In the future, there will be a revolution in mathematics. Some are already happening: the rapid development of computer technology, the increasing influence of big data and artificial intelligence, and the new challenges brought by life science and financial industry. Of course, there will be others, and many things are unpredictable.
In some cases, mathematical proof has replaced observation and experiment in other sciences? In other words, mathematics avoids being led astray by personal wisdom through proof, and avoids believing untrue things because of liking. The invention of microscope can't replace biological experiment, and computer can't replace mathematical proof. In the analogy of disciplines, we can see that the computer strengthens the technical means of proof, but does not change the consistency of logic, and deduces new theorems from known theorems, and the route of deduction must stand the strict examination of experts. The concept of proof will remain as the most basic thing in mathematics, just as Chen Jingrun proved Goldbach's conjecture.
The power of mathematics comes from the confluence of two sources.
What's the first one? The real world? . Johannes kepler, galileo galilei and isaac newton tell us that many aspects of the outside world can be understood through subtle mathematical laws (laws of nature). Sometimes physicists modify the form of these laws. Newtonian mechanics gives way to quantum mechanics and general relativity, and quantum mechanics gives way to quantum field theory, and quantum gravity or superstring leads the direction of future theoretical revision. Problems in the real world have stimulated the emergence of new mathematics. Even if the theory that produced it has changed, mathematics still exists and is still important.
The second source of strength of mathematics is human imagination: pursuing mathematics for the sake of mathematics. Brave pioneers often break away from the mainstream in the pursuit of personal fantasies and then find a better route. The value of mathematicians' exploration is obvious, which is their motivation. They don't need more reasons than the meaning of mathematical verification itself.
For example, Fermat's last theorem has been a huge problem for more than 300 years. Its mathematical expression is,? N is greater than 2 and an integer, and the equation x n+y n = z n about x, y and z has no positive integer solution? . It attracted many mathematicians and was finally proved by British mathematician wiles in 1995. He turned Fermat's expression into a kind of? Elliptic curve? Proposition (a completely different field of number theory).
Today, the method of pure mathematics has brought new vitality to applied mathematics. The problems in applied mathematics have stimulated the new development of pure mathematics. The golden age of mathematics is no longer ancient Greece, Renaissance Italy or Newton England, but today.
Speaking of today's mathematics, we have to mention seven famous unsolved mathematical problems in 2 1 century. 1900, David Hilbert, the greatest mathematician of that era, put forward 23 mathematical problems to be solved in the future, and most of them have been solved today. 100 years later, the Clay Institute of Mathematics (CMI, Massachusetts) in Cambridge, USA, publicly collected the answers to seven mathematical questions at the Millennium Annual Meeting held in France in May 2000. These seven questions were carefully selected by CMI's Scientific Advisory Committee, and the answer to each question will be rewarded with $6,543,800+0,000.
1, Burch and swinton Dell conjecture (BSD).
For any elliptic curve in rational number field, the zero order of its L function at 1 is equal to the linear rank of Abel group formed by rational points on the curve.
The BSD conjecture has made a breakthrough in recent years. For example, a mathematician from the Institute of Mathematics of the Chinese Academy of Sciences proved a special case and made substantial progress on this issue.
2. Hodge conjecture
This is an important unsolved problem in algebraic geometry, and it is a conjecture about the algebraic topology of nonsingular complex algebraic clusters and the correlation between them and the geometry represented by polynomial equations defining subgroups.
On the number family of nonsingular complex projection algebraic space, any? Hotch circle? It is actually a rational linear combination of algebraic closed chains. Together with Fermat's last theorem and Riemann conjecture, it becomes the carrier and tool of M-theory structural geometry topology which combines general relativity and quantum mechanics.
3. Naville-Stokes equation.
This is an equation of motion describing the conservation of momentum of viscous incompressible fluid. Although it has been 100 years since it was put forward as a viscous hydrodynamic equation, scientists still have a shallow understanding of it, hoping to understand turbulence from the mathematical theory of this equation and prove its existence and smoothness. This also involves quantum field theory? Quality gap hypothesis? .
4.P and NP problems (P versus NP problems)
Whether the problem class P of deterministic polynomial time algorithm is equal to the problem class NP of uncertain polynomial time algorithm. The answers to some questions are easy to check, but it takes almost infinite time for computers to do this. This is the so-called NP problem, where p is a polynomial and NP is a nondeterministic polynomial. The P/NP problem is about computers, but it can't be solved by computers. Go, which we are familiar with, is a NP-hard problem.
In 20 10, Vinay Deolalikar, a mathematician from Hewlett-Packard Laboratories in the United States, claimed to have solved the P/NP problem and published the manuscript of the paper. His draft paper has been recognized by complexity theorists, but its final draft has not passed the examination of experts.
5. Poincare conjecture.
In topology, any simply connected and closed three-dimensional popular and three-dimensional spherical homeomorphism. Poincare proposed more than 0/00 years ago that a two-dimensional sphere (such as the earth's surface) is simply connected and can be shrunk into a point. What about the three-dimensional sphere? This is a topological proposition, which is helpful for human beings to study three-dimensional or even multi-dimensional space.
