What Lies Behind Pythagoras’ Theorem

Pythagoras (570-500 BC) was born on the island of Samos in what is now Greece. On the advice of Thales of Miletus, he travelled to Memphis in Egypt where he came into contact with priests renowned for their wisdom. At the age of 40, he fled the tyranny of Polycrates to Croton in Southern Italy, where he established a philosophical and religious community. Those who entered the community’s inner circle were governed by a strict set of ascetic and ethical rules, forsaking personal possessions, assuming a mainly vegetarian diet, and—since words are so often careless and misrepresentative—observing the strictest silence. Some of the community’s more eccentric rules, such as ‘do not break bread’ or ‘do not poke the fire with a sword’ may have been riddles or allegories that required interpreting. Pythagoras’ brotherly community has been hailed as a prototype for later philosophical institutions such as Plato’s Academy, Aristotle’s Lyceum, and Epicurus’ Garden, and, indeed, for the monastic life and associated early universities.

Music played an important role in Pythagoras’ community. Pythagoreans recited poetry, sang hymns to Apollo, and played on the lyre to cure illnesses of both body and soul. It is said that, one day, Pythagoras passed by some blacksmiths at work, and found that their hammering on anvils produced especially harmonious sounds. He then found that the anvils were simple ratios of one another, one being half the size of the first, another two thirds of the size, and so on. This discovery of a relationship between numerical ratios and musical intervals led Pythagoras to believe that the study of mathematics was the key to understanding the structure and order of the universe. According to his ‘harmony of the spheres’, the heavenly bodies move according to mathematical equations that correspond to musical notes and form part of a grand cosmic symphony.

Pythagoras never separated religion from science and philosophy, which, even in his day, left him open to accusations of mysticism. No doubt under the influence of Orphism, an Ancient Greek mystery religion that arose from pre-Hellenic beliefs and the Thracian cult of Zagreus, he believed in the transmigration of the soul; that is, in the reincarnation of the soul over time into the bodies of human beings, animals, or plants (metempsychosis) until such a time as it became moral. He claimed to have lived four lives and to remember them all in great detail, and once recognized the cry of his dead friend in the yelping of a puppy. After his death, the Pythagoreans deified him, and attributed him with a golden thigh and the gift of bilocation (being in two places at once). But in his own lifetime Pythagoras had always been a paragon of modesty, declining to be called a ‘wise man’ or ‘sophos’, and preferring instead to be called ‘a lover of wisdom’ or ‘philosophos’—thereby coining the term ‘philosopher’.

It is said that Numa Pompilius, the second king of Rome, had been schooled by Pythagoras, whence his great wisdom and piety. This story is referred to and discredited by Plutarch and Livy, not least because the dates do not tally, with Pythagoras having lived from about 570 to 500 BC, and so considerably later than the semi-legendary King Numa. Even though Pythagoras and Numa never met, Pythagoras and the Pythagoreans exerted a strong influence on the Roman mind. In the Tusculan Disputations, Cicero indicates that Pythagoras rose to fame in southern Italy at just the same time that Brutus brought an end to the monarchy, and that many Roman usages derived from the Pythagoreans. Unfortunately, he does not elaborate on the nature of these usages. In his Natural History, Pliny the Elder tells us that, in 343 BC, during the war with the Samnites, the god Apollo ordered the Romans to erect one statue to the wisest and another to the bravest of all Greeks, with their choices falling upon Pythagoras for the former and Alcibiades for the latter. Pliny expresses surprise that they picked Pythagoras over Socrates, whom Apollo himself had called the wisest of all men. But the fact is that the Romans liked to think of the Greek-Italian Pythagoras as their very own philosopher, and spun all sorts of stories, such as the one about Numa, to better appropriate him.

Apart from this, Pythagoras also exerted a strong indirect influence on Roman thinking, and indeed on all philosophy and theology, through the teachings of Plato, the principal architect of the western mind. Aristotle, who was Plato’s pupil of twenty years, claimed that his master’s teachings owed much to those of Pythagoras; so much, in fact, that, in his History of Western Philosophy, Bertrand Russell upheld not Plato but Pythagoras as the most influential of all Western philosophers. Pythagoras’ influence is especially evident in Plato’s mystical approach to the soul and in his emphasis on mathematics and, more generally, abstract thinking, as a secure basis for the practice of philosophy.

