The Universal History of Numbers
A riveting history of counting and calculating from the time of the cave dwellers to the late twentieth century, The Universal History of Numbers is the first complete account of the invention and evolution of numbers the world over. As different cultures around the globe struggled with problems of harvests, constructing buildings, educating their citizens, and exploring the wonders of science, each civilization created its own unique and wonderful mathematical system. Dubbed the "Indiana Jones of numbers," Georges Ifrah traveled all over the world for ten years to uncover the little-known details of this amazing story. From India to China, and from Egypt to Chile, Ifrah talked to mathematicians, historians, archaeologists, and philosophers. He deciphered ancient writing on crumbling walls; scrutinized stones, tools, cylinders, and cones; and examined carved bones, elaborately knotted counting strings, and X-rays of the contents of never-opened ancient clay accounting balls. Conveying all the excitement and joy of the process of discovery, Ifrah writes in a delightful storytelling style, recounting a plethora of intriguing and amusing anecdotes along the way.
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What Hacker News says
- The book 'The Universal History of Numbers' by Georges Ifrah has a vast amount of detail on this from many civilisaitons. Some of it in incredible depth.
- Your claim was that “Romans didn't use subtractive shorthand notation”. This is a trivially falsifiable claim: There are numerous examples of subtractive Roman numerals used in Roman times. Wikipedia describes several.
> Wikipedia page merely suggests to me that innumeracy was a common problem across the Empire, and that people got it wrong a lot
This is comical deflection from my point, which was that Wikipedia directly refutes your claim (repeated from TFA) with numerous references.
People writing XL for 40 and XIIX for 18 and IC for 99 are not “getting it wrong”; they are using subtractive notation, which was both commonplace and well understood (but not standardized) at the time.
The author of TFA didn’t do any research whatsoever about the topic and is speculating based on some half-remembered trivia. Which is fine, it’s a blog post on a personal website. But you shouldn’t pretend it is a reliable source.
* * *
More generally though, if you want to learn about my main claims upthread that a counting board was used by Romans in preference to anything like written arithmetic, here are some references:
Chrisomalis (2020) Reckonings
> In fact, among the thousands and thousands of Roman and post-Roman medieval Latin texts that survive, on a wide variety of media, none line up Roman numerals in columns for arithmetic, or use Roman numerals directly for arithmetical purposes. Admittedly, much of the material and literate record of of the Romans is lost, but there is no evidence to support the idea that reckoners in the Roman tradition ever did what we might imagine them to have done, which is to use Roman numerals as computational aids by writing them down and manipulating them.
> This has not stopped Western scholars from trying to show that it could have been done. I know of no fewer than five attempts, largely independent of one another, to show how the Roman numerals could have been manipulated to perform basic arithmetic adequately (Anderson 1956; Krenkel 1969; Kennedy 1981; Detlefsen et al. 1976; Schlimm and Neth 2008). All of these are artificial—they use the structure of the system and then imagine different ways it might have worked. None of them directly present any evidence to show that they were used in this way, because there simply is no such evidence.
> At best, even when the conclusion is not so bad for the Roman numerals, it’s ahistorical and strange to insist on these comparisons. We are so attached to the idea that numerals are for arithmetic that it’s very hard to stop and ask whether they were actually used for doing calculations in a given society.
Turner (1951) “Roman Elementary Mathematics” https://penelope.uchicago.edu/Thayer/E/Journals/CJ/47/2/Roma... (who also describes Roman finger calculations)
> The other method of working mathematical problems available to the ancient Roman would be the use of the counting board or abacus. That this was the standard instrument for any type of complex calculation is, I believe, a safe assumption, but an assumption based on surprisingly little direct evidence. The very word "abacus" which I have been using does not seem to have been found in this sense in classical Latin literature. The most common terms used are rather calculi (pebbles) for counters, tabula for the table; calculos ponere for undertaking a problem. The widespread metaphorical use of calculus is ample evidence of the universal and early use of the device, but precise evidence on its nature or on the way it was used is remarkably scarce....
If you are interested in the general topic of number systems let me recommend Menninger’s classic book Number Words and Number Symbols. Or you could try Chrisomalis’s other book Numerical Notation. Or Ifrah, The Universal History of Numbers.
