[HN Gopher] Pythagorean Theorem found on clay tablet 1k years ol...
___________________________________________________________________
Pythagorean Theorem found on clay tablet 1k years older than
Pythagoras (2009)
Author : samaysharma
Score : 749 points
Date : 2023-10-05 03:19 UTC (19 hours ago)
(HTM) web link (link.springer.com)
(TXT) w3m dump (link.springer.com)
| pyninja wrote:
| So many red flags in the abstract alone. It's no surprise that
| the article itself looks like a middle school project.
| leke wrote:
| Kind of reminds me of that "Bro, I stole your code" - "It's not
| my code" meme.
| tiffanyh wrote:
| Mass publication.
|
| People typically wrongly attribute findings not to the person who
| first discovered it, but to the person who was able to most
| widely communicate/publish about it.
|
| This is a particular difficult challenge in ancient times.
|
| Knowledge was often shared verbally, not in written form.
|
| Or if it was in written form, the material used has long since
| deteriorated.
|
| So most of what we know about ancient thinking is based on
| knowledge that was so widely known and written that there's
| multiple copies of it; or the knowledge was communicated on a
| hard material like stone (egyptian hieroglyphics) ... but that
| doesn't mean it was the first ancients knew of it, it just means
| that the particular knowledge in written form has lasted the test
| of time the longest.
| eruci wrote:
| Prior art - Even Pythagoras didn't come up with Pythagoras's
| Theorem.
| zeroonetwothree wrote:
| The original example of Stigler's law of eponymy.
| fritzo wrote:
| Just curious, who came up with Stigler's law?
| mcphage wrote:
| Robert K. Merton
| chx wrote:
| As a side note Dijkstra has a wonderful generalization and it has
| a nice proof at https://www.cut-the-
| knot.org/pythagoras/Stevens.shtml and I was wondering whether
| this is old as well.
| bannedbybros wrote:
| [dead]
| xwowsersx wrote:
| > who uses the theorem two decades later for something about
| relatively
|
| I assume that should read relativity*?
| [deleted]
| abhinai wrote:
| It needs to be renamed as the Tablet Theorem.
| dylan604 wrote:
| [flagged]
| xxs wrote:
| As others have mentioned - the tablet doesn't provide a proof,
| so it's not a theorem.
| personjerry wrote:
| I know it was known in China at least before Pythagoras
| https://en.wikipedia.org/wiki/Zhoubi_Suanjing
| User23 wrote:
| Perhaps it was, but the article you link denies the certainty
| of that statement.
| Vt71fcAqt7 wrote:
| Has anyone found the source for that tablet? All I have found is
| this:
|
| >Note that quite a few descriptions on Babylonian tablets seem to
| cite a translation of a Pythagorean algorithm from a ca. 1900BC
| tablet by a Dennis Ramsey - I have not been able to find the
| original source of this anywhere.[0]
|
| The linked arctile cites wikipedia and bible-history.com. This
| book[1] misquotes the supposed tablet as being ycb 7289, probably
| because these tablets are referenced next to each other on
| wikipedia.
|
| This[2] website says it's in the British museum.
|
| [0]https://craftofcoding.wordpress.com/author/spqr/
|
| [1]https://books.google.com/books?id=XDBCEAAAQBAJ&pg=PT257&lpg=..
| .
|
| [2]https://mathshistory.st-
| andrews.ac.uk/HistTopics/Babylonian_...
| zestyping wrote:
| The source of the photo is cited in the article:
|
| https://personal.math.ubc.ca/~cass/Euclid/ybc/ybc.html
|
| This site has a detailed analysis and explains that it's from
| the Yale Babylonian Collection.
| Vt71fcAqt7 wrote:
| I am refering to this quote:
|
| > _4 is the length and 5 the diagonal. What is the breadth ?
| Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You
| take 16 from 25 and there remains 9. What times what shall I
| take in order to get 9 ? 3 times 3 is 9. 3 is the breadth._
|
| That doesn't appear to be in your link - am I wrong? I only
| see numbers there and it appears to be talking about
| something else entirely.
| renewiltord wrote:
| Stigler's law of eponymy
| https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy
| freework wrote:
| I've seen this tablet before, and an not convinced that it is
| actually the Pythagorean theorem. Its just a tablet with some
| tick marks inscribed onto it, along with a circular looking
| thing. It's very much a stretch to say the person who etched
| those markings intended to express the Pythagorean theorem.
|
| There was a point in time when I was very interested in ancient
| civilizations from Mesopotamia, but in more recent years I an way
| less interested in it. The scholarship in that field is just
| terrible. In my opinion, a lot of the stuff is on par with alien
| "investigators" and stuff like that, yet for some reason the
| general public sees the field as totally legit.
| [deleted]
| detourdog wrote:
| I don't mind the poor scholarship it offers opportunities to
| have better ideas. What I love about the ancients is that they
| are just like us with fewer objects.
| mandmandam wrote:
| I would argue with you, but you've just produced a bunch of
| squiggles.
|
| > It's very much a stretch to say the person who etched those
| markings intended to express the Pythagorean theorem.
|
| No it isn't.
|
| There are legit reasons to question a lot of the research on
| ancient civs, but _that_ isn 't one of them.
| freework wrote:
| How is this different from seeing a fuzzy video of some
| lights in the sky and then coming to the conclusion that it
| is definitely a UFO? If you're so convinced that this carving
| definitely proves that the carver was intending to express
| the Pythagorean formula, then what is the evidence?
|
| Some people's definition of "evidence" is different from my
| own. If somebody really wants to believe something, then just
| about anything qualifies as evidence. This is why UFO people
| consider literally every single fuzzy video as undeniable
| proof that aliens exist.
| orf wrote:
| ... because those "squiggles" are just "words" in a
| "language" you can't "read"?
|
| And if you could read it, you would find it contains a lot
| of relevant things, concluding with:
|
| > ... 1.414213, which is nothing other than the decimal
| value of the square root of 2, accurate to the nearest one
| hundred thousandth.
|
| You might then think to yourself:
|
| > The conclusion is inescapable. The Babylonians knew the
| relation between the length of the diagonal of a square and
| its side
|
| Which is all clearly explained in the article you're
| commenting on. Do you have anything else meaningful to add,
| beyond "it's nuffin' but squiggles mate" and "aliens"?
| pyninja wrote:
| You sound very confused.
| freework wrote:
| Here is a drawling of what they think this tablet says:
|
| https://commons.wikimedia.org/wiki/File:YBC_7289_sketch.s
| vg
|
| It's just a bunch of numbers scribbled onto a tablet. For
| all we know it could just be some guy writing down the
| number of sheep he is willing to sell to his neighbor or
| something. To say this tablet proves the Mesopotamian
| knew about Pythagorean's theorem is quite a stretch.
|
| To the people who want to believe, there is nothing that
| can be said. Believe what you want.
