# Where is the boundary between mathematics and physics?

”Archimedes, occupation: scientist, mathematician, physicist”;
”Descartes, occupation: philosopher, physicist, mathematician”;
”Isaac Newton, occupation: physicist, mathematician” “;
”Stephen William Hawking, occupation: physicist, cosmologist, mathematician”.
This is the professional description of several great scientists by Encyclopedia. From this, it is not difficult to find that physics and mathematics always appear at the same time. Many physicists are also mathematicians, and many mathematicians are also physicists. You must not ask “Physics or mathematics, which is more important?” This kind of question, otherwise you will cause a century war between the physics department and the mathematics department. But you can’m asking: “Where are the boundaries of mathematics and physics at?”
Predict and confirm the prediction
do a very simple thing: throw a ball. If you know the location and speed of its throwing, you can use physics to get the specific location of its landing. However, if you just write the mathematical formula for the trajectory of the ball after it is thrown, and solve this formula, you will get two answers: a positive number and a negative number, both of which correspond to the specific location of the ball, but the direction on the contrary. For another example, when I ask you “What is the square root of 4?”, you may subconsciously think that the answer is 2, but the answer may also be -2.
This may be the biggest difference between mathematics and physics: mathematics has a lot of uncertainty, it only predicts what the possible solutions are, and physics helps you get specific solutions. So mathematics is a tool to help solve physical problems, especially when you enter the world of general relativity or quantum theory, or even more distant universe inflation theory, extra dimensions and string theory, you will find that they all have mathematical models that describe themselves. .
On the other hand, physics is used to describe the real universe, so if you can’t come up with any physically observable quantities that conform to these mathematical models, these mathematical models will always be theoretical models, and they cannot be used. Describe the real universe. For example, although string theory may be a set of theories to solve the mystery of all things, it has never received relevant experimental predictions, that is, some realistic observations that conform to string theory, and even Einstein cannot prove it, so It has been firmly placed in the field of theoretical physics and cannot be called a conclusive conclusion.
Conversely, some theories that can be measured in reality may become conclusive. For example, the inflation theory is a theory to describe the universe proposed by American cosmologist Alan Guth in 1981. This theory points out that the early universe expanded exponentially during the period of 10-36 seconds to 10-32 seconds, and the expansion rate of the universe slowed down after the end of the inflation. In inflation theory, there are various mathematical predictions, one of which is quantum fluctuations.
In the process of inflation, “inflation” also occurs in quantum fluctuations (a temporary change in the energy of a point, that is, energy fluctuations). The universe before inflation is like an uninflated balloon, and the quantum fluctuation is a very small point on the surface of the balloon. When the universe is inflated, the balloon is inflated, and the inflation is like a magnifying glass, magnifying the dots on the balloon, that is, quantum fluctuations. Combined with the inflation theory, the amplified quantum fluctuations eventually caused the tiny temperature fluctuations of the cosmic microwave background radiation.
The cosmic microwave background is considered to be the radiation left over from the Big Bang, or the heat left over from the Big Bang. Initially, the temperature of the universe was extremely high, and it gradually decreased with the expansion of the universe. The currently observed temperature of the universe is only about 2.725 degrees Celsius higher than absolute zero. The temperature of the cosmic microwave background is almost uniform throughout the universe. Only very sophisticated detectors can observe tiny fluctuations. These fluctuations may be caused by quantum fluctuations.
Many years later, with the help of physical observations from instruments such as the Cosmic Background Observator (COBE), Wilkinson Microwave Anisotropy Probe (WMAP), and Planck satellite (Planck), it was only indirectly verified in 1981. A great prediction, so inflation theory is much more likely to be used to describe the real universe than string theory.
To put it simply, any mathematical model used to describe predictions or describe possibilities often needs to be linked to physical objective observations before it can be verified or used to describe reality.
Mathematical derivation of physical
laws of conservation is probably the most basic laws you can think of, such as energy conservation law, the law of conservation of momentum and angular momentum conservation law. But you are wrong. These conservation laws are the result of the derivation of a mathematical theorem. It is Noord’s theorem. At the same time, it is also one of the central theories of theoretical physics. In this way, mathematics and physics have a complicated relationship. Noord’s theorem tells us: every continuous symmetry corresponds to a conserved quantity.
What does it mean? Well, if there is no specific explanation, many people may not understand it. So, let’s explain now.
First of all, what is symmetry? Simply put, if you perform some operations on an object, and after these operations, it looks the same as before, then the object is symmetrical. Imagine placing a mirror in the middle of the face. The face in the mirror and your face should be symmetrical; some playing cards will be symmetrical with the original playing cards after being rotated 180 degrees. But these are discrete symmetry, they are only symmetrical along a single axis or under a special rotation angle. If the playing card is rotated 90 degrees, it will not be symmetrical with the original playing card. In Noord’s theorem, it is about continuous symmetry-symmetry under any axis and any rotation angle. For example, a perfect sphere will always be symmetric with the original sphere no matter how many degrees it rotates.
Let us take the law of conservation of momentum as an example. The law of conservation of momentum is derived on the basis of the symmetry of space translation-according to Noord’s theorem, because space translation is symmetric, there must be a conserved quantity, and this conserved quantity is momentum. Space translation symmetry means that the physical law does not depend on the choice of the origin of the space coordinate, and the physical law will not change after the space is translated. To put it vividly, if there are two laboratories in the school, one in the north and one in the south, and you do the same experiment in the two laboratories, the results you get must be the same. The simplest momentum conservation experiment is to let two balls of different mass collide at different speeds in different laboratories. Through precise calculations, you will find that before and after the collision, the sum of the product of the mass and velocity of the two small balls is constant, and the product of the mass and velocity is momentum-this is the law of conservation of momentum.
Mathematical tools to solve physics
mathematics in physics is useful, it is not surprising. When we need to measure and calculate physical formulas, mathematics is an indispensable tool.
Here is an interesting story about general relativity. In 1912, Einstein was brewing a subversive theory—general relativity, which asserted that massive objects would distort time and space. But Einstein encountered a problem in how to express it. At this time, Einstein discovered that the concept of curvature geometry proposed by the mathematician Bernhard Riemann was exactly what he needed. Riemannian geometry gave Einstein a powerful mathematical foundation, enabling him to construct an accurate equation of general relativity.
This story must be very proud of mathematicians. In this story, mathematics is like a beacon and guide for physics, bringing light and direction to physics in times of difficulty. But we can also see that mathematics is more like a tool to solve physics, and learning these tools will make it easier to solve physics problems. Sin30? You know, right? It is actually a mathematical tool. When you are calculating a physics problem, you must first calculate it before you can get other relevant specific quantities.
Therefore, mathematics is like nails, boards, hammers, and saws, and physics is like a house. In order to get this house, in addition to combining all the materials, tools such as nails, boards, hammers, and saws are also needed to fix it. , That is, mathematics is a tool of physics.
So do you know where is the boundary between mathematics and physics? If you can accurately describe the universe, and can objectively measure and observe it, you are physics. If your equations cannot be connected with any observations, then you will be firmly in the field of mathematics; mathematics can describe physics, derive physics, and solve physics problems…
What do you think about how to define mathematics and physics? Woolen cloth?