Advertisements

Are you an Epicurean?

Epicurus of Samos, who flourished not long after Aristotle died, founded a school of philosophy that convened at his home and garden in Athens and that dedicated itself to attaining happiness through the exercise of reason and the application of rational principles. According to Epicurus, reason teaches that pleasure is good and that pain is bad, and that pleasure and pain are the ultimate measures of good and bad. This has often been misconstrued as a call for rampant hedonism, rather than the absence of pain and tranquillity of mind that Epicurus actually intended. Indeed, Epicurus explicitly warned against overindulgence, because overindulgence so often leads to pain.

Epicurus wrote prolifically, but the early Christians thought of him as especially ungodly among the ancient philosophers, and almost none of his works survived their disapprobation. Epicurus held that the gods exist, but that they have absolutely no concern for, or even awareness of, humankind. Indeed, for them to get involved in the menial matters of men would be to perturb the supreme happiness and tranquillity that characterises and even defines them. Human beings should seek to emulate the gods in their supreme happiness and tranquillity, but they need not to fear them.

Neither need they to fear death, this for two principal reasons. (1) The mind of a person is a part of his body, and, just like other parts of his body (and everything else in the universe), it consists of atoms. The death of the person entails the death of both his body and his mind and the dispersion of their atoms. As there is no longer any person to be troubled, death cannot trouble the person after he is dead. And if death cannot trouble the person after he is dead, then nor should it trouble him while he is alive (this is the famous ‘no subject of harm argument’). (2) The eternity that comes before a person’s birth is not regarded as an evil. Therefore, nor should the eternity that comes after his death (this is the famous ‘symmetry argument’).

Epicurus himself died at the age of 72 from renal colic (kidney stones), which is associated with one of the sharpest and most intense of all bodily pains. On the last day of his life, he penned this remarkable letter to his friend and follower Idomeneus, which is nothing if not a testament to the overriding powers of philosophy.

I have written this letter to you on a happy day to me, which is also the last day of my life. For I have been attacked by a painful inability to urinate, and also dysentery, so violent that nothing can be added to the violence of my sufferings. But the cheerfulness of my mind, which comes from the recollection of all my philosophical contemplation, counterbalances all these afflictions. And I beg you to take care of the children of Metrodorus, in a manner worthy of the devotion shown by the young man to me, and to philosophy.

Epicurus agrees with Aristotle that happiness is an end-in-itself and the highest good of human living. However, he identifies happiness with the pursuit of pleasure and the avoidance of pain rather than with the pure exercise of reason. Pleasure is the highest good, and anything else that is good is so only by virtue of the immediate or deferred pleasure that it can procure. The behaviour of infants confirms that human beings instinctively pursue pleasure and that all of their actions, including those that may be construed as being either virtuous or altruistic, are ultimately aimed at obtaining pleasure for themselves. Just as human beings can immediately feel that something is hot or cold, colourful or dull, so they can immediately feel that something is pleasurable or painful. However, not everything that is pleasurable should be pursued, and not everything that is painful should be avoided. Instead, a kind of hedonistic calculus should be applied to determine which things are most likely to result in the greatest pleasure over time, and it is above all this hedonistic calculus that people seem unable to handle.

To help them a bit, Epicurus proceeds to distinguish between two different types of pleasure, ‘moving pleasures’ and ‘static pleasures’. Moving pleasures involve the satisfying of a desire, for example, eating a meal when hungry. Static pleasures on the other hand involve the state of having had a desire satisfied, for example, feeling sated after eating a meal. Static pleasures, says Epicurus, are better than moving pleasures because they free us from the pain of need or want. Epicurus also distinguishes between physical and mental pleasures and pains, and argues that anxiety about the future, especially fear of the gods and fear of death, are the greatest obstructions to happiness. To attain a state of perfect mental tranquillity or ataraxia, a person needs to avoid anxiety, which he can do by learning to trust in the future.

