*Ï€Ï‰Î»Î¹Ï„Î¹ÎºÏ‰Ïƒ*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 David Hilbert who had been born in Kà¶nigsburg, 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 Gà¶ttingen, 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.

*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]

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]

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”.

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]

In the end, it was a Platonist, Kurt Gà¶del, 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 Kà¶nigsburg and then in a formal paper published in 1931, ‘On Formally Undecidable Propositions of *Principia Mathematica *and Related Systems‘.

*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’.

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]

*à‰cole 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.

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

Ï€. [13, p 497]

by the end of the sixties, mathematics and physics departments were no longer on speaking terms. [13, p 496]

[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]

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]

*The Glass Bead Game*.

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.

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