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When I was a child, I was strongly attracted by the mathematical formulas in my father’s physics textbook. He is a physics teacher and avid math geek. I didn’t know anything about physics at the time, but I could feel that they were important, as if there was a kind of attraction. But it’s not just the mystery that attracted me. I was also struck by the visual form of these equations. For me, these forms have an abstract beauty in themselves, no matter what they represent, they are as beautiful as works of art.
Now I have also studied mathematics and physics myself, and I have a better understanding of the meaning of these equations and the beauty of the deeper ideas they convey. However, I found myself still attracted by their formal beauty and still liked their look. Some equations have a pleasant balance, reflecting the elegance of the ideas they represent in the visual aesthetics.
I don’t think this attraction is superficial at all: It represents an appreciation of form, and reflects a certain response of humans to the characteristics of the natural world. I believe many people feel this way. Appreciate the beauty of thought represented by great equations, the physical or abstract world they describe, and the visual beauty they present are not mutually exclusive. They are all different aspects of inner beauty, which are interrelated and complementary. In other words, beauty and truth must have common aspects.
Figure 1. Euler’s formula is called the most beautiful formula
In a 2014 neurobiology study on appreciation of mathematics beauty, , one group The mathematician was shown a series of formulas and asked to judge whether the formula was “beautiful.” In performing this task, functional magnetic resonance imaging was used to monitor the brain activity of each participant to determine the activity of brain regions when mathematical beauty was perceived. It is worth noting that the appreciation of mathematical beauty is related to the activity of the A1 area of the medial orbitofrontal cortex, which is the emotional area of the brain, and is also related to the beauty of other sources, such as visual arts and music. Especially, the equation presented in this test is a sample of a wide range of subjects across mathematics and physics-from number theory to mathematical biology, including famous and obscure equations, so participants may not understand all equation.
This shows that although there is a positive correlation between understanding and beauty, this correlation is not perfect. Some subjects will identify equations that they do not fully understand as beauty. The same is true for the results of the control group that includes non-mathematicians.
These experimental results show that the understanding of the beauty of mathematics is not entirely dependent on knowledge understanding. It is unclear why this happens. One possibility is that beautiful equations have a balanced form that is neither banal nor too complicated. It implies some deep importance, but whether they are fully understood or not, it will convey a positive awe Sense-Make people feel part of something huge and mysterious, but not completely overwhelmed. Perhaps it is mechanisms like this that attracted me to the beauty of equations when I was a child, and made me unconsciously realize that although I don’t understand them, their existence implies that the world is meaningful.
That’s why their mystery is exciting and comfortable, rather than disturbing. This possibility is related to “Aesthetic judgment is our meaningful emotional expression of something” The point of view is the same. In other words, we don’t like things that are too simple because they are too boring; we don’t like things that are too complicated because they often mean incomprehensible.
Beauty in physics and biology
Physicists often seek unified laws or ultimate theories that can succinctly explain experimental observations.
In this exploration, the elegance or simplicity of the theory is often used as the guiding principle. Many physicists believe that simplicity [3~5] will bring beauty, not just for Frugality is like giving a series of explanations and choosing only the simplest one, but it represents a deep-rooted belief-a basic belief like an axiom: Although the world looks complicated, the truth must be simple , Elegant and harmonious.
Richard Feynman (Richard Feynman) famously made this point of view in his speech on the characteristics of physical laws Summary: “You can understand the truth through simplicity and beauty… Inexperienced students often make very complicated guesses, which look good. But I know it is wrong at a glance, because the truth is always more than you think Simple”. Feynman’s enthusiasm for beauty is encouraging, and the meaning is very clear: If you think the world is complicated, it means you haven’t understood it correctly.
This view is not without basis. Many major discoveries in physics are basically made under the belief that beauty is the guidance. Murray Gellman(Murray Gell-Mann) once talked about how he and his colleagues boldly proposed a The new weak force theory. At that time this theory contradicted a lot of experimental evidence, but “because we think it is so beautiful, it must be right!” < /sup> Under this bolder claim, beauty can not only be used to guide truth, it may even outperform empirical observation. It is worth noting that the experiment proved wrong in the end, and they were right.
