Schopenhauer and the Geometry of Evil

At the beginning of the 18th century, Gottfried Leibniz took a break from quarreling with Isaac Newton over which of them had invented calculus to confront a more formidable adversary, Evil.  His landmark 1710 book Théodicée argued that, as creatures of an omnipotent and benevolent God, we live in the best of all possible worlds.  Earthquakes and wars, he said, are compatible with God’s benevolence because they may lead to beneficial consequences in ways we don’t understand.  Moreover, for us as individuals, having the freedom to make bad decisions challenges us to learn from our mistakes and improve our moral characters.

In 1844 another philosopher, Arthur Schopenhauer, came to the opposite conclusion, Schopenhauerthat we live in the worst of all possible worlds.  By this he meant not just a world is full of calamity and suffering, but one that in many respects, both human and natural, functions so badly that if it were only a little worse it could not continue to exist at all.   An atheist, Schopenhauer felt no need to defend God’s benevolence, and could turn his full attention to the mechanics and indeed (though not a mathematician) the geometry of badness.  He argued that if the world’s continued existence depends on many continuous variables such as temperature, composition of the atmosphere, etc., each of which must be within a narrow range, then almost all possible worlds will be just barely possible, lying near the periphery of the possible region.  Here, in his own words, is his refutation of Leibniz’ optimism.


To return, then to Leibniz, I cannot ascribe to the Théodicée as a methodical and broad unfolding of optimism, any other merit than this, that it gave occasion later for the immortal “Candide” of the great Voltaire; whereby certainly Leibniz s often-repeated and lame excuse for the evil of the world, that the bad sometimes brings about the good, received a confirmation which was unexpected by him…  But indeed to the palpably sophistical proofs of Leibniz that this is the best of all possible worlds, we may seriously and honestly oppose the proof that it is the worst of all possible worlds. For possible means, not what one may construct in imagination, but what can actually exist and continue. Now this world is so arranged as to be able to maintain itself with great difficulty; but if it were a little worse, it could no longer maintain itself. Consequently a worse world, since it could not continue to exist, is absolutely impossible: thus this world itself is the worst of all possible worlds. For not only if the planets were to run their heads together, but even if any one of the actually appearing perturbations of their course, instead of being gradually balanced by others, continued to increase, the world would soon reach its end. Astronomers know upon what accidental circumstances principally the irrational relation to each other of the periods of revolution this depends, and have carefully calculated that it will always go on well; consequently the world also can continue and go on. We will hope that, although Newton was of an opposite opinion, they have not miscalculated, and consequently that the mechanical perpetual motion realised in such a planetary system will not also, like the rest, ultimately come to a standstill. Again, under the firm crust of the planet dwell the powerful forces of nature which, as soon as some accident affords them free play, must necessarily destroy that crust, with everything living upon it, as has already taken place at least three times upon our planet, and will probably take place oftener still. The earthquake of Lisbon, the earthquake of Haiti, the destruction of Pompeii, are only small, playful hints of what is possible. A small alteration of the atmosphere, which cannot even be chemically proved, causes cholera, yellow fever, black death, &c., which carry off millions of men; a somewhat greater alteration would extinguish all life. A very moderate increase of heat would dry up all the rivers and springs. The brutes have received just barely so much in the way of organs and powers as enables them to procure with the greatest exertion sustenance for their own lives and food for their offspring; therefore if a brute loses a limb, or even the full use of one, it must generally perish. Even of the human race, powerful as are the weapons it possesses in understanding and reason, nine-tenths live in constant conflict with want, always balancing themselves with difficulty and effort upon the brink of destruction. Thus throughout, as for the continuance of the whole, so also for that of each individual being the conditions are barely and scantily given, but nothing over. The individual life is a ceaseless battle for existence itself; while at every step destruction threatens it. Just because this threat is so often fulfilled provision had to be made, by means of the enormous excess of the germs, that the destruction of the individuals should not involve that of the species, for which alone nature really cares. The world is therefore as bad as it possibly can be if it is to continue to be at all. Q. E. D.  The fossils of the entirely different kinds of animal species which formerly inhabited the planet afford us, as a proof of our calculation, the records of worlds the continuance of which was no longer possible, and which consequently were somewhat worse than the worst of possible worlds.* 


Writing at a time when diseases were thought to be caused by poisonous vapors, and when “germ” meant not a pathogen but a seed or embryo, Schopenhauer hints at Darwin and Wallace’s natural selection.  But more importantly his distinction between possible and impossible worlds may be the first adequate statement of what in the 20th century came to be called the weak anthropic principle, the thesis that our perspective on the universe is unavoidably biased toward conditions hospitable to the existence and maintenance of complex structures.  His examples of orbital instability and lethal atmospheric changes show that by an “impossible” world he meant one that might continue to exist physically, but would extinguish beings able to witness its existence.  At that time only seven planets were known, so, given all the ways things might go wrong, and barring divine assistance, it would have required incredible good luck for even one of them to be habitable.  Thus Schopenhauer’s principle, as it might better be called, was  less satisfactory as an answer to the problem of existence than to the problem of evil.

Returning to Schopenhauer’s  refutation of  Leibniz’s optimism, his  qualitative verbal reasoning can easily be recast in terms of high-dimensional geometry.  Let the goodness g  of a possible world   X   be approximated to lowest order as

g(X) = 1-q(X),

where  q  is a positive definite quadratic form in the d-dimensional real variable X. Possible worlds correspond to  X  values where   g  is positive, lying under a paraboloidal cap centered on the optimum,   g(0)=1,  with negative values of   representing impossible worlds.  Leaving out the impossible worlds, simple integration, of the sort Leibniz invented, shows that the average of  g  over possible worlds is  1-d/(d+2).   So if there is one variable, the average world is 2/3 as good as the best possible, while if there are 98 variables the average world is only 1% as good.  Thus, in the limit of many dimensions, the average world approaches  g=0,  the worst possible.   More general versions of this idea can be developed using post-18’th century mathematical tools like Lipschitz continuity.

Earthquakes are an oft-cited  example of senseless evil, hard to fit into a beneficent divine plan, but today we understand them as impersonal consequences of slow convection in the Earth’s mantle, which in turn is driven by the heat of its molten iron core.  Another consequence of the Earth’s molten core is its magnetic field, which deflects solar wind particles and keeps them from blowing away our atmosphere.   Lacking this protection, Mars lost most of its formerly dense atmosphere long ago.


One of my adult children, a surgeon, went to Haiti in 2010 to treat victims of the great earthquake and has returned regularly since. Opiate painkillers, he says, are in short supply there even in normal times, so patients routinely deal with post-operative pain by singing hymns until the pain abates naturally.  When I told him of the connection between earthquakes and atmospheres, he said, “So I’m supposed to tell this guy who just had his leg amputated that he should be grateful for earthquakes because otherwise there wouldn’t be any air to breathe?   No wonder people find scientific explanations less than comforting.”

*From R.B. Haldane and J. Kemp’s translation of Schopenhauer’s “Die Welt als Wille und Vorstellung”,  supplement to the 4th book  pp 395-397  On the vanity and suffering of life.