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.
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.
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.
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.
I think this is good to argue that my assertion is likely too strong. I fear this is close to arguing that early programmers were not familiar with map/flatMap. They did not discuss it as a first class thing, sure. Was it completely alien to all practitioners? I find that harder to swallow and it is likely that we are debating methods versus functions completely removed from the context in which the words were largely used.
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.
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.
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.
> 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.)
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.
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.
> 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”.
surely nobody thinks that zero spread from aryabhata 1500 years ago to the mesopotamians† 2700 years ago, and classical and hellenistic greece didn't even have a numeral for zero, much less the idea that zero was a number. (obviously the greeks have been using a numeral for zero for several hundred years now)
the hypothetico-deductive method that defines what we call 'mathematics' today does seem to have originated in the hellenistic period, although probably in alexandria rather than in greece proper. but menon and the timaios show that the pythagoreans already had the rudiments of it in classical greece, and there's no evidence that anything similar existed during the thousand-plus preceding years of mesopotamian mathematics. i think that's what you're saying
on the other hand, the majority of greek mathematics during the classical and hellenistic period was still mostly 'a tool, a method for business and construction [planners] to [derive] certain quantities', even if the calculations were being done on sand-tables instead of counting-boards. systems of axioms don't make an appearance in the surviving written record until roughly euclid; aristotle, plato, and pythagoras evidently weren't yet familiar with what we call 'mathematics' and in particular 'proof', although the later mathematicians, especially in the hellenistic period, did build on their work
so i think it's a mistake to talk about 'the greeks' as if they were a single person, like in a game of civ; all the greeks alive in plato's time were already dead by euclid's time, euclid didn't even live in greece, all the greeks alive in euclid's time were already dead by ptolemy's time, there are still greeks today, and in any time period, different greek people knew different things
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.
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.
maybe try to have a little more patience with other people; they may not find it as easy to understand you as you think it is, and it doesn't always mean they aren't trying. sometimes they're just dumb, sometimes what you wrote is actually pretty unclear (even though it's perfectly clear to you because you already know what you were trying to express), and sometimes they're actually aware of things you're not which introduce hidden contradictions into your narrative
the worst case, from my point of view, is when they're actually just dumb and you yell at them for their intellectual limitations. that's sad
I have no intention to argue with either of you, but from an outside perspective, it's you who is very hard to discuss with.
I don't understand what's wrong with rusticpenn original post and I don't think they argue in bad faith. Maybe you are right, but because of all the personal attacks you make, it's hard to see your point, and you also come of as very rude. I don't write this to criticize you - I'm just saying that if you change the way you argue, you will probably convince many more people to your ideas. Constructive discussion is, in general, a very useful skill in life.
> No. You are still ignoring that the mathematics of the Greeks and the mathematics of the Mesopotamians were radically different.
This doesn't matter. We're talking about 0. Whether mathematics are conceptualized as a tool or an academic discipline in its own rite wasn't the thrust of the article. It wasn't even part of your original post.
> Did you even read my original post?
It might surprise you, but yes -- yes, I did!
> My position is that a linear history of the concept of 0, which spreads from India to Mesopotamia to Greece is nonsense. That the Mesopotamians did some mathematics long before the Greeks did is irrelevant to my argument. It totally does not matter. What makes it even more ridiculous is that my OP obviously assumes that the Mesopotamians had mathematics before the Greeks, so pointing it out as some "gotcha" is just really stupid.
This is a better statement of your position than you originally posted, when you said "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." Here's why I think your initial statement is a weak argument for your refutation of a linear history of the concept of zero: Babylonian mathematics temporally happened first and the general opinion (when I was in school, at least) was that Babylonian mathematics likely influenced Greek mathematics. That's why I think this is a weak argument to argue against the linear history concept of 0.
> It genuinely makes me mad to have this low quality discussion, where someone barges in and gives you a trivial "gotcha" as if I wasn't completely aware of that fact. And when you try to point out why the person didn't understand what you were saying you are getting another person debating the stupid gotcha, as if it even mattered.
I'm sorry you're mad but I mostly disagree on your assessment of the quality of the discussion. Your top-level comment on the article about Mesopotamia creating nought was essentially that Greek geometers knew about it and that tying concepts and geographical locations seems odd.
I don't disagree with your second point, but we may be in the minority: lots of people are very interested in knowing who had what ideas first and where. Your first point doesn't stand well alone without the further elaboration you've made through the rest of this discussion. With the elaborations you've made since the rest of this thread might not have happened.
I wanted to give a seriously reply, but clearly there is no point.
I still don't get why you and the other guy think pointing out that "Mesopotamians did it first" is relevant. Even if they did and even if the Greeks were extremely influenced by them, their concept of a geometrical zero was still radically different to the Mesopotamians notion, so it is totally irrelevant who was first. The only counter argument could be that they had the same notion, which the Greeks adopted from the Mesopotamians as a complete package.
I don't think this is really complicated. If their understanding was radically different, then the concept couldn't have just "moved over", so disprove my thesis he discussion on "who was first", is obviously irrelevant.
> With the elaborations you've made since the rest of this thread might not have happened.
Well, the genuine curiosity and willingness to discuss the subject embodied in a single sentence dismissing my post because of an obviously true statement made me very glad to elaborate and discuss further.
I'm going to step back to a meta level and make a couple points:
- Several times in the context of this discussion you've made negative assumptions about someone or their motives replying to you (e.g. "I wanted to give a seriously reply, but clearly there is no point" and "What a stunning revelation, genuinely brilliant insight from you"); I don't think this fosters good discussion (and I also don't think it's particularly good for mental health)
- You seem particularly aggrieved by someone not automatically assuming you know something. Yes, we all understand now that you know the Mesopotamians predated the Greeks with regard to the mathematical notions we're discussing here. But you know what? There's nothing wrong with NOT knowing. And some people reading the discussion might not have known.
Now back to the main topic at hand.
> Even if they did and even if the Greeks were extremely influenced by them, their concept of a geometrical zero was still radically different to the Mesopotamians notion, so it is totally irrelevant who was first
Now I finally feel as though I understand the main thrust of your argument. Let me restate your hypothesis to see if I've got it: even though the Babylonians had the concept of zero, it was sufficiently lacking compared to the Greek concept as to be incomparable; the Greeks fundamentally independently invented the concept of a zero more mathematically powerful than the Babylonians rather than took the existing Babylonian concept of nought and developed it further. Is that right?
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).
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.