When the mind is held fast, preventing all flights of fancy, when it is benumbed as though night’s darkness had penetrated its deepest recesses, think of Kepler’s treatment of the heptagon. Here he culminates his rigorous investigations into the nature of plane geometry; he comes to the end of Book I having condensed the seemingly infinite into intelligible, finite order. “It follows therefore that the Concept, Knowledge, Determination, Description, and Construction of a figure serve to set up boundaries between the primary Orders to which the figures belong.” (Harmony of the World) There are an infinite number of regular polygons, unlike the very limited number of regular polyhedra, but this infinity is actually, bounded. Kepler’s exploration leads necessarily to an inconstructability proof which defines this universal boundary. He expends a great deal of effort in its demonstration: “this is a matter of importance, for it is on account of this result that the Heptagon and other figures of this kind were not employed by God in ordering the structure of the World, as He did employ the knowable figures explained in our preceding sections.” (Harmony of the world)
Arrive at this section for the first time as you work diligently through Kepler’s work and you will likely laugh with glee. Kepler’s boldness, his audacious certainty, is thrilling. “Here, indeed, we are concerning ourselves with Entities susceptible of knowledge; and we correctly maintain that the side of the Heptagon is among Non-Entities and is not susceptible of knowledge. For a formal description of it is impossible; thus neither can it be known by the human mind, since the possibility of being constructed is prior to the possibility of being known: nor can it be known by the Omniscient Mind by a simple eternal act: because by its nature it is among unknowable things.”(Harmony of the World) Not only does he assert that he knows God’s reason, he declares there are quantities which, being unknowable for Man, not even God knows. He claims the inconstructible heptagon’s inconstructability is both essential and universal. This is nothing short of a claim that it is possible to know the universe both essentially and universally. He dares to contain the universe in his own mind, an act so fundamentally human that its implications far outstrip any scientific discovery as such.
But then, a night creature emerges from the corner of your eye. If you listen, it will speak. “Is such knowledge truly possible? What are you to God? A mere nothing. What does it really mean to know? What is this knowledge in the face of the infinite? A droplet in the ocean. You are fooled to be so emboldened by your perception of Kepler’s mind. You like the idea of Kepler defying the littleness of constrained human creatures confining themselves to the chains of scholarship, and with hubris, proving his likeness to the creator. The nature of his personality imparts fire to your listless mind. But then, I plunge this rekindled mind into the shadows on an uncertain road again: Kepler is wrong. I will give you an example, one which any translator or commentator worth his salt jumps to point out. At the end of Book I of the Harmony of the World, Kepler attempts to generalize what he proves for the specific cases of the heptagon and nonagon. He tries to show that all odd sided polygons, except for the well defined cases of polygons with 3, 5, and 15 sides, are inconstructible. He makes a universal claim about prime numbers and he connects this to the fundamental difference between dividing a line and dividing an angle. All very intriguing. But ultimately, he is wrong. Admittedly the direction of his investigation is valid. But, for all his striving, Kepler doesn't reach the summit. He fails. What kind of universal statement would admit exceptions? Why didn't Kepler address them? He thought he was certain. But instead, he was certainly wrong.”
In the shadows, wonder: how do we reconcile this? How do we resolve this seeming inconsistency in Kepler’s work? Does Genius admit mistakes?
Overcome fear of the creature; let us retrace Kepler’s steps, finding the way marked out by Kepler’s mind’s eye, shining as it does for all wanderers on imagination’s dark paths.