Yes, but the point is the approach to proof (proving this for all cases) that represented a paradigm shift in mathematics, largely attributed to Thales.
I think this is the most important point modern readers need to make.
If we discover a proof earlier than Thales, regardless of whether it’s Greek, we might revise the discovery of trig. But this is about certain triples being used practically in Babylon (who recorded everything) without any recorded interest in the relationship between triples.
Because Greeks first recorded the important part —- the proof —- I think it’s factual and not at all offensive to refer to Ancient Greek culture as influential in that recorded innovation in geometry.
I’ve seen a movement to broadly give credit to other cultures for Pythagoras, but knowing about triples is just not enough to say that they promoted innovation leading to trig.
> this is about certain triples being used practically in Babylon (who recorded everything)
The difference isn't that Mesopotamia generated more records than other cultures. It's that Mesopotamian records become practically indestructible when burned.
(This didn't help much in Babylon itself, which didn't burn.)
With regard to Alexandria, if Euclid had access to all the resources of Alexandria, and Elements is the second-most translated and transcribed book (behind the Bible) then Euclid had the best known explanation of where the Greeks got the theorem (right or wrong) and also the historiographic influence to credit Pythagoras (also, of course, right or wrong). There might be a missing chapter of Euclid.
WRT Babylon, I don’t know how many tablets are left undeciphered. This article (or a comment) mentions the famous cosecant tablet. We have the tools to fill in the gaps for their language and math. I think extrapolating past that is at least as ahistorical as speculating about what was lost at Alexandria.
I think that there are several ideas that are both powerful and that people muddle when they talk about mathematics. Also the talk is often tinged with cultural pride.
The ideas are conceptualization, generalization, and rigor.
The Pythagorean triples are a conceptualization that for certain right triangles, the squares of the three sides obey a relationship.
The Pythagorean theorem is both 1. a generalization that that would be the case for all right triangles and 2. rigorously proved it. Furthermore, at least for the proof in The Elements, it was axiomatic rigor, proving this via a chain of logic starting from base concepts.
Now to the cultural pride aspect. Many cultures came up with concepts in mathematics. Many cultures generalized. Many cultures were rigorous.
The argument is/should really be that the Pythagorean theorem is on the far end of the rigorous, generalization axis. The better argument actually is that The Elements is.
My personal take is that the Pythagorean theorem isn't, at least in the long span of mathematics history, an exemplar of conceptualization.
And yes, there is some overlap and ambiguity with these ideas and you can argue about where the boundaries lie but I don't think we gain that much from that.
If people don't think that other cultures conceptualized, generalized, and were rigorous, just look at The Art of War, Bhagavad Gita, etc...
About conceptualization, my understanding is that ancient Greek followers of religious leader Pythagoras realized that this was not only a relationship between length and area, but a formula for irrational numbers. This might be a claim without evidence, and would not be the first time Western education has tried that.
Babylonians had no problem arriving at solutions to square roots, and a lot of their texts are yet undeciphered. If we find that Babylonians ruminated about irrational numbers, too, then we’re talking about a major revision in the history of math. Until then, to me, the concepts in the Greek treatments are more innovative than the practical usage in Babylon.
I think your take is supported if it turns out people have read way too much into the religion around Pythagoras. I think Pythagoras would be surprised to be famous for a proof when he’d rather be famous for musical scale or some weird ritualistic dance or something. In fact, as mathematicians we might be better served by later proofs, which is implied by your other points, I think.
Yes, irrational numbers were a great discovery (including how they were discovered) and the systematic treatment of math in The Elements was a better example of the advancement in approaches to knowledge and thinking. And yes, if the Pythagoreans actually thought that this was a formula for irrational numbers, that would indeed to quite impressive.
Also, rigor was romantacized. With the exception of The Elements, very little math till the last maybe 200 years had that level of rigor. Newton and Leibniz sure didn't.
This is one of my pet peeves, too, an obsession with referencing the first to try innovative math. Maybe Babylonian root finding is a simplified case of what we call Newton’s method. The minimum level of rigor for a concept to go from theorem to proof is both human and philosophical. We get very little recognition for the mathematicians putting in the hard work, and detrimental recognition for concepts that don’t exactly widen a field of math for innovation.
Recent movements to revise the recognition of Pythagoras are doing the right thing when they acknowledge the path dependence of unbroken written communication of ideas is often mistaken for awarding prestige to a particular institution, educational tradition, or culture.
