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Astronomy had no applications for centuries (unless you count astrology but do you really want to go there?). Saying that mathematics applies to it is disingenu

Astronomy had no applications for centuries (unless you count astrology but do you really want to go there?). Saying that mathematics applies to it is disingenuous in the context of a discussion around the applicability of mathematics to the daily lives of an ordinary taxpayer. Same goes for much of the theoretical work in geometry. Very little of it is applicable to daily life, with the notable exception of trigonometry and its uses in surveying and engineering. But the results I was alluding to by mentioning Euclid were those from his work in number theory, including his namesake lemma and his proof of the infinitude of the primes, and the fundamental theorem of arithmetic. Those results had no application to daily life until the advent of modern cryptography.

BlueTemplar 3y ago
Who do you think was making the calendars around the cycles of which agricultural societies were based ? And while we may think that astrology has no value, that certainly wasn't the opinion of most premodern societies ! (Though it all tended to be a bit lumped together, as the term "natural philosopher" indicates.)
jacobolus 3y ago
It seems like you are unfamiliar with technology before the past few centuries. Beyond astrology (which was important!), people used these theoretical tools from geometry and number theory for time keeping, calendars, cartography, navigation, surveying (you mentioned), city planning, architecture, construction and analysis of machines, hydrology, water distribution, crafts such as metalworking and carpentry, accounting, optics, music, any kind of engineering involving measurements, .... Obviously mathematicians were also excited about numbers and shapes for their own sake, and some of their activities (e.g. trying to square the circle or trisect an angle using compass/straightedge) were more like puzzles or games than engineering. But that doesn't mean the broad bulk of mathematics (even deductive proof-based mathematics) was practically worthless. Some of the most famous mathematicians of antiquity were also engineers. Fluency with prime numbers per se (and prime factorization) is pretty useful when you are trying to do substantial amounts of (especially mental) arithmetic. I'm not sure which "taxpayers" you are thinking about, but establishing an effective tax system and doing tax assessment and accounting is one example where having a more robust understanding of elementary arithmetic and number theory is pretty valuable. The development of the number and metrological systems and work on number theory in ancient Mesopotamia had quite a lot to do with tax accounting.

BlueTemplar 3y ago

Who do you think was making the calendars around the cycles of which agricultural societies were based ? And while we may think that astrology has no value, that certainly wasn't the opinion of most premodern societies ! (Though it all tended to be a bit lumped together, as the term "natural philosopher" indicates.)

jacobolus 3y ago

It seems like you are unfamiliar with technology before the past few centuries. Beyond astrology (which was important!), people used these theoretical tools from geometry and number theory for time keeping, calendars, cartography, navigation, surveying (you mentioned), city planning, architecture, construction and analysis of machines, hydrology, water distribution, crafts such as metalworking and carpentry, accounting, optics, music, any kind of engineering involving measurements, .... Obviously mathematicians were also excited about numbers and shapes for their own sake, and some of their activities (e.g. trying to square the circle or trisect an angle using compass/straightedge) were more like puzzles or games than engineering. But that doesn't mean the broad bulk of mathematics (even deductive proof-based mathematics) was practically worthless. Some of the most famous mathematicians of antiquity were also engineers. Fluency with prime numbers per se (and prime factorization) is pretty useful when you are trying to do substantial amounts of (especially mental) arithmetic. I'm not sure which "taxpayers" you are thinking about, but establishing an effective tax system and doing tax assessment and accounting is one example where having a more robust understanding of elementary arithmetic and number theory is pretty valuable. The development of the number and metrological systems and work on number theory in ancient Mesopotamia had quite a lot to do with tax accounting.