I have just begun reading Herbert Meschkowski’s Evolution of Mathematical Thought, and I have finished the first few chapters. The book is certainly an interesting read. The second chapter titled “Foundations of Greek Mathematics” provides some good insights into the way mathematics was understood by the Greeks during the classical Greek period. Based on that second chapter, I will write a little on why Plato laid so much stress on the study of geometry.
Tradition has it that the inscription above Plato’s Academy read, “Let no one enter who is ignorant of geometry.” We are, of course, not sure if that was really true, but we wouldn’t veer too far away from the truth if we do ascribe the above quote to Plato. And the reasons are quite compelling.
Platonic idealism (according to Wikipedia) is the theory that the substantive reality around us is only a reflection of a higher truth. To elaborate, in layman’s terms, there is a higher world (independent of us) out there and this world, according to Plato, consists of “ideal forms” or Ideals. These ideal forms are blueprints of all the tangible objects around us. So, in a sense, tangible objects are instances or copies of these ideal forms. For example, there exists the ideal circle (or the perfect circle) out there and all the circles we draw are copies that are close approximations of the ideal circle. In some sense, the approximations can only “aspire” to be similar to that ideal circle. Now, according to Plato, these ideal forms are more real than the instances of those forms, and he carries this idea over to the discipline of mathematics, thus providing a justification for the study of geometry.
In the Republic, Plato states:
And do you not also know that they (mathematicians) further make use of the visible forms and talk about them, though they are not thinking of them but of those things of which they are a likeness, pursuing their inquiry for the sake of the square as such and the diagonal as such, and not for the sake of the image of it which they draw?… The very things which they mold and draw, … they treat in their turn as only images, but what they really seek is to get sight of those realities which can be seen only by the mind.
In other words, geometers (or mathematicians) draw points, lines, squares, circles and so on for explaining theorems (or doing proofs), and the figures they draw are quite “inaccurate” ones, in the sense that no one can ever claim to have drawn a perfect round circle or a perfect straight line. However, this doesn’t prevent the mathematician from looking deeply into the “reality” behind those figures to obtain useful insights from the same. Such insights guide the mathematician in proving geometric results, which otherwise cannot be “extracted” from the (crude) figures.
So, for instance, no matter how one draws a triangle, if one measures all the three interior angles with, say, a protractor, the sum of the three angles will never equal two right angles (); the sum will always be a tad less or a tad more than two right angles. But, this doesn’t prevent the mathematician from proving that the sum is indeed equal to two right angles! (Of course, this is true only in Euclidean geometry.)
Thus, for Plato, knowledge of mathematics amounts to insights in the realm of ideas, and the pursuit of mathematics was a road to insights of a universal nature.
Thus, education through mathematical thinking frees man’s mind to “see” the world of ideal forms, which was more real to Plato than the tangible. This is further illustrated in Plato’s Seventh Letter in which he distinguishes among the different ways of viewing a geometric concept. For instance, consider a circle and let’s look at it’s different “interpretations.” First, a circle is something, “the name of which I just now uttered”; second, the concept defined verbally; third, the physical image of the circle as it is produced by the draftsman or the lathe-turner; and finally, the ideal circle that “approaches nearest in affinity and likeness to” the “real circle”, which alone is the object of scientific perception.
In conclusion, for Plato, the study of geometry (mathematics) was an intellectual exercise in training the mind in overcoming the “illusions” of the tangible world by learning to see the “ideal forms” or Ideals, which to him was more real than the physical reality around us.