Plato - Greek Mathematics - The Story of Mathematics
Apr 1, In fact, Pythagoras' ideas about rational philosophy set the stage for and influenced a lot of Plato's work. Which Greek is the bigger genius. Could Plato have appropriated Pythagorean dogma in his notion of forms, and if the direct relationship of numbers to physical and metaphysical phenomena. This study examined the relationship between the Pythagoreans and Pluto in order to further illirminate the derivations of Plato's views of physical educa- tion.
This author has no clue of precisely how this proof worked, but let it suffice to say that the scheme was supposed to tie number to word, and to thus enable a person to discover the true form of idea-number.
Aristotle rejected the direction of the Academy and many of the ideas that Plato had embarked on, but this is another story.
Our enquiry has provided some tantalizing insights into where the Pythagorean concept that tied numbers to things lead. Man was not the measure of all things after all, numbers were… and numbers were unambiguous, fixed and immutable. They were not only the essence of the forms but also words.
Unfortunately, we can not peruse this enquiry in any greater depth if this paper is to be kept to the requisite length. He came to believe that while words are ambiguous, numbers are not. He thus sought to tie numbers to words in his later years, seeking a truth that neither alone could provide him. These citations may lead to new lines of inquiry such as that detailed above.
There were many Pythagoreans who claimed that Plato plagiarized various Pythagorean documents. It was from him that Plato, in a letter, told Dion to buy the Pythagorean books… He wrote one book. This fragment shows only that Pythagoras read the writings of others, however, and says nothing about him writing something of his own.
Isn't there a similarity between Plato and Pythagoras? : askphilosophy
The second of these is a Sacred Discourse, which some have wanted to trace back to Pythagoras himself. The idea that Pythagoras wrote such a Sacred Discourse seems to arise from a misreading of the early evidence. Herodotus says that the Pythagoreans agreed with the Egyptians in not allowing the dead to be buried in wool and then asserts that there is a sacred discourse about this II.
For an interesting but ultimately unconvincing attempt to argue that the historical Pythagoras did write books, see Riedweg42—43 and the response by Huffman a, — The Philosophy of Pythagoras One of the manifestations of the attempt to glorify Pythagoras in the later tradition is the report that he, in fact, invented the word philosophy. This story goes back to the early Academy, since it is first found in Heraclides of Pontus Cicero, Tusc.
Moreover, the story depends on a conception of a philosopher as having no knowledge but being situated between ignorance and knowledge and striving for knowledge. Such a conception is thoroughly Platonic, however see, e.
For a recent attempt to defend at least the partial accuracy of the story, see Riedweg Even if he did not invent the word, what can we say about the philosophy of Pythagoras? For the reasons given in 1. The Pythagorean Question and 2.
There is general agreement as to what the pre-Aristotelian evidence is, although there are differences in interpretation of it. It is crucial to decide this question before developing a picture of the philosophy of Pythagoras since chapter 19, if it is by Dicaearchus, is our earliest summary of Pythagorean philosophy.
Porphyry is very reliable about quoting his sources. He explicitly cites Dicaearchus at the beginning of Chapter 18 and names Nicomachus as his source at the beginning of chapter The material in chapter 19 follows seamlessly on chapter Thus, the onus is on anyone who would claim that Porphyry changes sources before the explicit change at the beginning of chapter Wehrli gives no reason for not including chapter 19 and the great majority of scholars accept it as being based on Dicaearchus see the references in Burkert a,n.
Zhmud a, following Philipargues that the passage cannot derive from Dicaearchus, since it presents immortality of the soul with approval, whereas Dicaearchus did not accept its immortality.
However, the passage merely reports that Pythagoras introduced the notion of the immortality of the soul without expressing approval or disapproval. Zhmud lists other features of the chapter that he regards as unparalleled in fourth-century sources a, but, since the evidence is so fragmentary, such arguments from silence can carry little weight.
In the face of the Pythagorean question and the problems that arise even regarding the early sources, it is reasonable to wonder if we can say anything about Pythagoras. A minimalist might argue that the early evidence only allows us to conclude that Pythagoras was a historical figure who achieved fame for his wisdom but that it is impossible to determine in what that wisdom consisted. We might say that he was interested in the fate of the soul and taught a way of life, but we can say nothing precise about the nature of that life or what he taught about the soul Lloyd There is some reason to believe, however, that something more than this can be said.
