"It has become increasingly evident in recent times, however, that nature works on a different plan. Her fundamental laws do not govern the world as it appears in our mental picture in any very direct way, but instead they control a substratum of which we cannot form a mental picture without introducing irrelevancies." (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)
"Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field." (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)
"When an observation is made on any atomic system that has been prepared in a given way and is thus in a given state, the result will not in general be determinate, i.e. if the experiment is repeated several times under identical conditions several different results may be obtained. If the experiment is repeated a large number of times it will be found that each particular result will be obtained a definite fraction of the total number of times, so that one can say there is a definite probability of its being obtained any time that the experiment is performed. This probability the theory enables one to calculate." (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)
"It seems likely that this process of increasing abstraction will continue in the future and that advance in physics is to be associated with a continual modification and generalization of the axioms at the base of the mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)
"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalization of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)
"From our theoretical picture, we should expect an ordinary electron, with positive energy, to be able to drop into a hole and fill up this hole, the energy being liberated in the form of electromagnetic radiation. This would mean a process in which an electron and a positron annihilate one another. The
converse process, namely the creation of an electron and a positron from electromagnetic radiation, should also be able to take place. Such processes appear to have been found experimentally, and are at present being more closely investigated by experimenters." (Paul A M Dirac, "Theory of Electrons and Positrons", [Nobel lecture] 1933)
"I should like here to discuss the simpler kinds of particles and to consider what can be inferred about them from purely theoretical arguments. The simpler kinds of particle are: (i) the photons or light-quanta, of which light is composed; (ii) the electrons, and the recently discovered positrons (which appear to be a sort of mirror image of the electrons, differing from them only in the sign of their electric charge) ; (iii) the heavier particles - protons and neutrons." (Paul A M Dirac, "Theory of Electrons and Positrons", [Nobel lecture] 1933)
"If we accept the view of complete symmetry between positive and negative electric charge so far as concerns the fundamental laws of Nature, we must regard it rather as an accident that the Earth (and presumably the whole solar system), contains a preponderance of negative electrons and positive protons. It is quite possible that for some of the stars it is the other way about, these stars being built up mainly of positrons and negative protons. In fact, there may be half the stars of each kind. The two kinds of stars would both show exactly the same spectra, and there would be no way of distinguishing them by present astronomical methods." (Paul A M Dirac, "Theory of Electrons and Positrons", [Nobel lecture] 1933)
"Matter has been found by experimental physicists to be made up of small particles of various kinds, the particles of each kind being all exactly alike. Some of these kinds have definitely been shown to be composite, that is, to be composed of other particles of a simpler nature. But there are other kinds which have not been shown to be composite and which one expects will never be shown to be composite, so that one considers them as elementary and fundamental." (Paul A M Dirac, "Theory of Electrons and Positrons", [Nobel lecture] 1933)
"Now in practice the kinetic energy of a particle is always positive. We thus see that our equations allow of two kinds of motion for an electron, only one of which corresponds to what we are familiar with. The other corresponds to electrons with a very peculiar motion such that the faster they move, the less energy they have, and one must put energy into them to bring them to rest." (Paul A M Dirac, "Theory of Electrons and Positrons", [Nobel lecture] 1933)
"Of these, I shall deal almost entirely with the electrons and the positrons- not because they are the most interesting ones, but because in their case the theory has been developed further. There is, in fact, hardly anything that can be inferred theoretically about the properties of the others. The photons, on the one hand, are so simple that they can easily be fitted into any theoretical scheme, and the theory therefore does not put any restrictions on their properties. The protons and neutrons, on the other hand, seem to be too complicated and no reliable basis for a theory of them has yet been discovered." (Paul A M Dirac, "Theory of Electrons and Positrons", [Nobel lecture] 1933)
"To get an interpretation of some modern experimental results one must suppose that particles can be created and annihilated. Thus if a particle is observed to come out from another particle, one can no longer be sure that the latter is composite. The former may have been created. The distinction between elementary particles and composite particles now becomes a matter of convenience. This reason alone is sufficient to compel one to give up the attractive philosophical idea that all matter is made up of one kind, or perhaps two kinds of bricks." (Paul A M Dirac, "Theory of Electrons and Positrons", [Nobel lecture] 1933)
"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." (Paul A M Dirac, "The Relation Between Mathematics And Physics", Proceedings of the Royal Society of Edinburgh ", 1938-1939)
"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 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. [...] 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." (Paul A M Dirac, "The Relation Between Mathematics and Physics", Proceedings of the Royal Society of Edinburgh, 1938-1939)
"The main object of physical systems is not the provision of pictures, but the formulation of laws governing phenomena, and the application of these laws to the discovery of new phenomena. If a picture exists, so much the better; but whether a picture exists or not is a matter of only secondary importance." (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)
"One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful tool in its description. However, recent advances in physical science show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this, and one can get an appreciation of it only from a thorough examination of the various facts that make it up." (Paul A M Dirac, "The Relation Between Mathematics and Physics", Proceedings of the Royal Society Vol. LIX, 1939)
"The researcher 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. […] It often happens that the requirements of simplicity and beauty are the same, but where they clash the latter must take precedence." (Paul A M Dirac, "The Relation Between Mathematics and Physics", Proceedings of the Royal Society Vol. LIX, 1939)
"There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized. To do so physics will have to develop a long way to establish the details of how the correspondence is to be made. One hint for this development seems pretty obvious, namely, the study of whole numbers in modern mathematics is inextricably bound up with the theory of functions of a complex variable, which theory we have already seen has a good chance of forming the basis of the physics of the future. The working out of this idea would lead to a connection between atomic theory and cosmology." (Paul A M Dirac, [Lecture delivered on presentation of the James Scott prize] 1939)
"In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it’s the exact opposite." (Paul A M Dirac, [Bombay lecture] 1955)
"A physical law must possess mathematical beauty." (Paul A M Dirac, 1956)
"[…] it is more important to have beauty in one's equations that to have them fit experiment. […] It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress." (Paul A M Dirac, "The Evolution of the Physicist’s Picture of Nature", Scientific American, 1963)
"A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe." (Paul A M Dirac, "The Evolution of the Physicist’s Picture of Nature", Scientific American, 1963)
"It seems to be one of the fundamental features of nature the fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it." (Paul A M Dirac , "The Evolution of the Physicist’s Picture of Nature", Scientific American, 1963)
"Just by studying mathematics we can hope to make a guess at the kind of mathematics that will come into the physics of the future." (Paul A M Dirac , "The Evolution of the Physicist’s Picture of Nature ", Scientific American, 1963)
"I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it." (Paul A M Dirac [attributed])
References:
[1] Paul A M Dirac, "Theory of Electrons and Positrons", [Nobel lecture] 1933 [
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