Story placement

One small problem: maths has mostly been the preserve of white males throughout history, with some input from the Subcontinent (zero), and the Ottomans (Astronomy) around the first millennium, and a few notable women (Sophie Germain, Emmy Noether). All those disgusting racist white men during the Enlightenment came up with practically all the maths that underpins the tech we rely on every day.

And I found out recently that Gauss invented what is essentially the Fast Fourier Transform before even Fourier’s work.

It’s really sad when an institution like Durham sinks into this sort of crap.

Its not the fault of authors, it’s the fault of the evil press barons who created the machines that could churn out the western literature.

TG,
A little unfair on the sub-continent and Arabia. Indian maths was leading edge at the time, and not just for the zero. Until about 11-12th century Arabs led the world in maths, not in just the invention of algebra but the whole field. (I studied the history of maths as a divertissement on a maths degree.)

Western universites have spent centuries using western academics to teach western subjects, shock horror.

That’s a very western view of history, Natilie. North Africa was part of the Mediterranean world. The inheritors of Greek & Roman culture. It’s doubtful its goat fucking camel jockey conquerors from the sands of Arabia contributed much to mathematics. Although lopping off the odd hand got them counting past 10.

@Mark

Yes, some aspects of calculus were developed much sooner in East and South Asia and the non-European Mediterranean world than in Europe. Europe sorted out the fundamental relationship of integration and differentiation and discovered the hyperbolic functions. Also Eudoxus and Archimedes are worth mentioning as direct forefathers.

It took the Chinese until the 4th century to catch up with the ancient Greeks on this subject, which they used to create some excellent approximations to pi (up to 8 significant figures of accuracy and unmatched in Europe for another 1200 years), but by the 5th century they had extended the work to what we would call Cavalieri’s principle, using it to prove the volume of a sphere. So in calculus they quickly went from half a millennium behind the west to a millennium ahead.

In the Arab world, Alhazen used techniques we would interpret as integral calculus to find the volume of a paraboloid in the 11th century and the Persian Sharaf al-Din was using techniques of differential calculus to find the derivative function of certain curves and hence identify turning points. But these techniques don’t seem to have been generalised or the relationship between differentiation and integration identified.

The Kerala school in India was also far ahead in certain respects, finding Taylor polynomials and identifying infinite series as early as the 14th century, but also not quite bringing integral and differential calculus together in a cohesive way.

Similar stories for logarithms trigonometry or matrices – bits of development scattered across many different societies, with Europe really taking the lead by putting the bits into a coherent whole only during the Scientific Revolution. (Logarithms is an area where “Arabs stopped doing innovative maths by the 12th century” isn’t true, for what it’s worth.) But aside from the reabsorption of Greek knowledge back into Europe at the time of the Renaissance – which being via the Arabs, brought with it several Arabic and some Indian ideas into the European mathematical genealogy – many important discoveries of European mathematicians were rediscoveries of what had happened elsewhere. As a result, textbooks based on the progression of mathematical ideas tend to go in the order Greeks-Arabs and some Indians-early modern Europeans. The reality was far less linear, but the fact mathematicians elsewhere in the world often had priority by several centuries often didn’t feed in to the intellectual progression that got us to where we are today, because their ideas were not transmitted to Europe until it was too late. Textbooks in the places involved often look different, with an emphasis on achievements of the locals, though often this goes too far (as anyone who has had to endure the horrors of modern Indian nationalist “Vedic Mathematics” can tell you). Looking around a modern mathematics department in the West, and particularly at the typical enrolment in a US postgrad program, you’d be hard-pressed to say Europeans have some kind of inherent mathematical ability the rest of the world lacks. And that’s consistent with most of the last two millenia of history. But whatever socio-cultural, political, economic or technological forces were operating at the time of the Scientific Revolution, they must have been very powerful indeed – in almost all facets of mathematics, Europe leapfrogged the other centres of mathematical research.

@Anon,

“techniques we would interpret as integral calculus” Are you – for want of a better expression – just back projecting here? Oh, that looks a bit like calculus if we interpret it this way, so it must be – and would have become – what Newton and Leibnitz developed if it wasn’t for reasons?

