I think the most important lesson I learned from reading Jeff Suzuki’s A History Of Mathematics was that, for most of the time that humans have being doing serious math, it’s been all about the Geometry. The Ancient Greeks considered geometry more primary than numbers.
If an Ancient Greek mathematician had wanted to determine if two figures had the same area, he would not have assigned measurement-values to some or all of the sides, done the area-calculation for each figure, and then compared the results. He simply would not have thought in terms of numbers.
He would have, instead, set about determining if the figures could be sliced and diced in such a way as to make all one figure’s pieces exactly overlay the other figure. If so, than they had the same area. No pre-set measurement-values (centimeters, etc) need be involved.
When values were used, they were relational in nature… For example, a Greek might have determined that “side Alpha” was twice as long as “side Beta.” Or, he might have come up with a line-segment which would serve as the “unit” for that particular consideration. But this “unit” was without particular, real-world size– as I said, no centimeters or inches or any of that.
Philosophically speaking, the Ancient Greeks wanted their mathematics to be as abstract as possible. They were aiming at truths “higher” than a specific triangle here or a particular square there. The geometric relationships Greek mathematicians found in their demonstrations would be applicable to any triangles or squares or et cetera anywhere in the world, taking into account, of course, that Earthly triangles and squares would not be as perfect as the ideal ones of their mathematical philosophy. In this way, some Ancient Greeks believed that Mathematics was something higher, or more perfect, than mundane reality. Plato’s famous Ideals were assuredly inspired by the mathematical ideals he had been taught.
When writing of the Ancient Greek philosopher and mathematician, Thales, Suzuki relates that “it is virtually certain that Thales did not consider a right angle to be an angle of a particular measure, since the notion of the actual measurement of an angle never arose in the theoretical geometry of the ancient Greeks.” Thales may have been the first to find that a right angle is made when an angle is inscribed in a circle so that its vertex and its two endpoints are all on the circumference, with the two endpoints falling on opposite ends of a diameter.
Suzuki also states that the famous geometrist Euclid’s “definition of a Right Angle had nothing to do with the actual measure of the angle.” Instead, Euclid defined a Right Angle as occuring whenever two straight lines intersect in such a way that any two adjacent angles are congruent. Furthermore, Euclid never computes the AREA of a figure in isolation (never talks in terms of square feet or what have you), but only treats one area in comparison to another shape’s area.
Furthermore, the Ancient Greeks had this notion that the fewer instruments needed for solving a geometrical problem then the more genius and perfect was the solution. Says Suzuki: “this would eventually become the doctrine that the only instruments allowed in Geometry were the COMPASS and the STRAIGHT-EDGE.” And the Compass could be used ONLY to draw a circle with a given radius or to measure equal segments along an existing line, with the Straight-Edge allowed only for connecting points.
Another way geometry stayed front-n-center throughout most of the development of mathematics was the use by mathematicians of CONIC SECTIONS to explore the characteristics of curves. By slicing through different cones in different ways… ellipses, parabolas, and hyperbolas can be formed. The modern math student deals with these shapes frequently, but rarely thinks of them nowadays as the sliced cones they started out as.
I think it is very difficult for anyone born within the last few centuries to really comprehend just how different was the mathematical worldview of the Ancients. We moderns think in terms of numbers and pre-fab’d measurement-values. We can’t help ourselves. It’s ingrained in us from infancy. Numbers are how we “do” math, including Geometry, and we would think someone was joking if they talked about doing math without numbers. But for the Greeks, numbers were secondary to the logical relationships between component parts of figures.