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Is mathematics a science? On the one hand, the answer depends on what is "mathematics", on the other hand, it also depends on what is "science". Some respected philosophers of science will give a negative answer, while others will firmly believe that "the answer is yes".

So before we answer the question, we need to reach a consensus on the meaning of the words in the question.

What is science? Defining "science" is a thorny issue in itself. It is easy to talk about "science" (with or without prefixes and suffixes) and "scientific", but it is difficult for most people to provide a justifiable answer to what is a scientific question and what is not.

I asked a friend, Andrew Lias, who is a university professor, this question, and he gave me the following answer:

Science is a form of epistemology, just like any good epistemology, it tries to distinguish between true statement and false statement, so as to realize the accumulation of knowledge. One of the main differences between science and other epistemology is that it is a) systematic and b) non-dogmatic. A correct science must have a way to prove its claims and a way to identify and reject false claims. These methods should also be systematic.

Let us first accept this definition, and also agree that "true" and "false" here are consistent with the objective reality of the "external" world.

Mathematics does not seem to fit this definition: axioms are very close to dogma, it does not seem to care about truth or falsehood, or even the outside world; mathematics has proof, but there seems to be no experiment; mathematics does not follow scientific methods. But is that really the case?

In some history, most people have never come into contact with the study of modern mathematics. One may know that mathematics is a collection of methods, algorithms, and rules (such as the root formula of a quadratic equation, the rule of derivative, and the law of multiplication, etc.), or the axiomatic model of Euclid (Lemma, proof, theorem, inference, proof, all based on known abstract concepts and "self-evident" axioms). But in fact, mathematics is not both, although sometimes it is close to these intuitive impressions.

Until the 19th century, people engaged in "abstract mathematics" were called geometers. they were usually philosophers, amateurs or physicists and mathematicians who used their spare time to study related content. Major mathematical works are either about breakthroughs similar to Euclid's or summarizing methods and algorithms for solving specific problems (such as Diophantine and Fibonacci). Almost all the other people who do math are also doing physics. In fact, until the early 1800s, the most famous mathematicians were associated with physics in some way: Newton, Bernoulli, Fourier, and even Fermat; the official position of mathematician Prince Gauss was astronomer. He did a lot of work in physics. At that time, mathematics was mixed with physics.

Photo Source: pixabay's only math that is difficult to classify as above is number theory, which was considered leisure mathematics until Gauss's landmark "arithmetic research" was published. It's not serious research, it's a game.

In the following 19th century, there was a turning point. Mathematics began to develop as an independent discipline. On the one hand, it is related to some problems accumulated in the past: the basic problems in calculus, the difficulties caused by the use of childish and intuitive concepts, and the discovery of non-Euclidean geometry. On the other hand, it also depends on the outbreak of new ideas and methods: the beginning of group theory, complex analysis and algebra, and the development of naive set theory at the end of the century (later replaced by axiomatic variants) and the emergence of proof of non-constructive existence (most famous is Hilbert's finite basis proof).

In this crisis, there is a rift between physics and mathematics. Although most mathematicians are still committed to solving problems derived from physics, and most physicists still solve mathematical problems, their priorities are different. Generally speaking, physicists do not care much about the basic problems, because calculus and its derivation are obviously effective. These puzzles, paradoxes and contradictions may be interesting to philosophers, but they are not what people may encounter in "real" problems. However, mathematicians are very concerned about these problems and try to build their buildings on a solid foundation.

