Pythagorean Theorem proof in a 2100 year old Chinese book
fermatslibrary.com
fermatslibrary.com
But I think we should not be too eager to jump on to "science is only from laws" bandwagon. Courses on the philosophy of science take this dogma to absurd extremes, ignoring how current science is often done. There is enough science done today where the laws are only vague, but we do predict useful stuff from simulations - climate models and earthquake modelling being the most prominent - despite not having "simple" laws behind them. As Freeman Dyson often observes, physicists still don't have a good explanation of why a bicycle doesn't topple when in motion - but we've been riding them for a century and half [2].
I think trial-and-error, and simulations also count as sophisticated science.
It's more than a tad racist to insist that Egyptians, Babylonians, Chinese and Indians, as well as medieval Arab scientists were all child races incapable of truly knowing what they were doing.
I think modern western science discounts other sources of knowledge, because it is not just a platonic ideal but a sort of politics of the scientists, which they need to justify the choices they make.
"It's more than a tad racist to insist that Egyptians, Babylonians, Chinese and Indians, as well as medieval Arab scientists were all child races incapable of truly knowing what they were doing."
What on earth are you rambling about? Were have I made even the faintest suggestion that that is the case? Different civilizations has invented different things and advanced at different levels and different times through our history. Pointing this out isn't racist with respect to other people. Is is racist against Europeans to suggest that Indians invented the zero and our numerals. Am I then suggesting that those stupid childish Europeans couldn't come up with even basic stuff like numerals themselves?
Is it racist to point out that native Americans didn't invent quantum mechanics?
Anyway I don't believe there is such a thing as human races, but there definitely are differences between various cultures and civilizations.
It is an undeniable fact that western civilization was the main driver of the scientific revolution. Should it then not be interesting to understand why it was that the west advanced so much further than the others. One of the key observations which helps understand this is the western preoccupation with formulating laws of nature early on.
How else can you explain how a relatively poor and backwater region of the world could end up eclipsing, far richer, more populous and advance regions such as the Arab world, China, India?
Maya's totally knew about quantum, how else could they have had novelty theory and predicted the 2012 timewave zero collapse? It's not like they had access to the I-Ching[0] to be able to model it like Terrence McKenna did.
... on that note, a bit more serious; this is as good an opportunity to randomly ask as any; can anyone explain to me (as far as pseudo-science goes, going along with all of the assumptions) what the basic reasoning is in Timewave Zero, how they got from the King Wen sequence (which is a certain permutation of the 64 I-Ching hexagrams, but afaik they just use it as a particular 64-element permutation compared to the original order, without taking into further account the symbology of the invididual hexagrams), and turned it into that fractal line graph they call the Timewave? There is an article on deoxy.org that appears to go into technical detail but when I read it and tried to reproduce, it doesn't really say how that step is done. OTOH apparently there's software (screenshots) that plot this graph, so there must be some kind of deterministic algorithm, but I haven't been able to find any docs about it.
[0] ... OR DID THEY??
[1] https://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamic...
ps: I wonder if there's a skateboard dynamics.
Ibn_al-Haytham was one of the inventors of the "scientific method". https://en.wikipedia.org/wiki/Ibn_al-Haytham
https://upload.wikimedia.org/wikipedia/commons/5/5c/The_Scie...
What Dyson says is that it's hard to design a technological artifact, like a bycicle, from scratch, using theory only. We know perfectly how and why it works, but it's very hard to design something the behavior of which is actually governed cleanly by the fundamental explanatory principles in predictable ways. That doesn't make the principles of the explanation wrong: it's recognizing that reality is incredibly messy. It's about how hard technology is, not about science.
With our current focus on digital media and "cloud", there won't be anything left in 2100 years for people to have a blick of what we are doing.
"Feynman on pseudo science" https://vimeo.com/118188988
The problem with a lot of the ancient sources is the lack of proofs. What I am arguing is that lack of proofs in ancient mathematical texts doesn't mean that it is all bunkum, and the authors did not know what they were talking about.
I am not talking cultural relativism or "social science is a science" relativism.
Again, another example: Euler is the father of modern analysis, through his "Analysis of the Infinite" [1]. None of his proofs are rigorous according to modern standards. Do you think Feynman would've called Euler a fool?
