Visualize math! See the rules from the complex mathematical world – “Nature’s Mathematical Games”
I have another dream.
My first dream “Virtual Mirage Machine” is just a product of technology. It can help us visualize abstract mathematics, prompt us to build new intuitions, and allow us to ignore the tedious number structure in mathematical problems.
Not least, it can make it easier for mathematicians to explore the world of the mind. But since mathematicians occasionally create new landscapes as they linger in the mathematical gardens, the Mirage can also play a creative role.
In fact, the Virtual Mirage Machine or similar products will come out soon.
Classify complex operations of mathematics into simple patterns
I call the second dream “morphomatics” (morphomatics), it is not a technology, but a way of thinking. As far as creativity is concerned, morphological mathematics is of great significance. But I don’t know if it will happen, or even if it’s possible.
I hope the answer is yes, because we all need it.
The three examples in the previous chapter “droplet, fox and rabbit, and flower petal” have very different structures from each other, but they all show the same philosophical view of how the universe works. They do not derive simple patterns directly from simple laws, as the laws of motion lead to the elliptical orbits of planets. Instead, they run through a giant tree of leafy complexity that finally collapses into a fairly simple pattern at the right scale.
The simple narrative of “water dripping from the faucet” is accompanied by a series of extremely complex and incredible changes.
Although we have evidence from computer simulations, we still don’t know “why” these changes arise from the laws of fluids. This is a simple result, but the cause is not simple.
In the mathematical computer game consisting of foxes, rabbits and grass, many complex and random rules are included. However, the important characteristics of this artificial ecology can be represented by a dynamical system of four variables with an accuracy as high as 94%.
The number of petals is the result of complex interactions of all the primordia, but by virtue of the golden angle, these interactions just lead to the various Fibonacci numbers. Fibonacci numbers are the clue to every Sherlock Holmes, not the culprit hiding behind the scenes. Mathematical Moriarty in this matter is not Fibonacci, but dynamics; mechanisms of nature, not “numbers of nature.”
In these three mathematical stories, there is a common message: the patterns of nature are “emergent phenomena”, they burst out of the ocean of complexity, just like Botticelli (Sandro Botticelli, 1445-1510 )’s Venus suddenly appeared in the shell, without warning, and beyond the matrix.
They are not a direct result of the deep simplicity of the laws of nature, which do not apply at this level. They undoubtedly descend indirectly from the deep simplicity of nature, but the paths between cause and effect are so complex that no one can trace each step.
create a new kind of mathematics
If we really want to grasp the emergence of patterns, we first need to have a new scientific method that can keep pace with the traditional method that emphasizes laws and equations. Computer simulations are part of it, but we need more. It’s not satisfying just to have a computer tell us that a certain pattern exists, we want to know “why”.
This means that we must develop a new kind of mathematics that can treat patterns as patterns, and not just as accidental results of fine-scale interactions.
I’m not trying to change the existing way of thinking about science, which has taken us a long, long way, and I’m calling for another system that complements it.
One of the most striking features of recent mathematics is that it has begun to focus on general principles and abstract structures, and the focus has shifted from quantitative issues to qualitative issues. The great physicist Ernest Rutherford (Ernest Rutherford, 1871-1937) once said: “Qualitative is a poor quantitative description”, but this mentality is no longer justified.
Rutherford’s famous quote should be reversed: Quantitative is a poor qualitative description. Because numbers are just one of many mathematical properties that help us understand and describe nature. If we try to squeeze all the degrees of freedom into a limited numerical system, we will have absolutely no way of understanding the growth of trees or the formation of sand dunes.
The time was ripe for a new mathematics. Rutherford’s criticism of qualitative reasoning was mainly that it was sloppy; this new mathematics has considerable rigor, but it also includes more conceptual flexibility.
We do need an effective mathematical theory of the study of patterns, which is why I call my dream “morphological mathematics”. Regrettably, many branches of science are now going in the opposite direction.
For example, DNA is often regarded as the only answer to the form and pattern of organisms, but current theories of biological development are insufficient to explain why the organic and inorganic worlds share so many mathematical patterns. Perhaps DNA codifies the rules of dynamics, not just the patterns that govern developmental completion. If this is the case, current theories clearly ignore many key steps in the developmental process.
Establish an Appropriate Natural Mathematics System
The idea that mathematics is closely related to natural forms originated with Thompson, and, in fact, goes as far back as the ancient Greeks and even the Babylonians. However, it is only in recent years that we have begun to develop what could be called proper mathematics.
The previous mathematical systems themselves were too rigid, created to accommodate the constraints of pencils and paper.
For example, Thompson noticed that there are many kinds of organisms whose shapes are very similar to the shape of fluids, but if you want to simulate organisms, the equations used in today’s fluid mechanics are too simple.
If we look at a single-celled organism under a microscope, the most incredible thing is that its movements appear to have a definite purpose, as if it really knows where to go. In fact, it responds to its surroundings and inner states in a very specific way.
Biologists are gradually unraveling the mysteries of the mechanisms of cell movement, which are far more complex than traditional fluid mechanics. One of the most important features of cells is their so-called “cytoskeleton,” a sort of intertwined tubular network that looks like a bale of straw and functions as a rigid scaffolding inside the cell.
The cytoskeleton is an amazingly flexible and dynamic structure, under the influence of certain chemicals, it can disappear completely without a trace; wherever it needs support, it can grow there.
In fact, what the cells rely on to move is to disassemble some skeletons and replace them with other ones.
The main component of the cytoskeleton is the microtubules, which I mentioned when discussing symmetry. As I said in that chapter, this unusual molecule has the shape of a long tube and is composed of two units, alpha-tubulin and beta-tubulin, arranged in a black and white checkerboard pattern.
Microtubules can grow by adding new units, and can also curl back from the tip like a banana peel. Its curling rate is much greater than its growth rate, but both tendencies can be stimulated with appropriate chemicals.
——This article is excerpted from “Mathematical Games of Nature”, published by Tianxia Culture in November 2022, please do not reprint without consent.