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I am very unconvinced by this "history of zero". Definitely the Greek geometers were aware of that concept, they just expressed it geometrically not numerically

I am very unconvinced by this "history of zero". Definitely the Greek geometers were aware of that concept, they just expressed it geometrically not numerically. Putting the concept at some specific geographic location seems very strange to me. To me there is no doubt that in each mathematical culture there was some notion of it, just weaved into that conceptions of that particular culture. Of course then there is zero as a symbol and positional number systems. The first one seems very uninteresting, the later one definitely more so, but the question of how to express numbers is definitely more interesting and broad than the history of just one component of it.

rusticpenn 2y ago
Mesopotamians were at least a millennium ahead of Greeks though..
- constantcrying 2y ago
  No. They weren't even in the same category, even the comparison doesn't make sense. In Mesopotamia mathematics seems to have been a tool, a method for business and construction planers to drive certain quantities. Mathematics to the Greeks was "mathematics" in the sense we understand it today. From a system of axioms they derived a complex system of theorems, which allowed for an abstract description of reality. It was mathematics as a system of truth, which then could be used for other purposes. E.g. with Archimedes who discovered integration, but also was a prolific engineer. The Greeks were so far ahead of anything the Mesopotamians did, that even the comparison is unfair.
  - derstander 2y ago
    > No. They weren't even in the same category, even the comparison doesn't make sense. The comparison does make sense if you interpret the parent’s post simply to state that Babylonian mathematics (in Mesopotamia) were developed (and seemingly stagnated) before Greek mathematics began in earnest. Which seems to be pretty uncontroversial. There are extant clay tablets from 1800 to 1600 BC that would indeed predate the Greek Geometers by a millennia — and that’s if you’re counting Thales as the beginning. Ie. your parent post is using “ahead of” == “temporally before” not as in “more advanced”.
    - rusticpenn 2y ago
      Thank you.
      - constantcrying 2y ago
        [flagged]
    - constantcrying 2y ago
      [flagged]
      - rusticpenn 2y ago
        I am of the opinion that there is something to learn from everyone. However I come to HN for civil discussion. That was not a gotcha or are we trying to win something. Mesopotamians needed complex mathematics for taxes long before even Mycenian Greeks came into picture. You are talking about classical Greeks. Naturally there were advancements in mathematics during the Greek period, but that does not undermine the achievements of Mesopotamians.
        constantcrying 2y ago
        Your response was a single sentence, pointing out something which I obviously knew. That information was totally irrelevant to what I said, but you thought you had discovered something which I didn't know about. You also obviously didn't understand what I said, else you wouldn't have posted a single statement with no new information. Obviously this was a gotcha. You made no attempt to even try to understand me or post anything interesting or relevant, instead you took a cheap shot at a misreading of my post. If this is what "high quality discussion" is to you, what does mid level discussion look to you? Just a slur as a response? >Mesopotamians needed complex mathematics for taxes long before even Mycenian Greeks came into picture. You are talking about classical Greeks. Naturally there were advancements in mathematics during the Greek period, but that does not undermine the achievements of Mesopotamians. I am stunned by your knowledge. You really know that? That is breathtaking information. Genuinely thank you for posting this novel information that I was totally unaware of. Bringing that to my attention was extremely relevant and has radically changed my position. Seriously fuck off, you still haven't even bothered to read my OP.
  - kouru225 2y ago
    This is the difference between philosophy and number systems. The Babylonian number system was, by far, the superior system to the Greek number system (which they inherited from the Egyptians.) This is something even the Greeks acknowledged because they would use the Babylonian system to do their own calculations, and then convert it back into the Greek numbers (similar to how the USA scientists use the metric system and then convert back to the imperial system today).
empath75 2y ago
The question is when they used it or understood it _as a number_. The greeks didn't even consider 1 to be a number, since "number" implied a "multitude". It was controversial as to whether 1 was a number up until the 1400s or 1500s, let alone zero.
- kragen 2y ago
  thank you! i was wondering about that
- taeric 2y ago
  This feels wrong, at face value. Even in modern English, if I say I have a number of things, I almost certainly don't have just one. This does /not/ mean that one is not a number. It /does/ mean that the word "number" can have several uses.
  - empath75 2y ago
    more here: https://plato.stanford.edu/entries/aristotle-mathematics/#10 https://plato.stanford.edu/entries/aristotle-mathematics/#10
    - taeric 2y ago
      But this doesn't contend with my point? A sibling point brought up confusion of cardinal and ordinal numbers. In modern english, I challenge you to find a good understanding outside of advanced practitioners on the difference. I think it is fair to say that my statement is too strong to say it is wrong. My assertion would be that it is more complicated and almost certainly there is a lot lost in translation along the years.
  - wcarey 2y ago
    Euclid distinguishes the unit (VII, Def.1) and number -- a multitude composed of units -- (VII, Def.2). His definition of prime and composite numbers (VII, Def.11 and Def.13) clearly exclude one from the group of numbers, otherwise every number would be composite.
meroes 2y ago
