, without any examination, to the University of Moscow to study mathematics and physics. Kolmogorov did not restrict his studies to mathematics, but also took courses in metallurgy and history. His first research paper was in fact on landholding in late-medieval Novgorod, and Kolmogorov often told a story that his his teacher said to him “You have supplied one proof of your thesis, and in the mathematics that you study this would perhaps suffice, but we historians prefer to have at least ten proofs.”.[13]
In the midst of the momentous events of the Revolution and the restrictions of War Communism, Kolmogorov questioned the relevance of mathematics, but the attraction of mathematics would not let him go. In 1922, whilst still still only 19, Kolmogorov came up with an important technical result he became an international sensation. He became a postgraduate student in 1925, the same year as his first paper on probability, and received his doctorate in 1929, having published 18 papers on mathematics. For the summer of 1929, Kolmogorov and another young mathematician, Pavel Alexandrov, who had recently returned from post-graduate work at Princeton, travelled from Yaroslav to the Caucuses and then on to France and Germany. In 1931 Kolmogorov was back at Moscow, as a Professor of Mathematics. [4]
Whilst in the Caucuses, Kolmogorov started to think more seriously about probability, and seems to have discussed his ideas in France and Germany. His trip to western Europe must had clarified everything for Kolmogorov, because when he returned to Moscow he was able to put together the most important work in probability ever, his Grundbegriffe der Wahrscheinlichkeitsrechnung (‘Foundations of Probability’), which was published, by the German company Springer, in 1933.
The Grundbegriffe axiomatises probability theory firmly within mathematical analysis, synthesising Poincaré with the work of Borel and Hausdorf. Maistrov notes
The axiomatisation resulted in abstracting the notion of probability from its frequency interpretation, but at the same time made it possible to always pass over from a formal system to real-world processes [8, p 264]
In axiomatising probability Kolmogorov achieves two things, he frees probability from being tied to frequencies, a link first made by de Moivre and embedded by Laplace: a probability can be any measure of a set, concrete or abstract. This, on its own, would be useless, but Kolmogorov also ensures that the abstract mathematics is tied to the real-world. Poincaré had highlighted the importance of probability to science as the means for establishing inferences, in fact the importance of probability is much deeper, it links the physical world of events, to the formal, abstract, hyper—real world of mathematics. The importance of probability to science is much more fundamental than being a simple tool.
The purpose of the axiomatisation was to lay the foundations of probability, and the rest of the Grundbegriffe builds up a coherent, mathematical, theory of probability. One aspect was in clarifying the idea of conditional expectation, that the expectation is ‘conditional’ on what is known. This observation may not seem too revolutionary, but at the time it was. This is captured by the Bulletin of the American Mathematical Society review of the Grundbegriffe which appeared in 1934
Moreover, the theorem of Bayes, concerning whose validity there have been many controversies, is also an almost immediate consequence of the system of postulates, but the reviewer does not think this derivation of the theorem of Bayes settles the old contention relative to the validity of inferring the characteristics of a statistical population from a sample by means of the theorem of Bayes. [11]
In proving Bayes’s Theorem, by establishing the mathematical definition of conditional expectation, a proof that had rested on accepting that probability measures where abstract measure on sets and not defined in terms of relative frequencies, Kolmogorov made de Finetti’s subjectivist approach to probability, acceptable.
This, in Stalin’s Soviet Union, was not a politically neutral achievement. At exactly the same time as Kolmogorov was formulating the foundations of probability, 1929—1933, Stalin was, literally, executing his ‘Second Revolution’, the economic and social revolution that involved the extermination of the kulak, landowning peasants, and the forced collectivisation of the Soviet Union. At the time, the Marxist Dirk Struik, who was Dutch but in 1926 he was invited to both Moscow and MIT, he chose to move to the US to work with Norbert Weiner, was arguing that no subjectivist interpretation of probability was compatible with Marxist ideology, and that probability should be
a physical theory, and not a subjective theory, and a theory in which one investigates the relationship between causal and random events.[12, Quoting Struik, (1934) On the foundation of the theory of probability (in Russian).]