Figure 2. The cover of Nature Physics, published in November 2020, uses complex media to unravel entanglements.
Gellman’s brain activity deep in the orbitofrontal cortex has triggered insights into the basic structure of the universe. His appreciation and belief in the beauty of mathematics has evidence that this emotion is trustworthy and helps us understand the world better. It can be seen that beauty can be used as a more reliable guide than experience.
However, not all scientists see the problem in this way.
Because life systems are unusually complex, biologists tend to pay more attention to special situations than physics, and to the notion of generalized beauty (such as Elegant) Little attention is paid. At first glance this seems to mean a lack of imagination, and it boils down to the mere science of “philately” . But such colored glasses are unfair. Biologists’ preference for details is not due to a lack of aesthetic awareness. In fact, the imagination and creativity required for the best biology is no less than the insight required for the best physics. On the contrary, it is derived from beauty and truth. Another scientific perspective.
From the perspective of life sciences, two points are worth noting.
First, biology is usually pragmatic. Life systems are very different, and each has its own special details: bone cells are different from brain cells, they express their shared genomes in another way, and interact differently with the extracellular environment; ants and apes Different species, they have their own unique physiological functions, derived from their different ecological niches and specific ancestors.
These intricate issues can only be correctly understood through careful observation and clear definition. They are not easily classified as a uniform theory. Even in many biological scenarios, specific details are crucial, even more important than principles. For example, in drug discovery, the precise details of how the drug molecules act are more important than the principles that caused them. Meticulous life from generation to generationBiophysicians focus on pragmatic research that has greatly benefited our society-from advances in medicine and medicine to a better understanding of the services provided by biodiversity and ecosystems, these should not be devalued.
But beyond that, there is a deeper reason for appreciation. Biologists, like physicists, have a deep understanding of the beauty of nature, but they are more sensitive to other aspects of beauty. For example, in the eyes of many people, beauty does not first appear in abstraction, such as elegance, simplicity, or mathematical proofs, but in the natural moment, from the extraordinary diversity in front of them, like As Darwin described, “The most beautiful and wonderful shape of endless” . So physicists may see beauty in simplicity, but biologists may see beauty in interrelationships, mutually beneficial symbiosis, and complexity.
Undoubtedly, this is also related to the teaching methods of biological sciences. Compared with physics training, biological sciences usually pay less attention to mathematics. But this is not the only reason. Biologists are more accustomed to immersing in the intricacies of the natural world, so they are more comfortable with the ambiguities and complexity of nature than physicists who love to seek universal understanding. Moreover, this sense of immersion can cause an effect similar to the beauty or awe that physicists feel when they see elegance. So not only is it acceptable, we should also celebrate. Nature is so rich and beautiful and worthy of our cherishment.
Facts and consequences
The above is just a general summary. After all, many biologists also appreciate elegance, and many physicists also appreciate complexity. However, it is helpful to clarify these two points of view. They have many practical theoretical consequences.
Although there are overarching principles in biology that shape the way biologists treat science—for example, Nobel Prize winner Paul Nides(Paul Nurse) is in his latest book What is Life? “ outlines five such concepts, but people may still beTrust and skeptical, because these abstract summaries often do not fully consider biological details.
Figure 3. Nobel Laureate Paul Nides’ new book What Is Life?
In the words of neuroscientist Oliver Sacks(Oliver Sacks), many biologists “saw the rise of grand theory, yet again Overturned by solid facts”. Therefore, the elegant theories that physicists admire may be considered too simple by biologists. This dissent is not unreasonable. In fact, although the principle of simplicity is often used as a guide, it does not have a clear history. Its prominence in physics may be due to a kind of romanticism: preference for the rule of proof Instances, and exclude counter-examples of negative rules.
