Pythagoras was specifically known for accumulating the wisdom of diverse cultures—supposedly he met Thales, was initiated as Egyptian priest in Hermopolis, spent time in Babylon after being captured, and was initiated into every mystery cult he could. And as a boy on his home island of Samos, he would have been exposed to the building of the largest stone temple in Ancient Greece (to Hera) and the incredible engineering feat of the tunnel of Eupalinos.
Iamblichus’s “life of Pythagoras” [1] is worth a read as he had access to all the old sources now lost. The relationship between math and spirituality was very strong back then!
There are lots of fun stories that may be true but no one will ever know. In Diogenes Laertius’ “Lives of the Philosophers,” it is claimed that when Pythagoras made his discovery of what we call the Pythagorean theorem, he sacrificed 100 oxen (a hecatomb) [1]. As noted by Charles Dodgson (Lewis Carrol), “that would produce an inconvenient supply of meat” [3], especially for a vegetarian. Iamblichus, on the other hand, claims it was a single ox — and made of flour!
[1] Guthrie, K. S., & Fideler, D. R. (Eds.). (1987). The Pythagorean sourcebook and library: an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Red Wheel/Weiser.
[2] “he sacrificed a hecatomb, when he had discovered that the square of the hypotenuse of a right-angled triangle was equal to the squares of the sides containing the right angle.” DL, found in [1]
[3] Maor, E. (2019). The Pythagorean theorem: a 4,000-year history. Princeton University Press.
Oh how can you skip the most fun tale of them all. [1] Its authenticity is dubious, but there's probably at least parts of truth in it. In a nutshell, Pythagoras started a cult based around numbers, and the pseudo-divine purity of rational numbers - of which everything can be represented.
Hippasus, a member of said cult, however managed to compellingly demonstrate that the square root of 2 could not be a rational number. Pythagorus tried to swear him to silence. When that did no twork, he had him killed. Over the square root of 2.
I don’t know of any ancient sources claiming that Pythagoras killed Hippasus. Seems unlikely to me! In any case, Hippasus was a fascinating figure:
“Aristoxenus (Fr. 90 Wehrli = DK I 109. 31 ff.) reports that Hippasus prepared four bronze disks of equal diameters, whose thicknesses were in the given ratios, and it is true that, if free hanging disks of equal diameter are struck, the sound produced by, e.g., a disk half as thick as another will be an octave apart from the sound produced by the other disk (Burkert 1972a, 377). Hippasus, thus, may be the first person to devise an experiment to show that a physical law can be expressed mathematically (Zhmud 2012a, 310).” [1]
Also in [2] this experiment is claimed to be the first documented scientific experiment in history. After all, Hippasus took a mathematical model for a physical phenomenon (how consonance relates to the mathematical ratios of a musical string) and tests the generalization of that model in a another physical medium (viz. bronze chimes with the same ratios 1:2 and 2:3 make the octave and fifth).
I wish they’d put this stuff in elementary math books when kids learn about Pythagoras.
Well, they continue to introduce Pythagorean theorem through many stages of elementary math, so 8 is pretty early. But.. here could be a section of a textbook:
Pythagoras & The Pythagoreans: Mathematics, Music, and Mystery
Pythagoras was an ancient Greek mathematician and philosopher who lived around 500 BC. He traveled widely and gathered knowledge from diverse cultures, like Egypt. He also founded his own school of men and women in Italy. There, he taught that numbers held the key to understanding the whole universe. Pythagoras is best known for the Pythagorean theorem.
Fun Fact! There are stories that when Pythagoras discovered his famous theorem, he celebrated in a big way. Some say he sacrificed 100 oxen, while others claim it was just an ox made of flour. The Pythagoreans were famously vegetarian, so what do you think?
Math & Music: Pythagoreans explored the relationship between math and music.
They discovered that musical notes have mathematical relationships. Hippasus, a member of the Pythagorean community, used bronze disks to show that musical notes are connected to mathematical ratios. This is considered one of the first scientific experiments!
Activity: Using a stringed instrument, like a guitar, try plucking the strings when pressing at 1/2 the string or 2/3s the string. Try different fractions. Can you hear the mathematical relationships in the sounds?
I have thoroughly enjoyed all of your comments! You’re great at showing how learning can be fun. Do you have a blog where you compile these stories and anecdotes?
When I was 8 I would sometimes watch a Disney VHS tape that had Donald Duck exploring stuff like this. The golden ratio, octaves of sound/music, pythagorean theorem etc. I loved it.
Which is why hands-on learning combined with source material is best practice for children, imo. Taking several disks and demonstrating that the thicknesses double but diameter is the same, letting the kids try hitting them and noting the differences in tone.
