A good deal of my research in physics has consisted in not setting out to solve some particular problem, but simply examining mathematical equations of a kind that physicists use and trying to fit them together in an interesting way, regardless of any application that the work may have. It is simply a search for pretty mathematics. It may turn out later to have an application. Then one has good luck. At age 78.

A great deal of my work is just playing with equations and seeing what they give.

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.
At age 36.

God used beautiful mathematics in creating the world.

Hopes are always accompanied by fears, and, in scientific research, the fears are liable to become dominant.
At age 67.

I admired Bohr very much. We had long talks together, long talks in which Bohr did practically all the talking.
Recalling his Sep 1926-Feb 1927 stay in Copenhagen.

I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.

I do not see how a man can work on the frontiers of physics and write poetry at the same time. They are in opposition. In science you want to say something that nobody knew before, in words which everyone can understand. In poetry you are bound to say ... something that everyone knows already in words that nobody can understand.
Commenting to him about the poetry J. Robert Oppenheimer wrote.

I found the best ideas usually came, not when one was actively striving for them, but when one was in a more relaxed state… I used to take long solitary walks on Sundays, during which I tended to review the current situation in a leisurely way. Such occasions often proved fruitful, even though (or perhaps, because) the primary purpose of the walk was relaxation and not research.

I learnt to distrust all physical concepts as the basis for a theory. Instead one should put one's trust in a mathematical scheme, even if the scheme does not appear at first sight to be connected with physics. One should concentrate on getting interesting mathematics.

I think it is a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don’t have any physical meaning at all. Sometimes they do.
At age 60.

I think it is the general rule that the originator of a new idea is not the most suitable person to develop it, because his fears of something going wrong are really too strong…
At age 69.

It is more important to have beauty in one's equations than 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. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further developments of the theory.

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 ... If someone can hit on the right lines along which to make this development, it m may lead to a future advance in which people will first discover the equations and then, after examining them, gradually learn how to apply the ... My own belief is that this is a more likely line of progress than trying to guess at physical pictures.

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.

People were pretty well spellbound by what Bohr said… While I was very much impressed by [him], his arguments were mainly of a qualitative nature, and I was not able to really pinpoint the facts behind them. What I wanted was statements which could be expressed in terms of equations, and Bohr's work very seldom provided such statements. I am really not sure how much later my work was influenced by these lectures of Bohr's... He certainly did not have a direct influence because he did not stimulate one to think of new equations.
Recalling the occasion in May 1925 (a year before receiving his Ph.D.) when he met Niels Bohr who was in Cambridge to give a talk on the fundamental difficulties of the quantum theory.

Pick a flower on Earth and you move the farthest star.

The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.

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.

The methods of theoretical physics should be applicable to all those branches of thought in which the essential features are expressible with numbers.

The only object of theoretical physics is to calculate results that can be compared with experiment... it is quite unnecessary that any satisfactory description of the whole course of the phenomena should be given.

The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. He should 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.

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 generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation.

Theoretical physicists accept the need for mathematical beauty as an act of faith... For example, the main reason why the theory of relativity is so universally accepted is its mathematical beauty.

There are, at present, fundamental problems in theoretical physics … the solution of which … will presumably require a more drastic revision of our fundmental concepts than any that have gone before. Quite likely, these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker in the future will, therefore, have to proceed in a more direct way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities.
At age 28.

Well, in the first place, it leads to great anxiety as to whether it's going to be correct or not … I expect that's the dominating feeling. It gets to be rather a fever…
At age 60, when asked about his feelings on discovering the Dirac equation.

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. (1930)

When we make the photon meet a tourmaline crystal, we are subjecting it to an observation. We are observing whether it is polarised parallel or perpendicular to the optic axis. The effect of making the observation is to force the photon entirely into the state of perpendicular polarisation. It has to make a sudden jump from being partly in each of these two states to being entirely in one or other of them. Which of the two states it will jump into cannot be predicted, but is governed only by probability laws. If it jumps into the perpendicular state it passes through the crystal and appears on the other side preserving this state of polarisation.

When [Erwin Schrödinger] went to the Solvay conferences in Brussels, he would walk from the station to the hotel where the delegates stayed, carrying all his luggage in a rucksack and looking so like a tramp that it needed a great deal of argument at the reception desk before he could claim a room.

Dirac politely refused Robert's [Robert Oppenheimer] two proffered books: reading books, the Cambridge theoretician announced gravely, 'interfered with thought'.

If this is what the McCarran Act means in practice, it seems to us a form of organized cultural suicide.
In a letter co-signed with his Princeton University physics professor colleagues, Walker Bleakney and Milton G. White, protesting that Nobel Prize-winning, Cambridge professor, Dirac having been invited for a year's visit to Princeton, had been denied a visa by the U.S. State Department under section 212A of the Immigration and Naturalization Act (McCarran Act). Quoting a report in Physics Today, this regulation includes 'categories of undesireables ranging from vagrants to stowaways.' The real reason remains unclear, but was perhaps related to Dirac's prior science-related visits to Russia. Robert Oppenheimer's security clearance had recently been revoked, and this was the era of McCarthy's rabid anti-Communism hearings.

But, contrary to the lady's prejudices about the engineering profession, the fact is that quite some time ago the tables were turned between theory and applications in the physical sciences. Since World War II the discoveries that have changed the world are not made so much in lofty halls of theoretical physics as in the less-noticed labs of engineering and experimental physics. The roles of pure and applied science have been reversed; they are no longer what they were in the golden age of physics, in the age of Einstein, Schrödinger, Fermi and Dirac.