|
| Also, this tablet has no provenance. According to the
| wikipedia page on this tablet, it says "It is unknown
| where in Mesopotamia YBC 7289 comes from" Basically it
| just magically appeared one day. For all we know it could
| be faked. In any other field, this artifact would be
| ruled inauthentic. But in this field, for some reason it
| just doesn't matter.
| orf wrote:
| Imagine for a moment that the people who created the
| tablet used a different number system than us, and also
| imagine that we knew that number system and could convert
| it.
|
| Then those "bunch of numbers" becomes something else
| entirely. Specifically, they become a bunch of numbers
| that highly relate to the Pythagorean theorem.
| detourdog wrote:
| I agree Cuneiform tablets is a multistep process of recording
| an idea.
| maroonblazer wrote:
| I was surprised to learn that U.S. President Garfield devised a
| proof of Pythagoras' theorem. It got me wondering what other U.S.
| Presidents had an aptitude for math.
|
| Jefferson (3rd President) was quite fluent in geometry and
| surveying, designing his home in Monticello.
|
| Herbert Hoover (31st) was a mining engineer.
|
| Jimmy Carter (39th) was a nuclear engineer by training, having
| worked in the U.S. Navy's nuclear sub program.
| macintux wrote:
| Obligatory link for those unfamiliar with the story: Jimmy
| Carter averted a full-scale nuclear meltdown in Canada at great
| personal risk.
|
| https://www.snopes.com/fact-check/jimmy-carter-nuclear-meltd...
| kridsdale3 wrote:
| And clearly that dose of radiation, in mid-1950's comic book
| fashion, imbued his cells with unnatural longevity!
| kridsdale3 wrote:
| Jefferson also apparently published theorems on ideal geometric
| layouts of human settlement, which Salt Lake City is based on.
| gen220 wrote:
| George Washington was also a surveyor before his later
| career(s). It was a well-trodden path to wealth and status for
| people of middling backgrounds and aptitudes for math and law
| in the pre-revolutionary U.S.
| kridsdale3 wrote:
| Just as Software Engineering is today.
| Tainnor wrote:
| The article claims that the Babylonians "discovered" the
| Pythagorean theorem, but all it shows is that they (probably)
| believed it to be true.
|
| Until we have better evidence, it still seems to be the case that
| (at least in the "West", I'm unfamiliar with e.g. Chinese
| mathematics) the Greeks were the first to come up with the
| concept of a mathematical proof that is valid deductively, and
| not inductively.
| jaystraw wrote:
| Anyone in the threads I collapsed should buy a speed square and
| read the booklet. Possibly over beer or tea.
| stainablesteel wrote:
| i'm not surprised
|
| basic geometry was probably fundamental to the architecture
| necessary to make civilizations
|
| i'd bet that as far back as we can find large structure there
| would probably have been strong understandings of geometry to
| make them
|
| plus, humans have been around for 200-300k years, what we can
| find is from ~12k-25k years ago at the very fringe of our
| investigations. no doubt people have been mathematically capable
| for longer than they've been able to take full advantage of the
| concepts they understand
| voisin wrote:
| Why was this published in the Journal of Targeting, Measurement
| and Analysis for Marketing?
| dr_dshiv wrote:
| 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.
| massung wrote:
| 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.
| dr_dshiv wrote:
| 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.
| snerc wrote:
| "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?
| spaceman_2020 wrote:
| > initiated into every mystery cult he could
|
| And in turn, his followers created a strange cult after him!
| [0]
|
| 0: https://en.wikipedia.org/wiki/Pythagoreanism
| somenameforme wrote:
| 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.
|
| [1] - https://sciencefocus.ust.hk/the-square-root-of-two-at-
| the-co...
| imjonse wrote:
| 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.
| botanical wrote:
| The wiki page says he drowned:
|
| https://en.wikipedia.org/wiki/Hippasus
| moshun wrote:
| _was_ drowned was the version I heard in school.
| frankplow wrote:
| 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"."
| tjbiddle wrote:
| 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.
| hnuser123456 wrote:
| 4/1-4/3+4/5-4/7+4/9...
| dmoo wrote:
| That seems irrational...
| seasox wrote:
| At least he kept his cult real
| scarmig wrote:
| It seems pretty limiting for a cult, though. You can't
| count all the members he left out of it.
| continuitylimit wrote:
| well, if this cult doesn't work out for you there is
| always reddit.com with less "limitations" ..
| mjhay wrote:
| Sounds like a complex situation.
| selcuka wrote:
| I believe the whole story is imaginary.
| jaredsohn wrote:
| or at least more complex than described
| r13a wrote:
| Unless the story has some transcendental meaning ...
| Cyphase wrote:
| Only if you're primed to see that.
| nopurpose wrote:
| Don't you think this discussion is tangental to the
| topic?
| throwaway167 wrote:
| Do you assume a point exists?
| Cyphase wrote:
| I think that's hyperbolic. It seems like a natural
| digression. I think you're just being negative.
|
| Put another way, this discussion certainly has more than
| an infinitesimal relation to the original link.
| scarmig wrote:
| This seems like a pointless back and forth; you all
| aren't going to be able to square the circle here.
| brutusborn wrote:
| Discussing the history of Pythagoras under an article on
| the history of his most famous theorem?
|
| I think you have a very high bar for relevance.
| belter wrote:
| There must a right angle to look at it...
| RHSman2 wrote:
| Bravo
| dr_dshiv wrote:
| 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.
|
| [1]
| https://plato.stanford.edu/entries/pythagoreanism/#hippasus
|
| [2] https://www.sciencedirect.com/science/article/pii/S240587
| 262...
| detourdog wrote:
| one of my goals is a collection of educational activities
| based on the ground breaking earlier discoveries. Such as
| this.
| darkerside wrote:
| I'm pretty confused. How do you think an average 8 year old
| would feel?
| dr_dshiv wrote:
| 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?
| sotix wrote:
| 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?
| dr_dshiv wrote:
| That's very kind. I put work at https://Derek-Lomas.com
|
| (My Pythagorean math blog is yet to be, but I do have
| some Pythagorean blog posts at https://aixd.substack.com)
| pc86 wrote:
| They'd probably be fine.
| orochimaaru wrote:
| They would be fascinated. I have one at home and a 12 yr
| old. This is the kind of stuff they're always interested
| in.
| RHSman2 wrote:
| We should all be intrigued by this. Young or old. Not
| reels and shit....
| wishinghand wrote:
| 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.
| fishyjoe wrote:
| Donald Duck in Mathmagic Land.
|
| You can find the whole thing on youtube
| JKCalhoun wrote:
| Found Pythagoras:
| https://youtu.be/4m_ROtUQ5Vk?si=0V_Jt9lS_44w2QyL
| cdaringe wrote:
| HEATHENOUS DUCK! Off with his head!
|
| jk DD rocks
| [deleted]
| HideousKojima wrote:
| TIL Terrence Howard is secretly a member of Pythagoras' math
| cult:
|
| https://www.independent.co.uk/news/people/terrence-howard-
| th...
|
| "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."
