Mathematicians From Greece

"We may assume the superiority ceteris paribus [all things being equal] of the demonstration which derives from fewer postulates or hypotheses—in short from fewer premisses; for... given that all these are equally well known, where they are fewer knowledge will be more speedily acquired, and that is a desideratum. The argument implied in our contention that demonstration from fewer assumptions is superior may be set out in universal form..."

"The premises of demonstrative knowledge must be true, primary, immediate, more knowable than and prior to the conclusion, which is further related to them as effect to cause... The premises must be the cause of the conclusion, more knowable than it, and prior to it; its causes, since we possess scientific knowledge of a thing only when we know its cause; prior, in order to be causes; antecedently known, this antecedent knowledge being not our mere understanding of the meaning, but knowledge of the fact as well. Now 'prior' and 'more knowable' are ambiguous terms, for there is a difference between what is prior and more knowable in the order of being and what is prior and knowable to man. I mean that objects nearer to sense are prior and more knowable to man; objects without qualification prior and more knowable are those further from sense. Now the most universal causes are furthest from sense and particular causes are nearest to sense, and they are thus exactly opposed to each other."

"Knowledge of the fact differs from knowledge of the reason for the fact."

"Of things said without any combination, each signifies either substance or quantity or qualification or a relative or where or when or being-in-a-position or having or doing or being affected. To give a rough idea, examples of substance are man, horse; of quantity: four-foot, five-foot; of qualification: white, grammatical; of a relative: double, half, larger; of where: in the Lyceum, in the market-place; of when: yesterday, last-year; of being-in-a-position: is-lying, is sitting; of having: has-shoes-on, has-armour-on; of doing: cutting, burning; of being-affected: being-cut, being-burned."

"Nature does not do anything in vain."

"My lectures are published and not published; they will be intelligible to those who heard them, and to none beside."

"And surely to know what this good is, is of great importance for the conduct of life, for in that case we shall be like archers shooting at a definite mark, and shall be more likely to do what is right. But, if this is the case, we must try to comprehend, in outline at least, what it is and to which of the sciences it belongs. (Bk I, Ch I)"

"If, then, in the sphere of action there is some one end which we desire for its own sake, and for the sake of which we desire every thing else; and if we do not choose every thing for the sake of something else, for this would go on without limit, and our desire would be idle and futile, it is clear that this must be the supreme good, and the best thing of all. (Bk I, Ch I)"

"But it is clear there is a difference in the ends proposed: for in some cases they are activities, and in others results beyond the mere activities, and where there are certain ends beyond and beside the actions, the results are naturally superior to the activities. Now, as there are numerous kinds of actions and numerous arts and sciences, it follows that the ends are also various. Thus the end of the healing art is health, of ship-building ships, of strategy victory, of economy wealth. (Bk I, Ch I)"

"Every art, and every system, and in like manner every action and purpose aims, it is thought, at some good; for which reason a common and by no means a bad description of the good is, ‘that at which all things aim.’ (Bk I, Ch I)"

"Those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree."

"εἰ οὖν οὕτως εὖ ἔχει, ὡς ἡμεῖς ποτέ, ὁ θεὸς ἀεί, θαυμαστόν: εἰ δὲ μᾶλλον, ἔτι θαυμασιώτερον. ἔχει δὲ ὧδε. καὶ ζωὴ δέ γε ὑπάρχει: ἡ γὰρ νοῦ ἐνέργεια ζωή, ἐκεῖνος δὲ ἡ ἐνέργεια: ἐνέργεια δὲ ἡ καθ᾽ αὑτὴν ἐκείνου ζωὴ ἀρίστη καὶ ἀΐδιος."

"πάντων γὰρ ὅσα πλείω μέρη ἔχει καὶ μὴ ἔστιν οἷον σωρὸς τὸ πᾶν."

"τὸ γὰρ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ ὑπάρχειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτό （καὶ ὅσα ἄλλα προσδιορισαίμεθ᾽ ἄν, ἔστω προσδιωρισμένα πρὸς τὰς λογικὰς δυσχερείας）: αὕτη δὴ πασῶν ἐστὶ βεβαιοτάτη τῶν ἀρχῶν: ἔχει γὰρ τὸν εἰρημένον διορισμόν. ἀδύνατον γὰρ ὁντινοῦν ταὐτὸν ὑπολαμβάνειν εἶναι καὶ μὴ εἶναι, καθάπερ τινὲς οἴονται λέγειν Ἡράκλειτον."

"That which is desirable on its own account and for the sake of knowing it is more of the nature of wisdom than that which is desirable on account of its results."

