Why didn't the Romans contribute much to mathematics?
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The question should rather be, what made the Greeks (and, later, others who adopted their deductive, axiomatic method) so exceptionally productive at mathematics?
Or to paraphrase Wigner, why is Hellenistic mathematics so unreasonably effective?
Use of the term "dark age" is both dramatically inaccurate in many ways [1] and totally elides everything that happened outside of Europe, such as the establishment of algebra as an independent field of mathematical study (AD 800 in Baghdad), the creation of algebraic geometry (AD 1070 in Persia), and the discovery of ways to solve high-order polynomial equations (approximately AD 1200 in India and China).
[1]: https://slate.com/human-interest/2015/01/medieval-history-wh...
Edit: Fixed a location.
Just Google it:
where was algebra invented?
https://www.google.com/search?q=who.invented+algebra&ie=UTF-...
If we put ethnicity ahead of national origin America can’t claim anything to its name. Period.
It's silly to put ethnicity ahead of national origin, though, since one of America's biggest qualities it that it is a melting pot. This feels like saying "If we put learning curve ahead of type safety Haskell can't claim anything to its name. Period." It's true, but kinda vacuous.
Personally, I think if an immigrant of descendant of immigrants invents something prominent, it is fair for both countries to be proud of the accomplishment, and share that enthusiasm. It doesn't, or shouldn't hurt anyone. If anything, you are affirming accomplishment, not putting other people down.
Depending on the reasons for emigration, it might be fair to replace most of that pride with "dann, could have been 'ours'".
But other than not repeating those horrible conditions, the next step is to simply be proud of the accomplishment. In this case of algebra, Baghdad isn't the Baghdad of back then, nor Persia the Persia of then, all the people of the era are no longer with us, so what does it matter really? But hey, algebra is pretty cool. And Arabs/Persians happen to feel good about it. Good for them.
(Frankly, I doubt we know where Al-Khwarizmi was educated. We know he was originally from what's now Uzbekistan, and that he ended up in Baghdad, but not much more than that.)
Mesopotamia was conquered by Arabs in 633. Al-Khwarizmi was born in 780. Most the Arabic lands were gained by conquest. Even modern day Yemen and Oman were not under Arabic influence if you go back far enough. The "original" arabs spread from Hijaz and Najd.
edit: from wikipedia
> The Abbasid Caliphate first centred its government in Kufa, modern-day Iraq, but in 762 the caliph Al-Mansur founded the city of Baghdad, near the ancient Sasanian capital city of Ctesiphon.
My guess is that families from the Sassanid have immigrated to the new city and as a result Al-Khwarizmi was born there.
Source (a very enjoyable read at that!): https://going-medieval.com/2017/05/26/theres-no-such-thing-a...
For example, peasant households in Italy at the height of the empire had a wide variety of well-made pottery, some of it originating from production centers hundreds of miles away, along with tiled roofs; after the collapse, the variety and quality of pottery was greatly reduced, and tiled roofs disappear. (Tiled roofs provide better protection against rain, are less flammable, and require less maintenance and rebuilding than wooden shingles or thatch; but they also require large kilns, lots of clay and fuel, and a good transportation network.)
And ice cores from Greenland show extensive metalworking going on in Western Europe (in the form of copper and lead pollution carried through the air) during the Roman Empire; after the collapse of the Western Empire, the pollution levels don't return to Roman-era values until the 16th Century.
http://pzacad.pitzer.edu/~lyamane/ianmorris.pdf
Across a wide set of measures, there was a significant decline of the West after the fall of the Roman Empire.
Until the 15th century less than 5 people in Europe could do long division. For comparison, Aryabhatta discovered calculus in 5th century India and Madhava made large contributions in the 13th century (Newton/Leibniz were wrongly credited for similar work many years later).
That’s wildly inaccurate. Arithmetic was very much a subject at several schools with the addition, subtraction, multiplication, and division all being taught. https://en.m.wikipedia.org/wiki/Quadrivium
Long division being difficult in Roman numerals is one of the reasons for the switch.
I'm talking specifically about "long division" in Europe... and have heard this factoid from multiple mathematicians including Nasim Taleb. So I expect there is a basis for the claim...
However, if we are talking about the number of people being able to accurately do 1234 / 56 = 22 r 2. That was regularly taught in the Middle Ages with plenty of examples of such calculations being available.
He has bit of a habit exaggerating and elaborating on stories for the sake of making tasty reading. There was a story in Black Swan about two hospitals and a simple statistical riddle that he claimed professional statisticians would get wrong. It wasn't even a trick question, most people would get it right. I found this hard to believe but I was still sort of surprised when the source spoke of a much harder statistics question that was surveyed at a psychologists convention, not statisticians. That was the first (and the last) source I checked on him. I'd have checked a few more but I don't have the book any more.
But now I guess I can add this one to the list of examples.
Relevant (even though considered as pseudoscience):
> https://en.wikipedia.org/w/index.php?title=New_Chronology_(F...
