Hermann Weyl Famous Quotes
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Or, we may use Cartesian co-ordinate systems from the outset:
Without the concepts, methods and results found and developed by previous generations right down to Greek antiquity one cannot understand either the aims or achievements of mathematics in the last 50 years. [Said in 1950]
For mathematics, even to the logical forms in which it moves, is entirely dependent on the concept of natural number.
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.
Two possibilities present themselves for the analytical treatment of metrical geometry.
God exists because arithmetic is consistent - the Devil exists because we can't prove it!
With mathematics we stand precisely at that intersection of bondage and freedom that is the essence of the human itself.
No mathematician of equal stature has risen from our generation ... Hilbert was singularly free from national and racial prejudices; in all public questions, be they political, social or spiritual, he stood forever on the side of freedom.
The whole is always more, is more capable of a much greater variety of wave states, than the combination of its parts ... In this very radical sense, quantum physics supports the doctrine that the whole is more than the combination of its parts.
Numbers have neither substance, nor meaning, nor qualities. They are nothing but marks, and all that is in them we have put into them by the simple rule of straight succession.
By refraining from reducing multiplication to addition we are enabled through these axioms to banish continuity, which is so difficult to fix precisely, from the logical structure of geometry.
The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth.
The goal of mathematics is the symbolic comprehension of the infinite with human, that is finite, means.
Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory.
The objective world is, it does not happen. Only to the gaze of my consciousness, crawling along the lifeline of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time.
Besides language and music mathematics is one of the primary manifestations of the free creative power of the human mind.
We must here follow the first course so as to be able to pass on later to generalisations which extend beyond the limits of Euclidean geometry.
A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details.
The question for the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in which direction it will find its final solution nor even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization.
Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature.
the problem is then to develop a theory of invariance with respect to arbitrary linear transformations, in which, however, in contra-distinction to the case of affine geometry, we have a definite invariant quadratic form, viz. the metrical groundform once and for all as an absolute datum.
It is impossible to discuss realism in logic without drawing in the empirical sciences ... A truly realistic mathematics should be conceived, in line with physics, as a branch of the theoretical construction of the one real world and should adopt the same sober and cautious attitude toward hypothetic extensions of its foundation as is exhibited by physics.
The introduction of numbers as coordinates is an act of violence.
My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.
Our mathematics of the last few decades has wallowed in generalities and formalizations.
You can not apply mathematics as long as words still becloud reality.
Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.