Pleasure often arises from the satisfaction of desire and pain from its frustration. Thus, any desire should either be satisfied to yield pleasure or eliminated to avoid pain, and, overall, it is elimination that should be preferred. There are, Epicurus says, three types of desires, (1) natural and necessary desires such as those for food and shelter which are difficult to eliminate but naturally limited and both easy and highly pleasurable to satisfy, (2) natural but non-necessary desires such as those for luxury food and accommodation, and (3) vain desires such as those for fame, power, or wealth which are inculcated by society and which are not naturally limited and neither easy nor highly pleasurable to satisfy. Natural and necessary desires should be satisfied, natural but non-necessary desires can be satisfied but should not be depended upon, and vain desires should be entirely eliminated. By following this prescription for the selective elimination of desires, a person can minimise the pain and anxiety of harbouring unfulfilled desires, and thereby bring himself as close as possible to ataraxia. Given the prime importance that he attaches to the avoidance of pain, the elimination of desire, and peace of mind, Epicurus is far more of a ‘tranquillist’ than a hedonist. ‘If thou wilt make a man happy’, he says, ‘add not unto his riches but take away from his desires.’

Adapted from

Do you make time to think about death?

If I take death into my life, acknowledge it, and face it squarely, I will free myself from the anxiety of death and the pettiness of life – and only then will I be free to become myself.
– Martin Heidegger

In his influential paper of 1970, tersely entitled Death, the philosopher Thomas Nagel asks the question: if death is the permanent end of our existence, is it an evil? Either it is an evil because it deprives us of life, or it is a mere blank because there is no subject left to experience the loss. Thus, if death is an evil, this is not in virtue of any positive attributes that it has, but in virtue of what it deprives us from, namely, life. For Nagel, the bare experience of life is intrinsically valuable, regardless of the balance of its good and bad elements.

The longer one is alive, the more one ‘accumulates’ life. In contrast, death cannot be accumulated – it is not, as Nagel puts it, ‘an evil of which Shakespeare has so far received a larger portion than Proust’. Most people would not consider the temporary suspension of life as an evil, nor would they regard the long period of time before they were born as an evil. Therefore, if death is an evil, this is not because it involves a period of non-existence, but because it deprives us of life.

Nagel raises three objections to this view, but only so as to counter them later on. First, it is doubtful whether anything can be an evil unless it actually causes displeasure. Second, in the case of death, there does not appear to be a subject to suffer an evil. As long as a person exists, he has not yet died, and once he has died, he no longer exists. Thus, there seems to be no time at which the evil of death might occur. Third, if most people would not regard the long period before they were born as an evil, then why should they regard the period after they are dead any differently?

Nagel counters these three objections by arguing that the good or evil that befalls a person depends on his history and possibilities rather than on his momentary state, and thus that he can suffer an evil even if he is not here to experience it. For example, if an intelligent person receives a head injury that reduces his mental state to that of a contented infant, this should be considered a serious ill even if the person himself (in his current state) is unable to comprehend it. In other words, if the three objections are invalid, it is essentially because they ignore the direction of time. Even though a person cannot survive his death, he can still suffer an evil; and even though he does not exist during the time before his birth or during the time after his death, the time after his death is time of which he has been deprived, time in which he could have continued to enjoy the good of living.

The question remains as to whether the non-realisation of further life is an absolute evil, or whether this depends on what can naturally be hoped for: the death of Keats at 24 is commonly regarded as tragic, but that of Tolstoy at 82 is not. ‘The trouble,’ says Nagel, ‘is that life familiarises us with the goods of which death deprives us … Death, no matter how inevitable, is an abrupt cancellation of indefinitely extensive goods.’

Given the sheer pain of this conclusion, it is hardly surprising that philosophers throughout the ages have sought, more or less unsuccessfully, to undermine it. Death not only deprives us of life, but also compels us to spend the life that it deprives us from in the mostly unconscious fear of this deprivation. And it is precisely this unconscious fear that holds us back from exercising choice and freedom. In short, death is an evil not only because it deprives us of life, but also because it mars whatever little life we do have. While we may be able to somewhat postpone our death, there is absolutely nothing that we can do to prevent it altogether. In the words of the ancient philosopher Epicurus, ‘It is possible to provide security against other ills, but as far as death is concerned, we men live in a city without walls.’ All that we can do is to come to terms with death in the hope of preventing it from preventing us from making the most of our life.

Adapted from