| adolph wrote:
| _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._
| bzmrgonz wrote:
| Bro did Reddit/hacker-news on papyrus ...before the internet...
| :-D
| MrBuddyCasino wrote:
| > _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]
|
| [0] https://en.wikipedia.org/wiki/Seymour_Cray
| CommitSyn wrote:
| 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.
|
| https://maps.org/2004/08/08/nobel-prize-genius-crick-was-
| hig...
|
| > 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.
| MrBuddyCasino wrote:
| 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.
| CommitSyn wrote:
| 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 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.
|
| 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.
| ChrisMarshallNY wrote:
| 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."_
| 2devnull wrote:
| Cray sounds mildly autistic.
| ChrisMarshallNY wrote:
| A lot of these folks are.
| alganet wrote:
| For this particular story, he sounds more like a cynic
| making fun of some bureaucrats.
| pcdoodle wrote:
| That wiki was a fun read! It's fun to speculate about the
| elves and the tunnels.
| pavlov wrote:
| _> "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...
| carapace wrote:
| > Examples include transhumanism, Yudkowsky-style absurdly
| extrapolated rationalism (Basilisk etc), the simulation
| hypothesis, AGI salvation hopes...
|
| But those are all examples of ways to _avoid_ spirituality.
| pavlov wrote:
| 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.
| nerddadnear40 wrote:
| Always has been. My favorite example:
|
| 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.
| p-e-w wrote:
| > 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.
| dr_dshiv wrote:
| 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...
| wly_cdgr wrote:
| Impossible! Pythagoras wasn't born yet!
| layer8 wrote:
| > Why did the scribe choose a side of 30 for his example?
|
| Clearly because that makes the answer for the diagonal 42.
| bagels wrote:
| Diogenes said that Pythagoras hated beans: "One should abstain
| from fava beans, since they are full of wind and take part in the
| soul, and if one abstains from them one's stomach will be less
| noisy and one's dreams will be less oppressive and calmer."
| dekelpilli wrote:
| Maybe Pythagoras had a G6PD deficiency. It is more common in
| males and people with a Mediterranean background[0].
|
| [0] - https://www.healthdirect.gov.au/G6PD-deficiency
| mcpackieh wrote:
| I guess they hadn't yet figured out this little kernel of
| wisdom: _Beans, beans, they 're good for your heart. The more
| you eat, the more you fart. The more you fart the better you
| feel, so eat your beans with every meal._
| elteto wrote:
| I can definitely see this as Diogenes trolling Pythagoras.
| bzmrgonz wrote:
| This probably went up in smoke along soo many other knowledge
| gems at the Alexandria Library!!
| the_origami_fox wrote:
| For anyone wondering how they got the approximation
| sqrt(2)=1+24/60+51/60^2+10/60^3.
|
| It's based on the simple idea that: Z = (a +
| b)^2 = (a^2 + (2a+b)*b) => (2a+b)* b < Z-a^2
|
| Given an initial estimate "a", we need to find the largest "b"
| such that the term on the left is less than the term on the
| right. Therefore our estimate will always be slightly less than
| the actual answer and we can repeat the process to get slightly
| closer.
|
| For the first iteration, Z=2 and a=1. We choose b=x/60:
| (2+x/60)*x/60 < 2-1^2 120x + x^2 < 3600 x = 24 ...
| 3456 < 3600 x = 25 ... 3625 > 3600
|
| So our first term is 24/60.
|
| Repeat with a=1+24/60 and b=x/60^2:
| (2(1+24/60)+x/60^2)*x/60^2< 2-(1+24/60)^2 10080x+x^2 <
| 518_400 x = 51 ... 516_681 < 518_400 x = 52 ...
| 526_864 > 518_400
|
| Repeat multiple times.
|
| Writing this in code I can easily get:
| 1;24,51,10,7,46,6,4,44,50,28 = 1.4142135623730951
|
| This whole process can be codified into the long division
| algorithm for square roots which works quite neatly with base 10.
|
| Edit: formatting
| LombardBaker wrote:
| what evidence do you have that this is how they got it?
| PretzelPirate wrote:
| Why do you choose b=x/60 for the first iteration? That isn't
| obvious to me, but 60 must be an obvious choice to be chosen.
| zeroonetwothree wrote:
| You can choose any integer > 1
|
| For example if you choose 2 you will get the binary
| expansion.
| gleapsite2 wrote:
| Base 60 was the number system for the Sumarians/Babylonians.
|
| > Sexagesimal, also known as base 60 or sexagenary, is a
| numeral system with sixty as its base. It originated with the
| ancient Sumerians in the 3rd millennium BC, was passed down
| to the ancient Babylonians, and is still used--in a modified
| form--for measuring time, angles, and geographic coordinates.
| umeshunni wrote:
| Actual paper from 2009:
| https://link.springer.com/article/10.1057/jt.2009.16
|
| Link above is clickbait blogspam (like most things on IFLScience)
| dang wrote:
| Thanks! We've changed to that from
| https://www.iflscience.com/pythagorean-theorem-found-on-clay...
| above.
| codethief wrote:
| Would it make sense to add (2009) to the title? I came here
| thinking that people discovered something new.
| dang wrote:
| Ah yes. Added. Thanks!
| akrauss wrote:
| The journal is named ,,Journal of Targeting, Measurement and
| Analysis for Marketing", which may be the scientific equivalent
| of a clickbait blogspam site (but I haven't checked more
| deeply).
| gwervc wrote:
| The paper title itself could pass as-is an Buzzfeed.
| nologic01 wrote:
| Its only fitting that the now defunct "Journal for Targeting,
| Measurement and Analysis for Marketing" would have a clickbaity
| and misleading title on an article that seems completely off-
| topic and of very poor quality. A Springer sponsored precursor to
| the SEO driven abominations of today maybe?
|
| Confusing calculation with proof is an inexcusable mistake for
| any serious journal.
|
| There can be little doubt that proving theorems is a cognitive
| tool that developed on the basis of observed regularities. But
| both asking the question _why_ and, importantly, _answering it
| using logic_ are highly non-trivial developmemts.
| revskill wrote:
| I have a curiousity.
|
| Is there a theorem or conjecture like this ?
|
| "For any irrational number, like square root of 2, we can always
| find the approximation of at most 3 rational numbers with just
| +,-,* and /" ?
| alanbernstein wrote:
| Continued fractions can be used to find best rational
| approximations. But you can also just use floor(x*10^n)/10^n.
| Xophmeister wrote:
| If you're allowing division, you can get arbitrarily close with
| just two (potentially very large) integers.
| vanderZwan wrote:
| I guess the more interesting conjecture would be stating
| something about the size of the integers compared to the
| irrational number
| chud-guyver wrote:
| [flagged]
| mjfl wrote:
| did they know about the Pythagorean theorem, or did they know
| about the square root of two? It seems that people did puzzle
| over this quantity for a while.
| mindblown9 wrote:
| How did he live for so long?!
