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.
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.