In my previous post, I talked about the conceptual progression from the most basic intuitions about counting to the more general idea of number that transcends any particular physical situation.
As big a development as this is, I would say this not mathematics proper. In most civilizations where the number concept developed, the most complicated uses of number were related to measurements – something like geometry. Many ancient civilizations, for example, knew that a triangle formed out of ropes of lengths 3 units, 4 units, and 5 units formed a right angle (that is, a perfect 90 degree corner), and so that was often used to help place markers for rectangular fields. Many ancient peoples also had numerical approximations for some important numbers, though they probably did not think of them as “approximations” in the same way we do today. Details about ancient mathematics can be found without too much trouble via a search engine or a book on math history, it is fascinating to read about, but I won’t go into much more detail here.
The idea of precision we are used to today took a long time time to develop. Ideas involving number were presented until comparatively recently only as “word problems.” This is because the written symbols for numbers largely had not been developed, or were very cumbersome. (To see what I mean by cumbersome, imagine trying to multiply or divide using Roman numerals!) Most ancient peoples viewed numerical work as strictly practical, or in some cases spiritual elements may have been involved (as in some ancient astronomy, for instance). In any case, maximizing the precision was not really the goal, the goal was to be ‘good enough’ to use in everyday life. For example, consider the number we call pi today:
This constant is equal to the circumference of a circle divided by its diameter, and is very important in geometry and in every area of math. There were some civilizations, like ancient Babylon, that seem to have treated this number as being exactly the fraction 25/8, which written as a decimal is equal to 3.125 . We know today that pi is not equal to 3.125. In fact, today we know that the constant pi is something called an irrational number, which means there is no fraction equal to pi. But an ancient mathematician either didn’t know that 25/8 was not equal to pi, or they didn’t care that much. What they considered important was that the numerical methods got “close enough” to make something work correctly, and little else mattered. It is true that we use approximate values for pi all the time in calculations – computers cannot store irrational numbers exactly, and so they use approximations, and engineers will frequently round off their solutions. But even here, there is precision, because when an engineer does this, they are aware that they are using an inexact value, and they even have ways of keeping track of how much they have rounded off their answers. The awareness of the ‘inexactness’ of the rounded values either was not in the mind of these ancient civilizations, or they simply did not care.
While many peoples did make huge developments in advanced mathematics in ancient times, China and India stand out to me, the way mathematics was done in ancient Greece was a highly influential and new approach. Speaking as a mathematician, I would say that the huge change that began in Greece was the introduction of rigorous argument to mathematics. What this really means is, unlike peoples before them, the Greeks did not content themselves with application and approximation. They wanted to know why certain facts about numbers are as they are, and they wanted to know beyond any possible doubt. This was a revolutionary approach.
The way Greek mathematicians started this was by making unambiguous definitions of what objects like circles and lines actually are. They also laid out clearly a set of basic assumptions that serve as the rules of geometry. In that day, the tools of geometry were a compass – which is a tool for drawing circles, and a straightedge – which is any perfectly straight object (like a ruler, except a straightedge does not have markings on it like a ruler does). The straightedge tool came along with some rules – for example, every straight line segment can be extended. The compass also has an associated rule – every circle is defined by a point (its center) and a distance (its radius). These rules bring clarity about what circles and lines are. A circle isn’t some vaguely round thing, a circle is a shape all of whose points are exactly the same distance from some central point. In a similar fashion, the Greeks made explicit certain key concepts about angles, areas, and volumes.
Within a clear, structured framework, the Greeks could then find theorems, and used proofs to explain them. In mathematics, a theorem is just any true statement about math, and a proof is a bulletproof explanation for why a certain statement is true. It is worth noting that no other discipline, including physics and other sciences, have a concept of proof anywhere near this powerful. A scientist would usually be content to say a scientific theory is proven, roughly speaking, if it has a large body of empirical evidence in its favor, and very little evidence against it by comparison. This does not count as a mathematical proof, because empirical evidence could always have an exception that we just don’t know how to find. While the mathematician will frequently use empirical data to help them formulate their ideas, a mathematical proof relies only on logical argument. For this reason, a genuine mathematical proof literally cannot be mistaken, in the sense that if you accept that words like true and false actually mean something, and if you accept the conceptual framework of Greek geometry, then you cannot deny any theorem of Greek geometry.
To be fair on this point, you can always reject the framework. You can start with different definitions and assumptions if you want to, there is nothing to stop you from doing that. But by making different assumptions, you of course aren’t really doing Greek geometry anymore, you are doing some other kind of geometry that might turn out differently than Greek geometry, but it does not invalidate what the Greek geometers did.
Up to this point, I have yet to actually present a proof. I will do this in my next post, where we will talk about the most famous theorem of Greek mathematics today, the Pythagorean theorem. But before we can truly understand what the Pythagorean theorem is and what its proof means, it is important to have in mind this landmark intellectual achievement of the ancient Greeks, to which all who study mathematics today are indebted for their great contribution.
 Heaton, Luke. A Brief History of Mathematical Thought. London, Constable & Robinson Ltd. 2015.