| sn41 wrote:
| The earlier drafts of the proof were rejected by editors. He
| had to fight.
| bandrami wrote:
| The formula wasn't why people cared at the time; it had been
| empirically known for centuries. What caused such a stir was he
| was the first person to prove that for "most" right triangles
| there are no rational numbers P, Q such that PA = QC for leg A
| and hypotenuse C. _That_ was what was earth-shaking.
| saithound wrote:
| Every group who ever managed to build a building with a
| rectangular foundation figured out the relation between the side
| lengths and the diagonal. The Egyptians used Pythagorean triples
| to measure right angles way before the birth of Pythagoras.
|
| But that's not a theorem, just an observation. It becomes a
| theorem when you prove (i.e. explain why) this relationship
| always holds, based on more evident things. The Babylonian tablet
| mentioned in the article doesn't seem to do anything like that,
| whereas the Greeks definitely did (we don't know whether
| Pythagoras himself did it, as no writing of his survives, but
| later Greeks knew how to do it, and attributed it to Pythagoras).
| geraneum wrote:
| From the article:
|
| > Pythagoras is immortally linked to the discovery and proof of
| a theorem that bears his name - even though there is no
| evidence of his discovering and/or proving the theorem.
|
| Simply quoting. It seems like
|
| > Greek definitely did
|
| is not a universally held belief.
| saithound wrote:
| Your quote from the article does not suggest what you seem to
| think it suggests.
|
| The Greeks definitely were able to prove the Pythagorean
| theorem, and the Greeks definitely though Pythagoras
| specifically knew how to prove it: e.g. Proclus II states
| this in writing explicitly, as does Euclid 300 years later,
| and numerous ancient sources inbetween. It would be hard for
| this not to be the consensus. We can't know if Pythagoras
| really did, simply because we have no surviving writings
| directly from Pythagoras - as I stated explicitly in my
| previous post.
| geraneum wrote:
| You are actually right. You didn't claim that the Greek did
| it first and that is what I doubted in the first place. I
| misread your comment.
|
| That fact that the Greek could prove the theorem at some
| point is not unlikely even without evidence. It's as likely
| as others doing it well before them.
| fsckboy wrote:
| > _Pythagorean triples to measure right angles ... But that 's
| not a theorem, just an observation. It becomes a theorem when
| you prove (i.e. explain why) this relationship always holds,
| based on more evident things. The Babylonian tablet mentioned
| in the article doesn't seem to do anything like that_
|
| according to the article, this Babylonian tablet does come
| closer to theorem than you are suggesting. They weren't using
| Pythagorean triples, rather they figured out that the diagonal
| of a unit square is the square root of 2, and knew how to
| calculate that:
|
| 1 + 24/60 + 51/602 + 10/603 = 1.414213
| hyperthesis wrote:
| Agree it's closer but still an instance.
| fsckboy wrote:
| depends on the method they used to come up with the
| formula. It's not obvious that the sum of a "more
| complicated than you can do in your head" series of
| fractions would be either the square root of two or the the
| diagonal of a square, let alone both. This is a fragment of
| a single clay tablet. If they had a systematic method of
| coming up with this convergent series, there might have
| been a stack of tablets showing square root of three, five,
| etc. It seems pretty highly suggestive to me.
| posterboy wrote:
| Proof by construction, the Curry-Howard correspondance.
| ssener2001 wrote:
| I agree.Obtaining this formula by observing is more
| difficult than obtaining it by proving it. Besides, it is
| an issue that is already needed in field work. If you
| have pen, paper and a good number system, it shouldn't be
| too difficult to find the formula with shapes.
| loa_in_ wrote:
| I imagine back then things like number systems and
| methods of proving were just made up as they go.
| lazide wrote:
| Clay tablets and a stylus also less convenient than pen
| and paper.
| KMag wrote:
| That entirely depends on how close you are to high-clay-
| content muddy bank with a growth of reeds vs. how close
| you are to the nearest convenience store/office supply
| store.
| lazide wrote:
| Did I just find a new metric to put on my rental house
| evaluation checklist? Maybe! Haha
| [deleted]
| orra wrote:
| > It becomes a theorem when you prove (i.e. explain why) this
| relationship always holds, based on more evident things. [...]
| whereas the Greeks definitely did (we don't know whether
| Pythagoras himself did it
|
| Wait, really? I thought the proofs back then were geometric, so
| only proved specific instances. And so it wasn't until algebra,
| maybe trig, was discovered it was properly proven?
| n4r9 wrote:
| There are fully general and valid geometric proofs, eg
| http://mathandmultimedia.com/wp-
| content/uploads/2010/02/pyth...
| orra wrote:
| Despite labels $a$, $b$, $c$, I don't see how it's fully
| general. At best you could say it works for similar
| triangles, if you have that concept.
|
| But I don't see how it proves anything if $a$ were say
| doubled, and $b$ kept the same.
| usrusr wrote:
| Why would you need numbers for right angles? You need a
| straight line and a string.
|
| But I agree with the second paragraph: there's a huge
| difference between a procedure that is handed down as part of
| "this is how we estimate a building project" and a theorem that
| is declared universal truth and base for all kinds of other
| theorems. Even if they are exactly the same thing.
| andrewflnr wrote:
| I've personally worked on a project where we used a 3-4-5
| right triangle to lay it out on the ground. Straight lines
| alone do not get you right angles.
| mauvehaus wrote:
| You can certainly get close enough with straight lines. If
| you create a parallelogram, you can get it square by making
| the diagonals the same length.
|
| And if you can determine that the diagonals are the same
| length, you have what you need to get close enough to a
| parallelogram in the first place.
| chriswarbo wrote:
| IIRC, straight lines alone will give you projective
| geometry (which also has points, but they're the meet of
| two lines).
|
| > If you create a parallelogram
|
| Parallelism requires affine geometry, which you can't get
| just with straight lines (and their meeting points). Here
| are couple of explanations:
|
| - We can _also_ get projective geometry by using great-
| circles on the surface of a sphere (e.g. "equators" at
| different angles around the Earth), instead of straight
| lines on a flat plane: both situations give rise to
| exactly the same theory. Parallelism doesn't exist on the
| surface of a sphere, since all great-circles will meet at
| two antipodal points, so projective geometry (which
| describes great-circles _as well_ as straight lines)
| cannot be used to construct /ensure/check that two sides
| of a quadrilateral are parallel.
|
| - Alternatively, consider that projective geometry is
| invariant to changes in perspective, whilst parallelism
| is not. For example, we can get two straight lines by
| tracing over a photo of train tracks. If the photo was
| taken top-down, then the lines we traced will be
| parallel; but if the photo was looking along the track
| then our traced lines will converge (in the photo, they
| "meet" at the horizon). Projective geometry (and hence
| straight lines) can't distinguish between these two
| scenarios, due to this invariance.