Herodotus tells the story of the Thracian Zalmoxis, who taught his countrymen that they would never die but instead go to a place where they would eternally possess all good things IV. Among the Greeks the tradition arose that this Zalmoxis was the slave of Pythagoras. Ion of Chios 5th c. Although Xenophanes clearly finds the idea ridiculous, the fragment shows that Pythagoras believed in metempsychosis or reincarnation, according to which human souls were reborn into other animals after death.
According to Herodotus, the Egyptians believed that the soul was reborn as every sort of animal before returning to human form after 3, years. Without naming names, he reports that some Greeks both earlier and later adopted this doctrine; this seems very likely to be a reference to Pythagoras earlier and perhaps Empedocles later.
Many doubt that Herodotus is right to assign metempsychosis to the Egyptians, since none of the other evidence we have for Egyptian beliefs supports his claim, but it is nonetheless clear that we cannot assume that Pythagoras accepted the details of the view Herodotus ascribes to them. Similarly both Empedocles see Inwood55—68 and Plato e. Did he think that we ever escape the cycle of reincarnations? We simply do not know. The fragment of Ion quoted above may suggest that the soul could have a pleasant existence after death between reincarnations or even escape the cycle of reincarnation altogether, but the evidence is too weak to be confident in such a conclusion.
In the fourth century several authors report that Pythagoras remembered his previous human incarnations, but the accounts do not agree on the details. Dicaearchus Aulus Gellius IV. Dicaearchus continues the tradition of savage satire begun by Xenophanes, when he suggests that Pythagoras was the beautiful prostitute, Alco, in another incarnation Huffman b, — It is not clear how Pythagoras conceived of the nature of the transmigrating soul but a few tentative conjectures can be made Huffman Transmigration does not require that the soul be immortal; it could go through several incarnations before perishing.
It has often been assumed that the transmigrating soul is immaterial, but Philolaus seems to have a materialistic conception of soul and he may be following Pythagoras. Similarly, it is doubtful that Pythagoras thought of the transmigrating soul as a comprehensive soul that includes all psychic faculties.
His ability to recognize something distinctive of his friend in the puppy if this is not pushing the evidence of a joke too far and to remember his own previous incarnations show that personal identity was preserved through incarnations. Thus, it would appear that what is shared with animals and which led Pythagoras to suppose that they had special kinship with human beings Dicaearchus in Porphyry, VP 19 is not intellect, as some have supposed Sorabji78 and but rather the ability to feel emotions such as pleasure and pain.
There are significant points of contact between the Greek religious movement known as Orphism and Pythagoreanism, but the evidence for Orphism is at least as problematic as that for Pythagoras and often complicates rather than clarifies our understanding of Pythagoras Betegh ; Burkert a, ff. There is some evidence that the Orphics also believed in metempsychosis and considerable debate has arisen as to whether they borrowed the doctrine from Pythagoras Burkert a, ; Bremmer24 or he borrowed it from them Zhmud a, — Dicaearchus says that Pythagoras was the first to introduce metempsychosis into Greece Porphyry VP Moreover, while Orphism presents a heavily moralized version of metempsychosis in accordance with which we are born again for punishment in this life so that our body is the prison of the soul while it undergoes punishment, it is not clear that the same was true in Pythagoreanism.
It may be that rebirths in a series of animals and people were seen as a natural cycle of the soul Zhmud a, — One would expect that the Pythagorean way of life was connected to metempsychosis, which would in turn suggest that a certain reincarnation is a reward or punishment for following or not following the principles set out in that way of life.
However, there is no unambiguous evidence connecting the Pythagorean way of life with metempsychosis. It is crucial to recognize that most Greeks followed Homer in believing that the soul was an insubstantial shade, which lived a shadowy existence in the underworld after death, an existence so bleak that Achilles famously asserts that he would rather be the lowest mortal on earth than king of the dead Homer, Odyssey XI.
The doctrine of transmigration thus seems to have been extended to include the idea that we and indeed the whole world will be reborn into lives that are exactly the same as those we are living and have already lived. Aristotle emphasized his superhuman nature in the following ways: Kingsley argues that the visit of Abaris is the key to understanding the identity and significance of Pythagoras.
