Invitation to Mathematics： Mathematics, Just a Language？

Mathematical proposition is just a kind of grammar. -ludwig wittgenstein

In 1939, ludwig wittgenstein, 49, the pioneer of logical analysis philosophy, and alan turing, 27, the father of computers, the two greatest geniuses of mathematical logic in the 20th century and even in human history, respectively opened a course called "Mathematical Basis" at Cambridge University.

Turing, a 23-year-old junior, made a special visit to attend the courses offered by Wittgenstein. After talking about some trivial problems of mathematical philosophy and logic, Wittgenstein thinks that Turing has the "qualification" to attend his own courses. Although Turing has published an article "On Countable Numbers", the importance of which cannot be overemphasized by any adjective, it has laid the foundation of mathematics and logic related to the whole computer science.

The mathematical foundation that Turing devoted himself to teaching is the logical proof of the mathematical world. By selecting a set of strict and concise axioms as the logical starting point of the whole mathematics, according to certain rules, it developed into a huge mathematical building through deduction, and tried to verify what technical errors and limitations existed in this process.

The "mathematical basis" that Wittgenstein intends to discuss seems to be a philosophical principle that despises mathematics, at least at that time, most teachers and students who studied mathematics at Cambridge University tended to think so.

Wittgenstein also repeatedly said: "I want to explain again and again that the so-called mathematical discovery is more accurate as mathematical invention." Mathematical proof can not establish a conclusion as truth, but can only determine the meaning of some mathematical symbols.

Therefore, those seemingly unquestionable things in mathematics (for example, the sum of any two sides of a triangle is greater than the third side) are not a new fact about the world, but just a different statement (the sum of any two sides is greater than the third side is actually included in the definition of a triangle).

Wittgenstein said, "Mathematical proposition is just a grammar."

In fact, it touches a belief that all mathematicians refuse to face up to: the so-called rigor of mathematics and logic-some people like to call it rationality-is just a language structure of human beings.

The original purpose of the ancients to create the language or symbol system of mathematics was to try to describe the truth hidden behind the world and gain practical value. However, whether humans can finally find this truth or describe it accurately has nothing to do with the rigor of the language we use.

Unfortunately, its descendants mistakenly believe that strengthening the rigor of this language can lead mankind to the true nature of the world.

Therefore, when Cauchy, a French mathematics master, had not learned anything except Latin and Greek before he was 17 years old, he officially came into contact with mathematics after he was 17 years old, but he reinterpreted calculus by using linguistic methods. Although Cauchy’s linguistic analysis is undoubtedly successful, it is still considered by the mathematical community to destroy the artistic beauty of Newton’s calculus.

Mathematics is just a language, and what is more regrettable is that no matter how accurate and effective the language is in conveying the core information, it comes from irrational feelings at the beginning.

However, mathematicians after Cauchy still insist on perfecting the logic and rigor of this language, which is the purport of mathematics, and simply forget the original intention of ancestors to invent mathematical tools.

This view directly leads to the fact that the proof process overrides the conclusion, and pays more attention to the rigor of this language than the things described in the language itself.

Westerners habitually believe that mathematics originated in ancient Greece, because mathematics has never asked for proof in the endless years before.

The ancients who invented mathematics took it for granted that mathematical results were obviously much more important than a series of mathematical proofs. Sumerian mathematics is not based on evidence, which puzzles many modern mathematicians. This is normal for ancient society, and it is by no means an exception. The proof of process in mathematics of Egyptians, Chinese and ancient Indians is just some incidental interest. The proof process of drawing mathematical conclusions is only a tool to achieve practical goals, not the other way around.

The ancient Greeks changed their mathematical thinking. Because mathematics became a theology and religion in ancient Greece. This originated in Pythagoras. Russell said that the whole history of western philosophy is Plato’s footnote, and Plato may be Pythagoras’ footnote. Plato once entered the Pythagorean school to study, and especially wrote on the door frame of his own academy: Don’t enter the academy if you don’t know geometry.

When Pythagoras traveled to Egypt and the Middle East, Egyptians and Babylonians could already solve linear equations and quadratic equations, and Pythagoras’ number was known long before Pythagoras. Pythagoras was amazed by the mathematical achievements of the Egyptians and Babylonians. At that time, some people attributed the origin of the world to God and water to Qi. For example, Thales, the first western philosophy, thought that the origin of the world was water, while Pythagoras attributed the origin of the world to numbers. He believes that everything is counted, and turns the passion that mathematics admires infinitely into a deified superstition. This is a superstition, of course, is also a keen. He also organized a religious group about mathematics. When a faithful disciple found that irrational numbers threatened his faith, he brutally secretly led someone to kill him and threw him into the Aegean Sea to destroy his body.

Tracing back to the source, the way western mathematicians (including philosophers) pursue mathematical rationality is not rational. The ancient Greeks were amazed at the rigorous rationality of oriental geometric mathematics, which sublimated into a passion and belief. Instead of paying attention to geometric mathematics itself, we try to maintain and prove the correctness of geometric mathematics, and try to raise mathematical geometry, a limited practice mode of human beings, to the ultimate way to pursue it, which is a bit absurd in itself.

Therefore, a fundamental fatal problem of mathematics is that no matter how neat and rigorous the structure of the language of mathematics is, its foundation is not derived from reason.

No matter how accurate the mathematical theorem is, the mathematical axiom itself, as the premise of its inference, has not been proved. The definition of axiom in geometry is a theorem recognized by all people, and there is no need to prove it, nor can it be proved. It’s like a construction engineer has been proving the structural strength of reinforced concrete in a skyscraper, but he deliberately ignores the fact that this building is built on flowing sand.

The appearance of the language of mathematics, or it should be said that modern people misunderstand the language of mathematics, makes human efforts to explore the practical universe put the cart before the horse.

In the end, mathematicians worked hard to demonstrate the accuracy in the process of mathematical reasoning, only to find that mathematics could not prove its accuracy from the beginning.

The invention of any language is a result based on sensibility, because human beings need to communicate with reality through language.

Therefore, mathematics, as a language, can’t get rid of its irrational lineage in the process of its growth, no matter how precise and rigorous it is finally honed.

[Recommended by Book List]

These five books give you three minutes to understand mathematics.

As long as you stop for a moment for your life, you can have a close contact with mathematics.

Mathematics: the loss of certainty

Author: [America] M· Klein

Translator: Li Hongkui

Publishing House: Hunan Science and Technology Publishing House

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This is an epic work, which depicts a particularly vivid mathematical world.

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This monograph refutes this myth.

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Press: Zhejiang University Press

Subtitle: Genius is the responsibility.

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But also helps us to understand his philosophy.

Wang Yu, a translator, is fluent in both English and Chinese, and lives up to the original work. -Chen Jiaying

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The book has wonderful words and complete structure, which profoundly outlines the deep spiritual life of thinkers and can be called a first-class ideological biography.

What is mathematics: a basic study of ideas and methods

Authors: R. Courant, H. Robin and I. Stewart.

Translator: translated by Zhang Yici. Planned by Wang Yu.

Press: Fudan University Press, January 2012.

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What is mathematics: a basic study of ideas and methods, a popular mathematical expression, should really be adopted by all textbooks that pretend to be mysterious.

This book is written in the order of algebra, geometry and analytical mathematics, and it is based on a very simple foundation.

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Mathematics in Western Culture

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Publication year: May 1, 2012

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