(Edit: incorrectly said renaissance instead of Age of Enlightenment... the renaissance wasn’t Western European either)
https://aaron7roberts.files.wordpress.com/2015/04/roman_empi...
https://en.m.wikipedia.org/wiki/Demography_of_the_Roman_Empi... also shows larger populations in the “Latin West” than in the “Greek East”.
https://en.m.wikipedia.org/wiki/Demography_of_the_Roman_Empi...
edit terrible math on my part.
No one said that. The Roman Empire isn't Western Europe, and calling the period the "Dark Ages" is inaccurate.
That seems to imply that Rome isn’t part of the history of Western Europe at all (and neither did the Renaissance happen in Western Europe, apparently).
1. Roman empire dominion included, but did not limit to what we now refer to as Western Europe...
2... of whose languages are either mostly descended from, or heavily influenced by Latin...
3. ... And where the dominant religion is still the Roman state religion. Which was lorded over by the Roman state church for a thousand years before an angry German monk suggested that this was really weird and could we keep the book and get rid of the pope.
So, when you say Western Europe is not Rome, you need to say how it isn't, since there are quite a few things that link this geographic area culturally very strongly to that ancient empire.
To be fair, they said Rome is not Western Europe, which is true. Rome didn’t have cultural affinity with Western Europeans. Rome’s cultural affinity flowed to the east with the Greeks et al.
The Germanic Tribes were also taller and were acknowledged by Romans as formidable warrior.
Culturally tho, Romans were South Europeans and assimilated what we call South Italy and Greece now.
They also captured parts of middle east but culturally they never became Romans.
>3. ... And where the dominant religion is still the Roman state religion. Which was lorded over by the Roman state church for a thousand years before an angry German monk suggested that this was really weird and could we keep the book and get rid of the pope.
There were other Pagan religions at the time and in middle east Jews didn't accept Roman state religion.
But then again, people born in Roman Empire got Roman citizenship on Birth so it didn't matter what you were culturally or religiously or tribe/ethnicity.
I assume 'people' excluded slaves?
*I mean the Germanic tribes in North ate meat everyday.
Why the Downvotes? Anyone wants to challenge these facts, please post rebuttal. I'd love to educate myself more.
But then again, you say stuff about how romans vs germanic meat eating, but I don't see any citations for that. They'd be helpful.
That depends on when you're talking about. Until Caracalla made all free subjects citizens in 212 AD (supposedly so that he could tax them more heavily),[1] Roman citizenship was restricted and highly coveted.
> There were other Pagan religions at the time and in middle east Jews didn't accept Roman state religion.
From the late 300s onwards, pagans were persecuted in the Roman Empire. Nicene Christianity was the state religion. Also, Jews lived throughout the empire, not just in the Middle East. From about the 1st century AD onwards, there were more Jews living outside of the land of Israel than within it.
Was it Byzantine?
That really was my experience growing up in Western education. My understanding was that history focused on the important things and it wasn't until much later in life that I learned there was a lot happening globally after the fall of the Roman empire before the European renaissance.
After the USA was established, US history lessons completely ignore the rest of the globe until the early 20th century. I only learned yesterday that Italy wasn't a country until 1861 and I am almost 40.
That’s because they’re Western history lessons. I had world history as a separate class in high school.
https://slatestarcodex.com/2017/10/15/were-there-dark-ages/
> Although I sympathize with the feelings behind both positions, I say the Dark Ages happened. I think the best evidence we have suggests the fall of Rome (and the period just before) was associated with several centuries of economic and demographic decline, only reaching back to their classical level around 1000 AD. I think it was also associated with a broader intellectual and infrastructure decline, which in some specific ways and some specific fields didn’t reach back up to its Roman level until the Renaissance. I think that common sense – the sense you get when you treat the question of the Dark Age the same as any other question, and try to avoid isolated demands for rigor – says that qualifies for the phrase “Dark Age”.
Maybe the peoples of the future will only consider one particular mathematician as the relevant one? Maybe like Turing, because most likely computers will become a greater part of the world at large? Maybe we see programmers the way people used to see scribes some time back?
Wrong! If you mean times between 500 and 1200: how about arabs, iranians, indians, chinese? Algebra, Negative numbers, 0123456789 system of writing?
They had Archimedes of Syracuse
Many Greeks were part of Roman empire, what difference does it make?
If a Puerto Rican living in Puerto Rico invented something, we'd call it a Puerto Rican invention. If a European immigrated to the US, integrated themselves into the culture by learning English, adopting American cultural norms, and taking the oath of citizenship, we'd call that an American invention, despite the fact that both of them carry a passport that says "United States of America" on it.
This apocryphal story on economic incentives vs progressive incentives is as relevant today as it was 2000 years ago.
China, despite a much larger and more educated population, did not spark the industrial revolution. Their feedback loops were too stable, their elites too competent.
From the perspective of the old power hierarchy, the industrial revolution was a disaster.