Isn’t this what the BBC (sigh) did when they “followed up” Kenneth Clark’s “Civilization” a few years ago? Given that they employed the racist grifter David Olusoga to help, makes this assertion pretty well undeniable as far as I’m concerned.

“Europe leapfrogged the other centres of mathematical research”. Europe clearly had something, but whatever that is, it can’t be attributable to Europeans. Hence “critical race theory”, AKA cognitive dissonance.

Korea, for example, I understand beats the UK/US and Europe no doubt in maths as measured by exam scores, but if this down to the murderous pressure cooker of the suneung (look at this on youtube, for those who don’t know what it is) I’m not really that impressed.

@Mark

Calculus isn’t “just” Newton and Leibnitz though is it? There’s some evidence Descartes had a rough conception of the fundamental theorem of calculus, a version of it was known to Gregory and Barrow had produced a more general proof, even before the Big Two enter the scene. And they were building on foundations by Pascal, Wallace, Fermat, Cavalieri, Torricelli and others. But despite the time gap, Archimedes is also important and both Johann Bernoulli and Barrow were heavily directly influenced.

If you define calculus not just as a wider body of concepts and procedures, but very tightly in terms of a particularly modern Western understanding of functions and infinitessimals, using modern Western notation, then unsurprisingly you’ll see that it developed first in the early modern West and everyone else was close but no cigar. But that’s a pointless exercise, and if you took a snapshot of the world in, say, 600, then whatever term you’d like to apply to the calculus-like work of Chinese mathematicians was far ahead of others, and the likes of the ancient Greeks had had some similar ideas just not so good (after all they had some “wrong” intuition and used the “wrong” notation, as the Chinese would see it had they known the work, and hadn’t come up with some of the general formulas the Chinese knew).

The way people split up that body of knowledge into particular chunks and give those chunks different names depends a lot on their intellectual history and sometimes that’s distorting. We often view Eudoxus and Archimedes as separate from calculus because, although they were in direct intellectual lineage, there was a gap of over a millennium between them and the people we see as their successors. But their work was sophisticated and the conceptual gap is not so great, which is why many in the early development of calculus referred directly back to them. That enormous time gap is far more to do with the history of scientific thought in the Mediterranean and European world between the Greek and medieval eras, and far less to do with the deficiencies of their work. (This isn’t just a Dark Ages problem, Greeks’ intellectual distrust of infinitesimals – owing to some quite deep-rooted features of their dominant philosophies – was a hindrance too.)

The Chinese made the jump from Eudoxus-like reasoning to Cavalieri-like reasoning in a massively shorter period of time. Discussing what to call their body of knowledge and whether it is back-projection seems a totally sterile debate to me – they could use techniques of or akin to (take your pick) what we call “calculus” to figure out approximations to pi unmatched in accuracy in the West for over a thousand years. They had the conceptual understanding and were applying it to produce “useful” results (though not especially practical since this went way beyond the engineering precision available at the time). The flip side of this is that whatever superiority the West developed during the Scientific Revolution, clearly the West hasn’t enjoyed continuous superiority – even within facets of the body of knowledge we would generally call “calculus”, the West has sometimes been over a millennium behind. It would be wrong to conclude from this that westerners are idiots.

Another “back-projection” issue is we tend to think of concepts as more or less advanced depending on the order they were discovered in our own intellectual history. But that’s often a matter of chance. We tend to think of Taylor polynomials as a development that arose out of calculus but the Kerala School was using them hundreds of years earlier despite a rather patchy grasp of calculus that lacked the cohesion developed in Europe in the late 17th century onwards. Does that make them more ahead or more behind? You could argue if you want that “they may have had Taylor polynomials but they clearly didn’t truly understand them if their calculus was so spotty” yet that ignores the fact these Indians were happily performing calculations in a manner Europeans could not until hundreds of years later. And even European calculus had some rather ropey intellectual foundations until the development of mathematical analysis, which took centuries to sort out but didn’t invalidate the importance of the more intuitive work of Newton, Leibnitz, Barrow and the earlier gang.