"οὐ γὰρ δεῖν ἐπιτάττεσθαι τὸν σοφὸν ἀλλ᾽ ἐπιτάττειν, καὶ οὐ τοῦτον ἑτέρῳ πείθεσθαι, ἀλλὰ τούτῳ τὸν ἧττον σοφόν."

"For it is owing to their wonder that men both now begin and at first began to philosophize; they wondered originally at the obvious difficulties, then advanced little by little and stated difficulties about the greater matters, e.g. about the phenomena of the moon and those of the sun and of the stars, and about the genesis of the universe. And a man who is puzzled and wonders thinks himself ignorant (whence even the lover of myth is in a sense a lover of Wisdom, for the myth is composed of wonders); therefore since they philosophized order to escape from ignorance, evidently they were pursuing science in order to know, and not for any utilitarian end. — Metaphysics by Aristotle – Book 1, ClassicalWisdom.com"

"But as more arts were invented, and some were directed to the necessities of life, others to recreation, the inventors of the latter were naturally always regarded as wiser than the inventors of the former, because their branches of knowledge did not aim at utility. ... This is why the mathematical arts were founded in Egypt; for there the priestly caste was allowed to be at leisure."

"All men by nature desire to know. An indication of this is the delight we take in our senses; for even apart from their usefulness they are loved for themselves; and above all others the sense of sight. For not only with a view to action, but even when we are not going to do anything, we prefer sight to almost everything else. The reason is that this, most of all the senses, makes us know and brings to light many differences between things."

"Music directly represents the passion of the soul. If one listens to the wrong kind of music, he will become the wrong kind of person."

"Just as it sometimes happens that deformed offspring are produced by deformed parents, and sometimes not, so the offspring produced by a female are sometimes female, sometimes not, but male, because the female is as it were a deformed male."

"Concerning the generation of animals akin to them, as hornets and wasps, the facts in all cases are similar to a certain extent, but are devoid of the extraordinary features which characterize bees; this we should expect, for they have nothing divine about them as the bees have."

"Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end."

"The essential nature (concerning the soul) cannot be corporeal, yet it is also clear that this soul is present in a particular bodily part, and this one of the parts having control over the rest (heart)."

"The brain is not responsible for any of the sensations.. the correct view [is] that the seat and source of sensation is the region of the heart….the motions of pleasure and pain, and generally all sensation plainly have their source in the heart."

"The male has more teeth than the female in mankind, and sheep, and goats, and swine. This has not been observed in other animals."

"But voice is a certain sound of that which is animated; for nothing inanimate emits a voice; but they are said to emit a voice from similitude, as a pipe, and a lyre, and such other inanimate things, have extension, modulation, and dialect; for thus it appears, because voice, also, has these."

"It is not necessary to ask whether soul and body are one, just as it is not necessary to ask whether the wax and its shape are one, nor generally whether the matter of each thing and that of which it is the matter are one. For even if one and being are spoken of in several ways, what is properly so spoken of is the actuality."

"Sound is the motion of that which is able to be moved, after the manner in which those things are moved, that rebound from smooth bodies, when any one strikes them. Not every thing... sounds... but it is necessary, that the body which is struck should be equable, that the air may collectively rebound, and be shaken. The differences, however, of bodies which sound, are manifested in the sound, which is in energy; for, as colours are not perceived without light, so neither are the sharp and the flat perceived without sound. But these things are asserted metaphorically, from those which pertain to the touch; for the sharp moves the sense much in a short time, but the flat a little in a long time. The sharp, therefore, is not rapid, and the flat slow; but such a motion is produced of the one, on account of celerity, and of the other on account of slowness, that, also, which is perceived in the touch, appears to be analogous to the acute and obtuse, for the acute, as it were, stings; but the obtuse, as it were, impels: because the one moves in a short, but the other in a long time. Hence it happens that the one is swift but the other slow. Let it therefore be thus determined concerning sound."

"That body is heavier than another which, in an equal bulk, moves downward quicker."

"...suppose α without weight, but β possessing weight; and let α pass over space γδ, but β in the same time pass over a space γε,—for that which has weight will be carried through the larger space. If now the heavy body be divided in the proportion that space γε bears to γδ, ... and if the whole is carried through the whole space γε, then it must be that a part in the same time would be carried through γδ..."

"The least initial deviation from the truth is multiplied later a thousandfold."

"The bodies of which the world is composed are solids, and therefore have three dimensions. Now, three is the most perfect number,—it is the first of numbers, for of one we do not speak as a number, of two we say both, but three is the first number of which we say all. Moreover, it has a beginning, a middle, and an end."