The nobility floated on that great cruel ocean first charted by Malthus, an ocean which began to boil.
And then I guess a follow up question would be: do you have an opinion on why the scientific revolution happened in Western Europe and not in the Mediterranean, which is similarly fractured, but also better situated for exchange of technology and ideas (being connected to Asia and Western Europe).
And China hasn't really had too much stability. It's always been full of revolutions, competing empires, dynastic changes, warring states, shifts in power, changes in ethnic ruling class, and so on. Go through a list of deadliest wars and revolutions and a good chunk of them are in what is today China.
It's been invaded at least in part a few times (Mongols, 13th c., English, 18th c., and Japanese, 20th c., most especially), but retained its overall identity and either assimilated (Mongols) or repelled (English, Japanese) the invaders, eventually.
Contrast Europe which has seen vast shifts in control, and utter replacements or eradication of local culture or tribes multiple times, going to prehistoric times, to the present (past century certainly, past few decades quite arguably). There have been very few constant borders or identities, certainly not on the scale of China.
Even written language in China is still largely intelligible to moderns, from a thousand years ago or more. English, more than about 400 years ago is almost wholly foreign:
Hwæt,ic swefna cyst secgan wylle, hwæt me gemætte to midre nihte siþþan reordberend reste wunedon.
("Behold, I wish to tell the best of dreams which I dreamt at the middle of the night, after speakers remained in rest." https://www.arts.gla.ac.uk/stella/readings/OE/ROOD.HTM)
Watch that graphic to see the evolution of the extent of "China" since the Zhou Dynasty in 1000 BC. It occupies a fraction of the current extent of China. So there were lots of wars, and lots of revolutions before modern Chinese borders were established.
I get that Chinese have had more cultural and language cohesion, but that's at least partially because certain dynasties and regions won over these other regions in large swaths and history is written by the winners.
https://www.invidio.us/watch?v=UY9P0QSxlnI
Note that China was under a largely single and unified set of linguistic, cultural, and political organisation throughout the entire 3,000 year period you depict. The "modern" map of Europe has changed dramatically just in the past 30 years, let alone the past 100 - 200. Entire linguistic and cultural tribes have completely vanished, save for a few pitiful relic remains. Political continuity has been nil (Greece, Macedonia, Byzantine Empire, Rome, Moorish Spain, the Gauls, Britons, Celts, and Germanic tribes, the Rus? (Swedish invaders, BTW), Holy Roman and Austro-Hungarian Empires? Soviet Bloc? Independent Baltic and Balkan states? Ireland, under English subjugation since 1601, the north to this day. Poland as the pinball kid of Prussia, Russia, and Germany.
That's Europe.
Addenda: The extreme extents of the Chinese empire varied greatly. The bulk of the population, then as now, was in the eastern coastal region, largely the Yellow and Yangze river floodplains.
Here's a few hundred wars that occurred during the past couple millennia: https://en.wikipedia.org/wiki/List_of_Chinese_wars_and_battl...
The language, culture, identity, philosophical and religious beliefs, government, urban centres, technologies, currency, writing systems, etc., have persisted in a largely unbroken continuum or chain. There are points at which one or the other makes some jump from one form to another, but there's never been a wholesale wipe of one system to another. Over a period of 4,000 years of recorded history and arguably going back even further.
The West has nothing comparable to offer. Not in Egypt, not in Mesopotamia, not in Rome, not in England, not in France, not in Germany, not in Scandinavia. Maybe in Persia or India. Vietnam may have an older continuous civilisation. Japan does not, though it comes close.
The opening sentence of an arbitrarily selected webpage on ancient China makes this case: "Ancient China produced what has become the oldest extant culture in the world."
To be clear: I'm not some defender of the present regime in China. I'm only casually aware of its general history, geography, and culture. But as I study it the vastness of the culture, in both time and space, has impressed the hell out of me. And it seems that this isn't as widly known or appreciated as it could be.
Are you suggesting this text is from the 17th century? I'm no expert but this much Germanic influence seems much older.
"The poem is recorded in its fullest form in the Vercelli Book, a late-tenth-century West Saxon manuscript which was left in Northern Italy in Anglo-Saxon times. Vercelli is on the road to Rome; the manuscript was either abandoned or forgotten by a pious Anglo-Saxon pilgrim. An Old Northumbrian version of part of the poem also appears, carved in runic script, on the late-seventh-/early-eighth-century"
Source: https://www.arts.gla.ac.uk/stella/readings/OE/ROOD.HTM
That's largely because Chinese characters themselves didn't change, even though the underlying sound and grammar changed. In your English example, imagine "ic" is written as "I", "nihte" as "night", and so on, so that every word is written as the modern spelling of its descendant word. It will be much easier to guess its meaning.
Advantage West: a compact, typeset- and keyboard-friendly characterset with a writing style that generally allows sounding out of unknown words.
