The Mysterious Beauty of Mathematics---On "Is
2014-05-26
Mathematics is the language with which God wrote the Universe.
—-Galileo Galilei
There is something very mathematical about our universe. The more carefully we look at our universe, the more mathematics we seem to find. From financial modeling, weather forecasting, archetactures to electronic design, mathematical equations appear to be everywhere. However, thoughout history, no one was able to offer a clear explanation on why mathematics seems to be omnipotent in explaning everything. The nature of mathematics remains as a mystery. Even so, a great number of philosophers, mathematicians and physicists preserve in their journeys in exploring all facets of mathematics. Mario Livio is one of these people intrigued by the mysterious beauty of mathematics. In this short, accessible book titled Is God a Mathematician, Mario offers a profound exploration of the nature of mathematics. He centers his book around two questions: 1. Why is mathematics so effective in explaining the world around us that it even yields new knowledge? and 2. Is mathematics ultimately discovered or invented?
Unreasonable Effectiveness
Livio traces the development of mathematics from ancient Greeks, Archimedes, Pythagoras, Aristotle, Plato through the middle ages and the enlightenment to the present, and provides mini-bibliographies on Galileo, DeCartes, Newton, Gauss, Riemann, Boole and Russell etc.. He goes through various anecdotes about how these great figures came up with their great ideas that have reshaped the world. What truly fascinates Livio, however, is the role that pure math played in all these discoveries. He was bewildered by the amazing fact that concepts once explored by mathematicians with absolutely no application in mind have turned out decades later to be unexpected solutions to real-world problems. A long time ago when Pythagoras studied the construction of right triangles, he did not know that several years later, his theorem would play such an important role in the day to day life of engineers, architects and construction workers; When Euclid came up with his famous Euclidean Algorithm, he did not related that to the basis of musical intervals; When Newton deduced the law of gravity, he could hardly have known that the predictions made by these mathematical laws would perfectly overlap the empirical data he was originally trying to match; When Gauss wondered the sum of the nth odd numbers, he might not associate that with the motion of free fall bodies, which later becomes the core of classical mechanics. Livio also mentions the “group theory”, developed by Galois in 1832 to determine the solvability of algebraic equations, has become the language used by physicists, linguists, and even anthropologists to describe all the symmetries of the world (such as the structure of solids, the organization of elementary particles, etc.). Another often cited example is the Maxwell Equations, the most beautiful set of formulas ever that models the fundamentals of electromagnetism demonstrating the behavior of electrons to the nature of light. It is amazing that what guided Maxwell to derive such a powerful set of formulas was mathematical analogy, not empirical evidence. Just as all these stories, there are numerous instances in which mathematical principles, previously considered as purely logical curiosities, turns out to be uncannily productive in yielding new discoveries that are even unexpected on physical grounds.
Livio’s review suggests that mathematics cannot be separated from the physical world; they are essentially interrelated in some sense. It is impossible for us, for example, to learn physics without referring to mathematics. What is exactly the relationship between mathematics and physics then? Some people describe the situation by saying that mathematician plays a game in which he himself invents the rules, whereas physicist plays a game in which the rules are provided by Nature. However, as what Mario Livio tries to indicate in his book, mathematics and physics are highly interrelated, and it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. Mathematics also helps to correct physicists’ mistakes. There are many stories in which our ancestors made mistakes: Earth was once the center of the universe; heavy bodies were once supposed to fall faster than light bodies; chemical elements were once the basic units of all matters…It was the sense of mathematic principles that helped them correct their mistakes, and guided them towards the eternal truth.
Mathematics is not only applicable to areas of physics, but also to chemistry, biology and other disciplines such as geography, economics, psychology, not to mention our daily lives. The increasing use of computing machines testifies the idea that everything can be broken down to series of 0s and 1s. Mathematics appears as the fundamental units of nearly everything. This simple, undeniable fact gives rise to several philosophical questions that remain unresolved for years: What is mathematics? Why mathematical concepts have applicability far beyond the context in which they were originally developed? Why scholars in all disciplines find that they are unable to even state their theories without referring to abstract mathematical theories?
Challenged with these questions, Livio replies, “I don’t know”. It is not surprising that he is as baffled as other outstanding physicists who have attempted to answer these questions, most notably Eugene Wigner, Steven Weinberg and Mark Steiner. Wigner, in his The Unreasonable Effectiveness of Mathematics in the Natural Sciences, concluded his paper with the same question with which he began: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." The unreasonable effectiveness of mathematics seems to be inexplicable at the moment; however, I would like to offer one possible way of analyzing it. In my view, the unreasonable effectiveness of mathematics is in fact the unreasonable effectiveness of language. Mathematics constitutes a particular kind of description of the world, just like any language does. That mathematics does a better job than other natural languages in explaining physical reality is perhaps best explained by the predictive success of sciences based on mathematics. In other words, it is the predictive success of sciences based on mathematics that gives the most important validation of mathematics. As for why mathematics appears to be a perfect language is still a mystery. To figure it out we should perhaps first consider the nature of mathematics itself.