Dialectical materialism was, like logical positivism, opposed to metaphysical explanations of phenomena, effects had their causes that the materialistic scientist should discover. To get an idea of the politicisation of mathematics in the 1930s, Struik accused the logical-positivist Mises of taking a ‘metaphysical’ position in developing the frequentist approach to probability [12, Note 8 on page 351].
The risks Kolmogorov was taking were not abstract. In 1936 his PhD supervisor, Nikolai Luzin, became a victim of the second Revolution. Luzin’s mathematics had been critisiced in 1930 for being too abstract, and his PhD supervisor, Dmitri Egorov, was convicted for anti-Soviet activities because he opposed religious persecution. Luzin survived this purge but was criticised in 1936 for publishing abroad and of plagiarism in a campaign that was in part orchestrated by Pavel Aleksandrov under the guidance of Ernst Kolman. George Lorentz tells the story of this politically motivated attack
Famous mathematicians formed the interrogating commission at the Academy’s trial. Of these, Lyusternik, Shnirelman, and Gel’fond already belonged to the “initiating group” responsible for Egorov’s downfall. They were joined by Sobol’ev. Luzin’s former students were represented by Aleksandrov, Kolmogorov, and Khinchin. This revealed a split among Luzin’s students: Lavrentiev and P. S.Novikov were present, but did not say a word against Luzin, a sign of civil courage, while Menshov and Nina Bari (one of the best Soviet female mathematicians) were missing altogether. Actually, Kolmogorov said very little. [7]
Luzin was re-habillitated in 2012 but there is still some mystery as to why Aleksandrov, Luzin’s student, was a part of the group leading the prosecution. A widespread theory is that Aleksandrov was gay and the secret service had incriminating evidence on him, the deal was deliver a bigger fish and you can go free. There are views that Kolmogorov was drawn into the plot against his will because he had had a homosexual relationship with Aleksandrov. Or, Luzin was gay and corrupted Aleksandrov and possibly Kolmogorov; Kolmogorov famously and inexplicably slapped Luzin in public in 1946. Others argue that the homosexual context was invented after Stalin’s death to justify Luzin’s prosecution, which was motivated by Aleksandrov’s desire to advance in the bureaucracy.
At the end of the 1930s Kolmogorov turned his attention to biology, which was a courageous act in the Soviet Union, which was undergoing Stalin’s first purge that involved the disappearance of around 20 million [1, p 543]. At the time an agriculturalist, Trofim Lysenko, was advocating an idea, Lamarkism, that ‘acquired’ characteristics could be inherited, for example the muscles of a blacksmith would be inherited by their children (Darwin acknowledged this process as a possibility, which he called pangenesis, in Variation in Plants and Animals under Domestication). This was completely at odds with Medel’s mathematical genetics but fitted nicely with a Stalinist view that bad behaviour would disappear as people acquired socialist habits which would be inherited and reinforced. As a result, in the early 1930s Lysenko began to dominate Soviet biology, and by the end of the decade geneticists, such as Nikolai Vavilov, were being purged and sent to their death in prison-camps. In 1938 Kolmogorov, inspired by R. A. Fisher’s 1930 book The Genetical Theory of Natural Selection that integrated Mendelian probability with Darwinian biometrics, derived a differential equation that described how the ‘concentration’ of a species changed in time. In 1940 he went further and published a paper with the challenging title On a new confirmation of Mendel’s laws. ([13, p 899], [12, p 342])
During the war, like Wiener, Kolmogorov became involved in fire control and the best way to distribute anti-aircraft balloons around Moscow. However he worked on many other topics as well, including further work related to probability and analysis in general and the question of turbulence in fluids, an area in which his impact was as great as that in probability [4, Batchelor, p 47]. In the 1950s, Kolmogorov turned his attention to information theory, dynamical systems and complexity [4, Moffatt and Lorentz], including solving Hilbert’s 13th problem.
