While I was studying at Cambridge, I and a couple of other PhD students were invited to have tea in King’s College with the startlingly young Martin Rees (he seemed not much older than us, but was newly appointed to a prestigious professorship - just the start of a meteoric scientific career that took him to the posts of Astronomer Royal, Royal Society President and now a seat in the House of Lords). He had been given a fellowship at King’s College and the rooms formerly used by the famous economist Maynard Keynes. Some of walls of his large study were hidden by curtains, and in response to our curiosity he drew them back to reveal murals painted by Keynes' friends Duncan Grant and Vanessa Bell, who were artists associated with the Bloomsbury Group. The opposite wall also had some Grant murals including nudes of both sexes. Martin told us that a previous occupant had had them whitewashed over, having aesthetic and moral objections to displaying near life-sized nudes in a room used for undergraduates tutorials. (The college, however, in consequence of more recent enlightened opinions and posthumous reputation enhancement, had eventually insisted on a reveal-and-restoration. Martin nevertheless found them a little too much and requested the curtains. The work of Bell and Grant to my mind was rather successful in its aim of avoiding any obvious display of the conventional skills of the academic painter and personally I cannot say that I regretted the curtains.
There are few picture of this work, but see https://thecharlestonattic.wordpress.com/2016/04/11/collaboration-at-cambridge-bloomsbury-heritage-in-domestic-aesthetic-2/ and also https://inexpensiveprogress.com/2412/douglas-davidson/. I note that the curtain rail that I remember has been removed. Preparatory studies are known, which suggest that it was all entirely intentional and not, as was suggested to me at the time, the result of painting with a hangover after drunken revels.
This was, however, also my first strong awareness that the intellectual elites of the time were a small and closely interconnected group, with tendrils stretching throughout the upper layers of society. The Bloomsbury members were intellectuals in the old-fashion sense of being well-educated and widely read with extensive interests in many contemporary ideas, not confined to the arts.
One should not therefore feel too much surprise that Virginia Woolf’s diary reveals that she and other members of the Group were at one point discussing “The Crisis in the Foundations of Mathematics”, stimulated by philosopher and mathematician Bertrand Russell’s popular writings on the topic. (Russell was connected with the Group through overlapping membership of the Cambridge “Apostles”.) Some claim that structural experiments in her novels were perhaps influenced by contact with these abstract concepts.
So, bear with me while I spend some time talking about this so-called crisis. We will eventually get to a surprising connection with modern art.
The “Crisis” was a big philosophical issue at the time with potentially wide implications, even discussed in articles in the Times. Although it was generally assumed and seemed intuitively obvious that mathematics was the ultimate in logically rigorous argument, no one had ever managed to prove that the normal course of mathematical reasoning could not lead to contradictions (leaving the possibility that one might be able to demonstrate that the same proposition was both true and not true). From even one such contradiction anything could follow, and from that the whole tower of formal reasoning could fall. Hence, establishing those secure foundations was then felt to be a matter of considerable importance. Unfortunately, towards the end of the nineteenth century the increasing subtlety, complexity and highly abstract nature of the latest mathematics lead some mathematicians into considering the possibility that logical inconsistencies could be introduced into the foundations.
Russell was attempting to reduce all of mathematics to the most basic and intuitively incontrovertible principles of logic, so anything that could be derived by such formal reasoning would then certainly be accepted as valid. (Although Russell later became well known through his political activism and popular writings, his substantial academic reputation was originally built on this highly technical and ground breaking work in mathematical logic.)
Russell had, however, hit a snag, similar to the old paradox of the Cretan barber: this apocryphal hairdresser shaved every man on the island who did not shave himself - so who shaved the barber? It turned out that the foundations of mathematics were inescapably based on subtle ideas of infinite sets and to the great dismay of Russell and others involved in the quest a similar self-referential paradox, now famous as “Russell’s Paradox”, could then be formulated on those “intuitive” foundations. Russell had enormously clarified the problems, but his programme had ultimately failed.
At the same time in Berlin, David Hilbert, by common consent then the World’s leading mathematician, was attacking the same problem in an entirely different way, by claiming that mathematics was merely a game played with marks on paper according to certain formal rules, using symbols that had no fixed meaning. Yes, we could give those symbols a chosen interpretation at the start of the game (as long as we choose a consistent interpretation - that is we do not build in contradictions at the start) and at the end we could interpret the final line of a derivation as a mathematical “truth”. By intention, however, it might also be possible to give exactly the same symbols an entirely different but still consistent interpretation, with the same valid derivation then proving something entirely different. It is as if we had two different dictionaries and grammars, but the same words, so one sentence might have two very different meanings depending on which translation rules you were using.