|
| > And if you can determine that the diagonals are the
| same length
|
| If we extend our straight-line setup with some way to
| determine parallelism, we _still_ wouldn 't be able to
| compare the lengths of the diagonals, since they go in
| different directions. Projective geometry + parallelism
| is _affine geometry_ , which can only compare lengths _in
| the same direction_. Essentially, parallelism allows us
| to _translate_ : we can use this to compare two line
| segments by translating one so they share a common
| starting point, then seeing whether the other end has
| landed closer or further than the first line's. The
| latter comparison only makes sense if all the points end
| up colinear (i.e. the original segments were parallel,
| unlike a pair of diagonals).
|
| To compare the diagonals we also need some form of
| metric, e.g. like the distance between a pair of
| compasses.
| rcxdude wrote:
| You can get right angles with a straightedge and compass,
| both of which you can make on a construction site with
| string and pegs in the ground. It's just an instance of
| bisecting the angle. Which is more convenient in practice
| is situational.
| andrewflnr wrote:
| Fair, I didn't realize they were using the string as a
| compass.
| KMag wrote:
| Right. The big problem with accurately bisecting a 180
| degree angle is you need to have access to points rather
| far in both directions from both points. If you want an
| accurate right-angle corner near the edge of a property
| that's flanked on two sides by fences, rivers, busy
| roads, etc., then you might not have convenient access to
| one of the anchor points needed to perform your
| bisection. (Or semi trucks snagging your rope might be
| inconvenient.)
|
| The nice thing about Pythagorean triples for drawing out
| foundations is that you don't need access to any ground
| outside the foundation of your building. Being integers,
| you also don't need any measuring device apart from some
| rope. You just pace out a bit under 1/3 of the shortest
| side (or a bit under 1/5 the longest side, whichever is
| shorter) (call this an "'bout-right") length of rope. You
| then use your 'bout-right to make a 3'bout-right, a
| 4'bout-right, and a 5'bout-right piece of rope. Pull the
| three ropes tight in your perimeter, and you've got your
| right-angle for your foundation.
| detourdog wrote:
| How useful are universal truths compared to getting shit
| done?
|
| I only see theorems as useful for complex societies. The son
| of a gem dealer would have the time to work out universal
| truths.
|
| The reputation of everyone doing it without numbers would
| reveal the pattern of the universal truth.
|
| Finally this looks like it was all done using cuneiform.
|
| Which brings up questions of notation and the language to
| describe a square root.
| ordu wrote:
| _> How useful are universal truths compared to getting shit
| done?_
|
| As I understand Greeks invented proofs because "universal
| thruths" they exported from Babylon and Egypt sometimes
| explicitly contradicted each other.
|
| I believe such contradictions may be a great nuisance when
| you try to get shit done.
|
| Egypt and Babylon were sufficiently "complex" societies for
| proofs, but their tradition treated mathematics as a bunch
| of useful facts about numbers and shapes. New generation
| just memorized them. We should think it worked for them in
| most cases, and when it didn't work it was not so often for
| them to start thinking a lot of reforming mathematics. Plus
| they were indoctrinated by the math they learned (authority
| of a teacher is above of anything else, i suppose) and to
| reform math was not a natural idea for them.
| usrusr wrote:
| Getting shit done is the antithesis of progress because it
| goes hand in hand with "if it was good enough for my
| father, it will be good enough for me, who are you to
| disrespect (the methods of) my father!"
| detourdog wrote:
| I agree but dying for ideals isn't much progress.
|
| Progress is a luxurious goal.
| empath-nirvana wrote:
| What everyone is talking past here is that the Babylonians
| discovered what engineers call a "rule of thumb", and
| engineering is focused on getting things done using rules
| of thumb. The best possible rule of thumb is a mathematical
| proof or a scientific discovery, but it's by no means
| necessary. .
|
| Observing that something always holds and even having a
| formula for it is not the same thing as having a proof, and
| the proof is what makes it mathematics and geometry and not
| "just" engineering. The babylonians had a rule of thumb --
| the greeks discovered the theorem -- and more than that,
| they seem to have invented the mathematical/geometrical
| proof as a concept, along with formal logic.
|
| Without that mental framework, it's hard to say that the
| babylonians proved anything or had any theorems at all,
| only collections of rules of thumb. It's quite likely that
| lots of babylonians sort of independently and intuitively
| understood _why_ it must be true, but they don't seem to
| have ever written it down.
|
| Not that there's anything wrong with having rules of thumb
| -- it's a huge achievement to even notice and collect and
| teach those things, all the stuff around you is built
| relying on them.
|
| I encourage everyone to watch this series of videos.
|
| https://www.youtube.com/watch?v=_ivqWN4L3zU
| detourdog wrote:
| I think what you might be touching on is that the "rules
| of thumb" may have been so integral of the culture as to
| be hidden.
| [deleted]
| dclowd9901 wrote:
| 3,4,5
| sn41 wrote:
| Proofs are of course the gold standard. But even if others
| discovered the rule, their contribution should not be held to
| be trivial.
|
| A modern analogy: today, people are very happy to use LLMs and
| Transformers without anyone "proving" that they work. Right
| now, philosophically, they are at the level of empirical
| observations (perhaps not even that). Does that mean that
| today's AI researchers should get no credit when at a later
| date? I am not sure. Empirical discovery of a rule is also no
| trivial thing.
| saithound wrote:
| Yes, when people in the 4000s write clickbait holonet
| articles about how the Master Theorem of Neural Network
| Scaling was not actually invented by the Muskovites of Mars,
| but was found written on an Ancient American tablet
| containing the works of Hoffmann et al., I hope somebody will
| offer substantial corrections.
|
| That's not in any sense a value judgment of the empirical
| work done at DeepMind. Nor does it stop anybody from writing
| a better article which explains that the Babylonians (and
| many others) used the empirical observations underlying the
| theorem, while explaining that this did not constitute
| mathematical proof.
| outrun86 wrote:
| Found on an ancient tablet as the work of Schmidhuber et
| al. but ok
| Turing_Machine wrote:
| Pointing out that it's not a _theorem_ doesn 't mean it's
| been trivialized.
|
| If there's no proof, it's not a theorem, by definition. That
| doesn't make empirical observations worthless -- far from it!
| WalterBright wrote:
| > Every group who ever managed to build a building with a
| rectangular foundation figured out the relation between the
| side lengths and the diagonal.
|
| I don't see why that was necessary. You can get pretty far with
| eyeballing it and custom cutting to fit.
| fsckboy wrote:
| > _I don 't see why that was necessary. You can get pretty
| far with eyeballing it and custom cutting to fit._
|
| i'm not saying at all that I know the answer, but eyeballing
| and cutting is good for a patio, but on a ziggurat or pyramid
| scale it seems you don't do so much eyeballing or cutting,
| and more planning how much material and how many slaves
| you're going to need for how long, and where to put the doors
| so the passageways will meet up, that sort of thing.
| rebolek wrote:
| Please, stop this slave nonsense. Pyramids were built by
| highly skilled workers, it was prestigious job.
