Preamble

Previous article "What is Science? ” because a friend said that I had written an article on this topic before, but I couldn’t find it anymore. Not only the other party can't find it, but I can't find it myself. For various reasons, all the articles in my Baidu Tieba have been deleted by the bar owner.

So, no, I rewrote a "What is Science? " .

Then my friend said, I mentioned in it that I personally disapprove of calling a theory that has no experimental support but is mathematically self-consistent as a physical theory, so I want to know what mathematics and physics are, and, again, before writing Once again, articles on related topics can't be found.

So, no, I wrote another one.

First of all, it is necessary to explain that although I personally do not think that the mathematics is self-consistent and that there is no experimental support for the time being, a theory that cannot even be experimentally supported in the foreseeable future according to the requirements of the theory can be called a physical theory. , but this idea is obviously very conservative and old-fashioned, and does not represent the attitude of the mainstream physics community today - although the mainstream has not yet formed a unified view. In the eyes of radicals like superstrings/M-theory, why not have experimental support? The theory is beautiful and the math is beautiful enough.

This is a relatively anxious topic in the field of physical philosophy at present, and because the knowledge reserves related to mathematics and physics of philosophers of science are weak and their understanding is more philosophical than physics itself, on the other hand, physicists who are engaged in experiments are busy Doing experiments to push the verifiable boundary of the theory to the limit of human technology again and again, so it does not care about these metaphysical things, so the two sides of the argument are mainly theoretical physicists of different schools, such as circle quantum scientists who are more keen to think It's not physics without physical experiments to back it up, and the aforementioned superstring/M theorists staunchly reject this argument.

My personal position is, of course, inconsequentially on the former side, ie: physics without experimentation is nothing but a mathematical game that has nothing to do with reality .

So, here comes a question, what is the relationship between mathematics and physics ? Leaving the above controversial arguments aside, this question often leads to the following two sub-questions:

1. Why is the use of mathematics in physics so successful? Or why does physics depend on mathematics?
2. What are their similarities and differences?

Let’s briefly talk about this topic today.

What is a theory?

Before discussing mathematics and physics, let's take a look at what a "theory" and a "theoretical system" are in general.

Generally speaking, any set of theory has two crucial core elements:

1. Axiom set: it is the basis for all propositions to be established, including all a priori and transcendental presuppositions;
2. Inference rule: it is the key to the proposition that the proposition can be constructed from the axiom set;

The modern meaning of the terms transcendental and transcendental originates from Kant's Critique of Pure Reason, which appeared as synonyms in ancient Greek times, and were differentiated after Kant. The general idea can be that the so-called transcendental refers to that which exists prior to experience and can be applied to experience, while the so-called transcendental refers to that which is beyond experience and cannot be applied to experience. We can understand it this way: if experience is regarded as a convergent sequence, the prior is the limit of the sequence, and the limit is not in the sequence, and the transcendence is a number that is completely independent of the sequence and its closed cover. In the words of Professor Deng Xiaomang, it is:

The difference is that "transcendental" transcends "above" experience because it precedes and applies condescendingly to experience; while "transcendental" transcends "above" experience because Because it is completely beyond experience and cannot be applied to experience. Therefore, the transcendental nature can only belong to rational ideas and not to the category of understanding.

...

As long as we, by means of our concepts of reason, take as objects only the totality of the conditions in the world of senses, and what is available to reason in this totality, these ideas of ours, though transcendental, are still the universe on. But these ideas become transcendental once we place the unconditional (things really have to do with them after all) in that which is completely outside the world of senses, and therefore outside all possible experience, then these ideas become transcendental; they are not merely is used to carry out the empirical application of reason, ... but is completely separated from this application, and makes itself objects whose material is not taken from experience and whose objective reality is Not based on the completion of an empirical sequence, but on purely a priori concepts.

Therefore, in the theory of natural science, transcendental presuppositions are often empirical summaries, while transcendental presuppositions are hypotheses based entirely on ideas.

For different theories, their axiom sets are often different, and even the inference rules may be different. For example, in the natural sciences and mathematics, we use mathematical logic, while in other disciplines, we may use much weaker reasoning logic, or other completely different logic systems, such as the legal system we use in legal logic and mathematical logic. Not exactly the same, but just as powerful.