In 2006, the mathematical community finally confirmed that Russian mathematician grigory perelman had successfully solved Poincare's conjecture (he refused the bounty of $654.38+$00,000).
6. Young-Mills theory.
When Yang Zhenning-Mills gauge field theory is used to describe the strong interaction of elementary particles, a subtle quantum property is needed, and it is necessary to prove the existence of quantum Yang-Mills field, and there is a? Quality gap? . The equations of this theory are a set of nonlinear partial differential equations of great significance in mathematics.
Although classical waves move at the speed of light (mass is 0), quantum particles have positive mass. We can't understand this in theory at present.
7. Riemann conjecture.
This is the most famous unsolved problem in mathematics, which was first put forward by Georg Bernhard Riemann. This is a very special problem in complex analysis, and the answer of conjecture is likely to bring dawn to prime number theory, algebraic number theory, algebraic geometry and even dynamics.
Riemann found that all non-zero points of Zeta function lie on the straight line of Re(s)= 1/2 on the complex plane, that is, the real part of the solution of equation Zeta(s)=0 is 1/2. So Riemann conjecture can be expressed as:? All nontrivial zeros of the Riemannian zeta function lie on a straight line whose real part is 1/2. ?
This conjecture is related to many difficult problems about the distribution of prime numbers. For example, Goldbach conjecture is just a special case of it.
How important is it to prove Riemann conjecture?
It can be said that as the most difficult mathematical problem in today's mathematics, the correctness of Riemann conjecture directly affects the whole mathematical system based on Riemann conjecture. After all, we have more than 1000 mathematical propositions, all of which are based on Riemann conjecture and its extended form. Riemann conjecture, once proved, will become an indestructible mathematical theorem. On the other hand, if falsified, a large part of these mathematical propositions will inevitably become Riemann conjecture? Burial objects? .
Furthermore, Riemann conjecture studies the distribution of prime numbers in mathematics. It has been 160 years since it was put forward, and its vines have already crossed from mathematics to physics.
For example, the general theory of relativity stems from Einstein's realization that gravity is not a force, but a reflection of the geometric curvature of space-time caused by mass. However, there was no mathematical theory to support Einstein's idea until Einstein understood Riemann conjecture? Non-Euclidean geometry? The general theory of relativity came into being.
In 20 18, the British mathematician Atia claimed to prove the riemann conjecture, but it was strongly questioned by some scholars, and this proof was not valid. Nevertheless, his thinking may be helpful to later proof.
The seven mathematical problems mentioned above in 2 1 century will help mathematicians to promote the research and development of pure mathematics in the future.
Ian stewart, a professor of mathematics at the Royal Society, believes that in Newton's time, the main sources of mathematical problems were astronomy and mechanics, that is, natural science. In the future, more exotic subjects will flood into mathematics. One of them is quantum physics, which has been highly mathematical. Today, there are new connections among quantum field theory, geometry, topology and algebra. The future quantum fields, superstrings and new structures inspired by various theories outside them will open a brand-new world of algebra and topology.
19th century mathematicians put the traditional? Real? The number has expanded to? Reply? Count, let? - 1? With the square root, it brings infinite vitality to mathematics. Soon, all areas of mathematics? Get back together? The mathematics of generating complex numbers is as fruitful as the ancient real numbers. ? Quantify? Is it 2 1 century? Get back together? We will enter the world of quantum algebra, quantum topology and quantum theory.
The future life science will inspire a new mathematics: biomathematics. Scientists once thought that there were 654.38+ten thousand genes in the human genome, but the result was wrong, only 34,000. From gene to protein, the road map is much more complicated than we thought; In fact, there may not be such a map at all. Gene is a part of a dynamic control process, which not only creates protein, but also constantly revises protein, so that it can find its proper position in the evolutionary life and at the right moment in its life course. Understanding this process requires more than a series of DNA codes, but what we lack most is mathematics.
Bio-mathematics is a new mathematics that combines the dynamics of life growth with the genetic information process of DNA. The DNA password is still important, but not all. The new biomathematics may be a strange mixture of combinatorial biology, mathematics, analysis, geometry and informatics.
Different from the laws used by mathematics in physics to express quantification, real-world prediction is usually the result of big data plus artificial intelligence analysis. For example, in order to simulate the huge vortex of typhoon, engineers need to list the motion equations of warm and humid gases in thousands of small areas, and then solve these equations through a lot of calculations. Now with the help of computers and big data analysis? Vortex calculus? It is possible to liberate people from the endless entanglement of numbers. This is a qualitative and fluctuating mathematical theory formed by a dynamic model.
Another example is the futures and stock markets. Many intermediaries interact by buying and selling futures and stocks. This is how the financial industry stands out from the interaction. In the future, the mathematics of finance and business will also be produced in the revolution, and the popular ones will be abandoned? Linear? Model, so that the mathematical structure can reflect the market changes more accurately.
In the future, the development space of mathematics is still large enough. Is it a tool to help us re-understand the world? Through new models, not billions of magically beating numbers.