| PawgerZ wrote:
| I remember reading about the worker homes and tombs, and
| I know that skeletal records also showed the workers had
| much muscular strain. So, it's impossible to deny that
| there were skilled laborers working on the pyramids, but
| just because some of the laborers were proven to be
| skilled and not slaves, doesn't mean slaves weren't used.
|
| The scale doesn't seem right to me. I still can't fathom
| how that much could be done without using slave labor. If
| I remember correctly, they layed a block like every 6
| minutes for 20+ years.
|
| Do you know how many tombs of laborers were found, or
| where I could find more aobut that information? I'm very
| novice when it comes to Egypt
| WalterBright wrote:
| I've seen many Mayan structures in Mexico, and they all
| look like they were eyeballed.
| posterboy wrote:
| I'm currently in a brick build house which has a good ten
| percent slope, because the earth sank, but it hasn't
| collapsed. I'm pretty sure its not older than a hundred
| years and I'm amazed it still stands because it sure
| wasn't designed like this.
| danjc wrote:
| I built a tree house for my kids. I didn't exactly eyeball it
| but my amateur technique and planning left me having to do a
| lot of compensation for things being out of square.
|
| Everything becomes a custom cut, often in multiple dimensions
| and all earlier errors cascade all the way to the end.
| berkes wrote:
| > Everything becomes a custom cut, often in multiple
| dimensions and all earlier errors cascade all the way to
| the end.
|
| Unrelated, but that sounds exactly like many software
| projects I've been unfortunately part of.
| saithound wrote:
| A "you" can, and many individual "you"s presumably did that,
| to general satisfaction. But the implicit assumption you made
| is that eyeballing and cutting is easy, while triangles are
| hard, even though it's the other way around, especially with
| stone age tools.
|
| But we're talking about groups of people, "they", who all
| build a lot. Such groups tend to have a few people who make
| these observations, and then the observations proliferate,
| because they are both way easier to use than the alternative
| hack-work, and yield much more aesthetically pleasing
| results. Especially in the stone age when you nornally don't
| have easy access to anything at all with a right angle
| (unlike in modern construction) and you build stuff out of
| clay bricks, where minor inaccuracies inevitably add up and
| make your life much harder down the road. Tying together
| three pieces of string with prescribed ratios, pull it tight
| was a very easy way to get a right angle compared to anything
| that came before.
|
| It's necessary in the sense that stone age construction is so
| much easier if you know about it, and so much harder if you
| don't. Those who didn't come up with it didn't do such
| construction, because doing difficult things is harder than
| doing easy things.
| gregjor wrote:
| I think you mean bronze age.
| saithound wrote:
| I don't think I do. I'm fairly confident that the people
| of the chalcolithic did enough large scale construction
| (and pottery and watching the night skies and so on) to
| have figured a whole lot of empirically accessible
| geometry out.
|
| The _article_ happens to be about written sources from
| the Bronze Age Babylonians, which lets us glimpse at
| their accumulated knowledge. But there's no reason to
| believe this knowledge was particularly new at their
| time, and this was a clay tablet equivalent of an arXiv
| preprint.
| dylan604 wrote:
| >A "you" can, and many individual "you"s presumably did
| that, to general satisfaction
|
| I was chatting with the person that was in charge of
| marking the fields for the local youth soccer league. I
| volunteered to help one weekend, and one of the first
| things he asked was if I knew what a 3/4/5 triangle was
| since it was the only way to know you'll be squared. I
| never did figure out to what level he was dead panning his
| joke or if he was even meaning for it to be a joke. Either
| way, I laughed.
| xxs wrote:
| 3/4/5 is considered a wood working trick - that many
| beginners would not know (according to youtube, at least)
| dylan604 wrote:
| I learned it in geometry in like 10th grade. No wood
| involved.
| WalterBright wrote:
| You can easily accurately lay out a large square without
| knowledge of Pythagorean Theorem.
|
| 1. lay out a straight line 2x in length.
|
| 2. find midpoint (easy by drawing an arc with a string from
| each end point. Basic compass & ruler technique.
|
| 3. draw another arc from the midpoint, of radius 1x. Try
| different spots on the arc until it is equidistant from
| each of the endpoints of the line.
|
| 4. voila! an accurate right angle. Laying out the rest of
| the square is now trivial.
| ReptileMan wrote:
| Ancient Mesopotamian laws sometimes flayed people alive for
| incompetent or corrupt work. Those types of mild punishments
| are good incentive to use the right angles. Pun intended.
| lynguist wrote:
| The mainstream is and was _enamored_ with Ancient Greece.
|
| Our Western culture made Ancient Greek into the vocabulary root
| of our sciences.
|
| We have a lineage of philosophy from Ancient Greece to the 19th
| century.
|
| Only in the 19th century with archeology (again a neo-word made
| from Ancient Greek roots - it suggests to the mainstream that the
| Ancient Greek had a concept of archaeology, which they obviously
| didn't have) we saw the truth: History goes thousands of years
| deeper, the origin of everything is thousands of years older.
|
| Only 30 years ago the capital city of Hattusa was discovered; and
| only in the 20th century we gained an understanding of the
| multiple levels of the historic city of Troy.
|
| Only recently we understand that "it didn't start with Ancient
| Greece", but the mainstream still follows the tradition of
| medieval grammar schools and doesn't look beyond Ancient Greece.
| nequo wrote:
| The Greek still did something remarkable that to my knowledge
| we don't have record of from before: they started a culture of
| free thinking that grew into philosophy, logic, and scientific
| inquiry. They also sentenced some of their free thinkers to
| death for not worshipping the gods but that's a separate issue.
| goodbyesf wrote:
| > they started a culture of free thinking that grew into
| philosophy, logic, and scientific inquiry.
|
| Culture of free thinking? Socrates, the most iconic free
| thinker of all time was forced to commit suicide by the
| greeks. So much for a culture of free thinking.
|
| > They also sentenced some of their free thinkers to death
| for not worshipping the gods but that's a separate issue.
|
| It isn't. It directly contradicts and refutes your assertion.
| nequo wrote:
| > Socrates, the most iconic free thinker of all time was
| forced to commit suicide by the greeks.
|
| See my last sentence.
|
| > It directly contradicts and refutes your assertion.
|
| Maybe Greek society was not a homogeneous mass, and some
| parts of it nurtured free thinking while others reacted
| against it?
| User23 wrote:
| The Ancient Greeks didn't even think it started with Ancient
| Greece. Plato believed in Atlantis for example.
| gen220 wrote:
| Is it not a simpler explanation that Greece simply did a better
| job of disseminating its own literature than prior cultures?
|
| My understanding is that the "through line" of modern
| scientific progress begins in Greece because that's where
| "trivially-legible to contemporary scholars" written history
| began.
|
| Like, we always knew that it didn't start with Ancient Greece
| (the Greeks themselves mention this), but because abundant
| primary sources prior to Ancient Greece don't exist, there
| isn't much we can do other than light a candle for their sake
| and use its light to read their thoughts as filtered through
| Plato, etc.
