Why didn't the Romans contribute much to mathematics?

I "belivie" it is effect of cumulative knowledge from all their trade partners, who when put together is pretty much everyone on the planet at the time, since they were physically at the center of trade routes.

The general form of this question, or at least one version of it ("why did the Industrial Revolution occur in England and not in China, which had developed a vastly larger set of technologies far earlier") is known as the Needham Question, after sinologist Joseph Needham, author of Science and Civilisation in China, an epic 30+ volume work covering Chinese invention and technology, begun in 1954 and still in process. (Needham himself died in the 1990s.) There's a fascinating Wikipedia article on the topic, and if you can find a copy of the completed volumes (many college/university libraries have it), it's a treat.

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.

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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

> You can do long division just fine on a counting board

I searched around a bit, and didn't find much of anything you could do with a counting board beyond division.

To be fair I think the severity of climate change has only come into mainstream focus in the past 5-10 years. Just as one data point, when I was shopping for a car in 2012, I did not even consider CO2 emissions. Now, climate change is top of mind for me. I now own an electric car, stopped eating beef, mostly eliminated travel, etc... I'm not saying this will make a difference overall, but I imagine others feel similarly, and a lot of young people will hopefully be in a position to work on solving some of these problems.

Counting boards / the abacus are about as good as you get for calculating methods until mechanical calculators and logarithmic slide rules. So it's not really fair to say the Romans had a disadvantage here when no one had anything better. (Granted, the wire abacus was faster than counting boards / jetons).