[1] http://www.maa.org/press/periodicals/convergence/mathematica...
Ancient science is science. Ancient pseudo science is ancient pseudo science.
However I think that first making observations and then the crucial step of extrapolating and making predictions based on those is as good as any formally named 'law'.
Extrapolation isn't that easy to do with e.g. astronomy. Orbits don't go in straight predictable lines. They are ellipses where the speed will vary depending on where you are in the orbit. Kepler found that the are covered per time unit was constant, thus implying that when the orbits were closer to the sun they would speed up.
I guess you could say extrapolation was what Kepler tried to do but it took him a lot of work to figure it out. The Chinese while very advance in many respect did not try doing that. What I've read is that they did not have the Judeo-Christian belief that the mind of God could be learned. The laws governing the motions of the heavens was assume inscrutable to the human mind.
Now what this book shows is that 2100 years ago some level of proof was also a requirement in Chinese mathematics; this makes classical Greek/western thought look not quite as unique as it is portrayed.
The Greeks wrote their mathematical ideas in a different way than other cultures, so the answers to these questions aren’t completely clear-cut.
The Sumerian/Babylonian style of written records did not match, say, Euclid’s Elements. (They didn’t write books full of precise definitions and proofs. They didn’t leave behind textbooks, only problem sets and worked examples.) But note that neither did the style of the Pythagoreans necessarily, and the notion that Pythagoras or his disciples made a proof comes from written records from hundreds of years later.
> Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.
As humans we always lived a kind of ups and downs, regarding use of technology, relationships among countries, how open/closed the societies used to be, how travelling was seen and so on.
For example, in the ancient world there was already a kind of Erasmus, as Greek medicine students used to spend a few years with Egyptian doctors before getting back home and start practising.
But also, we can't trust geometrical diagrams: https://en.wikipedia.org/wiki/Missing_square_puzzle
I see the illustration, but I don't understand how that's a "proof."
That's an illustration of the proof. Here is a more modern, animated example: https://upload.wikimedia.org/wikipedia/commons/2/29/Pythagor... The Chinese proof looks more like this one though: https://i.stack.imgur.com/Ylzhg.gif There is also explanation in the annotations.
0. https://encrypted.google.com/search?hl=en&q=proofs%20without...
1. Image results: https://encrypted.google.com/search?q=proofs+without+words&n...
The biggest difference was that nothing was assured about the safety of the travel.
> In India, the Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle.
https://en.wikipedia.org/wiki/Baudhayana_sutras
> A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together.
> The lines are to be referring to a rectangle, although some interpretations consider this to refer to a square. In either case, it states that the square of the hypotenuse equals the sum of the squares of the sides.
> If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem.
These sorts of discussions always seem to devolve into "10,000 generations ago, someone who lived near where some of my ancestors later lived might have known more mathematics than people who lived near where your ancestors lived." [The 'That means I'm better than you.' is usually not explicit] Which is boring.
A far as paper writing, I found [1] which said roughly 100 A.D. some paper production began.
[1]: http://www.chinaheritagequarterly.org/tien-hsia.php?searchte...
Looks like they were using colour as variable/labels.
> The way I read it is that the first diagram says that(a+b)^2 = 4(ab/2)+c^2 and the second says that c^2 = 4(ab/2)+(a-b)^2. Both of these simplify to c^2 = a^2+b^2.
I only have a small proposed stylistic change that is to keep the are of the triangles as unknown values, let's call it T. So the first diagram shows that (a+b)^2 = 4T + c^2 and the second that c^2 = 4T + (a-b)^2. And simplifying you get simplify to c^2 = a^2 + b^2.
Is there a transcript and translation of the Chinese text?
Copy of your previous comments in other thread for context:
> Also while those are Han, the language is sophisticated to regular Chinese readers. I can't even make sense of the words (individually yes) as a native Chinese speaker.
> It's traditional chinese but it's a challenge. This is my best guess:
> Looks like they were using colour as variable/labels.
--
Most of my previous guess is wrong :( .
In the second diagrams, they use that the area of the triangles is 6, they are not ignoring the number.
