I have read some of Pythagoras' commandments; they are strange. Any idea why he considered that eating beans was a sin?
The book from which I took the data is 'History of Western Philosophy' by Bertrand Russell, p 50&51 and 74.
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Beans were anciently used in casting votes by balloting, the white beans for affirmative and the black ones negative. When Pythagoras said to his disciples, "Abstain from beans," he had no reference to them as an article of diet, for he ate them himself. What he did mean, and what his immediate followers already understood, was that they should abstain from the intrigues of politics as being antagonistic to a philosopher's pursuits.
It also couched a warning of the danger of criticising the popular government.
—HL Sumner, "The Beans of Pythagoras" The Path – February 1888
I defer to Gareth Rees much elaborated, well-sourced answer. I'm not sure how I found the above quote, but the Theosophical Society is an embarrassingly terrible source.
TL;DR: No-one knew the reason for the rule in antiquity (if there was indeed such a rule), and certainly no-one knows it now.
Aristotle had a lot of theories, which indicates that he did not know the reason and was guessing or passing on guesses:
According to Aristotle in his work On the Pythagoreans, Pythagoras counselled abstinence from beans either because they are like the genitals, or because they are like the gates of Hades … as being alone unjointed, or because they are injurious, or because they are like the form of the universe, or because they belong to oligarchy, since they are used in election by lot.
Diogenes Laertius (3rd century). Lives of Eminent Philosophers 8.1. Translated by R. D. Hicks. Perseus Digital Library.
Cicero related a theory that beans had been forbidden because they disturbed the mind, but also indicated some skepticism:
Now Plato’s advice to us is to set out for the land of dreams with bodies so prepared that no error or confusion may assail the soul. For this reason, it is thought, the Pythagoreans were forbidden to indulge in beans; for that food produces great flatulence and induces a condition at war with a soul in search of truth.
Marcus Tullius Cicero (c. 44 BCE). On Divination 1.62. Translated by William Armistead Falconer (1923). Perseus Digital Library.
Cicero’s theory was repeated by Diogenes Laertius:
He [Pythagoras] is said to have advised his disciples as follows: […] To abstain from beans because they are flatulent and partake most of the breath of life; and besides, it is better for the stomach if they are not taken, and this again will make our dreams in sleep smooth and untroubled.
It was also repeated by Pliny, who added a theory about beans “containing the souls of the dead”, perhaps a reference to the theory of metempsychosis, though he seems to have been skeptical:
Beans are mostly eaten together with other food, but it is generally thought that they dull the senses, and cause sleepless nights attended with dreams. Hence it is that the bean has been condemned by Pythagoras; though, according to some, the reason for this denunciation was the belief which he entertained that the souls of the dead are enclosed in the bean.
Pliny the Elder (c. 79). Natural History 18.30. Translated by John Bostock & H. T. Riley (1855). Perseus Digital Library.
Plutarch expanded on Aristotle’s guess that Pythagoras was not offering dietary advice but advising his followers to keep out of politics. (I assume this was the main source of the text quoted in the other answer.)
This duty Pythagoras also has enjoined in the form of allegories which I shall now quote and explain. For they contribute no small influence towards the acquisition of virtue. For example […] ‘Abstain from beans’; means that a man should keep out of politics, for beans were used in earlier times for voting upon the removal of magistrates from office.
Plutarch (1st–2nd century). The Education of Children 17. Translated by Frank Cole Babbitt (1927). Perseus Digital Library.
But Plutarch can’t have had much confidence in this theory, because elsewhere he wrote:
Why is it the customary rule that those who are practising holy living must abstain from legumes?
Did they, like the followers of Pythagoras, religiously abstain from beans for the reasons which are commonly offered, and from vetch and chickpea, because their names (lathyros and erebinthos) suggest Lethê and Erebus?
Or is it because they make particular use of legumes for funeral feasts and invocations of the dead?
Or is it rather because one must keep the body clean and light for purposes of holy living and lustration? Now legumes are a flatulent food and produce surplus matter that requires much purgation.
Or is it because the windy and flatulent quality of the food stimulates desire?
Plutarch (1st–2nd century). The Roman Questions 95. Translated by Frank Cole Babbitt (1936). Perseus Digital Library.
Lucian repeated a couple of Aristotle’s theories and added a third, not attested elsewhere:
First Dealer And why no beans? Do you dislike them?
Pythagoras No. But they are sacred things. Their nature is a mystery. Consider them first in their generative aspect; take a green one and peel it, and you will see what I mean. Again, boil one and expose it to moonlight for a proper number of nights, and you have—blood. What is more, the Athenians use beans to vote with.
Lucian (2nd century). Sale of Creeds. Translated by H. W. Fowler and F. G. Fowler (1905). The Works of Lucian of Samosata, volume 1, p. 192. Oxford: Clarendon.