Advantage East: a pronunciation-independent mapping of ideas allowing mutually unintelligible spoken language across either space (Japan, to some extent Korea, as well as modern Chinese provinces, particularly Mandarin vs. Cantonese) or time (modern vs. ancient) to at least largely understand one another.
Much of the shift in written English parallels changes in spoken English. Chinese doens't have that problem. But it's far harder to learn and type or encode in modern computers, where it is not letterforms but encodings standing for letterforms which are stored and projected via fonts and Unicode codepoints.
Chinese, as a tonal language is so dependent on inflection that my own primitive attempts to pronounce even simple phrases are not understood based on my incorrect tonality.
Contrast with various Germanic languages (German, English, Dutch, Danish, Norwegian, Swedish), which vary greatly in pronunciation and orthography. German is spelled and spoken logically. Danish is spelled much like German but not pronounced the same. Norwegian pronunciation differs, but its spelling at least corresponds to that pronunciation. Swedish is pronounced very differently, and the spelling doesn't much conform.
All share common linguistic roots.
Each approach of writing and alphabet appears to be brittle in its own ways.
Except you wouldn't contrast that. Those same events have happened in the region of China on the same scale as continental Europe as a whole. Entire societies have risen, conquered lands, and vanished.
The Manchu are the most notable example for coming in from the Northeast and conquering China. They had their own unique culture and language. Native speakers of their language are now countable on one hand.
The Tangut are another interesting example. They had a very unique culture and writing system unlike anything else and had their own empire in the middle of China. They were wiped out/eventually absorbed by Mongols and were completely forgotten for centuries until their writings were recently rediscovered.
Written Chinese today is an evolution of only one of the many, many cultures that formed what China is today. Even Japanese people can read classic Chinese texts with just a little practice, but they're definitely a separate society and always have been.
You could also argue western European cultures are all just Rome since they retain parts of the Latin language and writing system and can make out some words if they squint enough.
This is Shakespeare, from 1599:
Friends, Romans, countrymen, lend me your ears; I come to bury Caesar, not to praise him. The evil that men do lives after them; The good is oft interred with their bones; So let it be with Caesar. The noble Brutus Hath told you Caesar was ambitious: If it were so, it was a grievous fault, And grievously hath Caesar answer’d it.
Hardly foreign
Friends, Romans, Countrymen, lend me your ears:
I come to bury Cæfar, not to praife him:
The euill that men do, liues after them,
The good is oft enterred with their bones,
So let it be with Cæfar. The Noble Brutus,
Hath told you Cæfar was Ambitious:
If it were fo, it was a greeuous Fault,
And greeuoufly hath Cæfar anfwer'd it.
Heere, vnder leaue of Brutus, and the reft ...
https://iiif.bodleian.ox.ac.uk/iiif/image/74391696-b1e6-4b56...
Even taking the tack of following Germanic Anglo-Saxon culture buys you little -- spoken and written language would be unintelligible.
A Western college graduate of ancient languages today might understand Latin, which could get you to 240 BCE, or possibly, with a different alphabet, Greek, whose earliest writings, the Illiad and the Odyssey, date to the 6th century BCE in written form. But that's not a common skill.
Greek and Latin had been taught, as part of the high school curriculum, through the 1940s or 50s, but not generally any longer. Keep in mind that high school graduation rates in 1900 in the US were about 6%, lower than Ph.D. attainment rates today.
You're pretty much equating English and Europe when talking about language, which in this case is very wrong. I, as a spaniard, can read poetry from a thousand years ago perfectly fine, and we do indeed do it in school.
Though Spanish, Castillian, etc., were spoken. Well, back until 210 CE:
https://en.wikipedia.org/wiki/History_of_the_Spanish_languag...
(The logic of the Meiji "Restoration" was essentially "Let's get rid of the archaic Shogunate system that has dominated Japan in one way or another for the last seven hundred years, and pretend we're handing power back to the Emperor where it really belongs.")
The general question of the how, why, and when of the Industrial Revolution has fascinated historians, technologists, and economists for ages. Karl Polanyi's The Great Transformation, Gregory Clark's A Farewell to Alms,[1] and numerous other works address this.
Geographic determinism has become tremendously unpopular among historians, though elements of it carry weight with me. Of China and Japan vs. Europe and Britain, there are several factors:
- Political unity vs. disunity, as you note.
- Crops. Wheat is suitable to individual, independent farming. Rice requires community coordination.
- Hydrology. The Chinese empire effectively started as a large civil water works management bureaucracy. Outside Egypt and Rome's aqueducts, there was no similarly-scoped coordination in the West.
- Watersheds. Europe's rivers diverge from the interior, China's flow in parallel to relatively proximate mouths. Political boundaries in Europe have typically conformed reasonably well to watersheds, though allied / opposed alignments have changed with time. Even today, many county-level jurisdictions correspond to local watersheds. And in both China and Europe, virtually all heavy transport until modern times was along rivers or canals, if not sea or lakes.