Discovered or Invented?
It is already stunning that mathematics provides the solid structure that holds together any theory of the universe. It is even more interesting as we realize that the nature of mathematics itself is not entirely clear. Where does mathematics come from, after all? There is much to suggest that mathematics comes from our observations of the physical world and that there is an innate way our brains making sense of those observations through mathematics. However, we also have the impression that mathematicians tend to work in very abstract areas with no intention of seeing applicability in the physical world. They regard mathematics as purely intellectual enjoyment until they see its direct application to our physical world in an unexpected way.
The unreasonable effectiveness of mathematics thus creates another set of puzzles regarding the nature of mathematics itself: Does mathematics have an existence that is entirely independent of human mind? In other words, are we merely discovering mathematical verities, just as chemists discover previously unknown elements? Or, is mathematics a human invention, a pure product of human reasoning? Was the equation “2+2=4” an invention by ancient mathematicians to best organize quantitative relationships, or is it an objective truth that can never be violated experientially? Was Fibonacci Sequence created on purpose to explain the growing pattern of sunflowers, or that they just happen to overlap?
In discussing the nature of mathematics, many scholars have taken the Platonic view, the idea that mathematics has its own existence regardless of whether human knows it or not, and we just have to discover it. Once a particular mathematical concept has been grasped, say, 2+2=4, then we are up against undeniable facts. For example, if two apples are placed together another two apples, there would be four there, even though no humans were there to observe it, or that we do not think there are four. Similarly, the growing pattern of sunflowers is not a subjective interpretation made by the human mind, but an inalterable, pre-determined fact once the sunflowers were born. In the Platonic view, mathematics indeed exists in some abstract fairyland that is part of the physical reality. This view, however, remains arguable. If mathematics is part of the physical reality, then why does it appear to be immutable while the physical world is constantly changing? How does the human brain, with its limitations, get access to such an immutable, abstract and mysterious fairyland?
Opponents to the Platonic view thus argue that mathematics does not arise from the physical world but from the human brain. Human invented mathematics by idealizing and abstracting elements of the physical world. Such a view is able to explain why mathematics appears heavenly, but it fails to explain why math happens to fit the physical world so well. If mathematics has absolutely no existence outside our mind, how can we explain the fact that invention of so many mathematical truths remarkably anticipated real-world problems not even occurred until many years later?
Livio’s answer to this puzzle is both—Mathematics is both discovered and invented! Some people argue that Livio simply failed to answer the question as he offered no exact conclusion about the nature of mathematics. Though his conclusion is indeed a bit of a cop out, I do think, however, it is the best attempt and the most convincing answer that Livio could offer. What Livio suggests is that humans invented certain basic concepts, and then they discover the implications of those axioms. He demonstrated this idea using the evolution of golden ratio, an example which I find quite convincing. Euclid, in his monumental work The Element, defined the “golden ratio” as the length ratio of two segments that takes its algebraic form as (1+√5)/2. Livio considers Euclid’s definition of the concept “golden ratio” as an invention because Euclid’s inventive act singled out this ratio and attracted the attention of later generations to it. It came as no surprise when people later discovered that dodecahedron has the golden ratio written all over it, that the five-pointed star has its segment divided by the golden ratio, that leaf arrangements and structure of crystals embody patterns that resemble the golden ratio. These discoveries are only true because of the initial acceptance of the foundational axioms that Euclid invented. That is why, for Livio, mathematics is a combination of invention and discoveries.
Mystery As Ultimate Beauty
Livio does a remarkably good job in showing that historical flights of mathematical imagination, no matter how trivial and random they once were, are often found to be incredibly useful in describing the mysteries we are curious about several centuries later. As for why mathematics acts as such a prophet, Livio perhaps was not “successful” enough to provide us with a definite, convincing answer. But to quote from Einstein, “The most beautiful experience we can have is the mysterious-—the fundamental emotion which stands at the cradle of true art and true science”. We should be grateful for the fact that there are so much we do not know, as mysteries give rise to our willingness to learn as well as the pleasure we get from it.
I want to end with a quote from Bertrand Russell—-the influential philosopher, mathematician, logician, social critic, the person whom I most admire. To me, this quote speaks to the reason for studying almost everything:
"Philosophy is to be studied, not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conception of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation; but above all because, through the greatness of the universe which philosophy contemplates, the mind is also rendered great, and becomes capable of that union with the universe which constitutes its highest good.”
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