Therefore, the propositions in any set of theoretical systems can be obtained through a limited number of axioms in the axioms set after applying the inference rules, which reflects the completeness of the theoretical system—if the system is incomplete, then Shows that there are some propositions that cannot be proved or falsified in the system, that is, cannot be obtained from the axiom set of the system by applying the rules of inference a limited number of times, then from another point of view, the proposition itself can be added To the axiom set (for example, ZF set theory becomes ZFC set theory after absorbing the choice axioms that cannot be proved or falsified in ZF).

In formal logic, there is also a special proposition called an axiom schema , which gives a set of axioms rather than a single axiom through a set of external variables, so a theoretical system containing an axiomatic schema must first contain a number system , such as natural numbers, or more complex real numbers, hyperreal numbers, and so on. Of course, we will not make a distinction between theoretical systems that contain axiomatic patterns and those that do not.

Of course, we can use more "advanced" mathematical objects such as hypergraphs or Turing machines to describe theoretical systems, but we don't need to care too much here, we only need to know that any theoretical system is basically such a tree structure. .

In fact, the "theoretical system" given here is actually very close to the definition of a formal system. Because a formal system is composed of "symbols", "axioms" and "derivation rules", and the "symbols" are actually implied but not explicitly stated, the "theoretical system" given here is very close to the form in terms of definition system. But the difference still exists, that is, we do not require that the derivation rules must be as strict as in the formal system. The "inference rules" we require are actually a much weaker version, even allowing the logic of life used in everyday spoken language.

After understanding what a theoretical system might "look like", let's look at what mathematics and physics "look like".

(For more information on the structure of theoretical space, see this article )

What is math?

Mathematics itself is clearly not a science, mathematics is mathematics.

This is because the definition of science can be roughly written as follows (see the previous article "What is Science?" for details):

It is based on a measurable phenomenon to explore the underlying laws behind it, and can give testable predictions for phenomena that have not been measured, and the results of the prediction can be within the tolerance range of errors recognized by the scientific community and the new phenomenon. The measurement results are consistent, and the logical system on which the predictions are based is self-consistent, such a family of theories, and the intellectual activities around this family of theories.

Obviously, mathematics is not satisfied with the scientific research objects and verification methods: the research of mathematics is not only based on observable phenomena, and there is no possibility or necessity to do experiments to verify. Its research objects cover a much wider area. category, including quantity, structure, space, relationship, and their variations. Obviously, in terms of the research object, mathematics covers a very rich class of reality and non-reality, existence and non-existence, measurable and unmeasurable things.

According to logical positivism , "any unverifiable statement is neither true nor false, but has no real meaning", or it can be expressed as "a sentence if and only if the proposition it expresses is either an analytic , or empirically verifiable, the sentence is literally meaningful”, so mathematics is clearly not a natural science.

Logical positivism, also known as neo-positivism or scientific empiricism , is no longer mainstream in the philosophical community, but has been believed by everyone in the scientific community for a long time, including now. In the hottest era of scattering matrix theory, there is even a saying: "Don't talk nonsense, shut up and calculate", it is used to tell everyone that doing physics is two things: rigorous calculation, and doing experiments verify. Of course, with the prevalence of radical schools such as superstring/M theory, this view will obviously be abandoned sooner or later.

The phrase "shut up and calculate" originally came from an article written by Cornell University theoretical physicist Mermin in his column on "Physics Today", which was originally intended to spit out the prevailing Copenhagen interpretation school of quantum mechanics , but then I do not know why it became Feynman's words, and it also changed from complaining to positive "persuasion", and then became the mainstream view in the physics community for a long time.

Of course, not everything can be dealt with with mathematics, so the scope of mathematics is indeed broad, but not broad enough to cover everything.

There are many branches of mathematics, and among them can be called "the foundation of mathematics", it can be said that there are only two: mathematical logic and set theory .

The importance of mathematical logic can be said to be obvious, because we have mentioned before that a theoretical system can only derive other propositions from the set of axioms through the rules of inference, and in natural science and mathematics, the rules of inference are composed of Mathematical logic.