Wasn’t 2 the smallest actual number to Greeks? I think it’s clear from the history Greeks did a TON of math but missed this number system and abstraction. And if you say they still understood zero but didn’t have number system to tag along (they didn’t have Hindu Arabic numerals) well a better number system provides so much more math that the argument doesn’t have much oomph. There’s no doubt the Hindu Arabic numeral system, of which zero was apart, was the mathematical development bar none of its time. Did anyone really understand zero before it could do all these useful things in equations? Doubtful. And that’s the most charitable take for the Greeks regarding zero.
kouru225 2y ago
I’m in the process of reading this article, but I recently read Zero: The Biography of a Dangerous Idea by Charles Seife (which is a great book) so I feel like I can explain: The Greeks were ideologically opposed to the number zero. Aristotle outright refuses to acknowledge the existence of zero and of infinity. The Greeks were aware of the idea of zero, and they even used zero when they calculated using the Babylonian number system (which used zero as a placeholder number,) but they always converted the numbers back into their own system, and stubbornly refused to acknowledge its existence. The fact that the Greeks saw geometry and math as interchangeable was their weakness here. There’s no way to represent the number 0 geometrically, but the Greeks weren’t gonna give up their belief in Geometry because it provided them with social and political power. Pythagoras and Aristotle believed in a religious philosophy with logos at the center. Logos can be translated as “thought” or “word” (as it is in the Bible) or it can be translated as “ratio.” This is because they saw all these things as one (the Latin translation of the Greek word “logos” is “ratio.”) The ratio of numbers was thought the be the underlying mechanism that proved the order of the universe (which naturally saw the nobility as orderly and the peasantry as chaotic). This was a profoundly powerful sociopolitical tool that ended up spreading all across the world because Aristotles student just happened to be the greatest conqueror of the era: Alexander The Great. Anything that threatened the philosophy of logos was suppressed, violently. Hippassus and Zeno were both murdered for the crime of talking about irrational numbers and infinity. Zero was one of these threats. 1:0 = infinity, 10:0 = infinity, anything:0 = infinity. This was not logos and therefore it was suppressed. This philosophy extended beyond mathematics into the realm of astronomy and, weirdly enough, music (at the time, Pythagoras was actually most famous for his discovery of the golden ratio using an instrument called the monochord, which is a legend that seems to be false but nonetheless made him very famous.) This astronomical belief system was then later attributed to Ptolemy. This philosophy then was transplanted into Christian theology, and it took centuries for the monks to accept the existence of zero and infinity as a result. We even have cases of religious figures persecuting mathematicians about zero and infinity as late as the 1800s.
- jacobolus 2y ago
  > The Greeks were ideologically opposed to the number zero. I don't think this claim is supportable, certainly not as such a broad generalization. Do you have a specific statement clearly attributable to an ancient Greek author to that effect? > Greeks saw geometry and math as interchangeable You're going to have to define these terms more explicitly. I don't think this statement is right. Ancient Greeks spent quite a lot of effort studying areas of what we now call mathematics but which were not geometrical per se. > Greeks weren’t gonna give up their belief in Geometry because it provided them with social and political power Any claim that geometers in general had social or political power owing to their mathematical work seems exaggerated. While a couple of geometers happened to incidentally also be local political leaders (e.g. Eudoxus), geometers explicitly lamented how little their contemporaries cared about their work and how few colleagues they could find to share it with. > Aristotle outright refuses to acknowledge the existence of zero and of infinity. This seems like a reductive summary. Aristotle had a pretty sophisticated idea about this which finds echoes in modern mathematics: here is Aristotle: > Now there is no ratio in which the void is exceeded by body, as there is no ratio of 0 (οὐδέν) to a number. For if 4 exceeds 3 by 1, and 2 by more than 1, and 1 by still more than it exceeds 2, still there is no ratio by which it exceeds 0; for that which exceeds must be divisible into the excess + that which is exceeded, so that 4 will be what it exceeds 0 by + 0. For this reason, too, a line does not exceed a point-unless it is composed of points! Alternate translation: > But the nonexistent substantiality of vacuity cannot bear any ratio whatever to the substantiality of any material substance, any more than zero can bear a ratio to a number. For if we divide a constant quantity c (that which exceeds) into two variable parts, a (the excess) and b (the exceeded), then, as a increases, b will decrease and the ratio a :b will increase; but when the whole of c is in section a there will be none of c for section b; and it is absurd to speak of 'none of c' as 'a part of c.' So the ratio a: b will cease to exist, because b has ceased to exist and only a is left, and there is no proportion between something and nothing. (And in the same way there is no such thing as the proportion between a line and a point, because, since a point is no part of a line, taking a point is not taking any of the line.)" Later, about motion in a vacuum: > But if a thing moves through the thickest medium such and such a distance in such and such a time, it moves through the void with a speed beyond any ratio. (The physical premise here is wrong, but the concept of division by zero is clear.) See https://www.jstor.org/stable/pdf/2304187.pdf https://www.jstor.org/stable/pdf/2304187.pdf Aside: anyone telling you what Pythagoras thought about numbers is pulling your leg. We have no idea about this whatsoever, only what various people claimed like 5+ centuries later, most of which is nonsense.

