Mathematicians From Greece

"Mind is infinite and self-ruled, and is mixed with nothing, but is alone itself by itself."

"If one examines the reasons for the persecution of the best minds of different nations, and compares the reasons for the persecution and banishment of Pythagoras, Anaxagoras, Socrates, Plato, and others, one can observe that in each case the accusations and reasons for banishment were almost identical and unfounded. But in the following centuries full exoneration came, as if there had never been any defamation. It would be correct to conclude that such workers were too exalted for the consciousness of their contemporaries, and the sword of the executioner was ever ready to cut off a head held high.... A book should be written about the causes of the persecution of great individuals. By comparing the causes is it possible to trace the evil will."

"In mathematics... the Greek attitude differed sharply from that of the earlier potamic cultures. The contrast was clear in... Thales and Pythagoras, and it continues to show... in Athens during the Heroic Age. ...while Anaxagoras was in prison he occupied himself with an attempt to square the circle... the first mention of a problem that was to fascinate mathematicians for more than 2000 years. ...Here we see a type of mathematics that is quite unlike that of the Egyptians and Babylonians. It is not the practical application of a science of number... but a theoretical question involving a... distinction between accuracy in approximation and exactitude in thought. ...no more the concern of the technologist than those he raised... concerning the ultimate structure of matter."

"Wrongly do the Greeks suppose that aught begins or ceases to be; for nothing comes into being or is destroyed; but all is an aggregation or secretion of pre-existent things: so that all-becoming might more correctly be called becoming-mixed, and all corruption, becoming-separate."

"Anaxagoras... went back to ... settled at Lampsakos, and... founded a school there. Probably he did not live long after his exile. The Lampsakenes erected an altar to his memory in their market-place, dedicated to Mind and Truth; and the anniversary of his death was long kept as a holiday for school-children..."

"Living virtuously is equal to living in accordance with one's experience of the actual course of nature"

"Wise people are in want of nothing, and yet need many things. On the other hand, nothing is needed by fools, for they do not understand how to use anything, but are in want of everything."

"The universe itself is God and the universal outpouring of its soul"

"If I knew that it was fated for me to be sick, I would even wish for it; for the foot also, if it had intelligence, would volunteer to get muddy."

"We should infer in the case of a beautiful dwelling-place that it was built for its owners and not for mice; we ought, therefore, in the same way to regard the universe as the dwelling-place of the gods."

"He who is running a race ought to endeavor and strive to the utmost of his ability to come off victor; but it is utterly wrong for him to trip up his competitor, or to push him aside. So in life it is not unfair for one to seek for himself what may accrue to his benefit; but it is not right to take it from another."

"If I had followed the multitude, I should not have studied philosophy."

"It has been asserted that metaphysical speculation is a thing of the past and that physical science has extirpated it. The discussion of the categories of existence, however, does not appear to be in danger of coming to an end in our time, and the exercise of speculation continues as fascinating to every fresh mind as it was in the days of Thales."

"While [Thales] was studying the stars and looking upwards, he fell into a pit, and a neat, witty Thracian servant girl jeered at him, they say, because he was so eager to know the things in the sky that he could not see what was there before him at his very feet."

"[Thales] first went to Egypt and hence introduced this study [geometry] into Greece. He discovered many propositions himself, and instructed his successors in the principles underlying many others, his method of attack being in some cases more general [i.e. more theoretical or scientific], in others more empirical [...more in the nature of simple inspection or observation]."

"Since Alyattes would not give up the Scythians to Cyaxares at his demand, there was war [ Battle of Halys ] between the Lydians and the Medes for five years; each won many victories over the other, and once they fought a battle by night. They were still warring with equal success, when it happened, at an encounter which occurred in the sixth year, that during the battle the day was suddenly turned to night. Thales of Miletus had foretold this loss of daylight to the Ionians, fixing it within the year in which the change did indeed happen."

"According to tradition, Thales is the first to reveal the study of nature to the Greeks; although he had many predecessors, in Theopharastus' view, he so surpassed them as to eclipse everyone before him."

