Relationship between mathematics and physics - Wikipedia
The Relation between Mathematics and Physics I am a mathematician. But my first love was physics, particularly the physics in Astronomy. When I was young I. are used in Physics. That is, Mathematics is not only the "language"of Physics. Do you want to read the rest of this article? Request full-text. Physics and mathematics present two areas of intellectual activity deeply interwoven through the long history of science. Yet, they preserve two separate.
Physics of the Futureand The Future of the Mind Richard Phillips Feynman was an American theoretical physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics for which he proposed the parton model.
Feynman developed a widely used pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams. During his lifetime, Feynman became one of the best-known scientists in the world. In a poll of leading physicists worldwide by the British journal Physics World he was ranked as one of the ten greatest physicists of all time. Hope you enjoyed this short post on the relationships between physics and mathematics.
All May will be full of discussions and posts related to this topic. If you have any recommendations, let us know. Have a great day. I propose to deal with how the physicist's views on this subject have been gradually modified by the succession of recent developments in physics, and then I would like to make a little speculation about the future. Let us take as our starting-point that scheme of physical science which was generally accepted in the last century - the mechanistic scheme. This considers the whole universe to be a dynamical system of course an extremely complicated dynamical systemsubject to laws of motion which are essentially of the Newtonian type.
The role of mathematics in this scheme is to represent the laws of motion by equations, and to obtain solutions of the equations referring to observed conditions. The dominating idea in this application of mathematics to physics is that the equations representing the laws of motion should be of a simple form.
The whole success of the scheme is due to the fact that equations of simple form do seem to work. The physicist is thus provided with a principle of simplicity, which he can use as an instrument of research. If he obtains, from some rough experiments, data which fit in roughly with certain simple equations, he infers that if he performed the experiments more accurately he would obtain data fitting in more accurately with the equations. The method is much restricted, however, since the principle of simplicity applies only to fundamental laws of motion, not to natural phenomena in general.
For example, rough experiments about the relation between the pressure and volume of a gas at a fixed temperature give results fitting in with a law of inverse proportionality, but it would be wrong to infer that more accurate experiments would confirm this law with greater accuracy, as one is here dealing with a phenomenon which is not connected in any very direct way with the fundamental laws of motion. The discovery of the theory of relativity made it necessary to modify the principle of simplicity.
Presumably one of the fundamental laws of motion is the law of gravitation which, according to Newton, is represented by a very simple equation, but, according to Einstein, needs the development of an elaborate technique before its equation can even be written down. It is true that, from the standpoint of higher mathematics, one can give reasons in favour of the view that Einstein's law of gravitation is actually simpler than Newton's, but this involves assigning a rather subtle meaning to simplicity, which largely spoils the practical value of the principle of simplicity as an instrument of research into the foundations of physics.
What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature.
The restricted theory changed our ideas of space and time in a way that may be summarised by stating that the group of transformations to which the space-time continuum is subject must be changed from the Galilean group to the Lorentz group. The latter group is a much more beautiful thing than the former - in fact, the former would be called mathematically a degenerate special case of the latter.
The general theory of relativity involved another step of a rather similar character, although the increase in beauty this time is usually considered to be not quite so great as with the restricted theory, which results in the general theory being not quite so firmly believed in as the restricted theory.
We now see that we have to change the principle of simplicity into a principle of mathematical beauty. The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty.
He should still take simplicity into consideration in a subordinate way to beauty. For example Einstein, in choosing a law of gravitation, took the simplest one compatible with his space-time continuum, and was successful.
It often happens that the requirements of simplicity and of beauty are the same, but where they clash the latter must take precedence. Let us pass on to the second revolution in physical thought of the present century - the quantum theory.
Relationship between Mathematics and Physics – Life Through A Mathematician's Eyes
This is a theory of atomic phenomena based on a mechanics of an essentially different type from Newton's. The difference may be expressed concisely, but in a rather abstract way, by saying that dynamical variables in quantum mechanics are subject to an algebra in which the commutative axiom of multiplication does not hold.
Apart from this, there is an extremely close formal analogy between quantum mechanics and the old mechanics. In fact, it is remarkable how adaptable the old mechanics is to the generalization of non-commutative algebra.
Relationship between mathematics and physics
All the elegant features of the old mechanics can be carried over to the new mechanics, where they reappear with an enhanced beauty. Quantum mechanics requires the introduction into physical theory of a vast new domain of pure mathematics - the whole domain connected with non-commutative multiplication.
This, coming on top of the introduction of new geometries by the theory of relativity, indicates a trend which we may expect to continue. We may expect that in the future further big domains of pure mathematics will have to be brought in to deal with the advances in fundamental physics. Pure mathematics and physics are becoming ever more closely connected, though their methods remain different.
One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.