All the evidence quoted above suggests that the Pythagoreans did in fact have a rule forbidding the eating of beans, even if no-one knew the reason. But this is thrown into doubt by the discussion of the supposed rule in Aulus Gellius. He quoted the passage from Cicero, given above, and then continued:
But Aristoxenus the musician, a man thoroughly versed in early literature, a pupil of the philosopher Aristotle, in the book On Pythagoras which he has left us, says that Pythagoras used no vegetable more often than beans, since that food gently loosened the bowels and relieved them. I add Aristoxenus’ own words: “Pythagoras among vegetables especially recommended the bean, saying that it was both digestible and loosening; and therefore he most frequently made use of it.” […]
Furthermore, the reason for the mistaken idea about abstaining from beans seems to be, that in a poem of Empedocles, who was a follower of Pythagoras, this line is found:
O wretches, utter wretches, from beans [κυάμων] withhold your hands.
For most men thought that κυάμους meant the vegetable, according to the common use of the word. But those who have studied the poems of Empedocles with greater care and knowledge say that here κυάμους refers to the testicles, and that after the Pythagorean manner they were called in a covert and symbolic way κύαμοι, because they are the cause of pregnancy [αἴτιοι τοῦ κυεῖν] and furnish the power for human generation: and that therefore Empedocles in that verse desired to keep men, not from eating beans, but from excess in venery.
Aulus Gellius (2nd century). Attic Nights 4.11. Translated by John C. Rolfe (1927). Perseus Digital Library.
Plutarch passed on a similar theory, except that in his version, by “beans” the Pythagoreans meant “eggs” and not “testicles”. (Plutarch clearly did not believe the theory.)
When upon a dream I had forborne eggs a long time […] some at Sossius Senecio’s table suspected that I was tainted with Orpheus’s or Pythagoras’s opinions, and refused to eat an egg (as some do the heart and brain) imagining it to be the principle of generation. And Alexander the Epicurean ridiculingly repeated,—
To feed on beans and parents’ heads
Is equal sin;†
as if the Pythagoreans covertly meant eggs by the word κύαμοι (beans), deriving it from κύω or κυέω (to conceive), and thought it as unlawful to feed on eggs as on the animals that lay them. Now to pretend a dream for the cause of my abstaining, to an Epicurean, had been a defence more irrational than the cause itself; and therefore I suffered jocose Alexander to enjoy his opinion, for he was a pleasant man and excellently learned.
Plutarch (1st–2nd century). Table-Talk 2.3. Translated by William W. Goodwin (1874). Perseus Digital Library.
† Evidently this was a well-known expression of the Pythagorean rule, though its author is not known. Lucian satirized these lines a couple of times. In Dialogues of the Dead 20 Pythagoras says, “I have here [in Hades] learnt, that beans and the heads of our parents have nothing in common between them”, and in The Cock 4 Micyllus addresses his cockerel Alectryon, who claims to be the reincarnation of Pythagoras: “you have broken your own law, and by eating beans have been guilty of as much impiety as if you had devoured your father’s head”.
Those are all the primary sources that I was able to locate. They show that many people believed that the Pythagoreans had a rule forbidding beans, but that no-one knew the reason for it. Aristoxenus’ denial of the rule is not dispositive: we can explain it away, if needed, by supposing that he had concealed the truth in order to defend the sect to which he belonged from accusations of eccentricity.
If I had to synthesize my own theory out of these sources, to explain why so many writers were confident that there was such a rule, and yet no-one knew the reason, I would start by noting that abstinence from beans seems to have been a traditional form of religious dietary restriction: this is described by Plutarch in Roman Questions, quoted above, and also by Herodotus: “The Egyptians sow no beans in their country; if any grow, they will not eat them either raw or cooked; the priests cannot endure even to see them, considering beans an unclean kind of legume” (Histories 2.37). Herodotus also comments on the Egyptian origin of some Greek religious beliefs:
The Egyptians were the first who maintained the following doctrine, too, that the human soul is immortal, and at the death of the body enters into some other living thing then coming to birth; and after passing through all creatures of land, sea, and air, it enters once more into a human body at birth, a cycle which it completes in three thousand years. There are Greeks who have used this doctrine, some earlier and some later, as if it were their own; I know their names, but do not record them.
Herodotus (440 BCE). The Histories 2.123. Translated by A. D. Godley (1920).Perseus Digital Library.
Pythagoreanism was a cult religion that seems to have propounded the immortality of souls, so perhaps some of their doctrines and practices were inherited from Egyptian religion as suggested by Herodotus, and this might have included the prohibition on beans: adopted, like many religious practices, because it was traditional and not because there was a rational explanation.