A Quote by Paul Dirac
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.
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You may object that by speaking of simplicity and beauty I am introducing aesthetic criteria of truth, and I frankly admit that I am strongly attracted by the simplicity and beauty of mathematical schemes which nature presents us. You must have felt this too: the almost frightening simplicity and wholeness of the relationship, which nature suddenly spreads out before us.
If nature leads us to mathematical forms of great simplicity and beauty - by forms I am referring to coherent systems of hypothesis, axioms, etc. - to forms that no one has previously encountered, we cannot help thinking that they are "true," that they reveal a genuine feature of nature... You must have felt this too: The almost frightening simplicity and wholeness of relationships which nature suddenly spreads out before us and for which none of us was in the least prepared.
Like music or art, mathematical equations can have a natural progression and logic that can evoke rare passions in a scientist. Although the lay public considers mathematical equations to be rather opaque, to a scientist an equation is very much like a movement in a larger symphony. Simplicity. Elegance. These are the qualities that have inspired some of the greatest artists to create their masterpieces, and they are precisely the same qualities that motivate scientists to search for the laws of nature. LIke a work of art or a haunting poem, equations have a beauty and rhythm all their own.
It is quite clear that beauty does depend on one's culture and upbringing for certain kinds of beauty, pictures, literature, poetry and so on...But mathematical beauty is of a rather different kind. I should say perhaps it is of a completely different kind and transcends these personal factors. It is the same in all countries and at all periods of time.
The mathematical is that evident aspect of things within which we are always already moving and according to which we experience them as things at all, and as such things. The mathematical is this fundamental position we take toward things by which we take up things as already given to us, and as they must and should be given. Therefore, the mathematical is the fundamental presupposition of the knowledge of things.
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.
For he who would proceed aright... should begin in youth to visit beautiful forms... out of that he should create fair thoughts; and soon he will of himself perceive that the beauty of one form is akin to the beauty of another, and that beauty in every form is one and the same.
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.
Not only in geometry, but to a still more astonishing degree in physics, has it become more and more evident that as soon as we have succeeded in unraveling fully the natural laws which govern reality, we find them to be expressible by mathematical relations of surprising simplicity and architectonic perfection. It seems to me to be one of the chief objects of mathematical instruction to develop the faculty of perceiving this simplicity and harmony.
When I speak of the beauty of a game of chess, then naturally this is subjective. Beauty can be found in a very technical, mathematical game for example. That is the beauty of clarity.
I now want to tell three stories about advances in twentieth-century physics. A curious fact emerges in these tales: time and again physicists have been guided by their sense of beauty not only in developing new theories but even in judging the validity of physical theories once they are developed. Simplicity is part of what I mean by beauty, but it is a simplicity of ideas, not simplicity of a mechanical sort that can be measured by counting equations or symbols.
It is cruel, you know, that music should be so beautiful. It has the beauty of loneliness of pain: of strength and freedom. The beauty of disappointment and never-satisfied love. The cruel beauty of nature and everlasting beauty of monotony.
Grandeur and beauty are so very opposite, that you often diminish the one as you increase the other. Variety is most akin to the latter, simplicity to the former.
Beauty of style and harmony and grace and good rhythm depend on simplicity - I mean the true simplicity of a rightly and nobly ordered mind and character, not that other simplicity which is only a euphemism for folly.
Color contributes to beauty, but it is not beauty. Color should have a minor part in the consideration of beauty, because it is not color but the structure that constitutes its essence.
Capablanca's play produced and still produces an irresistable artistic effect. In his games a tendency towards simplicity predominated, and in this simplicity there was a unique beauty of genuine depth.
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