| digging wrote:
| > Is it not a simpler explanation that Greece simply did a
| better job of disseminating its own literature than prior
| cultures?
|
| That's _a_ simpler explanation but it 's not necessarily
| right.
|
| What we can say for sure is that Greek thought has been
| _easier_ for Western scientists and pseudo-scientists to
| learn. Availability and language are parts of that for sure.
| Geography is, too - it 's easier for Europeans to excavate
| Europe than Iran.
|
| But what does it mean to say "the Greeks" disseminated their
| knowledge better when virtually all of what we have comes
| from Roman citizens living centuries later?
| gen220 wrote:
| > But what does it mean to say "the Greeks" disseminated
| their knowledge better when virtually all of what we have
| comes from Roman citizens living centuries later?
|
| That Greek knowledge and culture survived the collapse of
| their prominence? Similar to the Romans after them and
| Babylonians before them.
|
| Greek had been a lingua franca in the major ports of future
| empires for centuries [1], and Greek remained a spoken and
| written language in the Byzantine empire (the same was not
| true of ancient Egyptian or Babylonian - which were
| supplanted by Greek or other Aramaic languages during the
| hellenistic period).
|
| I think at least as much credit is owed to the inheritors
| of Greek culture (Romans, Byzantines, the various Arabic
| empires), for preserving source material and references.
|
| But I think Greece was seen as the original "filter" for
| civilization because it was both "successful" and
| comparatively extroverted to the great civilizations that
| came before.
|
| Basically, it's what you said in your middle paragraph -
| wide availability, accessibility of language and culture.
| In other words, "better dissemination". :)
|
| [1]: https://en.wikipedia.org/wiki/Greek_colonisation
| digging wrote:
| > I think at least as much credit is owed to the
| inheritors of Greek culture (Romans, Byzantines, the
| various Arabic empires), for preserving source material
| and references.
|
| That's what I'm saying, though. Preservation is the work
| of the preservers. Many of the great thinkers of Greece
| didn't preserve a single word. Someone else did. Often
| other Greeks, often Romans (who obviously spoke Greek as
| you say, because _they_ believed it to be a superior
| language).
|
| > wide availability, accessibility of language and
| culture. In other words, "better dissemination".
|
| But B is a subset of A here. Not all of those facets that
| I mentioned are due to the Greeks themselves, not even
| indirectly.
|
| Sometimes, Western civilization sees "The Greeks" as a
| progenitor civilization because... we believe they're a
| progenitor civilization. It's a tradition to believe so,
| and it may well have started by mistake or for reasons of
| xenophobia or other bad motivations.
| mcphage wrote:
| > suggests to the mainstream that the Ancient Greek had a
| concept of archaeology, which they obviously didn't have
|
| Why is that obvious? The ancient Egyptians & Mesopotamians had
| a concept of archaeology, why not the Greeks?
| smokel wrote:
| I sometimes wonder how people will remember the invention of
| computers, smartphones, and the internet in a few centuries.
|
| People will most probably learn that either Bill Gates or Elon
| Musk invented it all in one evening when an apple fell from a
| tree.
| rkagerer wrote:
| This may be a bit forced, but you missed a golden opportunity
| to capitalize the "A" in that sentiment and accerate my
| reaction.
| bambax wrote:
| I wonder the same thing. See this video about "the Beatles in
| 1,000 years" for instance:
| https://www.youtube.com/watch?v=3Z2vU8M6CYI
|
| It's a parody but I wonder how close to the truth it is...
| RetroTechie wrote:
| History is written by some combo of historians & surviving
| records.
|
| With "records" being any physical objects that outlast oral
| history: writings, clay tablets, the pyramids of Egypt, etc.
|
| (and note the "surviving" bit!)
|
| This says nothing of what _really_ happened in the past. Only
| what exists in the present to support particular
| reconstructions of past events.
| Beijinger wrote:
| My math is a bit rusty, for a moment I was wondering about the
| proof of his theorem but then I realized that I messed up his
| theorem with the angles in a triangle that sum to 180deg, which
| does not always hold up.
| seanhunter wrote:
| If the angles in a triangle don't add up to 180o then I think
| that means the triangle is not on a flat plane. An example
| would be if you took a point on the equator of the earth, moved
| East or West 1/4 of the earth's circumference and took another
| point and then projected those two points North until they met
| at the pole, you would have a triangle with 3 interior right
| angles.
| Beijinger wrote:
| Yes. Pythagoras theorem holds up in non-euclidian space but
| the sum of the angles is 180 degree does not.
| cobbzilla wrote:
| According to Shaquille [1], it's still an unsolved problem.
|
| [1] https://www.brainyquote.com/quotes/shaquille_oneal_381872
| Perenti wrote:
| I recall seeing that there was a 14 000 year old dear scapula
| found in China that had the "simple proof" diagram engraved on
| it. The article seems to say that Pythagoras' Theorem was
| discovered by the Mesopotamians, but neglects mentioning it may
| be _much_ older.
|
| Someone above commented that just by building ancient peoples
| would have discovered this relationship.
| loondri wrote:
| It's interesting that Pythagoras gets credit for a theorem he may
| not have discovered, especially when there's proof that
| Babylonians knew it 1000 years earlier.
|
| This challenges the idea that ancient Greek mathematicians were
| always ahead of others.
| drexlspivey wrote:
| It's interesting that people in a CS forum don't seem to know
| what a theorem is. The tablet shows that the relationship was
| known, a theorem needs a _proof_
| seanhunter wrote:
| It's in the tradition of maths that things are named after the
| second person to prove them/make them well known. There are
| numerous examples, but one of my favourites is Venn diagrams,
| which were called (by Venn) "Eulerean Circles" because he took
| them from Euler. Everyone else is like "Nope - Venn Diagrams."
|
| It's probably just as well because otherwise just about
| everything would be named after Gauss which would make learning
| maths even more difficult than it already is.
|
| The Pythagoreans did make a number of important discoveries to
| do with number theory, the ratios between string lengths for
| various musical notes (eg twice as long is an octave lower
| etc), cosmology and some other results in geometry to do with
| the properties of various 3-d shapes and stuff.
| loondri wrote:
| Interesting, thanks for sharing this
| Chiba-City wrote:
| [dead]
| zestyping wrote:
| https://personal.math.ubc.ca/~cass/Euclid/ybc/ybc.html
|
| This is the original site about the tablet and has more detail
| and analysis than the Springer article.
|
| https://personal.math.ubc.ca/~cass/Euclid/ybc/analysis.html
| loganc2342 wrote:
| > _The Pythagorean Theorem is arguably the most famous statement
| in mathematics, and the fourth most beautiful equation._
|
| Just out of curiosity, which equations are considered the top
| three "most beautiful?"
| mauvia wrote:
| I know the top one is generally considered to be Euler's
| Equation.
|
| e^(i * pi) + 1 = 0
|
| It's considered incredibly elegant because it manages to
| combine multiple fundamental mathematical concepts into a
| single equation.