In this crisis, there are two main schools of mathematical thought, the Cronecke School and the Hilbert School. They all agree that mathematics needs to be built on a more solid foundation. The Cronecke School believes that algorithms and processing methods are at the core of mathematics, which come from clearly defined concepts based on empirical reality; the "experience" here must be vaguely understood: Cronecker's famous motto is "God has given us integers; the rest is human work", which means that he believes that the (possibly infinite) set of integers is an "empirical reality". They are called "constructivists", "intuitionists" or "formalists". For the Hilbert School, self-consistency and interest are the highest standards. A mathematical theory should be based on clearly stated axioms and rules, but it is meaningless to ask whether an axiom is "true" or "false". In their view, the only necessary question is: (I) is it possible to use axioms and rules to prove a proposition and its negative proposition? (ii) is the resulting theory interesting? If the answers of "no" and "yes" are given respectively, then the theory will be considered acceptable. The reason for asking the first question is that under the rules of classical logic, if a proposition and its negation can be proved, then anything can be proved. Such a theory is obviously both boring and useless. )

David Hilbert, however, these axioms need not have anything to do with "reality". Hilbert once said: "in a geometric axiom system, 'tables', 'chairs' and 'beer mugs' must always be used instead of 'points', 'lines' and 'faces'." In other words, the actual meaning of axioms is irrelevant; their semantic content does not play a role in mathematics.

Finally, most mathematicians accepted the view of the Hilbert school. Mainstream mathematicians describe mathematics as a model that follows the classical Euclidean axiom. Perhaps some typical examples can be found in Burbaki's works. It also greatly affects the writing of mathematical papers and the teaching methods of higher mathematics. It will be explained in more detail later.

Today, mathematics is roughly divided into two types: applied mathematics and pure mathematics. Applied mathematics is mathematics abstracted from practical problems, such as statistics and differential equations. Pure mathematics deals with problems arising from the theoretical framework, usually only about mathematics. However, this distinction is largely artificial. For example, number theory was once considered to be the purest pure mathematics, a branch of mathematics that could never be applied in practice. However, in recent years, it has become the cornerstone of modern cryptography and developed a very powerful application branch.

Why talk about history? What's the point of introducing the above? Well, the point is that the Hilbert School had a great influence on mathematics in the 20th century and today. When one studies mathematics, this influence in turn helps to produce and promote a particular writing style. This is the boring definition-Lemma-theorem-inference style that many people know.

The "problem" with this style is that it masks how math is done. Mathematicians engaged in professional research do not first write definitions, and then write theorems and their proofs, separating some key steps as lemmas. The way of reporting mathematics in research papers and books is quite different from the actual research way of mathematics.

As a result of this popular style, everyone except professional math researchers tends to be biased against the way mathematics is studied. This in turn leads some philosophers to conclude that mathematics is not a science because its methods are (obviously) so different from those of other empirical sciences. I will argue in my next article that once we go beyond the appearance provided by the writing style and really come into contact with the research methods of mathematics, then this conclusion is actually groundless.

Another influence of this style is that from the very beginning, especially after the work of Goedel, Turing and Church, the Hilbert School abandoned the concepts of "true" and "false" and supported the concepts of "provable", "provable" and "undeterminable". We put so much emphasis on truth in the definition of science that it seems to be possible to conclude that as long as mathematics does not seem to care about truth or falsehood, it cannot be regarded as science or scientific exploration.

Is mathematics a science after all? Gauss called mathematics "the queen and servant of science". Almost everyone agrees with at least the servant part: there is no denying that mathematics plays an important supporting role in science. This support is becoming more and more obvious not only in the fields of physics or chemistry, but also in other scientific fields. Statistics is the cornerstone of the transformation of medicine from art to science. For most social sciences, the more mathematics they use, the better and more scientific the research repeatability will be. However, there seems to be a lack of support for the claim that mathematics is a science, let alone the "queen of science".

First of all, does mathematics follow scientific methods of observation, hypothesis, experiment, test and verification?

Although some people may be surprised, the answer is yes. This is why the popular style of mathematical papers is not conducive to the public's accurate understanding of research mathematics. Mathematicians who are really engaged in research do not write a theorem first and then prove it. She usually gropes in the unknown like a natural scientist. She will think about some specific examples (observation) and check to see if they have a special nature; then she will ask some general and specific questions and try to answer them for a specific case; next, she will give a general statement (hypothesis) and continue to try to prove it (experiment); sometimes, if she fails, she will try to construct a counterexample (falsification and testing).

