The role of mathematics in nature and why in sciences it is so effective in catching the laws that govern the phenomena that we observe, has been the subject of speculations and conjectures throughout human history. As more recently we were able to demystify some of the elementary building blocks that make up life and we also got some insight into the processes by which our brain functions, some circle is being closed. Indeed, it is with the help of mathematics that scientists were able to unravel the mysteries of life, and to model our brain while on the other hand mathematics is an abstract construction of the brain and the brain/mind defines the identity of a living organism. This vague circularity starts shimmering faintly through but it is still largely mystical and far from being understood.
In 2011 the Fondation Cartier pour l'art contemporain organized an exhibition A Beautiful Elsewhere in Paris. Part of the exposition was a Library of Mysteries. This work was realized by David Lynch in collaboration with rock icon Patti Smith and the geometer Misha Gromov. Using quotes from great scientists, Lynch visualized the mysteries of time, space, matter, life, knowledge, and mathematics, presented subsequently as
- the mystery of physical laws
- the mystery of life
- the mystery of the mind
- the mysterey of mathematics
An excellent report on this exhibition by Michael Harris was published in the Notices of the AMS 2012 vol 59, no.6, p.822-826.
Gromov was asked to prepare this book as a "naive mathematician's version" of what Lynch had done in Paris, and this is how this book came about. If you know some of the films by David Lynch in which he creates his unearthly mysteries, then you know that they can take some shocking bends that lift the spectator out of reality wondering what had just happened. A sense of humour is not completely absent. It seems like an impossible task to translate this atmosphere into a book format. I was not able to attend the 2011 exhibition in Paris, but I know some of the work of David Lynch and with Harris' detailed report, I can imagine what it must have been like, and I assume that Gromov succeeds well in keeping the original ideas. After all he knows them very well because of his involvement in the 2011 project. This being said, you may not expect to read a linearly structured straightforward book. It uses different colours for (parts of) sentences and different fonts to give a precise meaning to words. Some would classify it as a painting with words and ideas or even poetry. It is full of quotes and ideas and it is seasoned with this this particular slight humorous undertone. Every section, and almost every sentence is an invitation to contemplate about its deeper meaning and the consequences. Some explanation is required, and a lot of these extras are included, not only in the text but also with several footnotes that can be found on each and every page. There are many illustrations, most are borrowed from the web, but their role is more to "give some air" to the text and make it lighter to work your way through. The hope is that the reader will play with the ideas and opinions and whatever is in between those two, so that he or she will experience the Beautiful Elsewhere called Mathematics.
It is difficult to summarize the content because it is multi-branched like a fractal tree that floats on a stream of conciousness. A first part of some seventy pages is mainly a discussion of quotes that reflect ideas of scientists of all times who gave an opinion on science in general, on numbers, laws (physics and others), truth, life, evolution, the brain, and the mind. All these are elements in the chain that eventually will form the circle. The conclusion is that mysteries remain. We do not know much about how space/time/matter/energy is transformed into life/brain, how the latter brings about what we can classify as mind/thought. It seems that we cannot conceive these relations but by using mathematics. This closes a circle because we therefore need to understand how mathematics can be the result of this brain/mind/thought complex and mathematics is precisely the instrument that tells us something about space/time/matter/energy. Only in this last relation we know something as shown by the results of physicists. The mathematics(?) to describe the other connections/transformations/mechanisms are still unknown. This book wants to be a first attempt to throw a hook at these missing mathematics.
The second part is called Memorandum ergo that one wants to complete spontaneously with the missing sum. The ideas/opinions/conjectures in the remaining 130 pages are analysing the finer structure of the components in the volatile circle of mysteries sketched above. For example, there is definitely a difference between brain and mind. The term "ergo" appears for the first time in the conjecture that there is something like an ergo-brain that is not accessible by introspection and that is responsible for unconscious thoughts. It contains structural patterns that we can recognize for example in natural language. The idea is that this ergo-brain is an instance of a wider ergo-system that hopefully can be analysed using mathematics and that eventually will also shed light on mathematics itself. This is the ergo-project and it requires to investigate all the components of this very complex system. The ergo-brain is alert to the unexpected and is bored by the ordinary, repetitive impulses. It is alert to the signals that are "interesting", not the ones that are "obvious" or "logical". It is responsible for a child learning a language or to read and write, or even walk. How do we learn things? It is not sufficient to understand the electrochemical system at the level of brain cells to explain how we attach a meaning to signals that we receive when seeing or hearing something. A strong ergo-brain is probably also responsible for children being gifted for mathematics or music or chess. It is responsible for goal-free learning, meaning that it is not the result of evolution which defines behaviour as maximizing the chance of survival. Evolution has a big hand in forming our ego-mind. The ego-mind is rational and intelligent. It plays by the rules and has common sense, while the ergo-brain wants to be free and wanders around always looking for the new and interesting. A cave-man with a super-ergo-brain would probably not survive, but it made Ramanujan fill up his notebooks with remarkable formulas. The ergo-system is responsible for our agility with our language, for finding pleasure in playful and "useless" activity like solving sudokus, for getting bright ideas, for progress in science and mathematics. The problem is that the ego-mind has no access to the ergo-system. Direct observation is impossible. Moreover it is not logical in the usual sense but requires some ergo-logic to deal with it.
All the elements that play a role in the whole process are analysed in the book: how external signals arrive in the brain, how language has to be analysed, how do we recognize structure, etc. This allows to formulate some principles (16 rules of the ergo-learner) of how we learn through the ergo-system. The trailing part of the book is more technical. It describes in terms of category theory how the ergo-system can analyse language and give meaning to words and sentences. More generally it has to recognize structure and units, classify them through partitioning and clustering and identify connections and relations between units. This is only a first attempt to formalise how an ergo-system can make sense of a text being read or being heard. Although formal in a framework of categories and functors, the description is still more qualitative than quantitative,
The fact that the text is more or less written as a freewheeling stream of consciousness to, in the end, arrive at some result that is still somewhat fuzzy, is a perfect illustration of how our ergo-brain works. This is how it gives meaning to observations and thus how it is feeding the knowledge of our ego-mind. Reading the book is a strange experience that will certainly keep your ergo-brain on alert since what you read is "interesting" and certainly not boring like a standard text is. The book unfolds its ideas following an ergo-logic and therefore should be read by an ergo-brain.