And set theory is the basis of almost all branches of mathematics. In principle, all mathematical propositions can be expressed as propositions in set theory, and even mathematical logic itself can be expressed as set operations in set theory.

But what is more interesting is that set theory itself is not simple, it needs to be built on a lot of preliminary knowledge, this - partly because the simplest set theory (ie Contor's set theory ) has its own irreconcilable problems, and partly because This is because there is no theory that does not need the antecedents. The antecedents of the theory here are the axioms sets, and the axioms cannot be proved within the theory, but can only be based on preliminary knowledge.

The irreconcilable problems in Contor's set theory are all caused by self-referential (referred to as self -referential) mechanisms, such as the famous Russell's paradox. The development of set theory after Canto's set theory solves this problem in different ways.

For example, in the well-known ZF and ZFC set theory, the axiom of classification model ensures that self-referential and self-contradictory sets cannot be constructed; while in NBG set theory, the method used in Russell's paradox is based on the class generalization axiom The pattern can only give classes but not sets, so it does not constitute a self-referential paradox (in fact, we can continuously construct sets, classes, classes of classes, classes of classes, and so on by this method, but The new basic set theory is similar to the NBG set theory, the Russell paradox can only create objects of type n+1 from type n, thus avoiding the same type of objects. self-referential problem.

It can be seen that there are no more than two commonly used methods in modern set theory:

1. Use a hierarchical structure to avoid intra-layer self-reference;
2. Declare a statement that may constitute self-referential as syntactically invalid.

Although set theory itself is not "simple" enough to require no prior knowledge, it can nonetheless be called the foundation of all mathematical theories, because, as mentioned earlier, propositions in all other mathematical theories are almost invariant. An exception can be expressed as a proposition in set theory and proved or falsified by means of set theory.

Since then, using set theory, people have developed several core branches of mathematics, including proof theory, model theory, and recursion theory. The well-known Gödel completeness theorem (not the more famous Gödel incompleteness theorem ) is a useful theorem in model theory, and the halting paradox is a well-known problem in recursion theory. One of the theoretical branches is triggered by set theory to study how the proposition is proved, while the other is triggered by mathematical logic to study what kind of proposition can be proved. Does it feel very close?

Of course, there are also some very interesting low-level mathematics branches, which have gradually been paid attention to in recent years. For example, inverse mathematics is devoted to the study of how to deduce axiom sets from theorems. This process is actually very similar to natural science, that is, from phenomena Reverse the essence.

On all these underlying mathematical foundations, we can build more majestic mathematical edifices, such as group theory, ring theory, and category theory in algebra (of course, recently some people think that category theory should replace set theory as the foundation of mathematics , this is a matter of opinion), algebra theory, representation theory, etc. In analysis, there are functional analysis, functional analysis, differential geometry, etc., and the two will merge under the Langlands program, such as algebraic geometry and the like.

The content of these higher-level mathematical branches is very rich and complex. In modern times, almost no one dares to say that they can understand all of these branches. The last person in history to be known as a "mathematical generalist" was the famous French mathematician Panga Lai , his most famous achievement is to prove that chaotic systems are sensitive to initial values. In fact, the branch of mathematics of chaos theory was opened by him.

Poincaré was not only a mathematician, but also a physicist. His chaos theory was developed when he studied the trajectory of the three-body system. In addition, he and Lorentz studied the contradiction between electromagnetic theory and Newtonian space-time theory independently of Einstein. The special theory of relativity was proposed, but unfortunately it gave a wrong understanding of the physical interpretation of the final correct mathematical result. So far, the space-time coordinate transformation formula in the special theory of relativity is still called the Lorentz transformation, and the space-time symmetry group is still called the Poincaré group.

However, it is worth noting that it is possible for the superstructure to change if we choose a different foundation. For example, when we choose ZFC set theory as the set theory used by the system, we can construct the sphere separation paradox, but if we choose ZF set theory, that is, if we do not admit that the axiom of choice is an axiom, then there is no sphere separation paradox Argument. From this, we can also be regarded as a glimpse of the important impact of the foundation on the whole.

Now, let's go back and see what mathematics is really about.

