Trivial in Mathematics

“[…] the mathematician learns early to accept no fact, to believe no statement, however apparently reasonable or obvious or trivial, until it has been proved, rigorously and totally by a series of steps proceeding from universally accepted first principles.” (Alfred Adler)

"No one should pick a problem, or make a resolution, unless he realizes that the ultimate value of it will offset the inevitable discomfort and trouble that always goes along with the accomplishment of anything worthwhile. So, let us not waste our time and effort on some trivial thing." (Charles F Kettering)

“Where the line is to be drawn the important and the trivial cannot be settled by a formula.” (Benjamin N Cardozo)

“Nowhere is intellectual beauty so deeply felt and fastidiously appreciated in its various grades and qualities as in mathematics, and only the informal appreciation of mathematical value can distinguish what is mathematics from a welter of formally similar, yet altogether trivial statements and operations.” (Michael Polanyi, “Personal Knowledge”, 1962)

“One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful tool in its description. However, recent advances in physical science show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this, and one can get an appreciation of it only from a thorough examination of the various facts that make it up.” (Paul A M Dirac)

“Mathematics is good if it enriches the subject, if it opens up new vistas, if it solves old problems, if it fills gaps, fitting snugly and satisfyingly into what is already known, or if it forges new links between previously unconnected parts of the subject It is bad if it is trivial, overelaborate, or lacks any definable mathematical purpose or direction It is pure if its methods are pure - that is, if it doesn't cheat and tackle one problem while pretending to tackle another, and if there are no gaping holes in its logic It is applied if it leads to useful insights outside mathematics By these criteria, today's mathematics contains as high a proportion of good work as at any other period, and as any other area, and much of it manages to be both pure and applied at the same time.” (Ian Stewart, “The Problems of Mathematics”, 1987)

“We decided that ‘trivial’ means ‘proved’. So, we joked with the mathematicians: “We have a new theorem - that mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial.” (Richard P Feynman, “Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character”, 1985)

“The difference between mathematicians and physicists is that after physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is trivial.” (Lucien Szpiro)

"Mathematics is trivial, but I can’t do my work without it." (Richard Feynman)

"Everything is trivial when you know the proof." (David V Widder)

Science upon Mind

“To a scientist a theory is something to be tested. He seeks not to defend his beliefs, but to improve them. He is, above everything else, an expert at ‘changing his mind’.” (Wendell Johnson, 1946)

"It is the mark of an educated mind to rest satisfied with the degree of precision which the nature of the subject admits and not to seek exactness where only an approximation is possible." (Aristotle)

“The sole aim of science is the honor of the human mind, and from this point of view a question about numbers is as important as a question about the system of the world.” (Carl G J Jacobi)

"The truth of a theory is in your mind, not in your eyes." (Albert Einstein)

“It stands to the everlasting credit of science that by acting on the human mind it has overcome man's insecurity before himself and before nature.” (Albert Einstein)

"The man of science will acts as if this world were an absolute whole controlled by laws independent of his own thoughts or act; but whenever he discovers a law of striking simplicity or one of sweeping universality or one which points to a perfect harmony in the cosmos, he will be wise to wonder what role his mind has played in the discovery, and whether the beautiful image he sees in the pool of eternity reveals the nature of this eternity, or is but a reflection of his own mind." (Tobias Dantzig)

“It has never yet been supposed, that all the facts of nature, and all the means of acquiring precision in the computation and analysis of those facts, and all the connections of objects with each other, and all the possible combinations of ideas, can be exhausted by the human mind.” (Nicolas de Condorcet, “Outlines Of An Historical View Of The Progress Of The Human Mind”, 1795)

"Nothing has such power to broaden the mind as the ability to investigate systematically and truly all that comes under thy observation in life." (Marcus Aurelius)

On Mind: Mathematics upon Mind

"This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth." (Proclus Lycaeus, cca 5th century)

 “If a man’s wit be wandering, let him study mathematics; for in demonstrations, if his wit be called away never so little, he must begin again.” (Francis Bacon)

"Mathematical knowledge adds vigour to the mind, frees it from prejudice, credulity, and superstition. This it does in two ways: 1st, by accustoming us to examine, and not to take thigs upon trust. 2nd By giving us a clear and extensive knowledge of the system of the world […]." (Dr. John Arbuthnot, “Usefulness of Mathematical Learning”, 1745)

“Mathematics is the science that yields the best opportunity to observe the working of the mind. Its study is the best training of our abilities as it develops both the power and the precision of our thinking. Mathematics is valuable on account of the number and variety of its applications. And it is equally valuable in another respect: By cultivating it, we acquire the habit of a method of reasoning which can be applied afterwards to the study of any subject and can guide us in life's great and little problems.” (Marquis de Condorcet)

"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." (Richard Courant & Herbert Robbins, “What Is Mathematics?”, 1941)

"For me mathematics cultivates a perpetual state of wonder about the nature of mind, the limits of thoughts, and our place in this vast cosmos." (Clifford A Pickover, “The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics”, 2009)

"I shall not attempt to prove that mathematics is useful. I will admit it and so save myself the trouble that here is a great and respected discipline where all is impossible yet much is useful. The usefulness largely flows from the impossibility. Mathematical concepts have been simplified and generalized until they describe an imaginative world no part of which could possibly exist outside men’s minds." (Billy E Goetz, “The Usefulness of the Impossible”, 1963)

“Everything that the greatest minds of all times have accomplished toward the comprehension of forms by means of concepts is gathered into one great science, mathematics.” (Johann F Herbart)

"Mathematics is a model of exact reasoning, an absorbing challenge to the mind, an esthetic experience for creators and some students, a nightmarish experience to other students, and an outlet for the egotistic display of mental power." (Morris Kline, “Mathematics and the Physical World”, 1959)

“Mathematical inquiry lifts the human mind into closer proximity with the divine than is attainable through any other medium.” (Hermann Weyl)

"Mathematics is a spirit of rationality. It is this spirit that challenges, simulates, invigorates and drives human minds to exercise themselves to the fullest. It is this spirit that seeks to influence decisively the physical, normal and social life of man, that seeks to answer the problems posed by our very existence, that strives to understand and control nature and that exerts itself to explore and establish the deepest and utmost implications of knowledge already obtained." (Morris Kline)