Wikipedia:Reference desk/Archives/Mathematics/2023 April 13

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April 13

Saved by zero?

How were ancient Greek mathematicians able to accomplish subtraction without a numeric zero? I understand that they subtracted by adding negative numbers; what happens when "adding" negative seven to seven, for example. 136.56.52.157 (talk) 16:49, 13 April 2023 (UTC)

Ancient Greek mathematicians had no concept of negative numbers. When asked to subtract 20 from 10, they would have said it is impossible. It is like asking, if Socrates has ten apples and he eats twenty of them, how many are left? They would have had no problem in figuring out that if Socrates has ten apples and he eats all ten, he has none left. They just did not think of "none" as a number.  --Lambiam 18:21, 13 April 2023 (UTC)

Greek mathematics outside of basic arithmetic (i.e. countable things) was essentially all geometry-based. They tended to think of mathematical operations in a geometric sense: addition involved combining two lines of the requisite length, subtraction involved cutting off a piece of a line, multiplication involved measuring the area between two orthogonal lines, etc. Concepts like negative length and negative area don't make sense in basic geometry, so the Greeks didn't even think in those terms. They would have taken such questions as nonsense; like "how do you cut 5 meters off of a 3 meter log". See Greek mathematics for more background. It is telling that the father of Greek philosophy is the semi-mythical Thales of Miletus, who is most well known for Thales's theorem, which is a geometric theorem about triangles and circles, the "fundamental units" of Greek mathematics. The other key figures in Greek mathematics were all known for geometry, Pythagoras (whose philosophy, Pythagoreanism, held that all reality was based on numbers and geometry, Euclid, whose Elements formalized almost everything there was to say about greek mathematics, etc. Indeed, western algebra as we know it today, in terms of the abstract manipulation of symbols devoid of any physical meaning, really doesn't come into fruition until the Islamic Golden Age with people like Muhammad ibn Musa al-Khwarizmi. --Jayron32 18:45, 13 April 2023 (UTC)

Although I accept all that, it is hard to imagine simple math without zero. Let's say a merchant needs to keep a ledger of daily sales in order to determine when, and how many whatsits to order. He currently has 36 whatsits; day one, he sells seven, day two: twelve, day three: [zero?] ... he keeps a daily tally by adding the negative values and eventually will have zero in inventory. -- Anyway, you get the idea; these folks didn't come from a cultural backwater. 136.56.52.157 (talk) 19:16, 13 April 2023 (UTC)

I mean, it happened that way, so your imagination is not particularly relevant. It's not that these cultures didn't have the concept of "nothingness" or "a lack of anything" or like that, they just didn't have any mathematical need for "zero" beyond "nothing". I mean, if you asked a Greek "I have three apples and you take all three away, how many do I have", he had some way to say "You don't have any apples at all", and could understand that, but that's not what "zero" is as a mathematical concept. Zero serves two important functions in mathematics, neither of which were relevant to Greek mathematics. 1) Zero serves as the boundary between positive and negatives numbers, in that way it is a place on the number line. If you don't have negative numbers, you have no use to treat that place as a number. It's superfluous. 2) Zero serves as a placeholder in decimal number system, so you can write something like three hundred and seven as 307, meaning three hundreds, no tens, and seven ones. Greeks didn't use such a numeral system either. 0#History explains the history of the development of the zero. --Jayron32 13:15, 14 April 2023 (UTC)

This seems dramatically overly reductive. Greek culture had a rich tradition of both number theory and calculation, with sophisticated accounting, engineering, astronomical prediction, etc. You are right though that there was not much concept of "arithmetic" per se: concrete calculations would have been done with a counting board. Take a look at e.g. Netz "Counter Culture". It's hard for me to imagine someone making something like the Antikythera mechanism without extensive calculations. Or go read The Almagest. ––jacobolus (t) 21:36, 13 April 2023 (UTC)

The Reviel Netz article looks interesting; especially the concept of cognitive history and the cultural divide between numerical record and numerical manipulation. I've given it a cursory scan and will give it a thorough read when I get the chance. --Thanks 136.56.52.157 (talk) 22:14, 13 April 2023 (UTC)

At no time did I ever say they didn't have all of that. I said they didn't have a zero, and they didn't have algebra, and they didn't use negative numbers. There's any of thousands of pages of text I could have written about Ancient Greeks, the kind of forum this is necessitates making choices about what material to cover and what to leave out, and just because I didn't write the entire thousands of pages worth of text doesn't mean I am denying those things existed. --Jayron32 13:18, 14 April 2023 (UTC)

More recently than Ancient Greece, Chinese writing didn't have a symbol for zero – or at least the one used today, 零, wasn't adopted until 1248. Even today it is little used as numbers in Chinese are written as they are said without zeros: 406 is 四百六 or "four-hundred-six". Modern Chinese has adopted 〇, i.e. the zero from Arabic numerals, for use in dates, telephone numbers where zero is needed, though you see wholly Arabic numbers in China a lot too.

China was very advanced by then, in writing but also in mathematics. They used an abacus for calculations which maybe explains why they did not need a written zero for even advanced calculations as zero could be represented on the abacus mid-calculation, but was not needed to announce or record a result.92.41.64.109 (talk) 21:01, 13 April 2023 (UTC)

It is a bit surprising about the Greeks because the Egyptians had zero and negative numbers for their surveying and accountancy. They didn't have a place number system though like was being talked about there for 406, zero was simply the origin for measurements or to say the accounts balanced out. I like the way they showed negative by writing the number with a sign for steps going backwards. NadVolum (talk) 22:45, 13 April 2023 (UTC)

• It should also be noted that "negative numbers" is not equivalent to "subtraction". Both the Greeks (and the Egyptians) had the concept of subtraction, as a process of taking something away from what is already there. That doesn't require understanding negatives. It just means that I can calculate what I have left once I have taken something away. If I have seven kilos of flour, and you take three kilos away from me, I have four left. Negative numbers aren't necessary there. Negative numbers are only necessary to express when something is less than nothing, and neither the Greeks or the Egyptians thought that was a meaningful idea. Like, if I posed the problem "If I have seven kilos of flour, and you take nine kilos away from me" is posed as a problem, you can say "Well, I'd have none, but you'd have not gotten your full nine" I can even say "I would still owe you two", but that's still a positive number. I owe you two real kilos of flour. Negatives aren't needed to make that statement or balance that account. Negatives would certainly help deal with that concept in a different way, but they aren't strictly necessary. They just make math a lot easier, but you can take more convoluted paths to get to the same end points, which is what these cultures did. --Jayron32 14:24, 14 April 2023 (UTC)

The Egyptians had measurements which were below ground point as well as above ground point and could cope with amounts being missing in their accountancy. Is it right to say these are not negative amounts? NadVolum (talk) 20:30, 14 April 2023 (UTC)