rusticpenn 2y ago

Mesopotamians were at least a millennium ahead of Greeks though..

constantcrying 2y ago
No. They weren't even in the same category, even the comparison doesn't make sense. In Mesopotamia mathematics seems to have been a tool, a method for business and construction planers to drive certain quantities. Mathematics to the Greeks was "mathematics" in the sense we understand it today. From a system of axioms they derived a complex system of theorems, which allowed for an abstract description of reality. It was mathematics as a system of truth, which then could be used for other purposes. E.g. with Archimedes who discovered integration, but also was a prolific engineer. The Greeks were so far ahead of anything the Mesopotamians did, that even the comparison is unfair.
- derstander 2y ago
  > No. They weren't even in the same category, even the comparison doesn't make sense. The comparison does make sense if you interpret the parent’s post simply to state that Babylonian mathematics (in Mesopotamia) were developed (and seemingly stagnated) before Greek mathematics began in earnest. Which seems to be pretty uncontroversial. There are extant clay tablets from 1800 to 1600 BC that would indeed predate the Greek Geometers by a millennia — and that’s if you’re counting Thales as the beginning. Ie. your parent post is using “ahead of” == “temporally before” not as in “more advanced”.
  - rusticpenn 2y ago
    Thank you.
    - constantcrying 2y ago
      [flagged]
  - constantcrying 2y ago
    [flagged]
    - rusticpenn 2y ago
      I am of the opinion that there is something to learn from everyone. However I come to HN for civil discussion. That was not a gotcha or are we trying to win something. Mesopotamians needed complex mathematics for taxes long before even Mycenian Greeks came into picture. You are talking about classical Greeks. Naturally there were advancements in mathematics during the Greek period, but that does not undermine the achievements of Mesopotamians.
      - constantcrying 2y ago
        Your response was a single sentence, pointing out something which I obviously knew. That information was totally irrelevant to what I said, but you thought you had discovered something which I didn't know about. You also obviously didn't understand what I said, else you wouldn't have posted a single statement with no new information. Obviously this was a gotcha. You made no attempt to even try to understand me or post anything interesting or relevant, instead you took a cheap shot at a misreading of my post. If this is what "high quality discussion" is to you, what does mid level discussion look to you? Just a slur as a response? >Mesopotamians needed complex mathematics for taxes long before even Mycenian Greeks came into picture. You are talking about classical Greeks. Naturally there were advancements in mathematics during the Greek period, but that does not undermine the achievements of Mesopotamians. I am stunned by your knowledge. You really know that? That is breathtaking information. Genuinely thank you for posting this novel information that I was totally unaware of. Bringing that to my attention was extremely relevant and has radically changed my position. Seriously fuck off, you still haven't even bothered to read my OP.
- kouru225 2y ago
  This is the difference between philosophy and number systems. The Babylonian number system was, by far, the superior system to the Greek number system (which they inherited from the Egyptians.) This is something even the Greeks acknowledged because they would use the Babylonian system to do their own calculations, and then convert it back into the Greek numbers (similar to how the USA scientists use the metric system and then convert back to the imperial system today).