In both pages they ignore the outer triangles. They are not painted in neither graph. If they were using "my" solution I'd expect that the outer triangles in the first diagram were painted with the same color as the inner triangles in the second diagram.
The first diagram makes no sense. It looks like a graphic to show a numerical equality, but the geometric properties give no insight of the numerical properties. Perhaps it was a usual writing style?
There is no 49 in the first graphic (nor in the second) so my previous explanation doesn't fit with the text.
--
My second unsupported attempt of translation is:
[Second diagram]
The area of each red triangle is 6 [because 4 * 3 / 2 = 6].
The side of the inner yellow square is 1 [because 4 - 3 = 1], so the area is 1 [ 1 * 1 = 1].
The area of the [rotated] square is the sum of the four red triangles and the small yellow square, it is 25 [because 6 + 6 + 6 + 6 + 1 = 25].
So the side of the [rotated] square is 5.
Then the sides of the triangle at the bottom are 3, 4, 5.
[First diagram]
If we subtract the area of a square with side 5 and a square of side 3 [painted in cyan], we get the area [painted in yellow] that is 16.
[Notice that 16 is the area of a square of side 4, that we never draw here.]
--
My current opinion:
The second diagram is a proof that in a right triangle with legs 3 and 4, the hypotenuse is 5. I think that this can be extended to any other Pythagorean triple, but one at a time. This doesn't look like a general proof for all of them. This looks more like the verification of an example than a general proof.
The first diagram is an auxiliary construction for the second diagram. It's not very enlightling.
The explanation of the side added in Fermat's library is no a transcript or an explanation of the proof in the main text. It's another proof of the Pythagorean Theorem with a somewhat similar graphic, but it's unrelated.
The first diagram I can't get my head round at all, or is it proving something different?
If we label the sides a=3, b=4, and c=5 :
The first diagram shows that the big outer square of length (a+b) equals the area of the 4 triangles plus the rotated square of side c. (a+b)^2 = 2ab + c^2
[I'm not sure why they drew an inner 3x3 square on its own].
The second diagram puts the same four triangles inside the rotated square of side c, which leaves an extra smaller square of side (b-a) inside.
c^2 = 2ab + (b-a)^2
However paper itself was invented around the same time. How come bookbinding already existed then?
"The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift." - Albert Einstein
"The people in the Indian countryside don’t use their intellect like we do, they use their intuition instead, and the intuition is far more developed than in the rest of the world… Intuition is a very powerful thing, more powerful than intellect, in my opinion. That’s had a big impact on my work.
Western rational thought is not an innate human characteristic, it is learned and it is the great achievement of Western civilization. In the villages of India, they never learned it. They learned something else, which is in some ways just as valuable but in other ways is not. That’s the power of intuition and experiential wisdom." - Steve Jobs
Carl Sagan was really keen on seeing 'the bigger picture' too, can't find the particular quote I was looking for. Maybe later.
The key difference is western medicine is tested objectively, while traditional medicines are based on belief. A million people getting sick and taking a treatment and recovering doesn't mean the treatment does any good, but they'll believe it does because everyone else believes it too.
A good read about this is Orientalism by Edward Said.
I don't know much about Eastern medicine outside of India to be honest. One exception: Chinese people buy tiger bone Viagra, elephant tusk art, rhino horn something, and shark fin soup.
I was born Hindu but became closer to Buddhist as I learned more about the scientific revolution and how it built the modern world.
Ayurvedic medicine clearly does more harm than good. It has never been shown to be significantly more effective than placebo. Ayurveda poses a major risk of heavy metal poisoning, due to the pseudoscientific theories of rasa shastra.
I think I should explain my thoughts on HN a little clearer going forward.
Cars kill more people per year than heroin, therefore heroin is healthier than cars.
Let's wait until pseudo-scientific treatments become multi-billion dollar industries and then we'll talk about what does more harm to what.
1.) A meat-based diet is directly linked to higher cancer rates.
