The purpose of mathematics The original purpose of mathematics, as well as other human activities, is to economize. For mathematics, it is to simplify. Geometry means land (geo) measurement (metry). With the knowledge of geometry, we can measure many things which are difficult or impossible to measure directly. For example, ancient Greeks already were able to measure the radius of the earth with remarkable accuracy. Modern mathematics is no different. Shannon’s information theory greatly simplifies our understanding of coding and greatly reduces the size of data in communication. That is why we can watch video online. Other mathematical tools, such as Fourier transform, greatly simplifies our understanding of waves, such as sound waves. As a result, we can transmit voice message at much lower cost and much more beautifully. Right now, the voices of old people are more beautiful over the phone than in person. Simplicity is beauty. AI is a massive application of mathematical tools, especially linear algebra. The success of linear algebra in AI greatly simplifies our understanding of languages and mind. We now understand languages mostly from the correlation between words. Some say mathematics is more about rigor. Rigor gives us confidence about our measurement. We can prove, with rigor, that three angles of any triangle add together to 180 degrees. After we measure two angles of a triangle, we can confidently know the size of the third angle without actually measuring it. Mathematical rigor simplifies our measurement. Mathematics has been extremely useful for us to simplify the understanding of our world. Like any other successful professions, the number of mathematicians has grown tremendously. Most mathematicians now are surrounded by other mathematicians instead of facing real world problems. Increasingly, mathematicians are being measured by fellow mathematicians instead of by their ability to solve real world problems. This situation is not unique to mathematicians. Paul Samuelson once said, "the economic scholar has a special responsibility to maintain his detachment. Not for us is the limelight and the applause. But that doesn’t mean the game is not worth the candle or that we do not in the end win the game. In the long run, the economic scholar works for the only coin worth having—our own applause." (Samuelson, 1962) Samuelson was nudging the profession away from popular, narrative political economy and toward the technical, mathematical modeling that came to dominate late 20th-century economics. What was the consequence of such an approach? Paul Romer, a prominent economist himself, called the phenomenon, mathiness (Romer, 2015). Mathematics in economic papers is often used not to inform, but to intimidate. Similarly, in mathematical research, works are increasingly valued not by simplicity, but by complexity. Researchers often pride themselves for producing hundred-page long papers that few could understand. Modern mathematics has become more and more detached from real life. The pursuit of rigor and complexity in mathematics has squeezed life out of it, turning main branches of modern mathematics into cold, dead tombstones. Recently, Hilbert Sixth Problem has been mentioned frequently in the press. It is an attempt to axiomatize physics theory. In particular, it attempted to establish a connection between Newtonian mechanics, which is time reversible, and statistical mechanics, which is time irreversible. Newtonian mechanics is about individual particles. Statistical mechanics is about the statistical distribution of a large number of particles. Once the ensemble theory was established by Gibbs (1902) and understood by others, there is really no more puzzle. From the standard statistical mechanics, it is very simple to understand why time is irreversible. In the pursuit of rigor by Hilbert, and the mathematicians after him, the mathematics becomes extremely elaborate, and hence extremely prestigious. Does esoteric mathematics help you gain a deeper understanding of nature? References J. W. Gibbs,1902. Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics, (New York: Dover Publications, 1960 [1902]). Romer, Paul (2015). "Mathiness in the Theory of Economic Growth". American Economic Review. Papers & Proceedings. 105 (5): 89–93 Paul Samuelson, 1962. "Economists and the History of Ideas", The American Economic Review, Vol. 52, No. 1 (March 1962), pp. 1–18.
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