For particularly thorny problems, the presentation of specific examples is considered to be a valuable pursuit, similar to the experimental confirmation of theoretical difficulties. For example, checking the perfection of all odd numbers over a wide range may not be mathematical evidence that odd perfect numbers do not exist, but it is still helpful. Confirming some of the inferences implied by the ABC conjecture does not prove the conjecture itself, but it will give mathematicians some "street credibility" (and make people more interested in proving the conjecture). There are many similar examples.

On the other hand, there are some significant differences between mathematics and sciences such as physics: although there are observations, there seems to be no observation of the "real world". Mathematicians always ask for "proof", which is a much stricter standard than any other science! Take Newton's law of gravitation as an example. Claiming that the law applies anywhere (its "universality") will never satisfy mathematicians. For such a statement, it is not enough to have no counterexample. Only the proof that meets the mathematical standard is correct. For example, comparing it with Fermat's Great Theorem, Fermat's Great Theorem cannot be found or proved for 350 years, but this cannot be regarded as true (mathematical) evidence. This is really annoying. The conjecture is accepted only when the proof is completed (and withstands inspection and verification). Another example is Goldbach's conjecture, which has been heavily validated; although indicative, it is not enough. The same is true of the existence of odd perfect numbers. There is a joke about the rigor of mathematics. A mathematician, a physicist and an engineer traveled across Scotland by train when they saw a black sheep in the distance. The engineer immediately asserted that "in Scotland, sheep are black." The physicist replied, "No, in Scotland, some sheep are black." Then the mathematician gently corrected him: "in Scotland, at least one sheep is black." at least one side is black. "

Source network: how should we deal with the need for "proof" and the obvious lack of practical observation?

On the first point, I don't think the need for strict proof in mathematics really distinguishes mathematics from other sciences. It's just that mathematical results must meet stricter standards than physics. But other sciences also set their own thresholds in order to be accepted: no more than a certain number of errors, statistically significant confidence to a certain extent, and a wide variety of observations and predictions. Mathematical standards are only different in degree (because they seem to be stronger and have higher thresholds), but there is no fundamental difference.

Let's look at the second point. Is the axiom arbitrarily stated? Are they followed dogmatically and never questioned or modified? Do mathematicians really care about "true" and "false"? What if it is the "true" and "false" of the outside world?

The idealized mathematical model put forward by the Hilbert School holds that the answers to the above questions are "Yes, they are" and "Yes, but they are arbitrary. We can change the axiom to other systems,"do not care" and "do not care" at will. But like all idealized models, this is not an accurate representation.

Axioms can be arbitrary statements, but they are almost never. Usually mathematicians have some reason to propose a specific set of axioms rather than others. They represent not so much "arbitrary statements" as "basic rules" of specific developments, and minimum agreed assertions that serve as the basis for the work of mathematicians. Usually, these axioms are a refinement of actual observation, or an attempt to abstract the real world in a way suitable for mathematical processing. Based on centuries of work and observation, the idea of calculus (an obvious empirical development that aims to provide tools for studying movements) has been refined into a series of "axioms" about real numbers. These axioms are a compromise made by mathematicians in the 18th and 19th centuries to avoid paradoxes, utility and usefulness.

To some extent, these axioms are "beyond doubt", because from a mathematical point of view, there is almost no problem with their "authenticity" in the sense of experience. In contrast, when a mathematical theory comes from the real-world situation it tries to abstract and study, its axioms are rarely unquestioned or unmodified, because people always try to ensure that abstract theories are as close to reality as possible. There is continuous feedback and fine-tuning between mathematical theory and the real world.

In addition, although mathematics does not usually say "true" and "false", but rather "provable" and "falsifiable", this does not mean that it has no connection or application with the outside world. Mathematical theorems are never simple statements; on the contrary, they always imply. All mathematical theorems are in the form of "if (certain conditions are satisfied), then (this conclusion will be drawn)". In addition, whenever we explain the theory in a specific model, it is correct.