empath75 2y ago

The question is when they used it or understood it _as a number_. The greeks didn't even consider 1 to be a number, since "number" implied a "multitude". It was controversial as to whether 1 was a number up until the 1400s or 1500s, let alone zero.

kragen 2y ago
thank you! i was wondering about that
taeric 2y ago
This feels wrong, at face value. Even in modern English, if I say I have a number of things, I almost certainly don't have just one. This does /not/ mean that one is not a number. It /does/ mean that the word "number" can have several uses.
- empath75 2y ago
  more here: https://plato.stanford.edu/entries/aristotle-mathematics/#10 https://plato.stanford.edu/entries/aristotle-mathematics/#10
  - taeric 2y ago
    But this doesn't contend with my point? A sibling point brought up confusion of cardinal and ordinal numbers. In modern english, I challenge you to find a good understanding outside of advanced practitioners on the difference. I think it is fair to say that my statement is too strong to say it is wrong. My assertion would be that it is more complicated and almost certainly there is a lot lost in translation along the years.
- wcarey 2y ago
  Euclid distinguishes the unit (VII, Def.1) and number -- a multitude composed of units -- (VII, Def.2). His definition of prime and composite numbers (VII, Def.11 and Def.13) clearly exclude one from the group of numbers, otherwise every number would be composite.

meroes 2y ago

Wasn’t 2 the smallest actual number to Greeks? I think it’s clear from the history Greeks did a TON of math but missed this number system and abstraction. And if you say they still understood zero but didn’t have number system to tag along (they didn’t have Hindu Arabic numerals) well a better number system provides so much more math that the argument doesn’t have much oomph. There’s no doubt the Hindu Arabic numeral system, of which zero was apart, was the mathematical development bar none of its time. Did anyone really understand zero before it could do all these useful things in equations? Doubtful. And that’s the most charitable take for the Greeks regarding zero.