"The World Health Organization has determined that dietary factors account for at least 30 percent of all cancers in Western countries and up to 20 percent in developing countries. When cancer researchers started to search for links between diet and cancer, one of the most noticeable findings was that people who avoided meat were much less likely to develop the disease. Large studies in England and Germany showed that vegetarians were about 40 percent less likely to develop cancer compared to meat eaters."
http://www.pcrm.org/health/cancer-resources/diet-cancer/fact...
2.) A meat-based diet is directly linked to clearing wild habitat for more farm animals.
"Both the meat-based average American diet and the lactoovovegetarian diet require significant quantities of nonrenewable fossil energy to produce. Thus, both food systems are not sustainable in the long term based on heavy fossil energy requirements. However, the meat-based diet requires more energy, land, and water resources than the lactoovovegetarian diet. In this limited sense, the lactoovovegetarian diet is more sustainable than the average American meat-based diet.
The major threat to future survival and to US natural resources is rapid population growth. The US population of 285 million is projected to double to 570 million in the next 70 y, which will place greater stress on the already-limited supply of energy, land, and water resources. These vital resources will have to be divided among ever greater numbers of people."
http://ajcn.nutrition.org/content/78/3/660S.full
3.) U.S. eats more meat than any nation on Earth and also has the most expensive health care in the world too, 20% of the GPD. We have a society whose delusions cause diseases and destruction around the world. If I was wrong, there wouldn't be so many brilliant scientists, doctors, artists, and entrepreneurs on this same train of thought.
4.) Yoga, meditation, plant-based diet have all played a major part in creating the modern world including computers and the internet. If people don't have discernment to know true from false, that is their shortcoming and not Dharma. ;)
> We have a society whose delusions cause diseases and destruction around the world. If I was wrong, there wouldn't be so many brilliant scientists, doctors, artists, and entrepreneurs on this same train of thought.
Argumentum ad populum.
> Yoga, meditation, plant-based diet have all played a major part in creating [...] computers and the internet.
This is probably the most absurd statement I've read all week.
I was interested in your argument about meat and then I got to this, which immediately unraveled any point you had for me.
Investments in military technology played a major part in creating the modern world including computers and the Internet. That hippies were so intimately involved at a few important points is more a reflection of the time period and culture than causality, more an observation of the type of people drawn to the work for a season. Stating that we have the existence we have in large part because of yoga and meditation and not eating meat is so revisionist and misguided that I don't even know how to dispute it. I'm sorry, but that's just blatantly ridiculous because it's such a complete inversion of the actual truth.
That you take it further and imply someone is defective and invoke Dharma for believing otherwise is quite something. You should really bring it down a bit if you want your point to make sense, because you're coming across as far more arrogant than I think you intend.
Semantics. What I typed, how he reacted, how I reacted, and now your reaction. We are so far from my original thought that I don't know how to respond. In fact, I think I'm going to stop here because I am wasting time talking about nothing of consequence.
Sorry for the harsh words, but I've seen the damage, and I am tired of how alternative medicine is handled with silk gloves and shown infinite respect, in the name of being open minded.
B, the original comment was focused on taking what works from both and leaving the rest behind. You're making the same mistake as a billion other people...clinging to the paradigm you are currently in thinking there is nothing beyond it.
C, I'm tired of people thinking they are smart because they used the word 'orthogonal' and say things like 'sorry for the harsh words bro', which doesn't apply in this context and this is a pretty typical reddit/HN thread. Do you have any original insights into any of this or are you just gonna drop more vocab words and 'harsh words'?
The first question was fine of course but the political flamebait in the rest predictably took over.
It is. And it's nothing new and well-known[1]. OP is Fermats library's high resolution picture of the 17th century reproduction.
District 13
Says Americans, biggest lovers on this planet for their own national flag. /s
This is the claim that China and Taiwan should reunite since the PRC drove the Nationalist Party out of mainland China. It isn't fictitious, it's merely a political wish to merge the two from the PRC side. Similarly, some Nationalists in Taiwan are happy to take back mainland China, as much as the country is divided on whether Taiwan should remain as it is, or should declare its full independence.
I'm very skeptical of purely historical claims of sovereignty. If the people of Taiwan legitimately vote to join the PRC, that's their right, but the PRC has no legitimate right to demand it.
[1] https://commons.wikimedia.org/wiki/File:ROC_Administrative_a...
