My most-puzzling why-did-this-take-so-long example is the base-ten system for writing numbers, using zero*. Wikipedia tells me this was invented in India in the 7th century AD and spread gradually into Europe after that. But this seems to be millennia late. There were plenty of highly organised empires trying to administer everything from military logistics to tax systems to pyramid-building with Roman numerals or worse. See here for the Babylonian version, for example.
So far as I can tell, once you have writing and some basic concept of writing-down-numbers (I can’t describe Roman numerals as mathematical notation), there are no further pre-requisites for the invention of zero. And the existence of the abacus, possibly invented as far back c2,700 BC, presumably helped. And yet, we have circa 3,400 years from inventing the abacus to figuring out how to write down a numerical system that actually made sense.
Why not?! Looking at your list of factors 1. Total number of researchers. 3,400 years times every civilisation across Eurasia that needed to administer a large polity or project. The number of person-hours of people calculating stuff must have been astronomical. 2. Speed of research. OK, this is before the printing press, but still. 3,400 years is an excessive delay. 3 size of opportunity. Just huge. 4 social barriers—I don’t think many civilisation treated math as a controversial topic.
*It doesn’t have to be base-10, a base-12 or −20 or whatever system would work fine too. Just not freaking Roman numerals!
I suspect it might have to do with (the representation of the thing) and (the thing) tending to blend together in people’s minds. Once you’ve learned to read fluently, seeing a string of writing will make you think of the meaning of the words rather than the underlying letters. And especially someone who is only familiar with one writing system is likely to see things not as a property of the writing system, but as a property of the words themselves. So instead of thinking “this writing system makes this word hard to spell”, they’ll just think “this word is hard to spell”.
In a similar way, I would expect the average person only familiar with Roman numerals to think not “our number system makes it hard to write down numbers efficiently”, but just “it’s hard to write down numbers efficiently”. In order to realize that the difficulty is a property of number system, you first need the idea that it’s possible for a number system to represent numbers more efficiently than you are currently doing, which is exactly the idea that you are missing if nobody has invented a better number system yet.
That explanation does still leave it a bit confusing why the abacus didn’t work as an example of an alternative number system. The one thing that comes to mind is that the abacus is a device for doing calculations by physically manipulating the beads, while Roman numerals are something that you write down. There are a lot of mathematical equivalencies that seem obvious to us but needed to be explicitly learned—it’s not immediately obvious to all children that 2 times 4 and 4 times 2 are the same thing, for instance. Likewise, if a culture doesn’t have the abstract concept of “a representational system” yet, it may not be very obvious to them that an abacus and a system for writing down numbers have anything to do with each other. “They’re different things for different purposes” may be the default thought.
East Asia used counting rods for calculation for thousands of years. Counting rods use true positional numeral system. It’s just that East Asia didn’t use it for writing. In other words, there were separate systems, one to calculate numbers which was efficient, and one to write numbers which was traditional. If that sounds weird, consider that we calculate in binary but write in decimal.
It is weird and it’s extra-weird that everywhere from Carthage to Greece to China failed to use an efficient system for writing numbers. It’s not like there was just one outlier which kept a traditional system.
And I wonder if the use of traditional systems for writing delayed the development of calculus and advanced mathematics too.
Epistemic status: thinking out loud
My most-puzzling why-did-this-take-so-long example is the base-ten system for writing numbers, using zero*. Wikipedia tells me this was invented in India in the 7th century AD and spread gradually into Europe after that. But this seems to be millennia late. There were plenty of highly organised empires trying to administer everything from military logistics to tax systems to pyramid-building with Roman numerals or worse. See here for the Babylonian version, for example.
So far as I can tell, once you have writing and some basic concept of writing-down-numbers (I can’t describe Roman numerals as mathematical notation), there are no further pre-requisites for the invention of zero. And the existence of the abacus, possibly invented as far back c2,700 BC, presumably helped. And yet, we have circa 3,400 years from inventing the abacus to figuring out how to write down a numerical system that actually made sense.
Why not?! Looking at your list of factors 1. Total number of researchers. 3,400 years times every civilisation across Eurasia that needed to administer a large polity or project. The number of person-hours of people calculating stuff must have been astronomical. 2. Speed of research. OK, this is before the printing press, but still. 3,400 years is an excessive delay. 3 size of opportunity. Just huge. 4 social barriers—I don’t think many civilisation treated math as a controversial topic.
*It doesn’t have to be base-10, a base-12 or −20 or whatever system would work fine too. Just not freaking Roman numerals!
I suspect it might have to do with (the representation of the thing) and (the thing) tending to blend together in people’s minds. Once you’ve learned to read fluently, seeing a string of writing will make you think of the meaning of the words rather than the underlying letters. And especially someone who is only familiar with one writing system is likely to see things not as a property of the writing system, but as a property of the words themselves. So instead of thinking “this writing system makes this word hard to spell”, they’ll just think “this word is hard to spell”.
In a similar way, I would expect the average person only familiar with Roman numerals to think not “our number system makes it hard to write down numbers efficiently”, but just “it’s hard to write down numbers efficiently”. In order to realize that the difficulty is a property of number system, you first need the idea that it’s possible for a number system to represent numbers more efficiently than you are currently doing, which is exactly the idea that you are missing if nobody has invented a better number system yet.
That explanation does still leave it a bit confusing why the abacus didn’t work as an example of an alternative number system. The one thing that comes to mind is that the abacus is a device for doing calculations by physically manipulating the beads, while Roman numerals are something that you write down. There are a lot of mathematical equivalencies that seem obvious to us but needed to be explicitly learned—it’s not immediately obvious to all children that 2 times 4 and 4 times 2 are the same thing, for instance. Likewise, if a culture doesn’t have the abstract concept of “a representational system” yet, it may not be very obvious to them that an abacus and a system for writing down numbers have anything to do with each other. “They’re different things for different purposes” may be the default thought.
East Asia used counting rods for calculation for thousands of years. Counting rods use true positional numeral system. It’s just that East Asia didn’t use it for writing. In other words, there were separate systems, one to calculate numbers which was efficient, and one to write numbers which was traditional. If that sounds weird, consider that we calculate in binary but write in decimal.
It is weird and it’s extra-weird that everywhere from Carthage to Greece to China failed to use an efficient system for writing numbers. It’s not like there was just one outlier which kept a traditional system.
And I wonder if the use of traditional systems for writing delayed the development of calculus and advanced mathematics too.