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"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."
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Leonhard Euler"All the greatest mathematicians have long since recognized that the method presented in this book is not only extremely useful in analysis, but that it also contributes greatly to the solution of physical problems. For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves. Now there exist on every hand such notable instances of this fact, that, in order to prove its truth, we have no need at all of a number of examples; nay rather ones task should be this, namely, in any field of Natural Science whatsoever to study that quantity which takes on a maximum or a minimum value, an occupation that seems to belong to philosophy rather than to mathematics. Since, therefore, two methods of studying effects in Nature lie open to us, one by means of effective causes, which is commonly called the direct method, the other by means of final causes, the mathematician uses each with equal success. Of course, when the effective causes are too obscure, but the final causes are more readily ascertained, the problem is commonly solved by the indirect method; on the contrary, however, the direct method is employed whenever it is possible to determine the effect from the effective causes. But one ought to make a special effort to see that both ways of approach to the solution of the problem be laid open; for thus not only is one solution greatly strengthened by the other, but, more than that, from the agreement between the two solutions we secure the very highest satisfaction."
Leonhard Euler was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, music theorist and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical termin
"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."
"La construction dune machine propre à exprimer tous les sons de nos paroles , avec toutes les articulations , seroit sans-doute une découverte bien importante. … La chose ne me paroît pas impossible."
"Now I will have less distraction."
"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities."
"To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be."
"Madam, I have come from a country where people are hanged if they talk."
"O dread and silent mount! I gazed upon thee, Till thou, still present to the bodily sense, Didst vanish from my thought: entranced in prayer, I worshipped the Invisible alone."
"Mister Toombs was willing to dissolve the Union to save slavery, Mister Phillips, to save liberty; while Mister Seward, denounced and derided by both, declared that the deepest instinct of the American people was for union. Reserved rights. State rights, limited powers, the advantages of union and disunion, were the cucumbers from which we were busily engaged in distilling light, overlooking the fact of nationality in discussing the conditions of union. We were speculating upon costume. We gravely proved that the clothes were the clothes of a woman, or of a child, without seeing that whatever the clothes might be there was a full-grown man inside of them. "The Constitution is a contract between sovereign States", shouted Mister Toombs, "let Georgia tear it and separate". "The Constitution is a league with hell", calmly replied Mister Phillips, "let New York cut off New Orleans to rot alone". "Oh, dear! it"s a dreadful dilemma", whimpered President Buchanan. "States have no right to secede, and the United States have no right to coerce. Oh, dear me! it"s perfectly awful! I"m the most patriotic of men, but what shall I do? what shall I do?" Separate! Cut off! Secede! It was of a living body they spoke, which, pierced anywhere, quivered everywhere."
"Its never a matter of ethics versus pragmatism; its a question of which informs the other. Humans have shown themselves capable of almost unlimited imagination and innovation—qualities that could be said to define human beings. People have used this capacity to do both great good and great harm. The point is that when humans set their minds to doing something, its frequently possible. It makes sense to first ask what people want to do and why, from an ethical standpoint, and then get to the pragmatic how-to questions. The very process of asking whats right is how people fill out ethics in praxis, to meet new demands and dilemmas, new social conditions and contexts. Anarchism, then, brings an egalitarian ethics out into the world, making it transparent, public, and shared. It maintains an ethical orientation, while continually trying to put such notions into practice, as flawed as the effort might be. When other people come into contact with this ethical compass, they will hopefully "get it" and incorporate the same values into their lives, because it works. It offers directionality to political involvement and buttresses peoples efforts to remake society. It turns surviving into thriving. Thats the crucial difference between a pragmatic versus ethical impulse: people, in cooperative concert, qualitatively transform one anothers lives."
"Thats... the big discovery of this principle of computational equivalence of mine. ...This is something which is kind of a follow-on to Gödels theorem, to Turings work on the ... that there is this fundamental limitation built into science, this idea of computational irreducibility that says that even though you may know the rules by which something operates, that does not mean that you can readily... be smarter that it and jump ahead and figure out what its going to do."
"The art of leading, in operations large or small, is the art of dealing with humanity, of working diligently on behalf of men, of being sympathetic with them, but equally, of insisting that they make a square facing toward their own problems."
"The establishment of "The Times" newspaper is an example, on a large scale, of a manufactory in which the division of labour, both mental and bodily, is admirably illustrated, and in which also the effect of domestic economy is well exemplified. It is scarcely imagined, by the thousands who read that paper in various quarters of the globe, what a scene of organized activity the factory presents during the whole night, or what a quantity of talent and mechanical skill is put in action for their amusement and information."