If you want to know about medieval European counting boards, look up Barnard (1916) The Casting-Counter and the Counting-Board.
A neat paper about Greek use of tokens is Netz (2002) “Counter Culture” http://worrydream.com/refs/Netz%20-%20Counter%20Culture%20-%...
- 1) A2K Handbook, free online, http://a2knetwork.org/handbook
"Access to Knowledge (A2K) is the umbrella term for a movement that aims to create more equitable public access to the products of human culture and learning. The ultimate objective of the movement is to create a world in which educational and cultural works are accessible to all, and in which consumers and creators alike participate in a vibrant ecosystem of innovation and creativity."
2) Sci-Fi author L. Sprague de Camp's 1963 book, "The Ancient Engineers", covers the period from early Egyptian engineering up to Galileo. http://www.amazon.com/Ancient-Engineers-L-Sprague-Camp/dp/03...
“History, technology, culture, finance, and sociology intersect here. It’s not history from the top (kings and such, which some say is dry), nor history from the bottom (average people, which is necessarily endless and perhaps not very revealing). It’s history from the nuts-and-bolts middle – how structures were built, how materials were transported, how wars were fought. When you know this sort of foundational information, everything else becomes more real.”
3) Paul Calter, "Squaring the Circle: Geometry in Art & Architecture", http://www.amazon.com/Squaring-Circle-Geometry-Art-Architect...
"the combination of the subject knowledge of design, architecture, art, geometry, philospohy, music theory, and mathematics ... Calter includes the basic lessons and explanations of a regular Geometry course in his book, but then he interweaves an integrated classical curriculum (based on deductive reasoning)"
4) Georges Ifrah, "Universal History of Numbers", http://www.amazon.com/Universal-History-Numbers-Prehistory-I...
"the first complete account of the invention and evolution of numbers the world over ... Dubbed the "Indiana Jones of numbers," Georges Ifrah traveled all over the world for ten years to uncover the little-known details of this amazing story. From India to China, and from Egypt to Chile, Ifrah talked to mathematicians, historians, archaeologists, and philosophers."
- Can anyone recommend books on the cross-cultural history of mathematics? I've found these:
1) Georges Ifrah, "Universal History of Numbers", http://www.amazon.com/Universal-History-Numbers-Prehistory-I...
"the first complete account of the invention and evolution of numbers the world over ... Dubbed the "Indiana Jones of numbers," Georges Ifrah traveled all over the world for ten years to uncover the little-known details of this amazing story. From India to China, and from Egypt to Chile, Ifrah talked to mathematicians, historians, archaeologists, and philosophers."
2) Paul Calter, "Squaring the Circle: Geometry in Art & Architecture", http://www.amazon.com/Squaring-Circle-Geometry-Art-Architect...
"the combination of the subject knowledge of design, architecture, art, geometry, philospohy, music theory, and mathematics ... Calter includes the basic lessons and explanations of a regular Geometry course in his book, but then he interweaves an integrated classical curriculum (based on deductive reasoning)"
- I have a wonderful book in front of me called “From One to Zero: a universal history of numbers” by georges ifrah.
chapter 9, roman numerals: a vestige of primitive origins? starts like this:
——————————
They are obviously letters of the roman alphabet, but this does not mean they have an alphabetic origin. The signs L, C, D, and M are not the original forms of the numerals 50, 100, 500, and 1000; they are altered forms of much older numerals. known instances of the use of L, D, and M as numerals do not go back much farther the the first century AD […]I V X L C D M—————————
The chapter goes on to illustrate the resemblence with etruscan numerals and the tally marks of ancient shepards counting sheep, see the “tally marks” section of the wikipedia page, or borrow the book from archive.org, page 131 to 146 or so.
https://en.m.wikipedia.org/wiki/Etruscan_numerals#Tally%20Ma...
- This is discussed in The Universal History of Numbers, by Georges Ifrah.
You better check the book, because it has been a long time since I read it, but he explained that most people counts 3 by 3, sometimes 2 by 2, and exceptionally 4 by 4. Only very exceptionally people can count 5 by 5. What most of us do when counting 5 elements, is to divide them in a chunk of 3 and one of 2.
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