It is difficult to predict what the result of all this will be. Possibly, the two subjects will ultimately unify, every branch of pure mathematics then having its physical application, its importance in physics being proportional to its interest in mathematics. At present we are, of course, very far from this stage, even with regard to some of the most elementary questions. For example, only four-dimensional space is of importance in physics, while spaces with other numbers of dimensions are of about equal interest in mathematics.
It may well be, however, that this discrepancy is due to the incompleteness of present-day knowledge, and that future developments will show four-dimensional space to be of far greater mathematical interest than all the others. The trend of mathematics and physics towards unification provides the physicist with a powerful new method of research into the foundations of his subject, a method which has not yet been applied successfully, but which I feel confident will prove its value in the future.
The method is to begin by choosing that branch of mathematics which one thinks will form the basis of the new theory. So he wrote and thought like a physicist, not a mathematician, and there is a difference. Similarly, Feynman knew a tremendous amount of mathematics. But he was first and foremost a physicist and wrote and thought like a physicist. So what is the difference between mathematicians and physicists who are very competent with mathematics?
Theoretical physicists such as Einstein, Dirac and Feynman used mathematics as a symbolic language with great power for analyzing physical problems. Much of our mathematical ideas, probably most of them, grew out of the study of physical problems. However, when a mathematician begins to work with such ideas a process of logical refinement begins. Every concept must be rigorously defined. Logical relation to other mathematical ideas must be examined. Theorems relating to the concept must be formulated very precisely and proved.
Everything is done in a precise logical fashion. This can be very frustrating for a mathematics student reading papers written by physicists. I remember, as a graduate student at Carnegie Institute of Technology, many discussions with other grad students about this point.
- There was a problem providing the content you requested
- The unreasonable relationship between mathematics and physics
I still thought more like a physicist at that time. Some exposure to the requirement of rigor in mathematics is found at the undergraduate level, but as graduate students we found ourselves at another level.
We reached a point where less secure students were afraid to make any mathematical statements, for fear someone would see a logical error and announce it to the world. In studying mathematics I have trouble figuring out what it is that I am supposed to not know. In mod 4 arithmetic, two plus two equals zero, not four.
Of course, a physicist would have little hope of solving many physical problems if they spent too much time worrying about such things. Their purpose when using mathematics is to get some useful and meaningful physical conclusions.
So it is usually assumed that functions are continuous and differentiable as needed, well behaved, and all needed integrals exist. Many steps in their reasoning will always be questionable when viewed by a pure mathematician.
This is true even when the physicist is Einstein or his equivalent. This can be a problem when a mathematician attempts to read a paper written by an astrophysicist or a nuclear physicist for example.
Enrico Fermi once taught a course in nuclear physics at the University of Chicago. One of his students took careful notes, and published them, in a little paper back book I have on my shelf. The notes look very mathematical, but they are not very readable to a mathematician who tries to read them as mathematics.
The papers written by Dirac are very elegant and mathematical in appearance but they are not mathematics. Similarly, when a physicist reads a paper written by a mathematician he or she can become irritated by the time and space spent on logical fine points and proofs.
The Relation between Mathematics and Physics | Benjamin Plybon - fim-mdu.info
Physicists today have reasons for being interested in group theory, but reading a paper written by a group theory specialist is not easy for other mathematicians with a different specialty, much less a physicist.
When such a mathematician publishes a paper on application of group theory to a problem in quantum mechanics the paper is usually written in good mathematical style, and probably unreadable for most physicists. This situation has become very troublesome in current papers on relativity, for example. Einstein formulated his theory using Riemannian geometry and Tensor analysis. His discussions are not logically rigorous, in the sense mathematicians use the term.
He used mathematics in a purely computational sense combined with purely physical thinking. He began his discussion with a survey of some facts about tensor analysis, since most physicists at that time knew little or nothing about tensor analysis.
He explained the physical significance of the metric tensor and the Riemann curvature tensor. No attempt was made at a rigorous mathematical treatment of the subject. But the results were very important for physics. At about that time a very young Wolfgang Pauli wrote a beautiful introduction to relativity as an encyclopedia article. His work is available today in a small paperback book that should be read by all students of relativity.
Even this is a work written for physicists, not mathematicians. After mathematicians became interested in relativity an interesting change came about. Einstein has been quoted as saying, that he could no longer understand his theory after the mathematicians got hold of it.
Of course all that happened is that mathematicians began to examine his work from the viewpoint of a mathematician. Experts in differential geometry, non Euclidean geometries, and particularly in Riemannian manifolds began to connect his ideas with their knowledge of geometry. Even the concept of Minkowski space, which is fundamental to special relativity, was introduced by the mathematician, H. This was only the beginning. Today papers on Relativity are written in the context of Topology and Differentiable Manifold Theory.
The results are interesting but far removed from the physical theory introduced by Einstein.