Every culture develops mathematical concepts. Even the most primitive cultures develops a counting system, even if it is nothing more than “One, two, three, many”. The history of mathematical progress in different civilizations reveals much about that civilization’s reasoning processes. 

The first civilization for which we have evidence of mathematical progress is the Mesopotamian civilization. However, I shall begin with Egyptian civilization, because it is likely that its use of mathematics predates Mesopotamia’s, although the papyrus records of such work have perished while the clay tablets used in Mesopotamia have survived.

It is widely presumed that Egypt needed to develop mathematical concepts early on because the annual flooding of the Nile erased property lines, requiring an annual reconstruction of those lines, which in turn required the development of geometry. The surveying techniques that the Egyptians developed later expanded with the development of complex architectural designs and the pyramids.

The records we do have show that the Egyptians were pretty good at a number of mathematical problems. However, we must be careful here not to impose our own mathematical concept upon ancient civilizations; they did some pretty impressive things, but they did them within a very different context.

Let’s take the simple concept of numbers. Ancient civilizations used a primitive number system that make calculations of any sort cumbersome. The Roman number system is similar in basic architecture, but has two improvements. First, it has special symbols for 5, 50, 500, and so forth. Second, it uses a subtractive system, so the number four is not represented as IIII, but as IV. Similarly, 9 is IX, not VIIII. To appreciate the number systems used by all previous civilizations, imagine the Roman system with the symbols for 5, 50, 500, and so forth removed and the elimination of the subtractive system. For example, in such a system, the number 139 would be represented as CXXXIIIIIIIII. That’s how the Egyptians, Babylonians, and Greeks calculated. As you can see, calculations weren’t easy for them; this makes their achievements all the more remarkable.

Egyptians didn’t have multiplication as we know it; if you wanted to multiply 13 by 5, you added 13 to itself to get 26, then added 26 again to get 52, then added 13 one more time to get 65. Division was even weirder; they used a system based on tables of fractions. A table would show how a fraction was broken up into smaller fractions. For example, 2/5 was equal to 1/3 + 1/15. Basically, every calculation was performed by figuring out how to combine various table entries to obtain the result you desired. For example, a student exercise challenges the student to calculate how to divide one loaf of bread among ten men. You scoff at this; obviously it’s 1/10 of a loaf per man. But they didn’t calculate that way; instead, the calculation proceeded as follows: if one man receives 1/10 loaf, then two men receive 2/10 or 1/5, and four men will receive 2/5; but the scribe already knows that 2/5 is 1/3 + 1/15. This means that eight men will receive 2/3 + 2/15, which is 2/3 + 1/10 + 1/30. Add in what two men get (1/5), and you get a final result of 2/3 + 1/5 + 1/10 + 1/30. The scribe knows that this equals 1, so his result is correct. Yep, that’s the way they calculated it.

They knew the value of π, but in fact, they never thought of it as a number per se. Instead, they knew that a circular area with a diameter of 9 units had the same area as a square with a side of 8 units; all other calculations seemed to have just scaled that relationship. This implies a value of 3.1605 instead of the correct value of 3.1416, and error of only 0.6%.

The Mesopotamians got further than the Egyptians; they tackled some fairly complicated problems. Many times those problems involved areas of land. Sometimes these patches of land were rectangular, but they could just as easily be triangles, trapezoids, or parallelograms – and the scribes were called upon to figure out how much land comprised a single parcel (for tax purposes), or how to divide a parcel fairly, or re-arrange several parcels to reflect some new policy. All this took lots of geometry. They could easily measure distances, but they had no way to measure areas directly, so they had to calculate the areas of parcels with all sorts of weird shapes. Sometimes they had to work backwards, figuring out how long a trapezoid of land should be in order to encompass a specified acreage.

All these calculations got them into quadratic algebra: calculations involving squares and square roots of numbers. To handle such calculations they required multiplication and taking square roots, so they developed techniques for handling these problems. The methods of computation, while more advanced than those of the Egyptians, were still quite clumsy by our standards. Like the Egyptians, they relied heavily on tables of recalculated results. Their number system was sexagesimal (base 60) rather than decimal, but they seem to have developed facility with it. They even had to solve all the variations on the basic quadratic equation. But they never actually did what we call algebra; they had no x, y, z, or even any equations. Instead, they just had rules for how to put numbers together to get answers. We have oodles of cuneiform tablets (little squares of baked clay about 2 or 3 inches on a side) with sample problems, and, in some cases, their solutions. Here’s an example of the kind of problems they tormented students with: a ladder of length 10 feet is leaning against a wall; if the top of the ladder slips down 2 feet, how far out must the bottom of the ladder move? The solution requires the application of the Pythagorean theorem, which the Mesopotamians knew.

The Greeks learned the basics from the Egyptians and Mesopotamians, but then they went much further. Their first huge leap was to distinguish between calculations of physical quantities and theoretically pure mathematical notions. This notion of “pure mathematics” was absolutely revolutionary; they weren’t concerned with figuring out the price of wine in Alexandria or the area of somebody’s farm or anything so vulgar as that; they were concerned with mathematics as a set of ideas. Where did this come from? There is absolutely nothing by way of precedent for this concept; no other civilization ever came close to it.

Elsewhere in this hyper document, I offer my guess that it arose from the Greek dependence upon mercantilism, which profoundly affected their social development. But there was another factor at work, something about the Greek mind that distinguished between the ideal and the real, between theory and practice, between mathematics and numbers. True, other civilizations have sometimes mentioned this distinction, but it dominated Greek thought. 

Let’s consider, for example, the famous Pythagorean theorem. This concept was well known by the Mesopotamians, the Egyptians, and the Chinese; the Greeks certainly didn’t discover it. But Pythagoras appears to be the first person to prove it. Indeed, the very concept of proof was a Greek innovation. None of their predecessors even considered the question of whether they could be absolutely certain that the theorem worked in all situations; they just knew that it worked, and that was all there was to consider. Other civilizations had skilled number-crunchers, but the Greeks invented the very idea of the mathematician.

It is noteworthy that Greeks made greater progress in mathematics than any other field. Pythagoras proved his theorem a hundred years before Socrates faced the Athenian assembly. At almost same time that Aristotle was inventing the syllogism, Euclid was writing the definitive work on geometry, which remained in use for nearly 2,000 years. Euclid laid down all the basic ideas of modern mathematics: axioms, precise definitions of terms, theorems, lemmas, and so forth. Western science surpassed Aristotle 200 years before Western math outgrew Euclid.

After the Greeks, little progress was made in math for centuries. The Romans added almost nothing; Islamic scholars did make some significant additions; the word 
algebra comes from Arabic (many words beginning with al can be traced to an Arabic source: alcohol, algorithm, alchemy…). It is to Islamic mathematicians that we owe the idea of mathematical variables (x, y, and so on). Above all, we owe our number system, sometimes called Arabic numbers, to the Islamic scholars, although it was actually created by Hindu mathematicians. 

The development of mathematics in different cultures argues against my thesis about the relationship between mercantilism and democracy on the one hand and logical thinking on the other hand. Hindu scholars invented our numeral system, but their culture had almost nothing in the way of either mercantilism or democracy; Islamic civilization did include a certain amount of mercantilism, but it didn’t dominate their culture as it did with Greek culture, and Islamic civilization was most certainly not democratic. The spectacular achievements of the Greeks provide strong support for my thesis; everything else serves against it.

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