In mathematical logic, we study how the propositions should be deduced, that is, the relationship between the propositions. In set theory, we study the relationships between sets of related mathematical objects. In recursion theory we study the properties and classifications of computing power possessed by various formal languages and their corresponding computing devices. In complex analysis we study the properties of various complex functions, and in differential geometry we study the properties of geometric spaces with differential structures.

Thus, we can almost think of it as the study of the properties and structure of a set of objects with some common property, and the relationship between this set of objects and other sets of objects with some common property, through the wider common property the established structure.

In this sense, it also seems reasonable to use category theory as the new core foundation, because category theory is indeed more suitable than set theory for the purpose of studying the above.

Of course, this is only a personal summary of the question of "what is mathematics". People who don't use it can give different answers to this question. For example, Russell believes that mathematics is logic, and Hilbert believes that mathematics is a driving system. Brouwer believed that mathematics was the intuition of the mind, and Gödel used the incompleteness theorem he discovered to tell everyone that mathematics is much more complicated than everyone understands. The answer given by Professor Yuan Lanfeng from the University of Science and Technology of China is more philosophical, but I personally think it is very reasonable: mathematics is a transcendental truth system, not an empirical science .

Mathematics: Invention or Discovery?

About mathematics, people often ask a very interesting topic, that is, was mathematics invented or discovered ?

To answer this question, we must first understand what is an invention and what is a discovery.

In the Cambridge Dictionary, the definitions of invent and discover are:

• invent : to design and/or create something that has never been made before
• discover : to find information, a place, or an object, especially for the first time

That is, invention is the creation of something that did not exist in the past, while discovery is the making known of something that existed but no one has seen it before.

Hence a central distinction: if an object existed before, but was not known, the process of making it known is discovery; and if an object did not exist before, it cannot be known. known, then the process of making it known is invention. The key difference is: Does the object already exist before?

For example, dinosaur fossils exist whether people have excavated them or not, so the process of excavating fossils is discovery rather than invention. But the computer will never appear if humans don't make it, so the computer was invented rather than discovered.

Going back to mathematics, did mathematics and mathematical theorems, theories, existed before humans wrote it down and figured it out?

This problem is not very easy to understand, because mathematics itself does not exist in our empirical world. We can say there is an apple, a banana, but we can't say we got a "1". Even, one apple plus one apple is equal to two apples, which does not guarantee that "1+1=2" must hold, because from the point of view of group theory, addition can be given arbitrarily, as long as the conditions of group addition are satisfied. . You say that an apple plus an apple is the successor of an apple so the answer is naturally two apples from an ordinal and cardinality perspective? This is rather far-fetched.

Therefore, the question of whether mathematics existed before it was written down or conceived by people is tantamount to asking whether the object of mathematics, which does not exist in the empirical world, exists transcendental or does not exist at all. Since human knowledge comes from the empirical world, we obviously cannot answer this question with our knowledge, according to the transcendental definition.

That is to say, the question of whether mathematics is an invention (which does not exist) or a discovery (which exists transcendental) is an unanswerable question.

For example, the famous fifth postulate problem, we can of course say that all non-Euclidean geometries did not exist until Lobachevsky showed that it was not irreplaceable, but Lobachevsky coined them; but In the same way, we can say that non-Euclidean geometry has existed for a long time, but people have not found that they can be found by replacing the fifth postulate. Lobachevsky just tells us how to find these long-existing geometric theories .

Both of these claims are valid, and neither can be refuted, only believed.

Of course, this is not to say that there is no room for discussion on this issue at all. After all, as long as the axiom set and inference rules are determined, what kind of theoretical system can be produced is a determined matter. Therefore, before changing the axiom set or inference rules, any proposition that can be integrated into the theoretical system is only discovered rather than invented. This outlet was created by you, you just discovered it. When the axiom set and inference rules are determined, the structure of the entire system has actually been determined. It is already a definite maze. All we have to do is to try our best to explore every corner of the maze.

Of course, this statement only holds for beings beyond the world of experience. For the empirical world, there are other criteria for judging whether it exists or not, otherwise we can of course say that the fundamental laws of physics have long been established, so the existence of a computer is just an "unseen labyrinth exit".

But if the axiom set is moved like the fifth postulate problem, then whether it is an invention or a discovery at this time is a matter of opinion.