"[W]ith Thales... Aristotle ascribes the statement: "Water is the material cause of all things." This... expresses, as Nietzsche... pointed out, three fundamental ideas of philosophy. First, ...the material cause of all things; second, ...that this ...be answered in conformity with reason, without ...myths or mysticism; third, ...that ...it must be possible to reduce everything to one principle."

"Thales' statement was the first expression of... a fundamental substance, of which all other things were transient forms. The word "substance"... was... not interpreted in the purely material sense ...Aristotle ascribes to Thales also ...All things are full of gods. ...[We can] imagine ...Thales took his view ...from meteorological considerations. ...[W]ater can take the most various shapes... ice and snow... vapor, and... clouds. It seems to turn into earth where the rivers form their delta, and it can spring from the earth. Water is the condition for life. Therefore... [as] a fundamental substance, it was natural to think of water first."

"In committing himself to a form of materialism, Thales rejects a picture of the universe found in the Homeric poems, one which posits, in addition to the natural world, a supernatural quadrant populated by beings who are not subject to such laws as may govern the interactions of all natural bodies. If all things are composed of matter, then it ought to be possible to explain all there is to explain about the universe in terms of material bodies and their law-governed interactions. This simple thought already stands in sharp contrast to a world supposed to be populated by supernatural immaterial beings whose actions may be capricious or deliberate, rational or irrational, welcome or unwelcome, but which as a matter of basic principle cannot be explicated in terms of the forms of regularity found in the natural world. In Thales’ naturalistic universe, it ought to be possible to uncover patterns and laws and to use such laws as the basis for stable predictions about the direction the universe is to take; to uncover causes and to use that knowledge to find cures for illnesses or to develop strategies for optimizing our well-being; and, less practically, to find broad-based explanations to fundamental questions which crop up in every organized society. Such questions persist: Where did the universe come from? What, ultimately, is its basic stuff?"

"It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity. Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...there is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"It was not Zeno, the founder of the Stoics, alone, who taught that the Universe evolves, and its primary substance is transformed from the state of fire into that of air, then into that of water, etc. Heraclitus of Ephesus maintained that the one principle that underlies all phenomena in Nature is fire. The intelligence that moves the Universe is fire, and fire is intelligence. And while Anaximenes said the same of air, and Thales of Miletus (600 years b.c.) of water, the Esoteric Doctrine reconciles all these philosophers, by showing that though each was right, the system of none was complete."

"With regard to the Pythagorean theorem my conjecture is that... it was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles."

"It seems probable that the early Greeks were largely indebted to the Phoenicians for their knowledge of practical arithmetic or the art of calculation, and perhaps also learnt from them a few properties of numbers. It may be worthy of note that Pythagoras was a Phoenician; and according to Herodotus, but this is more doubtful, Thales was also of that race."

"Nothing is more ancient than God, for He was never created; nothing more beautiful than the world, it is the work of that same God; nothing is more active than thought, for it flies over the whole universe; nothing is stronger than necessity, for all must submit to it."

"Thales asserted Water to be the principle of things. For he saw that matter was principally dispensed in moisture, and moisture in water; and it seemed proper to make that the principle of things, in which the virtues and powers of beings, and especially the elements of their generations and restorations, were chiefly found. He saw that the breeding of animals is in moisture ; that the seeds and kernels of plants (as long as they are productive and fresh), are likewise soft and tender; that metals also melt and become fluid, and are as it were concrete juices of the earth, or rather a kind of mineral waters; that the earth itself is fertilised and revived by showers or irrigation, and that earth and mud seem nothing else than the lees and sediment of water; that air most plainly is but the exhalation and expansion of water; nay, that even fire itself cannot be lighted, nor kept in and fed, except with moisture and by means of moisture. He saw, too, that the fatness which belongs to moisture, and which is the support and life of flame and fire, seems a kind of ripeness and concoction of the water."