"The infinite... happens to subsist in a way contrary to what is asserted by others: for the infinite is not that beyond which there is nothing, but it is that of which there is always something beyond. ...But that pertaining to which there is nothing beyond is perfect and whole. ...that of which nothing is absent pertaining to the parts ...the whole is that pertaining to which there is nothing beyond. But that pertaining to which something external is absent, that is not all ...But nothing is perfect which has not an end; and the end is a bound. On this account... Parmenides spoke better than Melissus: for the latter says that the infinite is a whole; but the former, that the whole is finite, and equally balanced from the middle: for to conjoin the infinite with the universe and the whole, is not to connect line with line."

"Plato... introduces two infinities, because both in increase and diminution there appears to be transcendency, and a progression to infinity. Though... he did not use them: for neither is there infinity in numbers by diminution or division; since unity is a minimum: nor by increase; for he extends number as far as to the decad."

"[T]he infinite is in capacity. That, however, which is infinite in capacity is not to be assumed as that which is infinite in energy. ...[I]t has its being in capacity, and in division and diminution. ...[I]t is always possible to assume something beyond it. It does not, however, on this account surpass every definite magnitude; as in division it surpasses every definite magnitude, and will be less."

"[H]ow will one part of the infinite be above, and another below? Or how will it have extremes or a middle? Further still, every sensible body is in place; but the species and differences of place are upward and downward, before and behind, to the right hand and to the left: and these things not only thus subsist with relation to us, and by position, but have a definite subsistence in the universe itself. But it is impossible that these things should be in the infinite: and... that there should be an infinite place. But every body is in place; and therefore it is also impossible that there should be an infinite body. ...[T]herefore ...there is not an infinite body in energy."

"[I]t's gravity is the cause; and that which is heavy abides in the middle, and the earth is in the middle: in like manner also, the infinite will abide in itself, through some other cause... and will itself support itself. ...[T]he places of the whole and the part are of the same species; as of the whole earth and a clod, the place is downward; and of the whole of fire, and a spark, the place is upward. So that if the place of the infinite is in itself, there will be the same place also of a part of the infinite."

"[I]t is impossible that each of the elements should be infinite. For that is body which has interval on all sides; and that is infinite which has extension without bound."

"[T]hey pronounce absurdly who thus speak, as the Pythagoreans assert: for at the same time they make the infinite to be essence, and distribute it into parts."

"[B]ecause that which is finite is always bounded with reference to something... it is necessary that there should be no end... [N]umber also appears to be infinite, and mathematical magnitudes, and that which is beyond the heavens. And since that which is beyond is infinite, body also appears to be infinite, and it would seem that there are infinite worlds; for why is there rather void here than there? ...If also there is a vacuum, and an infinite place, it is necessary that there should be an infinite body: for in things which have a perpetual subsistence, capacity differs nothing from being. The speculation of the infinite is, however, attended with doubt: for many impossibilities happen both to those who do not admit that it has a subsistence, and to those who do. ...It is ...especially the province of a natural philosopher to consider if there be a sensible infinite magnitude."

"All those... who discourse concerning nature, always subject a certain other nature of... elements, to the infinite... But no one of those who make the elements to be finite introduces infinity. Such, however, as make infinite elements, as Anaxagoras and Democritus, say that the infinite is continuous by contact. ...Rationally, too, do all philosophers consider the infinite as a principle; for it cannot be in vain, nor can any other power be present with it than that of a principle: for all things are either the principle, or from the principle; but of the infinite there is no principle, since otherwise it would have an end. ...it is also unbegotten and uncorruptible, as being a certain principle: for... end is the corruption of everything. ...It likewise appears to comprehend and govern all things, as those assert who do not introduce other causes beside the infinite... It would seem also that this is divine: for it is immortal and indestructible, as Anaximander says, and most of the physiologists."

"Since the science of nature is conversant with magnitudes, motion, and time, each of which must necessarily be either infinite or finite...[we] should speculate the infinite, and consider whether it is or not; and if it is what it is. ...[A]ll those who appear to have touched on a philosophy of this kind... consider it as a certain principle of beings. Some, indeed, as the Pythagoreans and Plato, consider it, per se, not as being an accident to any thing else, but as having an essential subsistence... the Pythagoreans... consider the infinite as subsisting in sensibles; for they do not make number to be separate; and they assert that what is beyond the heavens is infinite; but Plato says that beyond the heavens there is not any body, nor ideas, because these are no where: he affirms, however, that the infinite is both in sensibles, and in ideas. ...Plato establishes two infinities, viz. the great and the small."