kouru225 2y ago

I’m in the process of reading this article, but I recently read Zero: The Biography of a Dangerous Idea by Charles Seife (which is a great book) so I feel like I can explain: The Greeks were ideologically opposed to the number zero. Aristotle outright refuses to acknowledge the existence of zero and of infinity. The Greeks were aware of the idea of zero, and they even used zero when they calculated using the Babylonian number system (which used zero as a placeholder number,) but they always converted the numbers back into their own system, and stubbornly refused to acknowledge its existence. The fact that the Greeks saw geometry and math as interchangeable was their weakness here. There’s no way to represent the number 0 geometrically, but the Greeks weren’t gonna give up their belief in Geometry because it provided them with social and political power. Pythagoras and Aristotle believed in a religious philosophy with logos at the center. Logos can be translated as “thought” or “word” (as it is in the Bible) or it can be translated as “ratio.” This is because they saw all these things as one (the Latin translation of the Greek word “logos” is “ratio.”) The ratio of numbers was thought the be the underlying mechanism that proved the order of the universe (which naturally saw the nobility as orderly and the peasantry as chaotic). This was a profoundly powerful sociopolitical tool that ended up spreading all across the world because Aristotles student just happened to be the greatest conqueror of the era: Alexander The Great. Anything that threatened the philosophy of logos was suppressed, violently. Hippassus and Zeno were both murdered for the crime of talking about irrational numbers and infinity. Zero was one of these threats. 1:0 = infinity, 10:0 = infinity, anything:0 = infinity. This was not logos and therefore it was suppressed. This philosophy extended beyond mathematics into the realm of astronomy and, weirdly enough, music (at the time, Pythagoras was actually most famous for his discovery of the golden ratio using an instrument called the monochord, which is a legend that seems to be false but nonetheless made him very famous.) This astronomical belief system was then later attributed to Ptolemy. This philosophy then was transplanted into Christian theology, and it took centuries for the monks to accept the existence of zero and infinity as a result. We even have cases of religious figures persecuting mathematicians about zero and infinity as late as the 1800s.

jacobolus 2y ago
> The Greeks were ideologically opposed to the number zero. I don't think this claim is supportable, certainly not as such a broad generalization. Do you have a specific statement clearly attributable to an ancient Greek author to that effect? > Greeks saw geometry and math as interchangeable You're going to have to define these terms more explicitly. I don't think this statement is right. Ancient Greeks spent quite a lot of effort studying areas of what we now call mathematics but which were not geometrical per se. > Greeks weren’t gonna give up their belief in Geometry because it provided them with social and political power Any claim that geometers in general had social or political power owing to their mathematical work seems exaggerated. While a couple of geometers happened to incidentally also be local political leaders (e.g. Eudoxus), geometers explicitly lamented how little their contemporaries cared about their work and how few colleagues they could find to share it with. > Aristotle outright refuses to acknowledge the existence of zero and of infinity. This seems like a reductive summary. Aristotle had a pretty sophisticated idea about this which finds echoes in modern mathematics: here is Aristotle: > Now there is no ratio in which the void is exceeded by body, as there is no ratio of 0 (οὐδέν) to a number. For if 4 exceeds 3 by 1, and 2 by more than 1, and 1 by still more than it exceeds 2, still there is no ratio by which it exceeds 0; for that which exceeds must be divisible into the excess + that which is exceeded, so that 4 will be what it exceeds 0 by + 0. For this reason, too, a line does not exceed a point-unless it is composed of points! Alternate translation: > But the nonexistent substantiality of vacuity cannot bear any ratio whatever to the substantiality of any material substance, any more than zero can bear a ratio to a number. For if we divide a constant quantity c (that which exceeds) into two variable parts, a (the excess) and b (the exceeded), then, as a increases, b will decrease and the ratio a :b will increase; but when the whole of c is in section a there will be none of c for section b; and it is absurd to speak of 'none of c' as 'a part of c.' So the ratio a: b will cease to exist, because b has ceased to exist and only a is left, and there is no proportion between something and nothing. (And in the same way there is no such thing as the proportion between a line and a point, because, since a point is no part of a line, taking a point is not taking any of the line.)" Later, about motion in a vacuum: > But if a thing moves through the thickest medium such and such a distance in such and such a time, it moves through the void with a speed beyond any ratio. (The physical premise here is wrong, but the concept of division by zero is clear.) See https://www.jstor.org/stable/pdf/2304187.pdf https://www.jstor.org/stable/pdf/2304187.pdf Aside: anyone telling you what Pythagoras thought about numbers is pulling your leg. We have no idea about this whatsoever, only what various people claimed like 5+ centuries later, most of which is nonsense.