That means we may be suffering from confirmation bias. Actually, as we understand the universe, things will become more and more complicated rather than simpler, and even the world of mathematics will become wilder . For example, quantum mechanics and relativity are much more complicated than classical mechanics. But they are also richer-explaining more data and making important predictions, giving us a deeper understanding of the universe, and it is this richness that makes more sense.
Physicist Paul Dirac(Paul Dirac) later realized this problem, and made a slight adjustment to the physical beliefs, thinking that the discovery of Einstein’s theory of relativity made it necessary to change the principle of simplicity into the beauty of mathematics.
Because of this, beauty is becoming more and more important, especially the different views on beauty, which will have a major impact on how we treat science and what we think is a good “interpretation.” Alan Turing’s (Alan Turing) morphogenesis theory is an example. Turing proposed that when the uniform state of space becomes unstable by the diffusion of morphogens (morphogens), complex organisms may appear Learning pattern  , this is a general theory that is beautiful and universal in mathematics and is recognized It is a milestone in the understanding of biological development. By elegant standards, this is a good explanation.
However, despite the power of Turing’s theory, it cannot explain the formation of any specific biological patterns. To do this, we need to provide specific details: Which morphogens are involved? How do they react? How are they produced? How do they degrade? How fast are they spreading? How does the cell respond to this?
The answers to these questions cannot be detached: They together constitute another good “explanation” in a specific context, and provide richness and depth that abstract theory does not.
Therefore, although the Turing pattern (Turing Pattern) can provide a unified framework to think about the general pattern, the Turing model (or in fact any general model) cannot provide a complete explanation in any particular situation. If you really want to fully verify it as a general theory, you must provideThe details.
In addition, we now know that various other mechanisms are also important in establishing and stabilizing biopatterns , Turing theory is just one of them. The reality from a biological perspective is inherently complex. Given any background, principles and details are important. A complete explanation is not just one of the components, but a balanced blend of the two.
It is difficult to conduct cross-research between physics and biology.
Part of the reason is the well-known difficulty of interdisciplinary communication: The two use the same terminology in different ways, which naturally leads to misunderstandings. However, as long as there is enough time and willingness, language problems are relatively easy to solve. I think the main difficulty in combining physics and biology is not the differences in language or culture, but what we think is attractive— —That is beauty, different aesthetics will affect the questions we ask and the answers we think are reasonable.
This very basic but unrecognized difference will lead each other on different scientific paths, and it is difficult to unify the two. In other words, the combination of physics and biology is a particularly difficult challenge because it involves connecting two different scientific and philosophical methods, which are rooted in two opposing aesthetics. Concept.
But this does not mean that we should abandon the guidance of beauty. The desire for beauty is an important part of human nature and plays an important role in science. Of course, we should be wary of taking beauty as the only criterion for success. Just defining science in terms of what we consider attractive may make us superficial, and different concepts of beauty may also make us subjective and divided. We must be careful to remain sincere and seek truth in an honest manner. We must strike a balance between beauty and empiricist choices.
Figure 4. Phoenix Reborn, the winner of the 2020 Science and Art Photo Contest, is a collection of lipid vesicles, nanotubes, and biofilm aggregates. Imaging under a focusing microscope (false color)
To do this, we may need to have a broader perspective to better appreciate the multiple aspects of beauty, recognize that different aesthetic concepts can help us clarify different aspects of life, and recognize even superficial views For example, our superficial appreciation of the beauty of mathematics when we were young is also valuable, and it will inspire us to explore deeper truths.
This requires us to develop a scientific empathy: the ability to understand and truly see the opinions of others, and to learn from others. Those with a background in physics can learn to appreciate the intricate, interconnected, and complex beauty better, rather than rushing to abstract it. This is a gift from biology to physics; on the contrary, those who come from biology can learn to. Better appreciate the beauty of coherence, elegance, and harmony, and accept the simplicity that may be hidden behind the complex surface of life. This is also a gift from physics to biology.
This will not only be of great benefit to science, it will also help us cultivate a richer and more beautiful outlook on life in all aspects.
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This article is from WeChat official account:Jizhi Club (ID: swarma_org), author: B D. MacArthur, Translator: ten-dimensional span> p>