"There is... something which is in energy only; and there is something which is both in energy and capacity. ...of relatives, one is predicated as according to excess and defect: another according to the effective and passive, and, in short, the motive, and that which may be moved... Motion, however, has not a substance separate from things... But each of the categories subsists in a twofold manner in all things. Thus... one thing pertaining to it is form, and another privation. ...So the species of motion and mutation are as many as those of being. But since in every genus of things, there is that which is in entelecheia, and that which is in capacity; motion is the entelecheia of that which is in capacity... That there is energy, therefore, and that a thing then happens to be moved, when this energy exists, and neither prior nor posterior to it, is manifest. ... [N]either motion nor mutation can be placed in any other genus; nor have those who have advanced a different opinion concerning it spoken rightly. ...for by some motion is said to be difference, inequality, and non-being; though it is not necessary that any of these should be moved... Neither is mutation into these, nor from these, rather than from their opposites. ...The cause, however, why motion appears to be indefinite, is because it can neither be simply referred to the capacity, nor to the energy of beings. ...[I]t is difficult to apprehend what motion is: for it is necessary to refer it either to privation, or to capacity, or to simple energy; but it does not appear that it can be any of these. The above-mentioned mode, therfore remains, viz. that it is a certain energy; but... difficult to be perceived, but which may have a subsistence."

"[I]t is impossible for motion to subsist without place, and void, and time."

"Since... nature is a principle of motion and mutation... it is necessary that we should not be ignorant of what motion is... But motion appears to belong to things continuous; and the infinite first presents itself to the view in that which is continuous. ...[F]requently ...those who define the continuous, employ the nature or the infinite, as if that which is divisible to infinity is continuous."

"[S]ince causes are four in number, to know them all is the business of the natural philosopher, who also referring to the cause why a thing is to all of them, viz. to matter, form, that which moves, and for the sake of which a thing subsists, physically assigns a reason. Frequently, however, three of these causes pass into one: for the cause why a thing is, and that for the sake of which it is, are one. But that which motion first originates, is in species the same with these... [T]here are three treatises; once concerning that which is immoveable; another concerning that which is moved, indeed, but is incorruptible; and a third concerning corruptible natures. So the cause of why a thing is, is assigned by him who refers to matter, to essence, and to the first mover... But there are two principles which are naturally motive; of which, one is not physical, because it does not contain in itself the principle of motion. And if there is any thing which moves without being moved, it is of this kind; as is that which is perfectly immoveable, that which is the first of all things, together with essence and form: for it is the end, and that for the sake of which a thing subsists. So that since nature is for the sake of something, it is also necessary to know this cause."

"Fortune... and chance, are said to be in the number of causes... [W]ith some it is dubious whether these things have subsistence or not. For, say they, nothing is produced from fortune, but there is a definite cause of all such things... For if fortune were any thing, it would truly appear to be absurd; and some one might doubt why no one of the ancient wise men, when assigning the causes of generation and corruption, has ever defined any thing concerning fortune. ...[M]any things are produced, and have a subsistence, from fortune and chance... They did not, however, think that fortune was any thing belonging to friendship or strife, or fire, or intellect, or any thing else of things of this kind. They are chargeable, therefore, with absurdity, whether they did not conceive that it had a substance, or whether fancying that it had, they omitted it; especially since it was sometimes employed by them. Thus Empedocles says that the air...Thus it then chanc'd to run, tho' varying oft.He also says that the greater part of... animals were generated by fortune. But there are some who assign chance to the cause of this heaven, and of all mundane natures... [W]e must consider... whether chance or fortune are the same... or different from each other, and how they fall into definite causes."

"[L]et us consider, with respect to causes, what they are, and how many there are in number... this also must be done by us in discoursing concerning generation and corruption, and all physical mutation... knowing the principles of these... Cause... is after one manner said to be that, from which, being inherent, something is produced... But after another manner cause is form and paradigm (and this is the definition of the essence of a thing) and the genera of this. ...But it happens... that there are also many causes of the same thing, and this is not from accident. ...seed, a physician, he who consults, and, in short, he who makes, are all of them causes, as that whence the principle of mutation, or permanency, or motion is derived. ...It is, however, necessary always to investigate the supreme cause of every thing ...Further still, it is necessary to investigate the genera of genera; and the particulars of particulars... We should also explore the capacities of the capabilities, and the energizers of the things affected by energy."

"According to one mode... nature is thus denominated, viz. the first subject matter to every thing which contains in itself the principle of motion and mutation. But after another mode it is denominated form, which subsists according to definition: for as art is called that which subsists according to art, and that which is artificial; so likewise nature is both called that which is according to nature, and that which is natural. ...that which is composed from these is not nature, but consists from nature; as, for instance, man. And this is nature in a greater degree than matter: for every thing is then said to be, when it is form in energy... entelecheia, rather than when it is incapacity."