Why the Greeks?

The Greek islands were geographically predisposed to democracy. The ritualised, antagonistic debates of parliaments and law courts were then generalised to all philosophical domains, creating a unique intellectual climate that put a premium on adversarialism and pure reason. This style of thought proved ideal for mathematics.

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Opinionated History of Mathematics

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Why the Greeks, of all people? Why did mathematics start there, on a few scrawny little islands in the Mediterranean?

The very idea that mathematics is about systematically proving things is an exclusively Greek invention. Axiomatic-deductive mathematics has been discovered only once in human history. No other culture independently developed anything like it.

The lettered diagram is another uniquely Greek invention. Triangle ABC, the line AB, stuff like that. Geometrical diagrams with the points denoted by letters. Only in Greece did they feel the need to do geometry this way. If you find it elsewhere, it’s because they copied it from the Greeks.

Not that the lettered diagram is a big deal in itself, of course. But it’s a symbol; it’s emblematic of how so many aspects of mathematics that we now consider so essential and indispensable were in fact discovered once and only once in human history, at a particular time and place.

So what was it about that time and that place that made it explode with intellectual progress?

You can make a pretty good case for geographical determinism. The seeds of excellence was not in the blood or the genes of these people, but it was in the land and the sea.

Islands. That is the key. Greece is a country of a thousand islands. In fact, you can hear this in the very names of the great mathematicians of that time.

Consider Pythagoras, for example. More fully you often see his name given as Pythagoras of Samos, his place of birth. Which is an island. One of those typical picturesque Greek islands.

The same goes for other great Greek mathematicians. Hippocrates of Chios, Aristarchus of Samos, Archimedes of Syracuse, Hipparchus of Rhodes: island, island, island, island. Everybody is an islander in Greek mathematics. There’s also Eudoxus of Cnidus, and Diocles of Carystus: those are technically peninsulas, but pretty nearly islands basically.

What’s with all these islands? Let’s see where this geographical argument leads us.

First of all, islands are excellent for trade. Back then, it was a thousand times easier to transport goods by water than by land. Even the Romans, centuries later, used to import huge amounts of grain from Egypt for example. And that’s the Romans, who are famous for their excellent roads. Even to them it was much more of a hassle to get grain from mainland Europe than to swish it across the sea with some efficient ships.

So the Greeks became tradespeople. Because they had so much access to the water.

And what did they have to trade? Think of the typical landscape of a Greek island. It’s hilly and full of slopes and kind of dry, rocky soil. Not the typical agriculture landscape you would have on the irrigated flats of mainland Europe or America. That kind of stuff would slide right off the Greek hills. In Greece you need tougher plants with roots that really dig in and hang on for their life as a rain shower threatens to wash the whole thing down with it down the hill.

Hence: olives and grapes. These plants love a good slope. They thrive there.

And what luck for the Greeks! These plants are perfect for trade. Think about it. You use them to make olive oil and wine: expensive, non-perishable luxury products.

Vegetable and fruit is highly perishable: by the time you get to your destination to sell it, half of it is rotten or eaten by worms. And it’s also very bulky: a big barrel of cabbage isn’t going to fetch you a whole lot of cash. It doesn’t have many calories. So it’s a lot of work to transport for so little payoff. The cabbage business isn’t very lucrative.

But olive oil and wine is perfect. Olive oil is a calorie bomb: a little goes a long way, so it’s easy to transport a fortune’s worth of it. And these products don’t mind being stored. Just stick them in an urn with a good cork on it and you’re set. Wine can even get better by sitting around. Unlike a sack of cucumbers that will spoil before you put your sandals on.

Olive oil and wine are also highly processed. A lot of work goes into the production. What are you gonna do with a bag of cucumbers? They are what they are, you just eat them. But the grapes and olives are processed by expert artisans. Lots of added value. The labor theory of value, you know, that Marx talked about and so on.

So the Greek islands are a recipe for wealth. Perfect products for trade, and perfect access to the sea for trading. This creates wealth, which creates a large middle class with lots of leisure time. That is certainly a precondition for intellectual culture.

Maybe also trade is itself a recipe for a certain open-mindedness and diversity of thought. There was no Internet back then. Travel was a good way to get exposed to other ideas, other ways of doing things. And therefore to start thinking more critically about the idiosyncrasies of your own habits and worldview.

Plus, a merchant needs to trade with whoever is paying. That may be people of different religions and so on. So you get used to dealing with people different than yourself. You develop and kind of tolerance for differences of opinion, and strategies for reasoning with people you disagree with.

All that from trade. But there is a second big consequence of the islands: independent city states. Islands are naturally isolated units. It will be much harder for a single despot to impose a unified rule on a bunch of scattered islands than on a solid land mass.

This is the geography of democracy. And democracy means debate. You don’t have “do this because I’m the king and I’ll chop your head off.” Instead you have one guy presenting reasons for this, the other guy presenting reasons for that, and people are weighing the arguments and making up their own minds.

This is going to be the setting that gives birth to mathematics and philosophy. Geography created this rich, democratic, cosmopolitan people who fell in love with clashes of ideas and took that concept to the extreme.