- While Britain and Japan are both large islands near continental empires, the geology is utterly different: sedimentary with vast coal deposits, and volcanic with virtually no fossil fuels. While each island was tremendously politically stable, resistant to invasions, England could fuel growth of iron, glass, and steam industries, Japan could not. England was generally relatively wealthy, Japan was one of the poorest countries prior to industrialisation.
- China has long been politically unified (if subject to occasional invasions), Europe has long been politically fractured. China could shut down innovation and foreign trade. No such global policies were possible in Europe.
Within Europe, the distribution of coal is almost wholly in the north: Wales, England, a tiny patch in northern Spain, some in France, and heavy deposits in Germany and Poland. Southern Germany is very fuel-poor, excepting petroleum (not very handy in pre-industrial times) in Silesia, Romania, and Baku (Russia). Coal fueled metalurgy, glassmaking, and eventually steam power in England.
England's flat terrain and ready access to the sea (no part of Great Britain is more than 60 miles from the coast) made transport of the bulky fuel by ship viable. Overland transport wasn't an option -- firewood fuel locally gathered was far more attractive. A similar situation existed in the US where coal didn't overtake wood as a fuel until the 1880s. Rail transport finally made hauling coal from mountain-based mines in Apallachia possible, but benefited greatly from advances in steelmaking (Bessemer process, 1860s), allowing stronger, less fracture-prone rails, and stronger, more powerful locomotives. Rail is effectively canals-on-land, the first truly viable overland freight tansport mode.
There are many other factors, there's tremendous dispute over all of this, and as I've hinted, there's a large literature. I obviously find the geological argument at least plausible in many regards. Given the lack of testability, final adjudication of the question is unlikely.
________________________________
Notes:
1. Clark teaches a course at UC Davis on economic history before the Industrial Revolution, which touches somewhat on this (the principle focus is Europe). A corrected playlist for the YouTube lectures, in proper order, is here: https://pastebin.com/raw/bgKkGyjt
Also (as already mentioned in this thread), the asset-sheltering tendency of established power groups, halting the emergence of potential rivals, as Bernhard Stern documents in other (non-Chinese) instances.
A local lake used to be an artificial reservoir to keep the canal filled.
The UK is full of canals that were viable to dig because of that flat terrain.
So large parts of England that were not reachable by river are still reachable by canal boat, and even more used to be before many of the canals were filled in or drained when no longer commercially viable for transport.
China though has its Great Canal which dates to 12 BCE and has had pound locks since the 10th century: https://en.wikipedia.org/wiki/Grand_Canal_(China)
In the US the Erie Canal opened up the West and cemented New York City as a transport, commerce, and financial hub, beginning in the 1830s.
https://en.wikipedia.org/wiki/Category:Canals_in_Japan
Among the earliest is the Takase River canal, in Kyoto, 9.7 kilometers, dug in 1611. Most of the remainder date to the 19th or 20th centuries.
https://en.wikipedia.org/wiki/Takase_River
The Tatsumi Canal was constructed in 1623, 11 km.
https://en.wikipedia.org/wiki/Tatsumi_Canal
Contrast the UK:
https://en.wikipedia.org/wiki/Canals_of_the_United_Kingdom
https://en.wikipedia.org/wiki/List_of_canals_of_the_United_K...
The Renaissance, printing, the Reformation, the Enlightenment, and the breaking of uniform Catholic control over thought and science. That's an area I've been exploring (I'm exploring a lot of areas, progress is slow), and it's fascinating.
The Bacons (Francis and Roger, no relation), free-thought movements especially in the north (Amsterdam and Copenhagen), and relative political freedom of inquiry in England all seem to have helped. It's interesting that the Enlightenment itself played heavily in Scotland, otherwise a hinterland (Adam Smith travelled to London for an education he didn't think much of, taking six weeks to make the journey by coach or on horseback, in the 1740s or 50s).
James Burke's television series Connections and The Day the Universe Changed cover much of this development (as well as earlier and later periods), and have been useful, though I'm starting to find holes in Burke's treatments (he wholly ignores Joseph Needham's work and Chinese invention, as examples).
Gregory Clark, mentioned above, has a generally excellent treatment of the question, one of the best I've read yet, though there's much I've yet to read.
Another excellent resource is the History of Information website. It's a useful place to explore questions such as this.
http://2oqz471sa19h3vbwa53m33yj.wpengine.netdna-cdn.com/wp-c...
The borders especially of Spain, France, the former Austro-Hungarian (better described as the Danubian) Empire, and much of Germany and Poland are reasonably well evidenced.
The Balkans are as fractured hydrologically as they are politically.
Currently contiguous peninsular regions are defined in significant part by their coasts rather than rivers, probably showing the significance of sea-borne transport. Norway and Sweden are divided along their fall line, as are Spain and France by the Pyreneese Mountains. Switzerland is an identifiable mountain valley.
