Dostoyevsky on the Problems of Evil and Geometry PDF Print E-mail
Written by Mitch Stokes   
Saturday, 19 December 2009 14:00

In The Brothers Karamazov, nineteenth century Russian novelist Fyodor Dostoyevsky parallels the problem of evil with the discovery of non-Euclidean geometry.  Despite the naturalness with which evil and mathematics go together, Dostoyevsky’s discussion of mathematics is still surprising.  His point isn’t that mathematics causes human suffering, as true as that might be.  He’s showing us, rather, one of the reasons the problem of evil is the most powerful objection to the existence of God; namely, it forces us to lay aside our preconceived notions of what’s reasonable, something we’re loath to do.  But the problem of evil isn’t an intellectual problem – or at least not primarily intellectual.  Rather, it’s in part a problem stemming from mistaken loyalties.  So too, the problem of geometry.

One of the novel’s characters, Ivan Karamazov, is an astute intellectual who in many ways represents modernism (a version of rationalism), where reason claims our ultimate devotion.  In a crucial conversation with his younger brother, Alyosha, Ivan divulges his beliefs about God.  He begins with geometry, of all things.  Ivan believes that reason constrains God to create the cosmos according to ordinary, Euclidean geometry.  (Euclidean geometry is the geometry that Euclid presents in his famous book the Elements, the geometry most of us learned in high school.)  But, Ivan says,

there were and are even now geometers and philosophers, even some of the most outstanding among them, who doubt that the whole universe, or, even more broadly, the whole of being, was created purely in accordance with Euclidean geometry.

The problem, according to Ivan, is that the real world may not be Euclidean, but non-Euclidean.  That is, physical space doesn’t answer to all the “truths” of ordinary Euclidean geometry.  In real life, for example, parallel lines can meet!  (Einstein’s general theory of relativity later seemed to confirm this possibility.)

Ivan responds with disgust to this prospect: “Let the lines even meet before my own eyes, I shall look and say, yes, they meet, and still I will not accept it.”  It’s not that Ivan would deny the fact that the lines meet.  The problem is that their convergence would be repugnant.  If, against all reason, parallel lines can meet, then space is just as twisted as the humans who inhabit it.  By Ivan’s lights, even if God has made the world in a non-Euclidean fashion, he shouldn’t have.

Speaking of twisted humans, it similarly seems to Ivan that God shouldn’t have allowed the occurrence of wanton evil.  The existence of a perfectly good, all-knowing, and all-powerful God seems – according to Ivan’s reasoning – incongruous with the existence of horrific suffering.  Again, it’s not the fact of suffering that Ivan denies – that much is all too obvious – it’s just that the fact is unacceptable, inappropriate, just plain wrong.  It flies in the face of reason.  So Ivan resists God, despite believing that He exists.  He’s like the demons, who believe but swear fealty elsewhere.  Ivan defers to what he can understand, to what seems reasonable.  In this sense he is a modernist, and in this sense he is an idolater.

Dostoyevsky’s pairing of mathematics and evil in this way, however, really does seem odd.  We’re aware that there’s a problem of evil; but that someone would suggest there’s an analogous problem of geometry strikes us as improper, insensitive even.  Human suffering analogous to a mathematical discovery?  Tell that to people in genuine pain.  But modernists saw the discovery of non-Euclidean geometry as a bona fide crisis.  Their fundamental belief that human reason is the ultimate epistemic authority was based on man’s mathematical successes, in particular, on the legacy of Euclid’s Elements. Since its appearance around 300 BC, the Elements has prodded the West’s search for intellectual certainty.  And the confidence it created in mathematics and reason was rewarded by the scientific revolution, a revolution that fulfilled Plato’s dream of mathematically describing the cosmos, thereby inaugurating the Enlightenment.

But in the 1800’s, a previously unthinkable mathematical theory (one in which parallel lines can meet and the sum of the interior angles of a triangle aren’t 180 degrees) suggested that this unqualified allegiance to the Elements – and therefore to reason – was misguided.  Through non-Euclidean geometry, mathematics had begun to undermine itself and so, in effect, reason toppled its own rule.  This is modernist’s problem of geometry.  It tends to undermine belief in reason not unlike the way the problem of evil undermines belief in God.

This is some of the background to Ivan’s struggle.  In spite of Ivan’s allegiance to reason, the discovery (invention?) of non-Euclidean geometry caused Ivan to question his loyalties.  He says that reason – our reason at least – is fundamentally Euclidean (Dostoyevsky is alluding here to a view held by the Enlightenment philosopher, Immanuel Kant).  And because our reason is Euclidean, it’s hobbled.  So Ivan is torn; who’s to be trusted?  The existence of non-Euclidean geometry is a mathematical discovery – a discovery of reason.  Reason at one time told us that the world was Euclidean.  Now it tells us, all apologies, that it was wrong.  Ivan despairs, eventually going mad.

Ivan’s struggle mirrors the West’s.  Since before Plato, the West held reason in high esteem.  (The modernist spirit is not, therefore, modern after all.)  But then – just when the Enlightenment was hitting its stride – reason threw itself into doubt with non-Euclidean geometry.  This discovery is one of the main causes of postmodernism’s suspicion of reason.

But much of Dostoyevsky’s commentary here would be lost on us without an appreciation of the non-Euclidean revolution.  And this is but one example of how widespread mathematics’ influence is.  Not putting too fine a point on it, mathematics is important.  But merely being able to do mathematics is insufficient, primarily because there’s much more to understanding mathematics than recipes and formulas.  To be sure, mathematics is a powerful means for describing, predicting, and controlling the physical world.  But its study is also required for understanding culture.  To allude to Kant: calculation without understanding is empty, understanding without calculation is impossible.  Our problem with geometry is not the modernist’s; our problem is that we don’t understand it.

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