Geoffrey Lloyd the Cambridge professor has written good stuff about this. I’m going to quote extensively from his works.

“The level of technology and economic development” in ancient Greece was high indeed. In fact, it was “far in advance of many modern non-industrialised societies” today. And “Aristotle [explicitly] associated the development of speculative thought with the leisure produced by wealth.” And not for nothing.

However, “Egypt and Babylonia were, economically, incomparably more powerful than any of the Greek city-states.” So the explanation for the “additional distinctively Greek factor” of “generalised scepticism” and “critical inquiry directed at fundamental issues” must be something other than wealth alone.

The answer may lie in “a particular social and political situation in ancient Greece, especially the experience of radical political debate and confrontation in small-scale, face-to-face societies. The institutions of the city-state put a premium on skill in speaking and produced a public who appreciated and the exercise of that skill. Claims to particular wisdom and knowledge in other fields besides the political were similarly liable to scrutiny, and in the competition between many and varied new claimants to such knowledge those who deployed evidence and argument were at an advantage compared with those who did not.”

The Greeks were so fond of debates and clashes of ideas that they developed a refined social machinery for it. They ritualised and institutionalised the concept of a philosophical debate. “Public debates between contending speakers in front of a lay audience” was a prominent part of ancient Greek culture. Science and philosophy were born on this stage. Many otherwise peculiar characteristics of Greek thought are explained by this format.

For example, the stage debate requires the speakers to proclaim bold and provocative theses, and to strive to avoid reconciliation with other viewpoints at all costs. This is why early Greek thought is rife with crackpot claims such as that motion is impossible or “that man is all air, or fire, or water, or earth.” Indeed, the format demands a multiplicity of such viewpoints in competition with one another, whence “the remarkable proliferation of theories dealing with the same central issues” that “may well be considered one of the great strengths of Presocratic natural philosophy.”

Indeed, this used to always puzzle me. How can anyone in their right mind genuinely believe themselves to have discovered that “all is fire” or “all is water”? What were these people smoking, right? And that’s just a couple of generations before peak Greek philosophy and its many very refined insights in mathematics and science. How can they have been such crackpots and then gone from 0 to 100 in the blink of an eye?

But in fact it makes sense in the stage debate setting. “All is fire” is perfect for that. It’s like a dangerous stunt. Jumping across a ravine with a motorcycle, or juggling with three chainsaws. To go on stage and say “all is fire,” now try to prove me wrong, I will answer any counterargument. If somebody pulls that off, credit to them. The crazier and the more implausible their initial thesis is, the more impressive it is if they manage to parry objections and defend their thesis with clever arguments.

Nobody ever actually believed that “all is fire,” but they admired the guts of someone who was prepared to argue as if they did believe it. They glorified the ability to argue unconventional ideas well. This was a great move for stimulating philosophy.

The stage debate setting also explains why these kinds of crazy theses were always defended by abstract deductive reasoning, not empirical investigation. “Given an interested but inexpert audience, technical detail, and even careful marshalling of data, might well be quite inappropriate, and would, in any event, be likely to be less telling than the well-chosen plausible—or would-be demonstrative—argument.” Hence we understand why “with the Eleatics logos—reasoned argument—comes to be recognised explicitly as *the* method of philosophical inquiry.” This “notion of the supremacy of pure reason may be said to have promoted some of the triumphs of Greek science.”

However, these triumphs of reason “were sometimes bought at the price of a certain impoverishment of the empirical content of the inquiry.” In early Greek science, “observations are cited to illustrate and support particular doctrines, almost, we might say, as one of the dialectical devices available to the advocates of the thesis in question.” Also, “observations and tests could be deployed destructively [to disprove an opponent’s thesis], as they were by Aristotle especially, with great effect.”

These uses of observation fit well within the stage debate format. However, “theories were not put at risk by being checked against further observations carried out open-endedly and without prejudice as regards the outcome.” We can understand why since “The speaker’s role was to advocate his own cause, to present his own thesis in as favourable a light as possible. It was not his responsibility to scrutinise the weaknesses of his own case with the same keenness with which he probed those of his opponent.”

Of course, everyone was well aware of the deceptive potential of sly rhetoric for “making the worse argument appear the better.” So much so, in fact, that “early on it became a commonplace to insist on your own lack of skill in speaking.” But the Greeks did not see this problem, the rhetoric problem, as a reason to abandon the stage disputation format altogether. Instead they focussed on explicating “the correct rules of procedure for conducting a dialectical inquiry,” to ensure the intellectual integrity of the debates.

What I just described is basically a summary of Geoffrey Lloyd’s book “Magic, Reason and Experience,” about the origin of Greek scientific thought. Also very illuminating is Lloyd’s later book contrasting the Greek contrarian climate of thought with its opposite paradigm: reverential, conservative thought, typified for instance by the ancient Chinese tradition. The book title hints at this division: “Adversaries and Authorities,” it is called. Here is the argument.