What's impressive about Russia is how unified it is by waterways. The Volga-Baltic Waterway provides contiguous maritime communication from the Black Sea to the Baltic:
https://en.wikipedia.org/wiki/Volga–Baltic_Waterway
A comparable watersheds map of the US, showing the immensity of the Mississipi-Missouri-Ohio-Arkansas-Red river system:
https://2oqz471sa19h3vbwa53m33yj-wpengine.netdna-ssl.com/wp-...
(Also the utter insanity of most current US state boundaries. At least if you're a fish.)
And the comparable watersheds map of China:
https://web.archive.org/web/20190825155610/https://s3.amazon...
Not the GP, but I think this one was very clearly caused by the English Revolution protecting people with weird ideas.
A large part of the early English intelligentsia was composed of people fleeing from persecution in Italy.
Original: https://archive.org/details/technologicaltre1937unitrich/pag... (faint copy)
Re-typed / markdown: https://pastebin.com/raw/Bapu75is (updated and corrected from April, 2019 version.)
After all, it's fairly well known that the characteristics of a particular programming language have a strong influence on the way the language is used. (For instance, people don't do OOP or FP using C.)
Oh, but they do. It’s just that their syntax is awful.
I'm sure you can compute sine and cosine tables using Roman numerals, too. But it would be so awfully ugly and tedious it's hardly a surprise that few would consider attempting it.
Ptolemy’s table of chords was calculated in base 60 (inherited from Mesopotamians), by probably a Roman citizen living in Egypt and writing in Greek.
It was probably done on some kind of a counting board analogous to the ones used for decimal calculations. Hellenistic mathematicians didn’t do written arithmetic either.
And yes, making such tables is inherently “awfully ugly and tedious”, unless you have an electronic calculator to do it for you.
> And yes, making such tables is inherently “awfully ugly and tedious”,
Yup, but people did make such extensive tables long before calculators, many thousands more entries than that chord table, and far more accurately.
You might be thinking of Hipparchus’s table (from a few centuries earlier) which only had multiples of 7½°.
Sounds like it.
I want to expand a bit on that. In the evolution of D, I have a front row seat on how people use it. It's hard to downplay how relatively minor syntactical changes can have a heavy influence on the programming paradigms people select. It's startling.
People will often say "but I can do that, too, in my favorite language X", and they are correct. But they don't actually do it in X because it's inconvenient.
For example, D has a built-in syntax for unittests. That gets pooh-poohed a lot as being pointless. But it's hard to argue with how transformational that has been for D programs. People expect unittests when writing D code. They didn't before. Unittests often occupy more lines of code than what they tested. The addition of a very minor bit of syntactic sugar changed the whole way people write D code.
Roman numerals are a recording tool, not a calculating tool. Romans did calculations using pebbles or other counters on a counting board. Roman numerals are just a way of recording the state of a counting board before/after performing some algorithm. The goal of them is to be as direct and faithful a record of the counting board state as possible.
I think you just made my point :-)
Have fun doing long division that way.
We don’t really know much about people’s calculation algorithms, because they were an oral tradition not written down, and only a very small number of counting boards (e.g. made of marble) survived; others were presumably made of leather, wood, cloth, lines scratched in the dirt, ....
Japanese soroban experts handily beat westerners at doing division, in both speed and accuracy. There is no reason to believe that calculation experts of the ancient world would not have been comparably competent.
I searched around a bit, and didn't find much of anything you could do with a counting board beyond division.
With an abacus or with written numbers, you can get 10 digits of precision (or 50) if you are willing to put the work in.
Do you have a good source to read more about this? Both about the distinction you're making (recording vs. calculation) and in general about historical capabilities around them? I'm very interested in learning more!
Gnome/GTK also encourage an object-oriented style in C, via GLib and GObject.
Also, Greek numerals were just as unwieldy as Roman ones (if not worse), and at any rate the Greeks did not use numbers in their mathematics. There's an enormous amount of math you can do without arithmetic and algebra.
(Edit: Saw your username after I posted this and I want to say that I respect you deeply -- I don't know of anyone else who has written a working C++ compiler almost by themselves. And I like D a lot, though have only had a very limited chance to use it professionally.)
But only once. And probably by someone who had learned OOP in another language that was built around OOP.
I was in the C business before, during, and after the OOP revolution. I never saw any OOP attempts in C before C++ came along and popularized OOP. Many C programmers didn't want to move to C++, and were determined to make it work in C. It did work, but the result was kinda hideous and terribly fragile (had to throw type safety out the window with all the necessary casting).
> I don't know of anyone else who has written a working C++ compiler almost by themselves
Neither do I :-) Thanks for the kind words.
The original answer that I read yesterday has since been removed by the mods for some reason.
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[-] ImOuttaThyme66 points 16 hours ago
[Their number system.](http://storyofmathematics.com/roman.html)
Mathematician in studying here with a side hobby of history. Sources are included throughout.
You may recall from elementary school or personal perusing of Roman history books that they had their own numeral system with letters instead of their own numerals, excerpts of the Latin Alphabet. I is 1, C is 100, and so on.