As Hilbert pointed out in his comments on geometry, an axiom system should not rely on any specific meaning of all undefined terms or axioms. However, this means that if we draw some mathematically correct conclusions from these axioms, then they are correct in any explanation. If we interpret the "point" in a geometric theorem as "table", "line" as "chair" and "plane" as "beer mug", then this theorem will provide us with a correct explanation of tables, chairs and beer mugs (assuming that the axiom is also true at this time). In this way, mathematics must have something to do with the real world, and it also has the ability to test and check the validity of this interpretation. In addition, even if we realize that the semantic meanings that we may assign to undefined terms such as "point", "line" and "plane" should not play a role in proof, these semantic meanings usually have implications for proofs and theorems. Even if "circle" and "line" are terms that should not contain any semantic content in the proof itself, mathematicians will draw a circle and a line to help identify ideas or enlighten proof.

The standard of proof required by mathematics actually ensures that as long as the premise (including axiom) is true, the conclusion is true; if the conclusion is wrong, at least one premise is false. Relying on verifiability rather than truth provides us with flexibility and certainty. By relying on abstract rather than concrete considerations, we ensure (or at least try to ensure) that our inferences do apply to any specific interpretation.

Most mathematicians usually have some specific explanations when doing mathematics. However, we may inadvertently use the specific properties of the interpretation in the proof and get results that are invalid in other interpretations. Euclid once fell into such a trap. Proposition 1 of his original Geometry Volume 1 depends on the obvious fact that two specific circles have one thing in common. However, this "obvious" fact is not actually derived from the axiom. Euclid's theory needs some new axioms to become true theorems, and his theory will be true only if all axioms (old axioms and new axioms) are true.

Proposition 1, text Source: Zhang Butian's translation of the original Geometry is precisely because of this danger that mathematics developed its standard of proof. Just like other sciences develop their own science based on their own experience. In this respect, mathematics also shows the characteristics of a science.

Conclusion is mathematics a science? I believe so. It follows a scientific method (although unfortunately, popular writing styles mask this fact). Although it seems (and sometimes claims) to live in its own small world and does not care about reality, the fact is that even in the name of "purest", it pays close attention to the application and inspiration of reality. There is no doubt that under the guise of "application", it is close to reality, and its hypotheses, problems and conclusions are constantly tested and improved in this context. However, it is also different from other sciences in that it has higher and more definite standards. But this part is the power of mathematics as a science, not an unqualified attribute.

So, going back to our definition of science, does mathematics meet the requirements? It tries to distinguish between true statements and false statements. However, we must understand here that "true statement" does not refer to a theorem or Lemma in isolation, but refers to an implied statement given by a theorem, that is, as long as all axioms and assumptions are true in a certain interpretation, then the corresponding interpretation of the conclusion of the theorem is also true. Similarly, "misstatement" means that there is at least one explanation that makes the axiom and hypothesis true and the conclusion false.

There is no doubt that mathematics achieves this through systematic proof. Anyone can check the certification process. They are encouraged to "repeat the experiment" by examining the certification process line by line and recognizing its effectiveness (or asking for clarification or even pointing out errors). What was once thought to be correct was suddenly pointed out by a mathematician as a flaw in the argument, which did happen. Sometimes the whole process of proof is denied, and sometimes it just needs to be "fixed".

Through the use of proof, mathematics has a very systematic way to verify its claim: errors can be found by presenting counterexamples or pointing out omissions in the evidence. The vast majority of people agree with this systematic approach.

Finally, it should be confirmed that mathematics does meet the requirements of scientific definition; its special interpretation of scientific methods, its special threshold, may be different from other sciences in quantity, but the same in quality. In addition, it plays a unique role in science and is an indispensable tool in many other sciences.

Original link: Is Mathematics Science?

This article comes from the official account of Wechat: Institute of Physics of the Chinese Academy of Sciences (ID:cas-iop). "Mathematics should be a science." Right? (top) (bottom) ", author: Arturo Magidin, translator: Tibetan idiot, revision: Yun Kai Ye Luo

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