The earliest known proof of Pythagorean theorem is due to Euclid. It is strange at all that any credit is attributed to Pythagoras, since Babylonians, Egyptians, Indians and Chinese all had discovered this [1]. And Pythagoras did not prove his theorem, to the best of our knowledge.
Also weird that (more generally) Euclid is only rarely mentioned by name, and more often nearly anonymously as the author of Elements. Yet, Pythagoras is named early and often.
From what I remember from my epistemology courses, it's due to historiographical reasons: there are a lot of cross reference of the existence of Pythagoras from other texts, so his historical context and existence is well anchored.
As for Euclid, apart from the Elements, there is pretty much no mention of him anywhere, which is even more troubling since the depth and reach of the Elements is massive and should have had an impact at the time. This led some people to think that Euclid may not be a person but a group of people.
This is what I remember from courses 15 years ago, I may be wrong or outdated on the subject ;)
This is close to what I was taught. I don’t know ancient Greek, so I wonder if that language draws a distinction between singular and plural in the way “the author” of Elements is structured.
Fwiw, I do recall that Pythagoras seems to have forbidden written records, so what survives are his followers’ notes, who apparently gave him singular credit where we might expect a collaboration.
> I don’t know ancient Greek, so I wonder if that language draws a distinction between singular and plural in the way “the author” of Elements is structured.
I don't know what you mean by "in the way 'the author' of Elements is structured", but yes, Greek draws a distinction between singular and plural. (And dual, though to a lesser degree.)
I mean the structure of the way later writers refer to Euclid. Apparently, from Wikipedia, "ὁ στοιχειώτης" but I hope an improved discussion/potential rabbithole occurs if we know whether that unambiguously translates to one male author.
Yes, it unambiguously translates to one male. That question is easy. A literal translation would be something like "the Elements guy".
But it doesn't mean the author was one male. Lots of texts have attributed authorship. The Homeric Hymns are attributed to "Homer". The Gospel of Luke is attributed to "Luke".
Pythagoras isn't named so much in his aspect as a mathematician. He's named because he is the center of a major Greek mystery cult. (The other major cult being centered around Orpheus.)
The Hindu mathematicians Baudhāyana certainly discovered the theorem before Pythagoras [1] and Bhāskarāchārya independenly proved the theorem [2] (though the latter seems it was after Pythagoras).
It is also possible to discover mathematical truths without providing a formal proof. Indian mathematician Srinivasa Ramanujan "independently compiled nearly 3,900 results (mostly identities and equations). Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research. Of his thousands of results, all but a dozen or two have now been proven correct." [3]
The mathematical statement versus rigorous proof was also a cultural difference: "Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous few decades the foundations of mathematics had come into question and the need for mathematically rigorous proofs recognised. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied very strongly on his intuition and insights." [3]
To the highly-material Western mind, Hindus can be a little weird sometimes: "A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said the mathematical knowledge he displayed was revealed to him by his family goddess Namagiri Thayar. He once said, "An equation for me has no meaning unless it expresses a thought of God."" [3]
Wanted to ask if JuliaDB is something that might get more development attention? Or will that remain a community project? (I see it’s been in need of a release for awhile.)
It is not in our current set of major products. That said, informal office discussions mentioned JuliaDB as recently as last week, so it's not forgotten. If there's a demonstrated market, say a need for new high-performance cloud data science tools as part of the pharmaceutical domains we work in, then something like JuliaDB could possibly be revived in the future (of course, this is no guarantee).
In general, the community has discussed reviving the project (or at least the ideas and some of its codebase). Julia computing will also be contributing as part of that revival.
Thank you both for the comments. I believe I remember early on there were some comparisons to kdb+/q. I think there is some pretty great potential with an offering like this (an in-memory database integrated with the language, coupled with solid static storage) from the Julia community going forward. I can envision some use cases in genomics/transcriptomics.
This is fantastic work. I had picked up The Body Electric over a decade ago, and maintain a strong interest in these ideas, though I currently work in a different field (synthetic biology). I hope to contribute in the future.
I was at Casa by The Sea in Ensenada Mexico from 2000-2002 [0]. These were run by WWASP (I wish I was making that up). What a trip to learn today that Paris Hilton was at one of the US based programs. As bad as these were in the US, the foreign ones could be very brutal (Casa was raided and shut down by the Federales).