|
| 1 is the multiplicative identity, 0 is the additive identity,
| pi is the circle constant, e is euler's number, i is the square
| root of -1, the basic building block of complex numbers.
| travisjungroth wrote:
| IMO, it's better with tau.
|
| e^(i * tau) = 1
|
| You lose the 0, but isn't it a bit odd the 0 is there in the
| first place? Normally we'd reduce it to:
|
| e^(i * pi) = -1
|
| Which obviously isn't as nice in this case.
|
| And hey, if we're allowed to break the conventions, you can
| have the 0 back easily.
|
| e^(i * tau) = 1 + 0
| mcv wrote:
| Is there an easy explanation of why these are true? I
| already struggle grasping e^i, and I completely don't
| understand what e to an irrational power even means, let
| alone why it would be -1. Why the circle circumference
| ratio has anything to do with this is completely beyond me.
| photochemsyn wrote:
| 1. e^x, sin(x) and cos(x) can each be expanded out into
| an infinite sequence of fractions (Maclaurin series),
| each fraction of the form x^n / factorial(n). This relies
| on differential calculus, Taylor series, theory of
| limits, convergence etc.
|
| 2. The e series fractions contain all the integers in the
| numerator (x^0, x^1, x^2, etc), while the sin series has
| only odd integers and the cosine series has only even
| integers. Also, e series terms are all additive while the
| trig functions series alternate adding and subtracting
| each successive fraction.
|
| 3. Introducing complex numbers (i = square root of
| negative one), we can generate the series for e^ix, which
| can be shown to be equal to sin(x) + i * cos(x). Note
| that introducing i into the e series means we generate a
| negative term for the even fractions in the e series
| (squaring i gives us -1), which is why i is so necessary
| here.
|
| 4. Solving e^ix for x = pi, using sin(x) + icos(x), we
| get -1.
|
| Mathologer:
|
| https://www.youtube.com/watch?v=-dhHrg-KbJ0
|
| and
|
| https://www.youtube.com/watch?v=DoAbA6rXrwA
|
| As far as why an exponential function like e^x should
| have anything fundamental linking it to trigonometric
| functions like sin(x) and cos(x), it is rather strange.
| seanhunter wrote:
| So that's the part I explain in my note. So if you
| combine the two explanations together I think we have the
| whole picture. Yours fills in the gap in my explanation.
| larschdk wrote:
| 3Blue1Brown:
| https://youtu.be/v0YEaeIClKY?si=BQ2W64DWIuhHoUuX
| seanhunter wrote:
| So I haven't done complex analysis yet which I think you
| need to get the whole thing but I can get you some of the
| way there with basic trig.
|
| If you take a unit circle and construct a radius to some
| point (x,y), if you drop a perpendicular line down to the
| x-axis, it's easy to see that the length of that
| perpendicular line is y and the distance you've gone
| across the x axis is x. So you have a right angled
| triangle where the hypotenuse is 1 (it's a unit circle)
| and the other two sides are x and y. Now consider the
| angle at the origin and call that theta.[1] You can do
| basic trig to show that the coordinates of your (x,y)
| point are (cos theta, sin theta), because sin is opposite
| (y) over hypoteneuse (1) and cos is adjacent (x) over
| hypotenuse. Ok cool. So x = cos theta and y = sin theta.
| If you measure in radians, then the angle of a full
| revolution is 2 * pi radians and the angle of a half
| revolution (180 degrees in other words) is pi. Now
| consider the point when you have gone around the unit
| circle 180o, Its coordinates are x=-1 and y=0. Remember
| this point - we'll come back to it in a minute.
|
| Now imagine instead of your unit circle being just any
| old circle it's in the complex plane. This means that the
| x axis is the real part of some complex number and the y
| axis is the imaginary part. We now know that the
| coordinates of points on this circle are (cos theta, sin
| theta), but if you have a complex number z= a+bi, these
| correspond to a and b. So z = cos theta + i sin theta.
| Here's the bit where my current mathematical ability runs
| out of gas and you're just going to have to trust Euler,
| who showed that cos theta + i sin theta = e^(i theta).
|
| Now remember our point from before where theta = 180
| degrees? What was the angle in radians? It was pi. So
| e^(i pi) = -1 (because the real part of the number is the
| x coordinate, -1 and the imaginary part, the y coordinate
| is zero).
|
| [1] Here's a diagram I made which will get you up to here
| https://www.geogebra.org/calculator/btz38m3c. My note
| about the trig of unit circle I made while studing is
| here https://publish.obsidian.md/uncarved/3+Resources/Pub
| lic/Unit...
| User23 wrote:
| That might be because it should be written:
| e^(i*th) = cos th + i*sin th
|
| The formula loses most of its beauty when you just present
| it with a single arbitrary real plugged in.
| mkl wrote:
| The 0 and the + are important: e^(i*pi) + 1 = 0 contains
| the 5 most important constants and the three most important
| operations. Getting them back with "+ 0" is quite
| inelegant.
| travisjungroth wrote:
| I don't know. Shoving the 1 over to the left because you
| don't like what it actually equals, -1, seems like an
| ugly hack.
|
| Arguments better than I can make: https://tauday.com/tau-
| manifesto#sec-euler_s_identity
| zabzonk wrote:
| not sure if you include physics, but newton, the gas laws, and
| einstien?
| zeroonetwothree wrote:
| Ask ten people you will get ten different answers.
| [deleted]
| fasquoika wrote:
| Well you're on a website called ycombinator.com so maybe the
| fixed point combinator Y =
| lf.(lx.f(xx))(lx.f(xx))
|
| https://en.m.wikipedia.org/wiki/Fixed-point_combinator
| Tao3300 wrote:
| A = Pe^(rt)
| tapotatonumber9 wrote:
| The most delicious is:
|
| "(-1)^0.5 2^3 S p."
| dotancohen wrote:
| You do realize that the volume of a Pizza with radius z and
| height a is: Pi [?] z [?] z [?] a
| boomboomsubban wrote:
| This seems to be the original source
| https://physicsworld.com/a/the-greatest-equations-ever/ but it
| doesn't actually rank the equations. The other source
| commenting on that does, but only the sample is available on
| Google Scholar. From that, first is Euler's identity, second is
| Maxwell's four electromagnetic field equations, and third isn't
| in the sample. The NYT article also commenting on it
| https://archive.ph/H7ujx suggests the theory of relativity,
| F=ma, or amusingly 1+1=2.
| helsinkiandrew wrote:
| Discussion on HN 7 years ago about parallel proofs after
| discovery in a 2600 year old Chinese book:
|
| https://news.ycombinator.com/item?id=13952265
| zacharycohn wrote:
| So Pythagoras also invented time travel??
| jeisc wrote:
| Names are important as they indicate things; maybe Triangle
| Theorems would be better to cover the properties of triangles;
| perhaps there are other ones which we missed discovering...
| hubris has no limits
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