author: Liu Ruixiang, [met] here to thank Liu submission support!
Some people say that mathematics is the study of infinity (infinite). In other words, as long as it is a little more "advanced" mathematics, you will definitely encounter infinity, which makes sense.
Physics, which is often compared with mathematics, is not infinite. I encountered this problem when I was in college: According to Coulomb's law , is the field strength near a point charge infinite? The answer given by the teacher is no, because when you are very close to the so-called "point charge", the physical model of "point charge" is no longer applicable. For another example, the electric energy stored in a capacitor is related to the electric charge, so the charging and discharging of the capacitor takes a certain amount of time. This is because the electric energy cannot change suddenly (the derivative of electric energy with respect to time cannot be infinite), and so on.
There is true "infinity" in mathematics, perhaps because mathematics is mainly a "thinking" game rather than a simple reflection of reality. Taking definite integral as an example, the basic idea is to divide the research object into an infinite number of copies, each of which becomes infinitely smaller, and then add so many small parts together. This makes no sense in physics at all: when we are far from reaching the " infinitesimal", matter has become molecules, atoms and all kinds of inexplicable "sons", how can there be the original continuity nature? Therefore, a term specially invented in physics class—macro infinitesimal, used to describe this “infinitesimal” that ignores the discontinuities of matter. The actual discontinuity will occur.
This approach has an advantage,That is, sometimes it may be difficult for us to count the finite things, while the infinite things are very simple. Sometimes things are so strange. Let me give another example. For example, if we want to calculate the interaction problem within a crystal, it is obvious that the size of the crystal is limited, but it is far less convenient to calculate it than to treat the crystal as infinite. Naturally, infinity also brings troubles. For example, some mathematical paradoxes are related to infinity, and paradoxes cannot be solved through experiments like paradoxes in physics.
There is no "infinity" in physics, and there are practical barrier problems. For example, speed has a natural upper bound of , but there is no such thing in mathematics. The size of the universe is of course a stronger limit. In mathematics, there are not only infinities, but also several infinities of different levels, and even some infinities have completely exceeded human imagination and have no counterparts.
Chinese mathematician Gu Chaohao In this book "Talking about the infinite in mathematics", he talked about his own experience in mathematics, and said, "There are several things that are very profound to me." The impression of ": cyclic decimal , parallel straight lines can be seen as intersecting at infinity, the whole number of integers is "as many" as the whole even number, infinite series. As a small book for a great mathematician, the author emphasizes "rich and rigorous thinking and imagination ability" rather than learning the skills of solving problems, and hopes that readers "can see some of mathematics concepts . The background and the creativity and imagination of past mathematicians". I think it might be more beneficial for you to read such a book than to immerse yourself in the questions, and it will help you understand what "real mathematics" is.
.[Mean Jun]: "Talking about the infinite in mathematics" should have two editions of 88 and 2000. Those who are interested in this book can borrow it from major libraries, or there are second-hand books available online. To.
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