In celebrating the bicentenary of Newton’s death, Kolmogorov’s assessment, as the Russian science historian A. P. Youschkevitch writes, contrasted the “sound brightness of Newton’s mentality” with the “mathematical mysticism of Leibnitz and Euler”. In Kolmogorov’s words
Newton not only made fundamental discoveries by applying mathematics to the natural sciences ...Newton was also the first to conduct a unified mathematical study of all mechanical, physical, and astronomical phenomena. Speaking about the time of Newton, one may also discuss the subordination of isolated fragments of the natural sciences to mathematics. Of course, Leibniz’s ideas about the possibility of the mathematization of all human knowledge were even more universal. But, precisely because of their absolute generality and abstraction they proved to be fruitless.[14]
Having focused on the applications of mathematics in the 1950s, in the 1960s he laid the theoretical foundations of information theory, in the context of algorithms, developing Turing’s ideas. In doing this work, Kolmogorov highlighted the usefulness of some of Richard von Mises ideas in probability. [4], [13] In 1960 Kolmogorov was appointed to be the first Director of the Laboratory of Statistical Methods that introduced modern statistical techniques into Russia, in particular those developed by Jerzy Neyman in the 1930s. As well as applying statistical methods in the physical sciences Kolmogorov turned his attention to the ‘Applications of mathematical probability and statistics to poetics’, an area that he continued to publish in up to 1985 [4, p 40].
In 1963, as the country rebuilt itself following the death of Stalin in 1953, the Education Ministry opened four specialist schools for mathematics and physics in Leningrad (St Petersburg), Moscow, Kiev and Novosibirsk. Kolmogorov became so involved with the Moscow school that it became known as ‘Kolmogorov’s school’ and for fifteen years, until he was seventy-five, he not only gave lectures but introduced students to the arts and broader extra-curricular activities.
Kolmogorov was asked how youngsters should be introduced to science, he observed that ‘celebrated scientists’ had been nurtured by teachers, lecturers and doctoral supervisors, but in addition had been surrounded by supportive friends.
Now, when our country is in need of many capable and well-educated researchers in the most diverse branches of science and technology it becomes imperative to establish a wide system of institutional measures with extracurricular lessons with the senior school children: specialized schools, various types of non-school activities, wide familiarization of the young with the specific nature of work in the universities and technical colleges of the new technology, proper organization of entrance examinations, and wide involvement in research of students in colleges, where the teaching of future researchers is subsidiary only.[13, p 927]
Kolmogorov advocated a holistic approach to education, School children should be taught beyond the classroom, being exposed to the work of research intensive universities and applied technical colleges and students of the universities should mix with those in the technical colleges. In 1970, as part of this programme, he created a magazine, Quantum, on mathematics and physics for school-students.
Pedagogy dominated Kolmogorov’s work in the 1970s-1980s and alongside his mathematical legacy, Kolmogorov has left a more tangible legacy in the many Russian mathematicians who occupy academic posts across the world and were taught by a system he moulded.[13]
Kolmogorov died in October 1987 of a lung condition and having been suffering from Parkinson’s disease for a few years. He had had been highly decorated by the Soviet State and his obituary was signed by Russia’s leaders. While Kolmogorov was a truly great mathematician, he does not fit the stereotype of an academic living in an ivory tower manipulating symbols in an abstract game. He was, like Poincaré, a mathematician guided by his intuition [4, Hyland, p 63], rather than Hilbert’s formalism, and motivated by practical issues, from topics as diverse as biology, geology, fluid dynamics and poetry. He attacked the false statistics of Lysenko, risking his life, just as Poincaré had risked his reputation in attacking Bertillion. While he did not leave an explicit philosophy of science, he did leave a legacy in how science should be taught.
Outside of mathematics and eastern Europe, Kolmogorov’s name is not well known, certainly it is not as familiar as Newton, Leibnitz, Gauss, Riemann or Poincaré. Kolmogorov’s reputation has been partly hamstrung by the fact that he worked beyond the Iron Curtain, and The West was unlikely to lionise a Soviet scientist, and partly by the attitude of many prominent mathematicians to the Grundbegriffe, which challenged the status quo surrounding the basis of probability.


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[8] L. E. Maistrov. Probability Theory: A Historical Sketch, Translated by S. Kotz. Academic Press, 1974.
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[11] H.L. Reitz. Review of Grundbegriffe der Wahrscheinlichkeitsrechnung. Bulletin of the American Mathematical Society, 40(7):522—523, 1934.
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[14] A. P. Youschkevitch. A. N. Kolmogorov: Historian and philosopher. Historia Mathematica, 10:383—395, 1983.