numbers

"In Greek theoretical mathematics (as distinguished from practical or commercial arithmetic) a fraction that we would write as a/b was not regarded as a number, as a single entity, but as a relationship or a : b between the whole numbers a and b. Thus the ratio a : b was, in modern terms, simply an ordered pair, rather than a rational number."

"In past centuries it was widely accepted that an understanding of, as well as a facility with numbers, is an essential part of an education. ...This book has been written with an intention of showing that numbers have been the centre of man's awareness of his surroundings since well before any times of which we have surviving records. It will show that numbers have provided an answer to man's cultural needs at least since any form of organized human society came into being."

"It is fair to claim that it is a student's understanding of mathematics, above all other subjects, which suffers most from unenlightened teaching methods. ...the troubles may well stem mainly from the first year or two of the child's encounter with numbers... if children come to fear them or to be bored with them, they will eventually join the ranks of the present majority for whom the word 'mathematics' is guaranteed to bring social conversation to an immediate halt. If, on the other hand, numbers are made a genuine source of adventure and exploration from the beginning, there is a good chance that the level of numeracy in society can be raised significantly. There is a real role here for the history of mathematics—and the history of number in particular—for history emphasizes the diversity of approaches and methods which are possible and frees us from the straightjacket of contemporary fashions in mathematics education. It is, at the same time, both interesting and stimulating in its own right."

"I esteem myself happy to have as great an ally as you in my search for truth. I will read your work … all the more willingly because I have for many years been a partisan of the Copernican view because it reveals to me the causes of many natural phenomena that are entirely incomprehensible in the light of the generally accepted hypothesis. To refute the latter I have collected many proofs, but I do not publish them, because I am deterred by the fate of our teacher Copernicus who, although he had won immortal fame with a few, was ridiculed and condemned by countless people (for very great is the number of the stupid)."

"My purpose is to set forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning which the books written by philosophers are neither few nor small; nevertheless I have discovered by experiment some properties of it which are worth knowing and which have not hitherto been either observed or demonstrated. Some superficial observations have been made, as, for instance, that the free motion [naturalem motum] of a heavy falling body is continuously accelerated; but to just what extent this acceleration occurs has not yet been announced; for so far as I know, no one has yet pointed out that the distances traversed, during equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity."

"It was Pythagoras who discovered that the 5th and the octave of a note could be produced on the same string by stopping at 2⁄3 and ½ of its length respectively. Harmony therefore depends on a numerical proportion. It was this discovery, according to Hankel, which led Pythagoras to his philosophy of number. It is probable at least that the name harmonical proportion was due to it, since1:½ :: (1-½):(2⁄3-½)."

"Some fundamental unity was surely to be discerned either in the matter or the structure of things. The Ionic philosophers chose the former field: Pythagoras took the latter. ...The geometry which he had learnt in Egypt was merely practical. ...It was natural to nascent philosophy to draw, by false analogies, and the use of a brief and deceptive vocabulary, enormous conclusions from a very few observed facts: and it is not surprising if Pythagoras, having learnt in Egypt that number was essential to the exact description of forms and of the relations of forms, concluded that number was the cause of form and so of every other quality. Number, he inferred, is quantity and quantity is form and form is quality."

"Now any one who was in the habit of intently studying the heavens would naturally observe that each constellation has two characteristics, the number of the stars which compose it and the geometrical figure which they form. Here, as a recent writer [, Lea étapes de la philosophic mathématique] has remarked, we find, if not the origin, a striking illustration of the Pythagorean doctrine. And, just as the constellations have a number characteristic of them respectively, so all known objects have a number; as the formula of Philolaus states, 'all things which can be known have number; for it is not possible that without number anything can either be conceived or known.'"

"But when particularly complex societies began to appear in the wake of the Agricultural Revolution, a completely new type of information became vital - numbers."

"A critical step was made sometime before the ninth century AD, when a new partial script was invented, one that could store and process mathematical data with unprecedented efficiency. This partial script was composed of ten signs, representing the numbers from 0 - 9. Confusingly, these signs were known as Arabic numerals even though they were first invented by the Hindus."

"In communicating information about different sorts of things in the world, primitive man first learned to substitute crude pictures for speech to record seasonal occurrences for future use. ...As time went on the pictorial character of writing became less recognizable. ...The broad division between two kinds of writing... has its parallel in mathematics. The literature of mathematics begins with the pictorial or hieroglyphic language which we call geometry. ...At a much later date people stopped using nothing but pictures to record how numbers behave. They began to use letters, and compiled dictionaries in which you can find the meaning of the words used. Such dictionaries are called tables. ...Dictionary language, or, as mathematicians call it, "analysis," came later than hieroglyphic language, and grew out of it; but it has never supplanted the need for it completely."

"Bourgeois society is ruled by equivalence. It makes dissimilar things comparable by reducing them to abstract quantities. For the Enlightenment, anything which cannot be resolved into numbers, and ultimately into one, is illusion; modern positivism consigns it to poetry."

"C'est de l'Inde que nous vient l'ingénieuse méthode d'exprimer tous les nombres avec dix caractères, en leur donnant à la fois, une valeur absolue et une valeur de position; idée fine et importante, qui nous paraît maîntenant si simple, que nous en sentons à peine, le mérite. Mais cette simplicité même, et l'extrême facilité qui en résulte pour tous les calculs, placent notre système d'arithmétique au premier rang des inventions utiles; et l'on appréciera la difficulté d'y parvenir, si l'on considère qu'il a échappé au génie d'Archimède et d'Apollonius, deux des plus grands hommes dont l'antiquité s'honore."

"Mathematics is the queen of the sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank."

"Most texts on number theory contain inserted historical notes but in this course I have attempted to obtain a presentation of the results of the theory integrated more fully in the historical and cultural framework. Number theory seems particularly suited to this form of exposition, and in my experience it has contributed much to making the subject more informative as well as more palatable to the students. ...for the understanding of a greater part of the subject matter a knowledge of the simplest algebraic rules should be sufficient."