Personally, since any axiom can always be considered to be of finite length, it can be finitely encoded as a finite natural number, and any finite natural number already exists (the Peano arithmetic system, and then adding the above Said labyrinth exploration), so all mathematical systems whose axiom sets are finite sets (the axiom patterns can also be considered to be encoded as a natural number rather than an infinite number of natural numbers from the perspective of a Turing machine) can be considered to have existed for a long time. . So is there a mathematical system where the set of axioms is infinite (note, again, that an axiom pattern is just one element)? I am afraid no one can answer this question now.

So, personally, I choose to believe that mathematics was discovered rather than invented.

In fact, there is another very interesting and even sophistry perspective on the question of discovery or invention.

There is a joke in quantum mechanics: When everyone closes their eyes, the moon does not exist . In the words of the Copenhagen interpretation school, before the observer makes an observation, the quantum state of the system does not collapse to the classical state, so that everything is not determined. In this sense, the moon may not exist until someone looks at it, even from the point of view of whether it is certain that the probability of a unique classical state rather than being in some classical state is strictly 0. When looking at the moon it just doesn't exist. Then, looking at this action is actually "creating" the state of existence of the moon, so that the moon is "invented" rather than "discovered".

This is the impact of quantum physics on philosophical concepts based on the experience of the classical world. We don’t need to discuss this too much, just need to know that philosophical concepts may also change under different physical perspectives.

More on this topic can be found in the book on the embodiment of mathematics and physics, some of which are very wild.

What is physics?

Unlike mathematics, which is invisible and intangible, we still have more intuitive feelings about physics.

The research objects of physics cover quite a wide range, from the smallest elementary particles to the largest cosmic space-time, almost all of them are the research objects of physics.

From the scale of the object, that is, the size, we can divide it into four parts: microphysics, mesoscopic physics, macroscopic physics and cosmoscopic physics.

The objects of microphysics research range from the most basic particles such as quarks and leptons (of course, including possible more basic physical structures such as strings) to macromolecules, that is, the scale range below 1 nanometer, which can be subdivided into particle physics. , nuclear, atomic and molecular physics.

The objects of cosmoscopic physics research include planetary systems (central stars and their surrounding planets, dwarf planets, asteroids, etc.), star systems (including binary stars and star clusters, etc.), nebulae, galaxies, galaxy clusters, superclusters, long Walls, voids, the universe as a whole, possible multiverses (not the multiverse in fictional works like Marvel and DC), spacetime itself, from their structure, properties, evolutionary laws, and interactions. It can be said that the objects at the scale of planets and above all need to be considered in the cosmoscopic universe.

The research object of macroscopic physics is from the micrometer to the planetary scale, while the mesoscopic physics is to study the nanoscale to the micrometer scale.

On the microscopic scale, quantum mechanics is the dominant physical law, on the macroscopic scale classical physics dominates, and on the cosmoscopic scale it is more the theory of gravity. But mesoscopically, it is more interesting and complex, because on larger scales quantum laws are averaged by the interaction of a large number of particles, and on smaller scales, it is not necessary to consider the phylogeny composed of a large number of particles, but only a few A few particles, but in the category of mesoscopic, neither can the quantum average be made classical, nor can only a few particles be considered without considering the phylogeny composed of a large number of particles, so the laws of mesoscopic physics are the most complicated.

Qian Xuesen once proposed that it can be divided into five parts in terms of scale. Among them, the microscopic is divided into two parts: the microscopic and the microscopic. The former is from atoms to macromolecules, and the latter starts from the nucleus, including quarks and leptons. , medium bosons, and smaller possible physical objects such as strings, p-branes, and so on.

Under this division, the dominant force in the microscopic field is the electromagnetic force, and the strong and weak forces have very little effect; while the dominant force in the microscopic field is the strong and weak force, and the electromagnetic force is relatively small.

From the perspective of energy, low-energy physics and high-energy physics are also very different. The former is mainly manifested in Newtonian mechanics and relativistic mechanics (the former is a low-speed approximation of the latter), while the latter is mainly manifested in quantum mechanics.