I also have tried starting a research institute (eonias.org), and am a member of ronininstitute.org. I’d be happy to have conversations with anyone interested in setting up an institute: grant@eonias.org (my name).
In parallel, people have asked whether external electric fields can be used to alter neurons' activity, which is even older: a Roman physician in 46 reportedly cured headaches by applying a live electric fish to patients' heads. The idea of using electricity to improve mental function has waxed and waned ever since, with the most recent peak around ~2015 or so. Terzuolo and Bullock collected some of the first data on this using crayfish axons in 1956 (https://www.pnas.org/content/42/9/687) and subsequent experiments by Deans et al. (2007), Radman et al. (2007-9), Ozen et al. (2010), and Frolich and McCormick (2010) found similar results using in vitro and small animal experiments. In parallel, people went absolutely wild with human studies of transcranial electrical stimulation (TES), a family of techniques including tDCS (w/ direct current) and tACS (alternating current). While some of the results have been exciting, they have not always been reproducible (Horvath et al, 2015ab) and some work suggested that the previous work relied on fields much stronger than those achievable in humans (Voroslakos et al, 2018).
Together with some awesome collaborators, I set up a non-human primate model that let us test tES under conditions that closely match those found in humans: like macaques (and unlike rodents), we have big, convoluted brains in thick bony skulls and comparatively sparse neural networks. We found that tDCS could affect neural circuits (i.e., LFP oscillations) and behavior (Krause et al., 2017) https://www.cell.com/current-biology/pdfExtended/S0960-9822(... and single neurons, even in deep brain areas (Krause, Vieira, et al. 2019) https://www.pnas.org/content/116/12/5747.abstract [0] The fields we used were much weaker than those produced by some parts of the brain itself (~0.3 - 1 V/m vs ~4-8+ V/m), so it suggests that ephaptic mechanisms are probably pretty common.
I'm pretty confident in those results, but--to bring things back to the original topic--our recent experiments suggest that getting tES to do exactly what you want, when and where you want it, will take some cleverness and a lot of simplifying assumptions tend not to hold up.
[0] The missing full references above are in these two articles' bibliographies.
I think it's worth bearing in mind that Hayek lived before the information age, and that there are real limits to the validity of the idea that design isn't useful (and especially in economics/finance).
UBI may not be a good idea, but the reason being that it's likely not the best design (I would say Minsky had it largely correct [0]).
> the US absolutely does not have a functioning legislative branch
I'm more concerned by this than anything else. In effect this results in calcified government, which can neither regulate tech companies (or anything) effectively, or serve as a check on executive power. People need to start moving to Wyoming and Alaska, yeah the weather sucks.....but we need to redefine 'civic duty'.
Isn't this the sort of meta-hack that should(or is?) be illegal? I thought selling your services at a loss to specifically drive competitors out of markets was illegal? If it's not, should it be? Once capital has accumulated, there is really no effective way to compete if these practices are allowed.
I don't think that's how it's presented. It's running at a loss now to grow and be profitable later. That could be by driving out your competition, or it could be like facebook, where a gazillion dollars went in before the ad money came online. The story they tell themselves it the latter.
The ecosystem it creates, however, doesn't matter whether it's the former or latter. It drives out competition because VC's can make up the difference in a lack of profit.
That's why the title doesn't fit. Unfair advantage is raw capitalism, not a distortion of it. What you're talking about is government interference in the markets, which is a form of socialism, which is a big no no in our society.
Why do so many people confuse supply-side economics propaganda with reality?
Government interference in the market isn't socialism, it's regulation. It's not socialism when the government ensures that companies have to compete fairly amongst eachother, minimise environmental hazards, safety test pharmaceuticals or respect labour laws. Even anti-trust laws or price controls aren't socialist. They're collectivistic, because they look after the greater good, but so are charities, some churches and , ideally, the justice system. None of those are "socialist".
Socialism (and only the statist variety, at that) is when government owns (part of) the means of production, and not even always in those cases, either. The US postal service isn't "socialist", it's a state-run public service. NASA isn't "socialist", it's a government agency funded with public funds.
Government intervention in our markets is the rule, not the exception. Whether we call it ‘socialism’ or ‘stabilization’ is simply a matter of who is making the sales pitch.