“Any acquaintance with early Greek natural philosophy immediately brings to light a very large number of instances of philosophers criticising other thinkers.” Being a philosopher means being “subjected to blistering attack.” That could pretty much be considered the definition of philosophy in Greek antiquity. “From the list of occasions when philosophers are attacked by name, one could pretty well reconstruct the main lines of the development of Hellenistic philosophy itself.” Nor is this limited specifically to philosophy only. On the contrary, “hard-hitting polemic” is the name of the game in mathematics, medicine, and art as well. There is a “lack of great authority figures”; even Homer “is attacked more often than revered.”

This Greek style of philosophy is connected to its social context. “Greek pupils could and did pick and choose between teachers. Direct criticism of teachers is possible, and even quite common. Argument and debate are one of the means of attracting and holding students, and secondly they serve to mark the boundaries of [schools of thought].” “The Greek schools were there not just, and not even primarily, to hand on a body of learned texts, but to attract pupils and to win arguments with their rivals. They may even be said to have needed their rivals, the better to define their own positions by contrast with theirs.” “Dialectical debate, on which the reputations of philosophers and scientists alike so often depended, stimulated, when it did not dictate, confrontation. The recurrent confrontations between rival masters of truth left little room for the development of a consensus, let alone an orthodoxy; [and] little sense of the need or desirability of a common intellectual programme.”

“It was the rivalry between competing claimants to intellectual leadership and prestige in Greece, that stimulated the analysis of proving and of proof.” “Many have assume that the internal dynamic of the development of mathematics itself would, somehow inevitably, eventually lead to a demand for strict axiomatic-deductive demonstration, and that there is accordingly no need to pustulate any external stimulus such as [this.] Yet the difficulty for that view is [that] other, non-Greek, ancient mathematical traditions — Babylonian, Egyptian, Hindu, Chinese — all got along perfectly well without any notion corresponding to axioms and the particular notion of strict demonstration that went with it.”

The underlying cause is perhaps captured by the dichotomy between “adversarial Greeks and irenic, authority-bound, Chinese.”

The different philosophical styles of ancient Greece and China reflect differences in their political systems. “Extensive political and legal debates, in the assemblies, councils and law-courts, were a prominent feature of the life of Greek citizens.” Democracy primes people for debate, for listening to and assessing different points of view and conflicting claims.

“Greek philosophical and medical schools used, as the chief means for the expression of their own ideas and theories, both lectures and open, often public debates, sometimes modelled directly on the adversarial exchanges so familiar in Greek law-courts and political assemblies.” They imported democratic practices and put them to work in the sciences.

It was very different in China. “Many Greeks seem to have positively delighted in litigation; [they developed] taste for confrontational argument in that context and became quite expert in [evaluating such arguments]. [The Chinese, by contrast,] avoided any brush with the law as far as they could. Disputes that could not be resolved by arbitration were felt to be a breakdown of due order and as such reflect unfavourably on both parties, whoever was in the right.”

“The typical target audience envisaged in Greek rhetoric is some group of fellow citizens,” just as “in Greek law-courts the decisions rested with [peers] chosen by lot [who] combined the roles of both judge and jury.” “In China, the [intended] audience for much philosophical and scientific work was very different: the ruler or emperor himself.” “The Chinese were never in any doubt that the wise and benevolent rule of a monarch is the ideal.”

“We often find Greek philosophers adopting a stance of fierce independence vis-a-vis rulers. With this independence came a disadvantage. Compared with their Chinese counterparts, Greek philosophers and scientists had appreciably less chance of having their ideas put into practice. Autocrats — as in China — could and did move swiftly from theoretical approval to practical implementation.” Not so in Greece. Greek philosophers had little hope of real power, and perhaps that’s why they liked to pretend that they didn’t want any anyway. “The superiority of theory to practice is a theme repeatedly taken up by scientists as well as philosophers in Greece: but that was sometimes to make a virtue out of necessity.”

“Unlike in classical Greece, the bid to consolidate a comprehensive unified world-view was largely successful in China.” “The prime duty of members of a Chinese Jia was the preservation and transmission of a received body of texts. In that context, pupils did not criticise teachers, and any given Jia did not see it as a primary task to take on and defeat other Jia in argument.”

While the Greeks “adopted a stance of aggressive egotism in debate, the tactics of Chinese advisers was rather to build on what could be taken as common ground, [and] certainly on what could be represented as sanctioned by tradition.” “The emphasis is not on points at which [earlier philosophers] disagreed, but rather on what each of them had positively to contribute, how each succeeded, at least in part, in grasping some part of the Dao,” the true or right way.

So there you have it. The source of Greek exceptionalism in intellectual history comes down to this: to glorifying extreme adversarialism; to waking up in the morning and going “today I’m gonna point out errors in other people’s arguments.” The Greeks lived for that stuff. And it was this that made them mathematicians, eventually. But that was not a planned child. Geography led to democracy, which led to this combative philosophical climate.

When some fragments of mathematics from Egypt and the orient were dropped into this petri dish, the reaction was explosive. These two were made for each other. Mathematics and argumentative debate was a match made in heaven. The Greek philosophical context triggered an avalanche of mathematical progress that took geometry from a set of obscure calculation rules to mankind’s best exemplar of perfect knowledge.