Why Geometry?

The entrance to Plato’s Academy was, somewhat famously, marked by a sign reading “Let no one ignorant of geometry enter here.” What’s so special about the study of geometry, that makes a student ready to approach philosophy?

At the level of content, geometry has a clear value for the Platonist. While the geometer is busy drawing and measuring particular lines, circles, triangles, and other figures, she is simultaneously being drawn out from these enmattered particulars, to consider the separate and eternal nature of the line, the circle, or the triangle itself. Yet this is not, I suggest, the only reason—or even the most important reason—to recommend geometry.

We can also consider a “geometric ethos”: what’s it’s like to study, to reason, to approach problems in a geometric way—an ethos that is equally applicable to doing philosophy well.

Here are just a few of the lessons, or benefits, of that geometric ethos:

  1. Anyone who has ever solved a difficult problem or puzzle in geometry knows the “click” of recognition, the undeniable realization that the problem has been solved, the truth has been found. This can happen both with individual cases, and with discovering or trying out an entire method of problem-solving. As philosophers, we’re also looking for that same sort of “click.” And of course it’s no accident that I chose the term “recognition.”
     
  2. Something about that experience of recognition, and also the process of searching, measuring, exploring possibilities, makes the truth tantalizing. There can be something erotic, in the best (and most Platonic) sense of that term, driving us forward. So too for philosophical enquiry.

These features can apply even when we’re geometrizing (or philosophizing) on our own. But in Plato’s day, both geometry and philosophy are activities that would normally be done with others, together in community. In such a communal context, there are some additional features of the geometric ethos that would be especially valuable to the aspiring philosopher:

  1. In exploring any particular problem, a group of geometers can start together, from a kind of aporia or unknowing that is initially shared by all, and nonetheless have a high degree of confidence when they arrive at a conclusive result, thanks to the power of the geometric method. While having a teacher who has already solved a given exercise can sometimes be helpful, that is not necessary once the basic geometric techniques have been mastered. Likewise, the philosopher’s dialectic method allows a group of students to begin examining any given question from a place of shared ignorance, common to all, while trusting in the method itself to verify their eventual results.
     
  2. Geometric knowledge is clearly and obviously a non-zero-sum endeavour. I lose nothing when I help a fellow student discover a solution or understand a proof. And likewise, when my fellow student helps me. Far more than any model that strives to one-up others, such a collaborative approach to investigation, learning, and discourse is appropriate to the philosopher.
     
  3. This collaborative environment, where fellow learners bring each other along, teaches us the virtue of confessing our ignorance. And it creates a safe space for doing so.
     
  4. At the same time, there are clear standards of refutation in geometry. If I’ve made a mistake, that can be pointed out clearly and unambiguously. That experience will prepare the philosopher for the Socratic sort of elenchus.
     
  5. Refutation in community, then, can serve as a kind of test of motivation. If I feel humiliated by receiving such a refutation, why is that? Am I ready to be a student? Or if I feel some kind of prideful arrogance in delivering a public refutation to someone else, why is that? Am I ready to be a teacher?

So to all those on this philosophical journey together: Let’s keep on (or begin) practicing geometry!

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