Number words and number symbols

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1. jacobolusMay 2022
   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-%...

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2. eternalbanApr 2021
   You probably have a Chinese abacus, the Suan Pan [1]. The (“16th century”) Japanese derivative, the Soroban [2] has a single pebble in the top registers. The former can do both quinary and decimal reckoning. The Russian Tchoty [3] does not divide the board, but rather uses 2 colors for beads.

   If you are interested in a ~comprehensive treatment of the history and types of ‘counting boards’, K. Menninger’s Number Words and Number Symbols [4] (where I read about Kiyoshi Matsuzaki aka “the Hands” beating Pvt. Thomas Ian Wood and his electronic calculator) is a good resource. quinary and decimal variants are addressed in fair detail.

   The section Counting Boards in Ancient Civilizations starts with the Salamis Tablet [5] and the Abax[6] depicted in use by a Persian calculator in the Darius Vase [7]. These boards didn’t have wires, so the ‘calculators’ used pebbles. Then we have next in progression the Roman hand-held abacus [8], which apparently found its way to East Asia and there it found a lasting appreciation and use, culminating in the two Chinese and Japanese variants noted above.

   One of the surprising aspects of the discussion of the use of these instruments in Japan has to do with how they are used. I always thought the abacus delegated all the math to the device, but this is apparently not correct. Per Menninger, the reckoned must “constantly work the problems in his head”

   [1]: https://en.wikipedia.org/wiki/Suanpan

   [2]: https://en.wikipedia.org/wiki/Soroban

   [3]: https://womenshistory.si.edu/object/tchoty-or-russian-abacus...

   [4]: https://www.goodreads.com/book/show/1598307.Number_Words_and...

   [5]: https://en.wikipedia.org/wiki/Salamis_Tablet

   [6]: https://en.wikipedia.org/wiki/Sand_table

   [7]: https://en.wikipedia.org/wiki/Darius_Vase

   [8]: https://en.wikipedia.org/wiki/Roman_abacus

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3. zeth___May 2018
   The last 100 years have been dealing with the fact that true and false are not enough to describe the world of computation. You also need undecidable, or null, for statements that can be proved to never terminate.

   It is very much still an open question if a statement is absolutely undecidable and if we need to add something like null in all logic [0].

   As for your arguments on why we don't need null, they sound exactly like the arguments against zero from the middle ages [1].

   >Just as the rag doll wanted to be an eagle, the donkey a lion and the monkey a queen, the zero put on airs and pretended to be a digit.

   [0] http://logic.harvard.edu/koellner/QAU_reprint.pdf

   [1] Menninger, Karl (1969), Number Words and Number Symbols. Cambridge, Mass.: The M.I.T. Press.

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