The object of this paper is to show that, in the sixth and fifth centuries B.C., two different and radically opposed systems of thought were elaborated within the Pythagorean school. They may be called respectively the mystical system and the scientific. All current accounts of Pythagoreanism known to me attempt to combine the traits of both systems in one composite picture, which naturally fails to hold together. The confusion goes back to Aristotle, who usually speaks indiscriminately of ‘the Pythagoreans,’ though (...) now and then the phrase ‘some Pythagoreans’ indicates that he was aware of different currents within the school. (shrink)

Zeller argued that the ‘innumerable worlds’ mentioned in accounts of Anaximander's system must be an endless succession of single worlds, not an unlimited number of coexistent worlds scattered through infinite space, some always coming into being while others are passing away. Zeller pointed out that a succession of single worlds is grounded in the principles of the system. ‘Things perish into that from which they had their birth… according to the order of Time,’ a cycle of birth, existence, and destruction. (...) A world ends, and the living divine stuff begets a new world to take its place. On the other hand, there is much in the system to contradict the idea of coexistent worlds. Anaximander's successors, Anaximenes, Anaxagoras, and Diogenes, show that this idea is not a necessary consequence of the unlimitedness of the original world-stuff. Nothing in the appearances of Nature suggests it. Anaximander is a monistic hylozoist, whereas Democritus is a pluralist with his innumerable independent atoms producing, by similar processes, independent world-systems in different parts of an infinite void. (shrink)

The object of this paper is to show that, in the sixth and fifth centuries B.C., two different and radically opposed systems of thought were elaborated within the Pythagorean school. They may be called respectively the mystical system and the scientific. All current accounts of Pythagoreanism known to me attempt to combine the traits of both systems in one composite picture, which naturally fails to hold together. The confusion goes back to Aristotle, who usually speaks indiscriminately of ‘the Pythagoreans,’ though (...) now and then the phrase ‘some Pythagoreans’ indicates that he was aware of different currents within the school. (shrink)

Zeller argued that the ‘innumerable worlds’ mentioned in accounts of Anaximander's system must be an endless succession of single worlds, not an unlimited number of coexistent worlds scattered through infinite space, some always coming into being while others are passing away. Zeller pointed out that a succession of single worlds is grounded in the principles of the system. ‘Things perish into that from which they had their birth… according to the order of Time,’ a cycle of birth, existence, and destruction. (...) A world ends, and the living divine stuff begets a new world to take its place. On the other hand, there is much in the system to contradict the idea of coexistent worlds. Anaximander's successors, Anaximenes, Anaxagoras, and Diogenes, show that this idea is not a necessary consequence of the unlimitedness of the original world-stuff. Nothing in the appearances of Nature suggests it. Anaximander is a monistic hylozoist, whereas Democritus is a pluralist with his innumerable independent atoms producing, by similar processes, independent world-systems in different parts of an infinite void. (shrink)

We can now approach the interpretation of the famous symbol called the Tetractys or Tetrad, which is a compendium of Pythagorean mysticism. The tetractys is itself a system of numbers. It symbolizes the ‘elements of number,’ which are the elements of all things. It contains the concordant ratios of the musical harmony. It might well be described in the Pythagorean oath as ‘containing the root and fountain of everflowing Nature.’ In one of the acousmata preserved in Iamblichus it is identified (...) with the cosmic harmony. It was also called κσμος, ορανς, πν. Theon says it was held in honour because it contained the nature of the universe. (shrink)

Anaxagoras’ theory of matter offers a problem which, in bald outline, may be stated as follows. The theory rests on two propositions which seem flatly to contradict one another. One is the principle of Homoeomereity: A natural substance such as a piece of gold, consists solely of parts which are like the whole and like one another—every one of them gold and nothing else. The other is: ‘There is a portion of everything in everything’, understood to mean that a piece (...) of gold , so far from containing nothing but gold, contains portions of every other substance in the world. Unless Anaxagoras was extremely muddleheaded, he cannot have propounded a theory which simply consists of this contradiction. One or the other proposition must be reinterpreted so as to bring them into harmony. Some critics attack one, some the other; some try to modify both. Mr. C. Bailey has recently published a fresh attempt to reconstruct the theory in a form consistent with itself. But the result, as he admits, is arbitrary and uneconomical. These defects, like all the contradictions and obscurities found in other accounts of the system, can be traced to the second proposition: ‘There is a portion of everything in everything’, or rather to the construction put upon it. The language is crude, vague, and ambiguous. What does ‘everything’ mean? Has the first ‘everything’ the same sense as the second ? If taken to mean that there is a portion of every material substance in every material substance, the proposition leads to a result for which ‘uneconomical’ is an indulgent epithet. (shrink)