The Romans did decent with their numeral system. They could add, subtract, multiply and divide with their numbers. However, it was missing two very important principles, if that's the right word, that today's Arabic numbers have.
The first is the idea of zero in mathematics. They knew the concept of nothing yes but they had no numeral for the number "zero." So essentially, try doing your own math homework without using the number zero in the one's place, the ten's place, and so on. Now, zero itself didn't come with Arabic numbers but they did come from an ancestor. [Origins of Zero](https://www.smithsonianmag.com/history/origin-number-zero-18...)
Which led to the second problem with Roman numerals. Their numbers do not work on a positional system. Today's Arabic numbers, 0-9 work on a positional system. That is, we have the one's place, go left one digit, ten's place, left again, hundred's place, and so on. Each digit in a number is the base raised to a certain power. And this is what makes addition, subtraction, multiplication and division so easy to do with Arabic numbers, especially with the concept of zero, a placeholder to place in a certain space when a number has no "tens" or "hundreds" or "ones". Like 109. It has 9 ones, and 1 hundred, but no tens. Without zero, we couldn't be able to write it out like this.
A little sidenote here, technically, the Romans did have a positional system. That is, the greater numerals were on the left and the lesser ones were on the right unless they were using two numerals to communicate numbers. So, it's not an explicit positional system like we do today, in which each digit means a certain number, they did order their numbers around based on size.
Now, in fairness, the Romans simplified their numeral system a little over the years by adding two principles, subtraction and multiplication. What the Romans did was that whenever they placed a smaller number before a bigger number, those two numbers communicated a new number entirely by subtracting the smaller from the bigger. So for example, 109, CIX, the IX becomes worth 9 because X (10) - I (1) is IX (9).
Multiplication, to indicate 1000's, they would put a line over the symbol, and that would be the same as multiplying it by a 1000.
Let's do a little Roman math.
For Roman numerals, 109 would be written out as CIX. Great, we get the amount communicated. Let's say we want to add 32 to this number in Roman numerals. 32 in Roman is XXXII.
CIX + XXXII
Immediately, there's a problem. We have to separate the various numbers. So CIX becomes C IX and XXXII becomes XXX II. Now we can add them together by going "Okay, IX plus I makes X, there's an I remaining, X joins the three other Xs so we now get CXXXXI." aka 141 in today's numbers.
Let's imagine as if we did that with our numbers. Let's add 109 and 32 again. Only, we're adding 100 and 9 and 30 and 2. We know 9 + 2 makes 10 and 1. We know 10 and 30 makes 40. We have 1 and 100 remaining, there's nothing else to add them together so they stay like that. The resulting number becomes 100 and 40 and 1. Exhausting.
So, addition is possible. Subtraction is also possible. You have to go through the entire grouping of bases to do it but it is possible. Multiplication is somewhat possible but very iffy, having to do all those grouping of bases manually.
Division. Division was the hell of the Roman numeral system. The Romans did not have decimals. They had fractions but they did it in duodecimal form, that is, 12ths. They did not have a 1/10th. There was no talking about that explicitly, they preferred to do everything in 12ths. Now, this makes sense, 12 has many factors compared to 10. You can divide 12 by 1, 2, 3, 4, 6 and 12 itself whereas you can only divide 10 into 1, 2, 5, and 10. There are plenty of people who believe we should move to a duodecimal system. Then again, the French attempted installing a decimal system for time during the French Revolution. Maybe not all ideas are popular.
Either way, Romans did not have very explicit fractions. They had the base fractions 1/12 through 12/12, then they continued on from there by dividing them further. Like 1/144 (1/12 * 1/12) or 1/8 (3/12 * 6/12). More information on Roman fractions can be found [here](http://dmaher.org/Publications/romanarithmetic.pdf). Another side fact of their fractions was that they always named them as fractions of something, such as 1/12 of as, which was a currency. Never 1/12 by itself.
So, now that we've gone through the clusterfuck that is Roman numerology, we can pretty much understand why they didn't advance the mathematics field too much. It was functional for their time. The Romans still did many grand engineering feats that were no doubt developed from this number system. However, when it came to further mathematics such as calculus, which would finally be found by Newton and Leibniz independent of each other in the mid 17th century, you can't get any further when you don't have a positional system that makes adding/subtracting a lot easier, no numeral for the number 0, and your dividing tactics do not allow you to do decimals very well, especially when your numerals are 1, 5, and then powers of 10. Decimals could be made possible with these numerals yes but it would be insanely difficult to understand and is made so much easier when you assign every basic integer, like zero through nine for instance, their own numeral.
And that's probably why the Romans didn't do very well in mathematical advancement.
https://priceonomics.com/how-maritime-insurance-built-ancien...
The Romans invented Roman numerals, and it's important to acknowledge that this was a mathematical achievement even though we don't use them as much any more. By putting smaller numbers in front of larger ones, they created a number writing system where you did not have to learn a large number of symbols yet any particular number was short and easy to write. Greek numbers had separate symbols not only for one through nine, but for each of the symbols 10 through 90, which meant you had to learn a lot more symbols for just one through 99.