"The more one studies the period of Thales—the more one compares the knowledge he bequeathed to prosterity with the one he had found when he began his work—the more does his mathematical stature grow, until one is impelled to range Thales with such figures as Archimedes, Fermat, Newton, Gauss and Poincaré."

"Thales had a motto — sophotaton chronos aneuriskei gar panta — which means time is wisest because it discovers everything. We still live by that motto — we mark the time and aid the discoveries by keeping the soul lines intact."

"Water is the first principle of everything."

"Hope is the only good that is common to all men; those who have nothing else possess hope still."

"Placing your stick at the end of the shadow of the pyramid, you made by the sun's rays two triangles, and so proved that the pyramid [height] was to the stick [height] as the shadow of the pyramid to the shadow of the stick."

"Thales the teacher produced the first geometers, even as Thales the thinker founded the first geometry worthy of the name."

"Do not ask who started it. Finish it"

"Ὅπερ ἔδει δεῖξαι."

"ὅπερ ἔδει ποιῆσαι."

"Inasmuch as many things, while appearing to rest on truth and to follow from scientific principles, really tend to lead one astray from the principles and deceive the more superficial minds, he has handed down methods for the discriminative understanding of these things as well, by the use of which methods we shall be able to give beginners in this study practice in the discovery of paralogisms and to avoid being misled. This treatise, by which he puts this machinery in our hands, he entitled (the book) of Pseudaria, enumerating in order their various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of error with practical illustration. This book then is by way of cathartic and exercise, while the Elements contain the irrefragable and complete guide to the actual scientific investigation of the subjects of geometry."

"Καὶ τὸ ὅλον τοῦ μέρους μεῖζον [ἐστιν]."

"Not much younger than these (sc. Hermotimus of Colophon and Philippus of Mende) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. He is then younger than pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says."

"There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid."

"Euclid... gave his famous definition of a point: "A point is that which has no parts, or which has no magnitude." …A point has no existence by itself. It exists only as a part of the pattern of relationships which constitute the geometry of Euclid. This is what one means when one says that a point is a mathematical abstraction. The question, What is a point? has no satisfactory answer. Euclid's definition certainly does not answer it. The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry? ...It cannot be answered by a definition."

"The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries."

"There is irrefutable evidence that a substantial portion of the material recorded in the Elements was known before Euclid, and there is nothing either in the style or in the plan of the treatise to suggest that it was intended as a collection of original contributions. Thus, on the whole... the chief objective... was to put system and rigour into the work of his predecessors."

"With the completion of Euclid's Elements... For the first time in history masses of isolated discoveries were unified and correlated by a single guided principle, that of rigid deduction from explicitly stated assumptions. ...If the Pythagorean dream of a mathematized science is to be realized, all of the sciences must eventually submit to the discipline that geometry accepted from Euclid."

"The term 'axiom' was used by Proclus, but not by Euclid. He speaks, instead, of 'common notions'—common either to all men or to all sciences."

"Euclid is said to have written the Elements of Music. Two treatises are attributed to Euclid in... the Musici, the... Sectio canonis (the theory of the intervals) and the... (introduction to harmony). The first, resting on the Pythagorean theory of music, is mathematical and clearly and well written, the style and the form of the propositions agreeing well with what we find in the Elements. Its genuineness is confirmed not only by internal evidence... Euclid is twice mentioned by name, in the commentary on Ptolemy's Harmonica published by Wallis... The second treatise is not Euclid's..."

"The Greeks elaborated several theories of vision. According to the Pythagoreans, Democritus, and others vision is caused by the projection of particles from the object seen, into the pupil of the eye. On the other hand Empedocles, the Platonists, and Euclid held the strange doctrine of ocular beams, according to which the eye itself sends out something which causes sight as soon as it meets something else emanated by the object."

"μὴ εἶναι βασιλικὴν ἀτραπὸν ἐπί γεωμετρίαν."

"Non est regia [inquit Euclides] ad Geometriam via."

"Δός αὐτῷ τριώβολον, ἐπειδὴ δεῖ αὐτῷ ἐξ ὧν μανθάνει κερδαίνειν."

"Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος."