7 months, 2 weeks ago

Is the mathematical law an objective reality or a game invented by mathematicians? This is a difficult question to answer. Mathematicians often hold two incompatible views on the objects they study.

For example, there is a relationship between prime numbers that is revealed by the Goldbach conjecture, and mathematicians are constantly discovering this relationship. But is this conjecture (mathematical object) independent of human existence?

If the mathematical objects are real objects, why can't they be touched and can't interact with them? These problems often lead mathematicians to make this assumption: in fact, the world of mathematical objects is fictitious.

Math concept game

One of the most surprising reactions when I told others that I was a mathematician was: "I really like math classes because everything here is right or wrong, there is no ambiguity. Or uncertainty." I always responded to this. In fact, not everyone likes mathematics, and I don't want to attack people's enthusiasm for mathematics. In fact, mathematics is also full of uncertainty, but mathematics itself hides this uncertainty well.

I certainly understand the idea that there is no uncertainty in mathematics. For example, if the teacher asks you if 7 is a prime number, the answer is definitely "yes". Because by definition, a prime number is an integer greater than 1 and can only be divisible by itself and 1, and 2, 3, 5, 7, 11, 13, etc. are prime numbers, so 7 is a prime number that is very certain.

In the past few thousand years, any mathematics teacher anywhere in the world, at any time, has to admit that the phrase "7 is prime" is correct and will not give you an answer. However, few other disciplines can achieve such incredible consensus as mathematics. However, if you ask 100-bit scientists what the nature of these mathematical propositions can be explained, you may get 100 different answers. The number 7 may really exist only as an abstract mathematical object, and the prime property is a feature of the object. Or, the concept of prime numbers may itself be a well-designed game by mathematicians. In other words, mathematicians can agree that a proposition is correct or wrong, but they cannot agree on the nature of the proposition.

To a certain extent, these controversies are a simple philosophical question: Is mathematics an objective law discovered by humans or an invention that relies on subjective wishes? Maybe 7 is a real object that is independent of us, but what is the essence of it is what mathematicians are still exploring. Perhaps it is a fictional object in people's imagination, its definition and attributes are flexible. In fact, this behavior of mathematical research stimulates a view similar to philosophical dualism, in which mathematics is the invention of human beings and the discovery of human beings.

All this seems to me a bit like an improvisational drama. The mathematician constructs a mathematical background stage consisting of a few characters or objects, as well as some interacting rules, and then see how these mathematical objects evolved in this context. The result is that these character actors are completely independent of the mathematician's intentions and quickly develop surprising features and relationships. However, no matter who directs the show, the ending is always the same. It is the inevitability of this ending that gives the mathematics discipline a strong cohesive force. The difficulty of acquiring the nature of mathematical objects and the acquisition of mathematical knowledge is still hidden and undiscovered.

Truth and proof

How do we judge whether the mathematical proposition is correct? With natural scientists often inferring the fundamentals of nature from the observation of natural phenomena, mathematicians start from the rules of mathematical objects and rigorously derive conclusions. This deductive process is called proof. This process usually starts with a simple premise and derives complex conclusions. At first glance, the process of mathematical proof seems to be a key factor in achieving consensus among mathematicians.

But the proof only gives the truth that mathematics is based on certain conditions, that is, the authenticity of the conclusion depends on the authenticity of the premise. There is a general view that the consensus among mathematicians is generated by a proof-based argument structure. Proof is based on certain core assumptions, and other conclusions depend on these assumptions. This raises the question: Where do these core assumptions and ideas come from?

In fact, the most important point of mathematics is usually usefulness. For example, we need numbers so that we can count the number of cows and measure the area of ​​the field. Sometimes the initial assumptions are aesthetically pleasing. For example, we can invent a new arithmetic system in which a negative number multiplied by a negative number is a negative number. However, in this system, those intuitive and ideal numerical axis properties will disappear. The mathematician's judgment of basic objects (such as negative numbers) and their properties (such as the result of multiplying them) needs to be self-consistent with a larger mathematical framework. Therefore, before proving a new theorem, mathematicians need to watch the development of this drama. Only then can the mathematician know what to prove: what is the true and inevitable conclusion.

Therefore, the development of mathematics has three stages: invention, discovery and proof.

Characters in mathematics are almost always composed of very simple objects. For example, a circle is defined as a collection of all points that are equidistant from the center point. Therefore, the definition of a circle depends on the definition of a point (this is a very simple object type) and the distance between two points. Similarly, the process of repeating addition is multiplication; a number repeats itself by multiplying itself by the power. Therefore, the power of the property inherits the properties of the multiplication. Conversely, we can also learn more complex mathematical objects by studying objects that are defined to be simpler. This led some mathematicians and philosophers to think of mathematics as inverted pyramids, many of which are derived from simple concepts at the bottom of a narrow tower.

In the late 19th and early 20th centuries, a group of mathematicians and philosophers began to think about what it was to hold up this heavy mathematical pyramid. They are extremely worried that mathematics has no foundation—nothing supports the authenticity of mathematical conclusions such as 1+1=2.

Some mathematicians hope to draw all mathematical truths from a relatively simple axiom set. However, the work of American mathematician Kurt Godel in the 1930s was often used to prove that this axiomatic system was impossible. First, Gödel shows that any reasonable axiom system is incomplete, and the mathematical expressions of this system can neither be proved nor refuted. Gödel's theorem on mathematical incompleteness gave mathematics a devastating blow. Originally, everyone thought that the basic system of mathematical axioms should be consistent, and there is no expression that can be both proved and refuted. More importantly, previous mathematicians felt that the mathematics system should be able to prove its own consistency. But Gödel's theorem points out that this is impossible.

The process of finding the basis of mathematics has indeed led to the discovery of a basic axiom system called the Zermelo-Fraenkel set theory, from which people can get the most interesting mathematics. Based on set theory, not only does mathematics become very simple and clear, but most of the mathematics knowledge has a solid foundation.

Throughout the 20th century, mathematicians debated whether Zemelo-Fremont's set theory should be extended, the so-called selection axiom: if you have a myriad of collections containing objects, then you can select an object from each collection. Form a new collection. For example, there are a row of buckets, each bucket has a set of balls, and there is an empty bucket. From each bucket lined up in a row, you can select a ball and place it in an empty bucket. Selecting axioms allows you to operate with countless rows of buckets. Not only is this method intuitively appealing, it can be used to prove some useful mathematical expressions, but it also hints at some weird things, such as the Banach-Tarski paradox, which shows that you can divide a solid ball into several parts and These parts are reassembled into two new solid balls, each equal in size to the original ball. In other words, you can get two balls. The choice of axioms contains many important expressions, but it also brings additional questions, including some strange bad expressions. But if there is no axiom of choice, mathematics seems to lack some key essential content.

Most modern mathematics uses a set of standard definitions and conventions that evolve over time. For example, mathematicians used 1 as a prime number, but not now. However, they are still arguing whether 0 should be understood as a natural number (sometimes called a count number, a natural number is defined as 0, 1, 2, 3... or 1, 2, 3... depending on who you ask). Which characters or inventions can be part of a mathematical classic usually depends on how interesting the results are, and such observations can take years. In this sense, mathematical knowledge is cumulative.

Discover or invent

As mentioned earlier, mathematicians initially consider defining mathematical objects and axioms under specific application conditions. However, over time, mathematics has reached a second stage – discovery. For example, prime numbers are the cornerstone of multiplication and are the smallest multiplication units. If a number cannot be written as the product of two smaller numbers, then this number is a prime number. All non-prime numbers (combined numbers) can be obtained by multiplying a unique set of prime numbers.

In 1742, the German mathematician Christian Goldbach assumed that each even number greater than 2 is the sum of two prime numbers. If you choose any of the even numbers, then Goldbach guesses that you can find two prime numbers to add this even number. If you choose 8, the two prime numbers are 3 and 5; if you choose 42, you can be 13+29. The Goldbach conjecture is surprising because although the prime numbers were originally designed to be multiplied, this conjecture suggests that there is an incredible relationship between the sum of prime numbers and even numbers.

A lot of evidence shows that the Goldbach conjecture was established. In the next 300 years, computer numerical calculations confirmed that this conjecture is correct for all even numbers less than 〖4×10〗^18. However, this evidence is not enough for the mathematicians to claim that the Goldbach conjecture is correct, because no matter how many even numbers the computer checks, but even numbers have infinite numbers, there may always be a counterexample lurking in the corner - one is not An even number of the sum of two prime numbers.

Imagine that the computer will record this even number each time it finds that the sum of two prime numbers is a specific even number. So far, this is a very long list of numbers, you can use it as a compelling reason to make everyone believe that Goldbach's conjecture is right. However, there is always someone who can think of an even number that is not on the list and asks you how to know that the Goldbach conjecture is still true for that number. Not all (infinitely many) even numbers will appear in the list, so it is only from the basic principle that it is sufficient to prove that the Gothbach conjecture holds for any even number through logical arguments. However, until today, no one has been able to provide such proof.