It's true that doing calculations with Roman numerals is a pain, especially division, but I don't think the Romans thought this was a big deal. Calculations were typically done using an abacus anyway, so you simply needed a simple way to record results.
It has a graceful theme which is perhaps not accidental for mathematical inspiration. The same symbolic methods, being symbolic could have been painted as "domination and sacrificing" but that might not temper a mindset as mathematically conducive as notions of "completion and balancing".
A cultures achievements in different areas could owe substantially to the spectrum of mindsets which it hosts and celebrates.
Or maybe what would be considered a mathematically conducive mindset is determined by whose mathematical tradition we are following? Maybe there exists a mathematical mindset out there where "domination and sacrificing" are the requirement for conductivity of ideas?
They had papyrus, parchments and wax tablets, none of them were as convenient or affordable as paper.
The adoption of paper was what really set things in motion in Europe, the Renaissance.
A lot of merit to share.
Papyrus is cheap, easy to make and affordable. It would certainly have been much cheaper than paper, which didn't become any good until substantial industrial and chemical advancements. It was used all over the place. We don't have much left from the Northern Mediterranean because it rots.
Wax tablets were used until the 19th century because it's also cheap and practical.
Velum didn't become mainstream until the Islamic invasion of Egypt cut the supply of papyrus.
>The adoption of paper was what really set things in motion in Europe, the Renaissance.
The idea that there was a Renaissance at all is dubious. Paper was certainly a step up from velum. But It would have been much better for everyone if we could just have used papyrus continuously.
A book made of wax tablets, or velum is not as good. I don't imagine vast libraries and universities running on books made of wax tablets or velum.
With respect to Papyrus, I agree. The Great Library of Alexandria was all papyrus. It can be produced at scale. But Europeans had to import it from far, far away.
We went fromom electricity to flight to nuclear to space flight to the internet in less than 150 years.
Its been less than 100 years since antibiotics were invented, so, sorry we aren't addressing antibiotic resistance at a fast enough pace for you.
Hope you don't mind all those vaccines that are keeping you healthy and providing here immunity: they've only been around for about 130 years.
It's the most innovative time in human history, despite your negativity.
1947?
Concorde was testing by 1969, but didn't carry passengers until 1976.
The USSR's Tu-144 actually beat Concorde on both dates -- 1968 and 1975. Though it also beat Concorde, by 25 years, on retirement from passenger service: 1978.
Interesting though you cite mainly engineering achievements, I had a similar feeling that we are in a phase of engineering/materials advancements, particularly sub-microscale in the post-WW2 period. But having said that, technological advances of this kind tend to end up as historical footnotes, no matter how amazing they seem at the time.
I think vaccines and antibiotics are probably our era’s truly great technological advance that kids will still learn about post 26th century. But quantum mechanics and relativity are really our Pythagoras’ theorem equivalents, and will stay in textbooks for a very, very long time.
The Kyoto Protocol was in 1997, but the US didn’t ratify it because the API managed to influence US politicians enough to cast doubt on the problem.
Granted, NZ's a small country, but if it was known about there pre Great War, I suspect many people knew about it elsewhere too.
It is one of those things that until very recently had been quite easy to ignore though I suppose (and many still try even now).
The failure of our times has been just this. Not realizing that when what you are doing has global impact you need to think hard about its effects and be crystal clear on the ethics of it all. Perhaps the original sin was not pushing for more hands-on ethics training for every child and college student.
Anyway, the top answer was still a cool read.
They have an explicit policy of removing any amateurish or unsourced answers, in order to keep quality high.
https://web.archive.org/web/20190825021639/https://old.reddi...
Answers must be accurate, researched, supported, and factual. Generally, "scholarly". It's not uncommon to find posts with a thousand or more votes, and 100s of comments, all removed. The mods are looking for good answers. As are most of the readers.
The result may be frustrating for commenters, but that's really not the point.
I actually had an opportunity to answer on a subject which I've some familiarity last week, and got mention in the weekly wrap-up for it, which was kind of nice. A bit more than I deserved, really.
(I was talking trash ... but at least that was the topic.)
Early science throughout the world came about because of religion, not despite it. Early Muslim proto scientists like Ibn Sina and Ibn Khaldun explicitly did their science through religion and were heavily involved the religious politics of the day. This was also true also of early scientists in europe like Issac Newton.
Also that statement implicitly kinda carries the idea that europeans took over role of the "advancers of science" due to their tolerance and secularism. That was certainly not the case, european christianity in the 16th and 17th centuries was not in any way tolerant/moderate.
Similarly even if we agree that a decline occurred there are a myriad of other factors that could potentially cause a drop in scientific output such as the various consequences of war, the policies of the Ottoman empire, a lack of the urbanisation that occurred in China and Europe etc.