We could have the same debate about matrix algebra. The Japanese had a well-developed theory of matrix determinants by the 17th century. Yet another area where Western “superiority” wasn’t established until rather late into the modern era? Or should we argue that because the Japanese didn’t see this in terms of scale factors of linear transformations between vector spaces, they didn’t have the correct/modern/Western understanding and their work “doesn’t count” or is nothing more than a back-projection? Far as I can see, they knew a concept, knew how to calculate it, and could do useful things with it (solving simultaneous equations for example). So I would be perfectly content to give them the win on this one.

Jumping to talk about the general high school leaving exam is not the best comparison here, as you point out that’s much more to do with the country’s education, social and family systems. General mathematical literacy in the West is not high, but judging from PISA scores can move about depending on education policy. If you’re talking about serious mathematics and the production of young mathematicians you might want to look at IMO and gold-medal olympians in particular. But makes even more sense to look at PhD students at elite universities. For serious research output, perhaps some of the more prestigious mathematical prizes (though there are certain biases at play in doing that). If you’re trying to argue white people have some brilliant mathematical gene that explains their total dominance in mathematics over the past 2500 years, then (a.) for much of the past 2500 years it hasn’t been true that Europe and its later colonies have been way ahead, though the West has been when it mattered most ie getting the Scientific and Industrial Revolutions done, (b.) mathematical literacy in the Great British public is not very high with most people failing to retain much of their school knowledge, so that the majority of adults are sub-GCSE levels (skill tests show similar results across Europe), (c.) try looking at a top-rung, internationally competitive PhD program like Berkeley or Stanford and you’ll see it’s not full to the brim with Europeans and their descendents.

We can all agree David Olusoga isn’t a reliable source on the history of mathematics without trying to turn that history into some kind of racial war. It in no way emasculates you as a human being, nor undermines the achievements of your ancestors, that a thousand years ago there were clever people with neat ideas in various corners of the world. Nor does it that there’s people on your street who can’t do long division and probably don’t even remember their seven times table. Nor does it that in the world’s leading centres of mathematical research, there are esteemed and productive professors who have ideas you or I will never grasp and who don’t look like you or me. Calculus is great, Barrow, Newton and Leibnitz were also unfathomably smarter than us, and it’s nice to live in a part of the world where some of these big ideas really got put together cohesively at a crucial period in world history. I’m happy to leave the race war to the grifters.

I suppose it is rather glib to say so, but one of the reasons that industrialisation never happened in China and India is that they didn’t need machines. With a large population of manual labourer who could live on a bowl of rice a day, there was no need to automate or mechanise so many processes ( apart from killing the enemy, which is why gunpwder became so widepsread).

There are lots of threads that came together to promote industrial capitaism in NorthWest Europe: the use of coal, discovery of steam power, agricultural improvement, access to large amounts of cheap cotton, double entry bookkeeping, destigmatisation of usury to allow people to borrow money and so on. Electricity was a phenomenon that was known in many lands, but it was a solution in search of a problem, until the 19th Century.

Anon is right in what he says. There is little value in arguing over who could measure the distance to the Moon first. The important thing is, who could use the knowledge and shoot a rocket to it first ?

I do have doubts about Archimedes. As far as Archimedes’ Principal’s concerned, anyone who casts metal is going to come up with it. It’s derived from how much stock one needs to melt to fill a mould. And I’d say most of his other stuff is also from crafts. He’s just the person who wrote them down. They likely predate him by centuries.

CRT and decolonization are not about the provenance of math. They are about western rigour. When the curriculum has been, well, emasculated, none of the dumfuks who got their doctorates in maths will ever be able to do anything with it beyond evangelizing it. They won’t be able to build space rockets or any kind of high technology. But they will demand to be appointed to positions of high status and authority in the organizations that build space rockets and Formula One racing cars.

@ Chernyy Drakon

Of course, which rather makes the point!

for ‘Sharan’ read ‘Sharan’ throughout