China really doesn't want to find out what might happen if countries start recognizing Taiwanese independence, so the US president poking at that bubble is seen as a threat. China responds with assertive words because fighting with words is better than fighting a war.
So it's not as if the ROC is free to amend their own constitution - as doing so would negatively affect their national security. It's safer for them to keep up the ridiculous notion that they have sovereignty over Mongolia and parts of the Russian far east.
https://en.wikipedia.org/wiki/Confederados https://www.vice.com/en_us/article/welcome-to-americana-braz...
There were also settlements in Belize and Mexico apparently. None of them lay claim to being independent nations and owning mainland US though.
The ROC as I see it retreated into an enclave of the former whole China and was never fully vanquished from Chinese land. So it's really more like if the Union stopped fighting after the Confederacy was defeated in all but one breakaway state. Even more accurately, since the PRC were the rebels, it'd be like the Confederates took over all the Union states except, say, Pennsylvania and then the war came to a halt. The Pennsylvania redoubt of the US would still be the US and may or may not make claims to the rest of the formerly united United States.
But what do I know about the situation? I'm just an American.
[1] https://en.wikipedia.org/wiki/Political_status_of_Taiwan
[2] https://en.wikipedia.org/wiki/Political_status_of_Taiwan#Arg...
Although to be realistic I don't think China will ever outcompete the US. Not because I don't want them to, but simply because so many fundamentals count in Americas favor and against China. China is facing a demographic catastrophe. The US can easily grow its population with immigration and has plenty of land and natural resources to do so.
China is facing a demographic catastrophe.
Strongly disagree. The view on the ground in China is great. Almost everyone has their material needs exceeded, many sectors of infrastructure are second to none, education and internationalization of tastes is increasing, and the society is generally very safe. E-vehicles and e-commerce are everywhere. Rent a bicycle for nothing. Visit the tropics, Himalayas, desert or snow with a cheap domestic flight. Cuisine is arguably the most extensive in the world. Mobile payments are ubiquitous. Internet is cheap to free and literally you can get a signal anywhere. Yes, of course like anywhere it's not perfect, but it's really a far cry from America's stagnant and elitist education system, crumbling or monopolistically privatized infrastructure (eg. transport, medicine), and widespread opiate, gun and race problems.
(FYI: minimum wage in most Chinese cities for an unskilled foreigner with any kind of get up and go is 200CNY/hr (~USD$28/hr) these days... a great many western workers are on less than half of that, and would do very well to migrate.)
so many fundamentals count in Americas favor
I would like to hear what these are supposed to be. Having elite universities doesn't count if you rent them out to all comers, default English doesn't count if more people speak it overseas, VC doesn't count if it's easier to get elsewhere, and land isn't needed if your population is already happy to urbanize with extreme density. China owns the world's supply chain and manufacturing, has a highly efficient centralized form of government that can get huge or long term projects done without recourse to political terms, and leads the world in output and deployment scale of renewable energy systems (hydro, solar), modern transport infrastructure, robotics, internet, mobile payment and e-vehicles. Those are some pretty big fundamentals, too. From my perspective all the US really has are systemic global financial and communications surveillance capabilities and a big fat army. Their diplomatic trajectory is nothing to write home about, either.
you didn't refute that. you just said "life on the ground today is good." That's why its called a cliff. Things are pretty good right up to the moment you fall off.
>FYI: minimum wage in most Chinese cities for an unskilled foreigner with any kind of get up and go is 200CNY/hr
But that's not the same thing as minimum wage. Many many Chinese people earn less than that. When you're a Chinese person you can't just hit up your mate on wechat for some bullshit English teaching job whenever you want beer money.
Obviously not everyone earns the same, then again local people tend to have more money than even wealthy foreigners at the high end, and at the low end at least tend to benefit from free housing, familial support networks, and do not have to fund international travel to maintain ties to home.
Furthermore, downvoting because you don't agree with someone is considered bad form.
Quoting wages is meaningless if you don't also mention cost of living.
Are you guys seriously not getting the joke? I don't usually buy into the stereotype that Hacker News is utterly humorless, but you're not making it easy.
http://www.scmp.com/business/global-economy/article/2081771/...