One needs to be ingenious, but it turns out to be possible to arrange that one interpretation would be mathematics while the other would talk about the mathematical process. Using such dual interpretations Hilbert hoped to use such meta-mathematical reasoning to demonstrate the consistency and completeness of the whole framework (i.e. every true proposition could be proved and all false propositions disproved). Be in no doubt, this is tricky, mind-bending stuff, even for trained mathematicians - a tightrope walk of mathematical logic where you have to think clearly about diverse interpretations at the same time, or fall on one side or the other.
Unfortunately for Hilbert, his programme was blown out of the water in 1931 by Kurt Godel, who used Hilbert’s own tools to construct a particular statement which in one interpretation was a perfectly straightforward and obviously valid (though rather long-winded) arithmetical assertion, while in an equally valid interpretation it said, essentially, “This statement cannot be derived using the rules of mathematics.” If the statement is true, it cannot be proved; if it is false, then mathematics can generate a contradiction - another apparent paradox. Godel later showed that no formal system can prove its own freedom from contradiction - and even the very idea of mathematical “truth” turns out to be rather slippy.
Godel’s theorem is probably one of the cleverest things anyone has ever done. A well-known popular book “Godel, Escher, Bach” attempted to give some insight into the considerable mental gymnastics by drawing heavily on analogies with M.C. Escher’s self-referential and logic-defying graphic art and Bach’s highly structured music, both of which require simultaneous understanding on several levels for full appreciation. (Yes, we are eventually getting closer to art.) The more you think that you are beginning to understand what is going on, the more you start to feel that your reason is slipping its moorings - which was part of Hofstadter’s point. You will not be surprised to find that Hofstadter’s mix also contains some Zen Buddhism: you do not achieve enlightenment until you can feel comfortable with paradox.
Hofstadter’s tour-de-force discussion of artistic concepts in “Godel, Escher, Bach” was an essential part of communicating the immensely subtle central idea to a non-technical audience. It works because ambiguity of symbolism with multiple layers of interpretation has a role in art - and art can make self-referential points about art.
The sculptures of Henry Moore are excellent examples: his nudes invite interpretation both as human figures, but at the same time as purely abstract positioning of volumes and spaces. Are you thinking about the figure represented by the art, or are you thinking about the process of representing a figure? Moore produced great art because he successfully walked the tightrope of diverse interpretations, showing that he understood and communicated both. It is perhaps not a coincidence that Moore had connections with Roger Fry, another Bloomsbury Group member and a Cambridge Art History Professor. Moore was indeed directly influenced by mathematical concepts. Moore said:
Undoubtedly the source of my stringed figures was the Science Museum. …
I was fascinated by the mathematical models I saw there, which had been made to illustrate the difference of the form that is half-way between a square and a circle. One model had a square stone end with twenty holes along each side making eighty holes in all. Through these holes strings were threaded and led to a circle with the same number of holes at the other end. A plane interposed through the middle shows the form that is halfway between a square and a circle. One end could also be twisted to produce forms that would be terribly difficult to draw on a flat surface. It wasn’t the scientific study of these models but the ability to look through the strings as with a bird cage and to see one form within another which excited me.
Unfortunately most of the relevant illustrative images are still under copyright, so I cannot include any. In 2012, however, the Royal Society hosted an exhibition exploring how Moore had become intrigued by 3D models produced - in the days before computer graphics - to help mathematics students visualise complex ideas. It is well worth looking at the catalogue - see https://www.newton.ac.uk/files/attachments/1063471/179161.pdf . Barbara Hepworth is also known to have experienced similar influences (and again copyright is a problem).
Godel’s theorem was an enormous shock to mathematicians and philosophers at the time, especially Hilbert, but it is now well established that some easily stated logical propositions can neither be proved nor disproved: they are formally undecidable. We have even managed to demonstrate that certain long-standing mathematical questions of great, even practical interest fall into that category: in a sense they are neither true nor false.
These ideas were not closely confined within the mathematical community. They were widely discussed amongst intellectuals, partly because of the excellent writings of experts such Russell (who eventually won the Nobel Prize for Literature because his exploration of deep and complex ideas was expressed in utterly lucid prose). Russell was also, of course, very much part of the ruling class - he eventually inherited an earldom - with many connections throughout intellectual society, including within the artistic world - but at times also a bohemian lifestyle.
What does this have to do with art? Well, avant garde movements in Russia in the early 20th Century - “Constructivists” and “Concretists” and so on - were exploring a radical abstract art using “meaning-free symbols” manipulated according to formal rules, and inviting multiple interpretations. Much of this came out of the political turmoil of the early 20th Century, but it is also true that many of the radical artists were drawn from the same elite sections of European society that produced the philosophers and scientists, and some of their language sounds remarkably similar to things being said by mathematicians such as Russell and Hilbert - perhaps not by coincidence, because you can, indeed, put your finger on direct connections. The ideas were “in the air”.