Abaris was a shaman from Mongolia part of what the Greeks called Hyperboreawho recognized Pythagoras as an incarnation of Apollo. The stillness of ecstacy practiced by Abaris and handed on to Pythagoras is the foundation of all civilization.
Whether or not one accepts this account of Pythagoras and his relation to Abaris, there is a clear parallel for some of the remarkable abilities of Pythagoras in the later figure of Empedocles, who promises to teach his pupils to control the winds and bring the dead back to life Fr.
There are recognizable traces of this tradition about Pythagoras even in the pre-Aristotelian evidence, and his wonder-working clearly evoked diametrically opposed reactions. Similarly Pythagoras may have claimed authority for his teachings concerning the fate of our soul on the basis of his remarkable abilities and experiences, and there is some evidence that he too claimed to have journeyed to the underworld and that this journey may have been transferred from Pythagoras to Zalmoxis Burkert a, ff.
It is plausible to assume that many features of this way of life were designed to insure the best possible future reincarnations, but it is important to remember that nothing in the early evidence connects the way of life to reincarnation in any specific fashion. One of the clearest strands in the early evidence for Pythagoras is his expertise in religious ritual. Herodotus gives an example: It is not surprising that Pythagoras, as an expert on the fate of the soul after death.
(6) Pythagoras vs. (2) Plato
A significant part of the Pythagorean way of life thus consisted in the proper observance of religious ritual. The earliest source to quote acusmata is Aristotle, in the fragments of his now lost treatise on the Pythagoreans. It is not always possible to be certain which of the acusmata quoted in the later tradition go back to Aristotle and which of the ones that do go back to Pythagoras.
Thus the acusmata advise Pythagoreans to pour libations to the gods from the ear i. A number of these practices can be paralleled in Greek mystery religions of the day Burkert a, Indeed, it is important to emphasize that Pythagoreanism was not a religion and there were no specific Pythagorean rites Burkert Pythagoras rather taught a way of life that emphasized certain aspects of traditional Greek religion.
A second characteristic of the Pythagorean way of life was the emphasis on dietary restrictions. There is no direct evidence for these restrictions in the pre-Aristotelian evidence, but both Aristotle and Aristoxenus discuss them extensively. Unfortunately the evidence is contradictory and it is difficult to establish any points with certainty.
One might assume that Pythagoras advocated vegetarianism on the basis of his belief in metempsychosis, as did Empedocles after him Fr. This makes it sound as if Pythagoras forbade the eating of just certain parts of animals and certain species of animals rather than all animals; such specific prohibitions are easy to parallel elsewhere in Greek ritual Burkert a, Some have tried to argue that Aristoxenus is refashioning Pythagoreanism in order to make it more rational e.
Certainly animal sacrifice was the central act of Greek religious worship and to abolish it completely would be a radical step. The later tradition proposes a number of ways to reconcile metempsychosis with the eating of some meat. Pythagoras may have adopted one of these positions, but no certainty is possible. For example, he may have argued that it was legitimate to kill and eat sacrificial animals, on the grounds that the souls of men do not enter into these animals Iamblichus, VP Perhaps the most famous of the Pythagorean dietary restrictions is the prohibition on eating beans, which is first attested by Aristotle and assigned to Pythagoras himself Diogenes Laertius VIII.
Aristotle suggests a number of explanations including one that connects beans with Hades, hence suggesting a possible connection with the doctrine of metempsychosis. A number of later sources suggest that it was believed that souls returned to earth to be reincarnated through beans Burkert a, There is also a physiological explanation. Beans, which are difficult to digest, disturb our abilities to concentrate. Moreover, the beans involved are a European vetch Vicia faba rather than the beans commonly eaten today.
Certain people with an inherited blood abnormality develop a serious disorder called favism, if they eat these beans or even inhale their pollen.
The discrepancies between the various fourth-century accounts of the Pythagorean way of life suggest that there were disputes among fourth-century Pythagoreans as to the proper way of life and as to the teachings of Pythagoras himself. The acusmata indicate that the Pythagorean way of life embodied a strict regimen not just regarding religious ritual and diet but also in almost every aspect of life. Some of the restrictions appear to be largely arbitrary taboos, e.
