Leonardo 4. Mathematics does not help

As I have mentioned in previous posts the book Leonardo da Vinci by Walter Isaacson helped me to solve Leonardo’s mystery – how he managed to learn so much, remain independent in times when being dependent was must for survival, did not finish many of his works, and did not accumulate much wealth.

A full answer would take a long time, so I will just mention a few things:

— Leonardo was lucky to have an uncle who was digging deep into what is going on around us and how the world works. Together they walked around the city and learned how the things were done. They also went in the fields and study methods of agriculture, how the plants grow, caught grasshoppers and frogs and studied them outside and inside. This gave Leonardo a solid base for his further studies and provided the sweetest taste of new knowledge and skills.
— It made him free mentally since he was able to understand everything from the first principle. It taught him to see things as they are, not how one would like them to be.
— And it made him free financially since he acquired mastery of making things, which provided him with a steady income.
— Having a steady income allowed him to do what he liked without the need to make more money.
— It is unfortunate that he did not encounter somebody who would become his secretary and helped him to publish his notes. Himself, he was more interested in learning new things than completing the projects. That’s why many of his discoveries remained unknown and were rediscovered 300 years later.
— So, he learned things doing. To finish his works was not the point for him. He turned down lucrative offers to paint a portrait because he had made several portraits that helped him to learn things (how life can be presented on a two-dimensional surface) and did not need to paint another one. He carried three paintings with him and improved them all his life (one of them was Mona Lisa).
— He wrote right to left not because he tried to hide something. That was how all lefthanded people were taught to write at the time.

I also got answers to the following two questions: 

— Why Leonardo had such a little interest in mathematics?

— What was Alberti‘s vision of the goal of art and the effect this vision had on Leonardo?

First, a few words about Alberti. He was, as Leonardo later, the “Renaissance Man”, proficient in many areas of art and science, including mathematics. He wrote several influential books, but it is not yet clear whether the ideas expressed in them were his original or he just managed to express whatever was already “in the air.” I would not be surprised that both were true. He learned something from others and added something to it too.

Anyway, his book Della Pittura (“On Painting”) contained the first scientific study of perspective in art, based on classical optics as a geometric instrument of artistic and architectural representation. By the way, his knowledge has as one of the major sources the work of the Arab polymath Alhazen (965-1041), transferred through optical workshops Roger Bacon, John Peckham, and Witelo. Note that Leonardo successfully studied optics too. It is such a shame he did not publish his results. Otherwise, many discoveries in optics would be attributed to him.

But back to Alberti. He insisted that “all steps of learning should be sought from nature” and that painters and sculptors should aim to make their work to “appear to the observer to be similar to the real objects of nature.” But it should not be just imitation, because, according to another Alberti’s statement, “in painting beauty is as pleasing as it is necessary.” And the beauty was for Alberti “the harmony of all parts in relation to one another,” so that “this concord is realized in a particular number, proportion, and arrangement demanded by harmony.” 

Alberti also has particular advice to the painters and sculptors. In order to reflect the nature and its harmony better, he suggests, one has to start from studying how the things are created structurally. If one would like to paint a human figure, for example, one has to know how the skeleton is constructed, how muscles and ligaments are hooked to it and control the motions, how they affect the shape under the skin and, ultimately, under the clothes. Only after all that, when the shape is correctly and naturally depicted, the painter can apply the knowledge of optics and add the correct shadows, colors, reflects, and perspective.

Leonardo followed this advice (taking it from Alberti or just from the vision shared by the art community) and spent long night hours in morgues cutting open the bodies and creating amazingly detailed anatomical drawings. In the process, he got interested and went deeper in his understanding of how the human body works, again making many discoveries that remained recorded only in his notebooks.

He followed the advice consistently and painted and sculptured human and animal bodies and everything around them to the amazing effect we are still enjoying today. Yet, he discovered nuances and sought to improve his works all his life. That is why he kept three paintings (including Mona Lisa) with him all his life and touched them and added and changed some details literally until his death.

And here comes the question about his using of mathematics. Why he did not follow Alberti’s lead and did not master it? There are signs that he tried but did not go deep. Why? Today we know how helpful mathematics can be for solving the mysteries of the Universe.

One possible answer is that contemporary mathematics was not developed enough to provide the answers to many immediate questions. And Leonardo was not oriented on abstract knowledge. He was a man of action and built his world view through the touch and feel – through his skin – the best and immediate way to understand how everything works (There are people who may disagree with me; well, I allow that there are such cases and examples of people like Goedel, Turing, Maxwell, even Einstein who did not need to touch things while having deep insights into the working of nature; but these examples are far from being typical; besides they significantly benefited from the knowledge accumulated by others through the touch and feel and built upon them).

But I like to think that there is another aspect that may be the factor on its own or, more probably, in addition to the first answer. I think that Leonardo, with his acute and unbiased ability to see the things as they are, observed that theory cannot capture all the variety of things that exist in the Universe. Nature has an uncanny ability to demonstrate an exception as soon as an “unbreakable” law of physics is established. Besides the soft human element in the middle of the hard measurable facts often adds completely unexpected dynamics.

That’s why, I think, Leonardo did not master mathematics. He envisioned its limitations, while his project was vast. He attempted to understand everything under the control of the mind. His project was to understand (read “to build”; he was the man of action) life in the minute details.

But such a project cannot be tackled by an individual. It is a project which can be taken on by all humanity only. After many thousands of years, we have made a lot of progress, but we are still far from success.

Besides, there is an embedded weakness in such an approach. Details can go down all the way (like turtles). Meanwhile, the life imitation and simulation (if not reconstruction) can be achieved starting almost from any level of details. And that is what Leonardo did, observing and reproducing the nature in his drawings.

As we do it today too, Leonardo looked intently around and tried to answer the fundamental questions:
What is reality?
How to distinguish it from an illusion?
What is the illusion? Can we recreate reality using an illusion?
Do we even need to try answering these questions? 
Will the answers make us happier?

It did not look like mathematics could help to answer these questions. That is why, I think, Leonardo did not master mathematics.