The Intimate Relation between Mathematics and Physics | badz.info
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. Physics and mathematics present two areas of intellectual activity deeply interwoven through the long history of science. Yet, they preserve two separate. Physics, as a science, has to be in accordance with the scientific method. Scientific laws are empirical - they must be in agreement with experimental evidence).
I first discovered him a while ago, after a good friend has recommended one of his books. Kaku has written several books about physics and related topics, has made frequent appearances on radio, television, and film, and writes online blogs and articles.
The book I have read is Physics of the Impossible Since then he has published two more books: 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.
The coevolution of physics and math | symmetry magazine
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. 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.
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.Relationship between mathematics and physics
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. 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.
Relationship between Mathematics and Physics – Life Through A Mathematician's Eyes
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 unreasonable relationship between mathematics and physics
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. One should be influenced very much in this choice by considerations of mathematical beauty.
It would probably be a good thing also to give a preference to those branches of mathematics that have an interesting group of transformations underlying them, since transformations play an important role in modern physical theory, both relativity and quantum theory seeming to show that transformations are of more fundamental importance than equations. Having decided on the branch of mathematics, one should proceed to develop it along suitable lines, at the same time looking for that way in which it appears to lend itself naturally to physical interpretation.
This method was used by Jordan in an attempt to get an improved quantum theory on the basis of an algebra with non-associative multiplication. The attempt was not successful, as one would rather expect, if one considers that non-associative algebra is not a specially beautiful branch of mathematics, and is not connected with an interesting transformation theory.
I would suggest, as a more hopeful-looking idea for getting an improved quantum theory, that one take as basis the theory of functions of a complex variable. This branch of mathematics is of exceptional beauty, and further, the group of transformations in the complex plane, is the same as the Lorentz group governing the space-time of restricted relativity. One is thus led to suspect the existence of some deep-lying connection between the theory of functions of a complex variable and the space-time of restricted relativity, the working out of which will be a difficult task for the future.
Let us now discuss the extent of the mathematical quality in Nature. According to the mechanistic scheme of physics or to its relativistic modification, one needs for the complete description of the universe not merely a complete system of equations of motion, but also a complete set of initial conditions, and it is only to the former of these that mathematical theories apply.
The latter are considered to be not amenable to theoretical treatment and to be determinable only from observation. The enormous complexity of the universe is ascribed to an enormous complexity in the initial conditions, which removes them beyond the range of mathematical discussion.
I find this position very unsatisfactory philosophically, as it goes against all ideas of the unity of Nature. Anyhow, if it is only to a part of the description of the universe that mathematical theory applies, this part ought certainly to be sharply distinguished from the remainder. But in fact there does not seem to be any natural place in which to draw the line. Are such things as the properties of the elementary particles of physics, their masses and the numerical coefficients occurring in their laws of force, subject to mathematical theory?
According to the narrow mechanistic view, they should be counted as initial conditions and outside mathematical theory. However, since the elementary particles all belong to one or other of a number of definite types, the members of one type being all exactly similar, they must be governed by mathematical law to some extent, and most physicists now consider it to be quite a large extent.
For example, Eddington has been building up a theory to account for the masses.
But even if one supposed all the properties of the elementary particles to be determinable by theory, one would still not know where to draw the line, as one would be faced by the next question - Are the relative abundances of the various chemical elements determinable by theory?
One would pass gradually from atomic to astronomic questions. This unsatisfactory situation gets changed for the worse by the new quantum mechanics.
In spite of the great analogy between quantum mechanics and the older mechanics with regard to their mathematical formalisms, they differ drastically with regard to the nature of their physical consequences. According to the older mechanics, the result of any observation is determinate and can be calculated theoretically from given initial conditions; but with quantum mechanics there is usually an indeterminacy in the result of an observation, connected with the possibility of occurrence of a quantum jump, and the most that can be calculated theoretically is the probability of any particular result being obtained.
The question, which particular result will be obtained in some particular case, lies outside the theory. This must not be attributed to an incompleteness of the theory, but is essential for the application of a formalism of the kind used by quantum mechanics. Thus according to quantum mechanics we need, for a complete description of the universe, not only the laws of motion and the initial conditions, but also information about which quantum jump occurs in each case when a quantum jump does occur.
The latter information must be included, together with the initial conditions, in that part of the description of the universe outside mathematical theory. The increase thus arising in the non-mathematical part of the description of the universe provides a philosophical objection to quantum mechanics, and is, I believe, the underlying reason why some physicists still find it difficult to accept this mechanics. Quantum mechanics should not be abandoned, however, firstly, because of its very widespread and detailed agreement with experiment, and secondly, because the indeterminacy it introduces into the results of observations is of a kind which is philosophically satisfying, being readily ascribable to an inescapable crudeness in the means of observation available for small-scale experiments.
The objection does show, all the same, that the foundations of physics are still far from their final form. We come now to the third great development of physical science of the present century - the new cosmology.