On the other hand, some aspects of the Pythagorean life involved a moral discipline that was greatly admired, even by outsiders. Pythagorean silence is an important example. The ability to remain silent was seen as important training in self-control, and the later tradition reports that those who wanted to become Pythagoreans had to observe a five-year silence Iamblichus, VP Isocrates is contrasting the marvelous self-control of Pythagorean silence with the emphasis on public speaking in traditional Greek education.
In addition to silence as a moral discipline, there is evidence that secrecy was kept about certain of the teachings of Pythagoras. Indeed, one would expect that an exclusive society such as that of the Pythagoreans would have secret doctrines and symbols. That there should be secret teachings about the special nature and authority of the master is not surprising. This does not mean, however, that all Pythagorean philosophy was secret.
Aristotle singles out the acusma quoted above Iamblichus, VP 31 as secret, but this statement in itself implies that others were not. For a sceptical evaluation of Pythagorean secrecy see Zhmud a, — There is some controversy as to whether Pythagoras, in fact, taught a way of life governed in great detail by the acusmata as described above.
Plato praises the Pythagorean way of life in the Republic bbut it is hard to imagine him admiring the set of taboos found in the acusmata Lloyd44; Zhmud a. Although acusmata were collected already by Anaximander of Miletus the younger ca. However, the early evidence suggests that Pythagoras largely constructed the acusmata out of ideas collected from others Thom ; Huffman b: Gemelli Marcianoso it is no surprise that many of them are not uniquely Pythagorean.
Moreover, Thom suggests a middle ground between Zhmud and Burkert whereby, contra Zhmud, most of the acusmata were followed by the Pythagoreans but contra Burkert, they were subject to interpretation from the beginning and not followed literally, so that it is possible to imagine people living according to them Thom, It is true that there is little if any fifth- and fourth-century evidence for Pythagoreans living according to the acusmata and Zhmud argues that the undeniable political impact of the Pythagoreans would be inexplicable if they lived the heavily ritualized life of the acusmata, which would inevitably isolate them from society Zhmud a, — He suggests that the Pythagorean way of life differed little from standard aristocratic morality Zhmud a, If, however, the Pythagorean way of life was little out of the ordinary, why do Plato and Isocrates specifically comment on how distinctive those who followed it were?
The silence of fifth-century sources about people practicing acusmata is not terribly surprising given the very meager sources for the Greek cities in southern Italy in the period. We would then have lots of people who followed the acusmata of the name in the catalogue appear nowhere else. Moreover, other scholars argue that archaic Greek society in southern Italy was pervaded by religion and the presence of similar precepts in authors such as Hesiod show that adherence to taboos such as are found in the acusmata would not have caused a scandal and adherence to many of them would have gone unobserved by outsiders Gemelli Marciano— Once again a problem of source criticism raises its head.
Pythagoras - Wikipedia
Zhmud argues that the split between acusmatici who blindly followed the acusmata and the mathematici who learned the reasons for them see the fifth paragraph of section 5 below is a creation of the later tradition, appearing first in Clement of Alexandria and disappearing after Iamblichus Zhmud a, — He also notes that the term acusmata appears first in Iamblichus On the Pythagorean Life 82—86 and suggests that it is also a creation of the later tradition. The Pythagorean maxims did exist earlier, as the testimony of Aristotle shows, but they were known as symbola, were originally very few in number and were mainly a literary phenomena rather than being tied to people who actually practiced them Zhmud a, — Indeed, the description of the split in what is likely to be the original version Iamblichus, On General Mathematical Science So the question of whether Pythagoras taught a way of life tightly governed by the acusmata turns again on whether key passages in Iamblichus On the Pythagorean Life 81—87, On General Mathematical Science If they do, we have very good reason to believe that Pythagoras taught such a life, if they do not the issue is less clear.
The testimony of fourth-century authors such as Aristoxenus and Dicaearchus indicates that the Pythagoreans also had an important impact on the politics and society of the Greek cities in southern Italy. Dicaearchus reports that, upon his arrival in Croton, Pythagoras gave a speech to the elders and that the leaders of the city then asked him to speak to the young men of the town, the boys and the women Porphyry, VP