The phenomena we encounter in our daily life are basically macroscopic and low-energy, so we can use the physical theory based on Newtonian mechanics, or use the theory of relativity in phenomena such as GPS and planetary motion in the solar system. The category of classical physics. In our particle physics experiments, we are dealing with high-energy microscopic phenomena, so we mainly use quantum theory such as quantum field theory.

Only in some very special cases, we need to consider macroscopic high-energy physics, such as the origin and end of the universe, and the formation and evolution of black holes. Here we need to consider both quantum theory and relativity. This field of combination has not yet been concluded. The best result is the quantum field theory of curved spacetime developed by Hawking.

Both superstring/M theory and loop quantum gravity theory are physical theories that aim to fully combine general relativity and quantum theory, but their starting points are completely different. The former uses strings and higher-order membranes to replace point particles, so that the graviton that transmits gravity is interpreted as a closed string with a special vibration mode, and then the motion of the string is used to explain the gravitational phenomenon. The latter quantizes space and time in a completely different way, and the angle is different from the traditional quantization scheme, so I won't go into too much detail here.

But there are many other attempts to combine general relativity and quantum theory, such as following the path of the standard model and using the Poincaré group as the symmetric nature of gravity beyond the standard model theory, or starting from the holographic principle and corresponding gravity to the universe Field-theoretical on conformal boundaries, and entropic force theories that treat gravity not as a fundamental force but as a phenomenon caused by a more fundamental force, etc. Of course, these theories are still just conjectures, and no one dares to say which one is more correct.

In terms of research objects, there are more categories, such as particle physics, atomic physics, molecular physics, solid physics, fluid physics, particle physics, material physics, astrophysics, cosmology, and so on.

But in any case, physics can be regarded as a science to find the basic laws behind natural phenomena . In the words of Einstein:

"It is necessary to establish connections between various empirical facts that enable us to foresee future facts from those already experienced. . ." ("The Collected Works of Einstein, Volume 1")

Of course, non-scientific theories such as theology also try to establish connections between empirical facts, but compared with physics, the main difference is that theology attributes the laws behind affairs to one deified or even multiple deified gods, so that the principles behind phenomena are not the same. It does not show a kind of constancy, but in the eyes of natural sciences such as physics, the law is constant and unchanging, and it is not explained by the emotions among several gods.

Historically, the ancients thought that the laws of nature were not immutable. This is actually quite normal. After all, the science and technology of the ancients were very underdeveloped, so it is natural to have such an idea. But in modern times, some people think that the laws of nature may not be constant. They think that everything we think of as laws is actually just a temporary phenomenon. What's more, they believe that all physical theories are nothing more than words manipulated by physicists, such as the philosopher of science Nancy Cartwright. For such arguments, if we study their theories carefully, of course, we can find some reasonable points. Some of them are helpful for us to think about science and physics, but in general, I don't think that there is any point in studying their theories too seriously. necessity.

Of course, the existence of a natural law does not mean that our theory gives this natural law. However, as in "What is Science? " As mentioned in the book, physical theories always aim at the laws of nature and make constant approximations. Although there may be detours at times, the goal is clear.

This is also why we believe that the laws of physics will one day find the ultimate laws of nature within a sufficiently valid range of precision .

Physics and Chemistry, Biology

From the perspective of spatial scale, the research objects of chemistry and biology are also within the scope of physics research, so people often wonder, isn't chemistry and biology a branch of physics?

In fact, with the development of physics, chemistry and biology, we are increasingly using physical research methods and theories in chemistry and biology (such as quantum mechanics, which can be used to explain the chemical bonds between atoms and molecules, thereby explaining many chemical phenomena and theorems). Therefore, Feynman once said that if he was given a sufficiently powerful computer, he could calculate all chemical processes using quantum theory.

Although there is some truth to such a statement, it has to be said that Meng Lang has many elements. If quantum theory gives the rules of the game of molecular activities, then chemistry studies various phenomena in the actual molecular game and the laws behind them. A person who is familiar with the rules of Go may not be able to play Go well, nor may he know the most correct way to play Go.

Physics and Mathematics

If we use language to make an analogy, then mathematics gives syntax and physics gives semantics . Only when syntax and semantics are combined can we say a sentence that is unambiguous, clear, and not nonsense.

