A Quote by Carl Friedrich Gauss
I have the vagary of taking a lively interest in mathematical subjects only where I may anticipate ingenious association of ideas and results recommending themselves by elegance or generality.
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Conventions of generality and mathematical elegance may be just as much barriers to the attainment and diffusion of knowledge as may contentment with particularity and literary vagueness... It may well be that the slovenly and literary borderland between economics and sociology will be the most fruitful building ground during the years to come and that mathematical economics will remain too flawless in its perfection to be very fruitful.
Mathematicians themselves set up standards of generality and elegance in their exposition which are a bar to understand.
It is important to notice that these badly functioning designs were praised for 'elegance.' But elegance as theoretical scientists apply it is quite different. The elegance of a mathematical formula is that it explains a phenomenon beautifully, with no parts left over. In design, elegance is more readily perceived as a property of product than of process. If we had more elegant theories, we might look to design for more than elegance.
The most ingenious men continually pretend to condemn tricking--but this is often done that they may use it more conveniently themselves, when some great occasion or interest offers itself to them.
When I think of the most able students I have encountered in my teaching - I mean those who have distinguished themselves not only by skill but by independence of thought - then I must confess that all have had a lively interest in epistemology.
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.
Can the difficulty of an exam be measured by how many bits of information a student would need to pass it? This may not be so absurd in the encyclopedic subjects but in mathematics it doesn't make any sense since things follow from each other and, in principle, whoever knows the bases knows everything. All of the results of a mathematical theorem are in the axioms of mathematics in embryonic form, aren't they?
The traditional mathematician recognizes and appreciates mathematical elegance when he sees it. I propose to go one step further, and to consider elegance an essential ingredient of mathematics: if it is clumsy, it is not mathematics.
Mathematics may be the only exception in the sciences that leaves no room for skepicism. But, if mathematical results are exact as no empirical law can ever be, philosophers have discovered that they are not absolutely novel - instead, they are tautological.
Self-confidenc e, poise, consciousness of possessing the power to accomplish our desires, with renewed lively interest in life are the natural results of the practice of Contrology [Pilates].
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.
Mathematical demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick are the only truths that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth according as their Subjects are more or less capable of Mathematical Demonstration.
Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates. Indeed, the beauty and elegance of the physical laws themselves are only apparent when expressed in the appropriate mathematical framework.
If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.
I do not believe that it can be too often repeated that the freedoms of speech, press, petition and assembly guaranteed by the First Amendment must be accorded to the ideas we hate or sooner or later they will be denied to the ideas we cherish. The first banning of an association because it advocates hated ideas - whether that association be called a political party or not - marks a fateful moment in the history of a free country.
The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.
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