Asking the children "why do you think these sound different?" without telling them why first, letting them explore it themselves, gently correcting them along the way. Then pulling out another example and encouraging them to apply the same logic to the new example.
No, instead for the most part, children _still_ get what we all had: memorise these facts without explanation and fill out the blank space on the exam where the memorised words get you points. Most of our entire schooling is just an exercise in memorisation, not in thought and it's sad.
Pythagoras definitely did not have him killed since he had been dead for ~50 years. So even if the story is true it is about Pythagoreans, the followers of Pythagoras.
"How can it equal one? If one times one equals one that means that two is of no value because one times itself has no effect. One times one equals two because the square root of four is two, so what's the square root of two? Should be one, but we're told its two, and that cannot be."
But his success has not stopped the actor from claiming he spends up to 17 hours a day creating nameless plastic structures, which are made of cut up pieces of plastic and either stitched together with copper wire or soldered, that he believes prove his new form of mathematics.
Howard studied chemical engineering at the Pratt Institute in Brooklyn until he fell out with one of his professors over the answer to the 1x1=1 conundrum.
It says he drowned... assumed because of the reason OP gave. Besides, moot point giving you're referencing Wikipedia to begin with.
> Hippasus is sometimes credited with the discovery of the existence of irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them. However, the evidence linking the discovery to Hippasus is unclear.
> Pappus merely says that the knowledge of irrational numbers originated in the Pythagorean school, and that the member who first divulged the secret perished by drowning.
It also says “one writer even has Pythagoras himself "to his eternal shame" sentencing Hippasus to death by drowning, for showing "that
2√2 is an irrational number”.”
> supposedly he met Thales, was initiated as Egyptian priest in Hermopolis, spent time in Babylon after being captured, and was initiated into every mystery cult he could
It should be noted that such claims were made about many ancient sages to boost their "wisdom pedigree". Plato is said by ancient sources to have traveled to Egypt and Italy, but my understanding is that most modern scholars doubt that those journeys really happened.
Pythagoras’s father was a gem trader from the Levantine coast so the likelihood of him traveling around the Mediterranean is pretty decent. His Egyptian connection is attested early by Herodotus and his devotion to diverse schools of wisdom is attested in a critical statement by his contemporary, Heraclitus.
Of course it’s hard to know. But Plato wrote several letters about his trip to Italy, because he was briefly enslaved there before being freed by his friend the Pythagorean Archytas (who is famous for creating a steam powered flying machine and wrote a work on mechanical engineering).
https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1...
> The relationship between math and spirituality was very strong back then!
On Seymour Cray:
Another favorite pastime was digging a tunnel under his home; he attributed the secret of his success to "visits by elves" while he worked in the tunnel: "While I'm digging in the tunnel, the elves will often come to me with solutions to my problem." [0]
That sounds like he was being figurative. He probably meant that he got ideas while working.
Every morning, I get up at 5, and take a 5K walk. During that time, I tend to "triage" the day ahead, and often solve problems that were vexing me, the night before.
Part of my walk is around a local high school track. There is a small flock of killdeer birds, that hang out there, and I guess they give me the ideas I have, as they often come to me, at that point in my walk.
I enjoyed this part:
> One story has it that when Cray was asked by management to provide detailed one-year and five-year plans for his next machine, he simply wrote, "Five-year goal: Build the biggest computer in the world. One year goal: One-fifth of the above." And another time, when expected to write a multi-page detailed status report for the company executives, Cray's two sentence report read: "Activity is progressing satisfactorily as outlined under the June plan. There have been no significant changes or deviations from the June plan."
I'm actually curious if "digging in the tunnel" is a euphemism for using psychedelics, which would have been severely frowned upon at the time. The elves are known for their wisdom in helping you see the world and solve problems in unique ways.
> FRANCIS CRICK, the Nobel Prize-winning father of modern genetics, was under the influence of LSD when he first deduced the double-helix structure of DNA nearly 50 years ago.
> The abrasive and unorthodox Crick and his brilliant American co- researcher James Watson famously celebrated their eureka moment in March 1953 by running from the now legendary Cavendish Laboratory in Cambridge to the nearby Eagle pub, where they announced over pints of bitter that they had discovered the secret of life.
> Crick, who died ten days ago, aged 88, later told a fellow scientist that he often used small doses of LSD then an experimental drug used in psychotherapy to boost his powers of thought. He said it was LSD, not the Eagle’s warm beer, that helped him to unravel the structure of DNA, the discovery that won him the Nobel Prize.
DMT is pretty famous for, oddly, inducing hallucinations of Machine Elves in people. Don't discard the building tunnels part though, it is a weirdly popular hobby among the very male-brained.