But in actual modern physics research, the situation may be that syntax precedes semantics, that is, we often try to find or find a set of mathematical models that "look good", and then reconstruct all physical concepts in this model, And try to make these physical concepts and their interrelationships can conform to the corresponding relationships in the chosen mathematical model.

This is obviously the opposite of what we have always thought of physics as a process of constructing by summarizing phenomena, because it first decides how language is to be said and then describes nature, rather than describing nature and then summarizing as we traditionally think out grammar.

The emergence of this status quo, I am afraid, can be traced back to the familiar Einstein.

Before it, the study of physics basically followed the pattern that has been continued since the ancient Greek times, that is, observation, experiment, and summary. Galileo's era added thought experiments to this model, and then, in Newton's era, mathematical computation began to be introduced into physics on an unprecedented scale.

Galileo's most famous thought experiment was rolling a ball on a ramp, which led to the discovery of the law of inertia. But in fact, his other thought experiment is more famous, that is, the free fall experiment on the Leaning Tower of Pisa - yes, this experiment Galileo did not do, although in 1654 his student Vincenzo Viviani (Vincenzo Viviani) The book "The History of Galileo's Life" wrote a book about this "history", but it didn't actually happen.

Of course, someone in history did do a falling body experiment on the Leaning Tower of Pisa, Galileo did do an experiment about falling bodies, and finally did a thought experiment about free falling bodies in a vacuum, but these three things are completely different.

In 1612, a believer of Aristotle did a free fall experiment on the Leaning Tower of Pisa. The result was that there was indeed a small difference in the landing time of the two balls, so it was an embarrassing manifestation of Ari. Stothe's theory (see "Story of Science" co-written by Zhang Guangxi and Song Jiali). Earlier in 1586, the Dutch physicist Simon Stephen also conducted a free fall experiment, but not on the Leaning Tower of Pisa. The result was that two balls landed almost simultaneously. In Viviani's book, Galileo performed a free fall experiment on the Leaning Tower of Pisa in 1590, and the result was that the two balls landed almost at the same time. Obviously, these two real historical events were merged together. Now the word is the little tailor of history.

In Galileo's own book "Dialogues on the Two World Systems of Ptolemy and Copernicus" published in 1638, he proposed the thought experiment of free fall in a vacuum environment, but did not really do it. experiment. What he really did was the water drop experiment mentioned in the same book to study the effect of drag on the falling body, and finally deduce the correct conclusion that the falling path is proportional to the square of the falling time.

Therefore, strictly speaking, the free fall experiment on the Leaning Tower of Pisa can only be regarded as a thought experiment for Galileo (he did not know the experiment of the two previous physicists), and it was "the most beautiful thought experiment".

Since Einstein, the importance of thought experiments and mathematical tools has been further enhanced. People began to construct theories purely from intuition and aesthetics when experimental evidence is not enough to give answers, and tell physical experiments where Under the conditions to verify the theory, to prove after the fact that their theoretical conception is correct or wrong.

Many physicists now tend to use the "beauty and ugliness" of the theory to measure whether the theory is correct. This practice probably goes back to Einstein, because he heard Eddington's verification My own theory later said: (If Eddington had not confirmed general relativity in 1919) I would be sorry for my dear God. Either way, the theory is still correct (as to whether Eddington actually verified general relativity, it is mentioned in this article ). Henceforth, physicists seem to be keen to "argument" how beautiful a theory should be in the absence of experimental evidence, such as superstrings and M-theory.

But in fact, I personally have always held an objection to this. The reason is that since we can't have the same aesthetics as monkeys, how can we believe that we have the same aesthetics as God? So I would feel sorry for my dear God, if he really had the same aesthetic as we do, and constructed the universe from it .

Of course, if you put aside the subjective judgment of "beauty and ugliness", "conciseness" may indeed be a criterion . Of course, what is considered here is not the philosophical Occam's razor , but from the perspective of Turing machine To consider, if we think that there is a completely random process to generate Turing machines that control the laws of nature, then it can be proved that the more "simple" Turing machines, that is, the shorter Turing machines, are generated in this random environment. In fact, the probability of such a Turing machine is proportional to the logarithm of the incompressible length of the Turing machine, and the fluctuation range is a constant determined by the Turing machine language. Of course, whether nature can use this random Turing machine generation to explain the laws behind it is an unknown. (See this article for more related content)

This has become an industry norm today because, in the words of physicists, our theories are at least 100 years ahead of experiments . Of course, this statement may be optimistic .