A third and perhaps the most radical antagonist in the debate was Leo Brouwer, who dealt with the Foundation problems by banning the very concept of infinite collections - but throwing away much of modern maths in the process. He was a major figure in topology, which is the mathematics of shape and form. He was also a rather eccentric character who produced one of the strangest - almost mystical - doctoral dissertation of any major mathematician, and choose to live in a small town noted for its artistic community (think Stroud etc.) rather than the conventional milieu expected for someone of his high academic status and distinguished reputation. His personal connections included people who influenced many artists including the founders of the Bauhaus in Germany. Theo van Doesburg, for example, founded De Stijl, a neoplatonic art movement based on the philosophy of the mathematician M.H.J. Schoenmaekers, out of which the art of Piers Mondrian emerged.
Both artists and scientists in that post-WWI context believed in breaking down the boundaries that kept them apart. Einstein himself (who was for a time on the governing body of the Bauhaus, until the rise of the Nazis) often claimed that his ability to visualise complex ideas before turning them into mathematics was of great important to his work, and similar abilities, and a good knowledge of topology, certainly helped Stephen Hawking to continue doing physics long after he was unable to use a pencil or a computer keyboard because of his motor-neurone disease. A recent Nobel Laureate, Roger Penrose, has often used his competence as a draftsman to illustrate complex physical ideas - and indeed some his work directly inspired Escher’s graphic designs.
At about this time the leading mathematical physicist Hermann Weyl was introducing deep and important ideas concerning symmetry into physics, but he was also writing very accessibly about science, nature and art in a popular book called, simply, Symmetry. Weyl also influenced his friend the well known Swiss artist Max Bill (a later Bauhaus director) and good deal of Bill’s art of the time explores subtle ideas of symmetry. D’Arcy Wentworth Thompson’s 1917 book “On Growth and Form”, an exploration of the mathematics underlying the natural world was also admired and used as inspiration by architects.
Some modern conceptual art seems to me to have deep roots in the early part of the 20th Century, when these ideas were floating around and explicitly engaging artists, and I think that there are clear lines of descent ultimately leading to piles of bricks in Tate Modern.
Ideas, of course, never get passed on without modifications, particularly if they are only partly understood and sometimes these misunderstandings are themselves creatively valuable. They can, however, also lead into a wilderness where neither the artist nor the public knows why the art was produced. I personally feel that there is a difference between the precise and intentional communication of ambiguities of meaning, which I find very intriguing (as in Henry Moore’s work) and using obscurity to hide a lack of a clear vision or profundity. There is also a distinction, in my view, between works where multiple layers of meaning are not immediately apparent but become clear after some intellectual effort, perhaps requiring an immersion in the context from which the art has evolved, and, in contrast, work in which the artist’s intention is merely incoherently expressed, or in fact just incoherent. The former is often a sign of great art that repays lasting engagement, the latter is just lack of inspiration and skill. The artist is certainly allowed to assume an educated audience with a knowledge of artistic context, but they still have a duty to communicate as clearly as they are able within the bounds of that context. Sometimes, unfortunately, the ideas are there, and indeed clearly expressed, but turn out to be essentially trivial. As Grayson Perry said of one instance of performance art: “Not many replays in that!
I have personally never found any contradiction between the artistic and mathematical sides of my personality and I do not think that I am out of tune with history.
Most obviously the science of linear perspective was systematically documented by the florentine artist, architect and goldsmith Filippo Brunelleschi. His book on linear perspective, which was a rediscovery of principles known to Greek and Roman artists, was also the foundation of an entire branch of modern mathematics, now known as Projective Geometry. In its current sophisticated form it has many practical applications in computer graphics but also rather deep and abstract uses in pure mathematics and theoretical physics. Leonardo da Vinci, who we must remember also regarded himself as a military engineer, agreed to illustrate a book on geometry in exchange for lessons in mathematics. Artists at the time really did need to know about what was then cutting edge mathematics.
So, why are we increasingly forced to choose between maths/science and humanities, or art and design as we progress through the education system? Partly, of course, there is an awful lot to learn. It is certainly the case that in maths and science that even four year degree courses are now barely sufficient to give an adequate technical grounding for the increasingly specialised professionals. I cannot speak from personal knowledge of art courses, though I know how much time and effort it takes me to acquire new skills as an amateur artist and I imagine that three years is all too little to acquire the range of skills with modern media that are now demanded from art college graduates.
But we do loose something though this specialism. Real creativity often arises from crossing boundaries, which was well understood by Max Bill, Moore, Weyl and Einstein. I resent that I was not allowed to study art at school because I was too good at “academic” subjects. Admittedly I do not regret my subsequent highly enjoyable scientific path, but I also think that I had recognisable artistic potential that could have been developed in a rewarding way. In the event, it took some years after graduating from university before I got round to exploring my artistic inclinations. I also, however, believe that artists who have little knowledge of science are missing insights into one of the major (perhaps the major) cultural influences of the age. Leonardo is a good model.