A Quote by Leopold Kronecker
All results of the profoundest mathematical investigation must ultimately be expressible in the simple form of properties of the integers.
Quote Topics
Quote Author
Related Quotes
Unlike physics, for example, such parts of the bare bones of economic theory as are expressible in mathematical form are extremely easy compared with the economic interpretation of the complex and incompletely known facts of experience, and lead one a very little way towards establishing useful results.
[On Archimedes mathematical results:] It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanation... No investigation of yours would succeed in attaining the proof, and yet, once seen you immediately believe you would have discovered it.
The properties of executability and universality associated with programming languages can be combined, in a single language, with the well-known properties of mathematical notation which make it such an effective tool of thought.
We shall see that the mathematical treatment of the subject [of electricity] has been greatly developed by writers who express themselves in terms of the 'Two Fluids' theory. Their results, however, have been deduced entirely from data which can be proved by experiment, and which must therefore be true, whether we adopt the theory of two fluids or not. The experimental verification of the mathematical results therefore is no evidence for or against the peculiar doctrines of this theory.
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.
Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.
Nature does not count nor do integers occur in nature. Man made them all, integers and all the rest, Kronecker to the contrary notwithstanding.
Research, as the college student will come to know it, is relatively thorough investigation, primarily in libraries, of a properly limited topic, and presentation of the results of this investigation in a carefully organized and documented paper of some length.
The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.
To treat programming scientifically, it must be possible to specify the required properties of programs precisely. Formality is certainly not an end in itself. The importance of formal specifications must ultimately rest in their utility -in whether or not they are used to improve the quality of software or to reduce the cost of producing and maintaining software.
I hold that space cannot be curved, for the simple reason that it can have no properties. It might as well be said that God has properties. He has not, but only attributes and these are of our own making. Of properties we can only speak when dealing with matter filling the space. To say that in the presence of large bodies space becomes curved is equivalent to stating that something can act upon nothing. I, for one, refuse to subscribe to such a view.
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.
No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof.
Perspective is an Art Mathematical which demonstrates the manner and properties of all radiations direct, broken and reflected.
The properties of people and the properties of character have almost nothing to do with each other. They really don't. I know it seems like they do because we look alike, but people don't speak in dialogue. Their lives don't unfold in a series of scenes that form a narrative arc.
Space is a laboratory, an experiment in all forms of all things, an infinity of possibilities, properties, and places that cry out for investigation and exploration.
Related Authors
Ada Lovelace Quotes
English
Mathematician
Alfred North Whitehead Quotes
English
Mathematician
Benoit Mandelbrot Quotes
French
Mathematician
Carl Friedrich Gauss Quotes
German
Mathematician
Charles Babbage Quotes
English
Mathematician
David Hilbert Quotes
German
Mathematician
G. H. Hardy Quotes
British
Mathematician
George Polya Quotes
Hungarian
Mathematician
Hannah Fry Quotes
English
Mathematician
Henri Poincare Quotes
French
Mathematician
Isaac Barrow Quotes
English
Mathematician
Isaac Newton Quotes
English
Mathematician
Norbert Wiener Quotes
American
Mathematician
Pythagoras Quotes
Greek
Mathematician
Rene Descartes Quotes
French
Mathematician
Seymour Papert Quotes
American
Mathematician