In fact, our theories are constantly talking about what physics would look like under the Planck energy scale, and to get to that level of energy, we might need to use the energy of an entire galaxy - I'm not talking about the solar system, but the Milky Way . Therefore, saying that it is only ahead of 100 years is obviously just to adjust the atmosphere. The actual situation is that there may not be hope in 100 centuries. The bright side, of course, is that people are always hopeful about the future, although no one knows where that confidence comes from .

Therefore, the relationship between physics and mathematics has undergone an interesting reversal in the hundreds of years since the Renaissance, from the original physical explanation before the mathematical model, and now the mathematical model precedes the physical explanation. In fact, there are many experts in mathematical physics who believe that every mathematical theory will eventually be applied to physics - so it took 7-8 years for a mathematician such as Mochizuki's Far Abelian Teichmuller Theory Mathematical theories that have not been fully understood by study and have been described as "so mysterious that there is absolutely no need to spend time studying" may finally have applications in physics? I am afraid that this physics is no longer available to human beings.

The name Inter-Universal Teichmüller Theory at first glance seems very middle-of-the-road, and some of the terms in it make people feel like they are in a hot-blooded second dimension, such as "Hodge Theate", "Alien Arithmetic Holomorphic Structures" and so on, but it's actually a very serious mathematical term. For example, the Teichmuller space refers to the deformation parameter space (modular space) of the complex structure of the Riemann surface, which is also related to string theory. The Universal Teichmüller Space is a generalization of the general Teichmüller space, and the intercosmos Teichmüller theory is actually a theory about the relationship between the Universal Teichmüller space, only because " The English words for "universal" and "universe" are the same, which is why such a middle-two name appears - of course, another part of the reason is that Mochizuki Shinichi is indeed full of bad taste when naming his term...

But in any case, language without grammar is ineffective, so it is understandable that mathematics is used extensively in the research of natural sciences such as physics, which also creates what Wigner calls " the incredible effectiveness of mathematics in natural sciences. "The miracle.

But we must face up to the relationship between mathematics and physics. Although the two are close, they are not meant to replace each other.

In fact, the same physical fact can often have a large number of different mathematical representations, and the same mathematical representation can often have a large number of different physical interpretations.

The latter is common in the developmental history of physics, such as the numerous different interpretations of the most famous quantum theory, which have almost identical mathematical representations of each other, but there are different opinions on what the mathematical formulas represent. The former is relatively rare in physical development, but it is not without. For example, the matrix representation, wave representation and path integral representation of quantum mechanics, the mathematical methods of the three representations are not exactly the same (but they can be proved to be equivalent to each other), and the corresponding The physical process is the same.

There is an interesting phenomenon here. For a first-order formal language system, according to Tarski's indefinability theorem , there is no so-called "strong self-expressive language" that can fully define grammar by semantics, that is to say, semantics cannot completely determine grammar; and according to [Löwenheim–Skolem theorem] ]( https://en.wikipedia.org/wiki/Löwenheim–Skolem_theorem ), if this kind of language has an explanation model M, then it must have an explanation model of potential smaller than M. The explanation model is the basic substructure of M, and the potential The explanatory model of potentials larger than M is an extension of M, that is, the grammar cannot fully determine the semantics. So for first-order formal languages, syntax and semantics are independent of each other.

Of course, the laws of nature are not necessarily first-order formal languages, and it is just a metaphor to say that mathematics is grammatical physics and semantics, so the above phenomenon is just an interesting footnote.

So far, about the relationship between physics and mathematics, my personal views are generally clear.

end

At the request of a friend, I finally finished this article.

The full text basically just starts from the personal content of mathematics, physics and related philosophy of science, and makes a brief discussion on what is mathematics, what is physics, what is the relationship between mathematics and physics, and what is the relationship between mathematics and science. .

Due to limited personal knowledge, I dare not say that it is more reasonable than other people's ideas.

Anyway, let me say it rashly, and you can rashly listen to it.