The object of this paper is to determine the relations between the two parts of Parmenides' poem: the Way of Truth, which deduces the necessary properties of a One Being, and the False Way, which contains a cosmogony based on ‘what seems to mortals, in which there is no true belief.’.

The earlier part of this paper yielded the result that the assertion ‘A portion of everything in everything’ has no place or function in the explanation of any sort of apparent ‘becoming’ or change. This conclusion is important because, ever since Aristotle, it has been assumed that the assertion was made in order to explain away becoming and change. But if , according to the best evidence, becoming and such sorts of change as Anaxagoras considered can be explained away without (...) using the assertion at all, it follows that the motive for it must be sought, not in the processes of apparent becoming and change, but in the permanent constitution of the ungenerated and unchanging things which figure as elements in the system. A visible mass of gold or of flesh is formed by the aggregation of smaller particles of gold or of flesh, and it is infinitely divisible into parts, every one of which is gold or flesh. But what is the constitution of a piece of gold of whatever size, and how does it differ from the constitution of a piece of flesh? The proposition we have so far successfully excluded comes in here as part of the answer to this question. (shrink)

The object of this paper is to determine the relations between the two parts of Parmenides' poem: the Way of Truth, which deduces the necessary properties of a One Being, and the False Way, which contains a cosmogony based on ‘what seems to mortals, in which there is no true belief.’.

That three of them were popularly regarded as correlated with the three ages of human life; that the structure of early Greek societies, as of other primitive societies, was based on the distinction of three main age-grades, of which the three virtues are characteristic; that Plato's own Ideal State has the same age basis underlying the other features peculiar to it, and is indeed transparently modelled on the Spartan constitution; that it is therefore probable that Plato started with the three (...) divisions of his State and their several virtues, and then, proceeding on the assumption that the ‘natural’ State must reflect on a large scale the constitution of the individual ‘nature,’ arranged the structure of the soul to correspond with his polity. (shrink)

That three of them were popularly regarded as correlated with the three ages of human life; that the structure of early Greek societies, as of other primitive societies, was based on the distinction of three main age-grades, of which the three virtues are characteristic; that Plato's own Ideal State has the same age basis underlying the other features peculiar to it, and is indeed transparently modelled on the Spartan constitution; that it is therefore probable that Plato started with the three (...) divisions of his State and their several virtues, and then, proceeding on the assumption that the ‘natural’ State must reflect on a large scale the constitution of the individual ‘nature,’ arranged the structure of the soul to correspond with his polity. (shrink)

Aristotle is asking why the heaven revolves in one direction rather than the other. His answer is based on his earlier proof that the Universe has a top and a bottom, a right and a left. The upper region, as the place of divinity, is prior or superior to the lower; so upward motion is prior to downward motion. Right is similarly prior and superior to left. The present problem can be solved by supposing that the world has also a (...) front and a back, and hence a forward motion, which will be prior and superior to backward motion. Since Nature always follows the best course, the reason for the direction of the heaven's revolution will be that that direction is forward and therefore better. (shrink)

The first of these passages states some simple principles of mechanics. The second uses one of these principles to prove that a finite mover cannot cause a motion that will occupy unlimited time. The argument there has given much trouble to commentators because the principle in question was not understood, owing to the choice of a false reading in the earlier passage.

The object of this paper is to supplement Dr. Zielinski's admirable articles on Hermes und die Hermetik by calling attention to a passage in Aristotle where the triad–Hermes, Pan, Logos –appears, and by showing that there is some probability that the passage refers to a lost work of the rhetorician Alkidamas, the pupil of Gorgias.

The object of this paper is to supplement Dr. Zielinski's admirable articles on Hermes und die Hermetik by calling attention to a passage in Aristotle where the triad–Hermes, Pan, Logos –appears, and by showing that there is some probability that the passage refers to a lost work of the rhetorician Alkidamas, the pupil of Gorgias.