Goldbach’s conjecture illustrates the important difference between the mathematical discovery phase and the proof phase. In the discovery phase, people seek mathematical facts and mathematical phenomena, while the nature of mathematics requires solid proof.

Mathematicians need to sort out mathematical findings and decide what to prove, but they can also be deceptive. For example, let's build a series of numbers: 121, 1211, 12111, 121111, 1211111, and so on. Let's make the following guess: all the numbers in the series are not prime numbers. It is easy to provide evidence for this conjecture. It can be seen that 121 is not a prime number because 121=11×11. Similarly, 1211, 12111, and 121111 are not prime numbers. This mode can last for a while, but then it suddenly goes wrong. The 136th number in this sequence (ie, the numbers 12111...111, where 136 "1"s follow "2") are prime numbers.

The stage of mathematical discovery is still extremely important. For example, it can reveal the hidden connection between the prime numbers given by Goldbach's conjecture. Before discovering this profound connection, mathematicians usually study two completely different branches of mathematics. A relatively simple example is the Euler's identity, eiπ+1=0, which relates the geometric constant π to the number i (algebraically defined as the square root of -1) by the number e (the base of the natural logarithm). These amazing discoveries are part of the aesthetic and curiosity of mathematics. They seem to point to a deeper infrastructure, and mathematicians are just beginning to understand these structures.

In this sense, mathematics can be both invented and discovered. Research objects are precisely defined, but they have their own lives and reveal unexpected complexity. Therefore, mathematical objects can be seen as both actual and artificial. As one philosopher wrote, “duality has no effect on the way mathematicians work”.

Reality or illusion

Mathematical realism seems to be the philosophical stance of the discovery phase: the objects of mathematical research, such as from circles and prime numbers to matrices and manifolds, are real and independent of human thought. Just like astronomers exploring distant planets or paleontologists studying dinosaurs, mathematicians are gathering insights into real entities. For example, to prove that the Goldbach conjecture is established, that is to prove the specific nature of the association between even and prime numbers by addition, just as paleontologists may indicate the origin of a dinosaur through the correlation between the anatomical structures of two species. Another kind of dinosaur.

Various manifestations of realism, such as Platonism, make it easy to understand the universality and practicality of mathematics. Every mathematical object has a property. For example, 7, it is a prime number, like a dinosaur with flying attributes. A mathematical theorem, such as the sum of two even numbers is even - this is correct. Because even numbers do exist and there is a specific relationship between them. This explains why people who cross time, geography, and cultural differences generally agree with these mathematical facts.

But some people have objections to realism. They believe that if mathematical objects exist, their nature must be very unique. First, mathematical objects are very abstract, so you can't really interact with them. This is a problem because dinosaurs can be broken down into bones that can be seen and touched. Planets can also pass in front of stars and be observed by astronomers, but the mathematical circle is an abstract object, independent of space and time. . In fact, π is the ratio of the circumference to the diameter of the circle and is not related to soda or donut; it points to a mathematically abstract circle in which the distance is precise and the points on the circle are infinitesimal. Such a perfect circle seems to be impossible to achieve in real life. So, if there is no special sixth sense, how can we understand the facts about the circle?

This is the difficulty of realism - it cannot explain how we know the nature of abstract mathematical objects. All of this can lead mathematicians to retreat from a realistic standpoint. Anti-realism can make mathematics frame a purely form of thinking practice or a complete novel, and it is easy to avoid epistemological problems.

Formalism is a form of anti-realism and a philosophical point of view. It argues that mathematics is like a game, and mathematicians are just playing the rules of the game—saying 7 is a prime number, as if to say that the knight is the only chess piece that can move in L form. Another philosophical point of view is fictionalism, which believes that mathematical objects are fictitious—saying 7 as a prime number is like saying that a unicorn is white. Mathematics has meaning in its fictional universe, but there is no real meaning outside it.

But if mathematics is only made up, how can it become an indispensable part of science? From quantum mechanics to ecological models, mathematics is a broad and precise scientific tool. Scientists do not expect elementary particles to move in accordance with the rules of chess. The burden of natural science description falls entirely on mathematics, which is quite different from game or fiction.

Finally, these issues do not affect the practical application of mathematics. Mathematicians are free to choose an explanation of their profession. In The Mathematical Experience, Philip Davis and Reuben Hersh have a famous saying: “A typical professional mathematician is a Platonist on weekdays. At the weekend it is a formalist."


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