Funny enough DMT is the only psychedelic that's ever given me real, true insight and legitimately changed my life after a single breakthrough dose, 99.5% of which I didn't remember even immediately after 'coming to' (and still don't). I was suicidally depressed and had been for a number of years, struggling with drugs and depression, had all but given up on life. I did it alone not meaning to break through.
Immediately after my breakthrough, I was so overjoyed and happy and couldn't help but bellow out "there is so much love in the universe!!" over and over -- right after a run to the washroom to purge a rainbow from my mouth. Thankfully I was home alone. I immediately called my loved ones to talk to them and tell them how much I loved them. My sister (psych major) thought I was going to end my life soon because I was suddenly so happy and unburdened. Suicidal ideation ceased for over 5 years, and still has only come out in the rare circumstance I was under extreme emotional distress. The majority of the lessons I learned -- many obvious in retrospect, I feel I was almost purposely blinding myself from seeing them because it was easier not to -- came in dreams and waking daydream 'visions' in the days after my trip, and my sudden ability to notice my problems weren't so serious, I just needed to look at them from a different viewpoint, of which there are many.
The only person who noticed (or mentioned) any difference right away in me, was the wife of a client -- a devout Hindu woman. As soon as she saw me a few days after my trip she said "MyName, are you alright?" with a concerned look on her face. I still wonder what she noticed or what concerned her.
I haven't felt the need to do it or any other psychedelics since, but for some reason I felt I'd share a quick tale of my story since the topic of machine elves and insights came up.
I was also, coincidentally (common as it is, apparently?) obsessed with digging a tunnel as far as I could as a 12-13 year old boy until it collapsed on me after about 5ft because I'd started on an unstable hill. That was the end of my tunneling, although I suddenly have a strange urge to grab a shovel.
I got another one - Kary Mullis, PCR inventor, revolutionized genetics and biochemistry:
„In it, he recounts his late-night encounter with a glowing raccoon that spoke to him, addressing him as "doctor" -- a raccoon that may or may not have been an alien. He tells of passing out after inhaling too much nitrous oxide and later learning that he'd been saved from a fatal overdose by a woman who traveled to him on an "astral plane."“ [0]
> "The relationship between math and spirituality was very strong back then!"
Seems to me this connection is having a resurgence now as people attempt to project the capabilities of computers beyond human capabilities, where the implications are necessarily bordering on spiritual.
Examples include transhumanism, Yudkowsky-style absurdly extrapolated rationalism (Basilisk etc), the simulation hypothesis, AGI salvation hopes...
George Boole (Who we get Boolean from), had a road to Damascus experience as a teenager and later wrote "An Investigation of the Laws of Thought" where we get our 0s and 1s from, and truth operations, largely to prove God is Good.
The concept of 0 and 1, absolute truth and false, was born from his spiritual views.
They are often ways to transmute spirituality into a form more palatable to contemporary sensibilities trained on technology.
Who runs the universal simulation? Another name for God. The eternal torment of Roko’s Basilisk? Another name for Hell. AGI will save us from ourselves with its incomprehensible intelligence? A cyber-Jesus. Moving your mind into a transhumanist body? Souls rising to Heaven. Etc.
"The relationship between math and spirituality was very strong back then!"
According to those whose communication ended up being indelible. I wonder how we'll be able to preserve digital info for millennia?
Now I’m very much interested in reading more about his life! Gotta say tho, sounds a bit like an ancient Marco Polo where maybe the myth is larger than the man at this point.
Still going to read more. Thanks for the citations.
Try to get a copy of the Pythagorean Sourcebook! It’s great reading original sources. So satisfying and often more interesting than modern scholarship about them.
Iamblichus’s “life of Pythagoras” [1] is worth a read as he had access to all the old sources now lost. The relationship between math and spirituality was very strong back then!
There are lots of fun stories that may be true but no one will ever know. In Diogenes Laertius’ “Lives of the Philosophers,” it is claimed that when Pythagoras made his discovery of what we call the Pythagorean theorem, he sacrificed 100 oxen (a hecatomb) [1]. As noted by Charles Dodgson (Lewis Carrol), “that would produce an inconvenient supply of meat” [3], especially for a vegetarian. Iamblichus, on the other hand, claims it was a single ox — and made of flour!
[1] Guthrie, K. S., & Fideler, D. R. (Eds.). (1987). The Pythagorean sourcebook and library: an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Red Wheel/Weiser.
[2] “he sacrificed a hecatomb, when he had discovered that the square of the hypotenuse of a right-angled triangle was equal to the squares of the sides containing the right angle.” DL, found in [1]
[3] Maor, E. (2019). The Pythagorean theorem: a 4,000-year history. Princeton University Press.