Schopenhauer and the Geometry of Evil

Gottfried_Wilhelm_von_LeibnizAt 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, as Alejandro Jenkins pointed out,  Schopenhauer’s 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 198 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.
Cf German original, pp. 2222-2227 of  Von der Nichtigkeit und dem Leiden des Lebens

9 thoughts on “Schopenhauer and the Geometry of Evil

  1. Dear Charles, What a lovely (first on this blog) post! I wonder if regarding the world (and reality) we live in as “so-so” could also be a serious option. People tend to give too much weight to polarized points of view, perhaps not only in philosophical discussions.

  2. In reply to Gil, I’m all for moderation in politics and interpersonal relations, but in high dimensional geometry the middle is small and the periphery large. However, in a sense, Schopenhauer’s principle, in a sufficiently large universe (larger than he imagined) acts to maintain things in a locally precarious but globally stable intermediate condition between Leibniz’ unrealistic optimum and what would be a true pessimum, a globally dead universe with no life, civilization, etc anywhere. Paraphrasing Schopenhauer, a large universe provides an enormous excess of the potentially habitable places, so that the extinction of life in one does not entail extinction of life in the universe, for which alone nature really cares.

  3. Pingback: A geometria do mal!!! Schopenhauer refuta Leibniz!!! | madfornumb3rs

  4. Taking time into account may matter. Life on the Earth evolved and can be adapted to given conditions during about 4-5E9 years and “goodness” may grow for some species if to consider them as part of the biosphere. Someone (Dawkins ?) even got some related analysis just for effects of natural accidents such as tsunami.
    Yet another point with “goodness” in such kind of models is question about possibility to estimate size and dimension of whole “manifold” – i.e. if it Big Bang or Big Bounce, universe or multiverse and if yes, if stringy multiverse may be distinguished from quantum one due to different cardinality.

  5. Thanks to Graeme Smith for reminding me that it was Alejandro Jenkins who drew my attention to the connection between Schopenhauer and anthropic reasoning.

  6. The idea that “it’s good we have plate tectonics because even though we get earthquakes we also have an atmosphere” seems more in keeping with Leibniz’s optimism. It’s a humble thought that we don’t understand the why’s behind many of our observations and experiences. Not comforting in the least since we humans want to think life is ‘fair,’ or narrowly deterministic, and not dominated by seemingly random events like earthquakes.

    Quite enjoyed the post. Thank you.

  7. Dear Charles,
    thank you opening this very interesting blog.
    I want to put the Théodicée into its historical context.
    Why did Leibniz write this very popular opus with its justification of god ?
    I believe his contribution was fundamental for an enlightened thinking in dealing with the evil, e.g. with natural phenomena like earthquakes and other disasters.
    The fundamental question in the metaphysics of the 18th century reads:
    Is there evil in our world, and if yes, how did it come into our world ? Is it better to be or not to be ? Why could both evil and bad manifest themselves so forcefully in a creation, which the creator has evaluated as good ?
    In 1669 Pierre Bayle published his sceptic view (1) (2). Religious statements can only be internalized by following the path of true faith. His argument reads: The bible confirms the coexistence of god and evil. This is why we simply have to accept this. No analyzing rationality exists which solves the problem of evil. The continued existence of the good in man always is under threat through evil, war, expulsion, famine, epedemics and religious persecution. History shows in its entirety a “pathetic sight of mankind”. Therefore he rejects the possibility of gnosis.

    Leibniz wrote 1710 in response to Bayle his “Essais de Théodicée sur la bonté de Dieu…”. He campaigns for a better understanding of god. He puts up with a minimum of evil, by solving the problem of optimization (mathematically spoken). Thus a maximum of goods can be obtained, by which an optimized creation comes into being and persists. Remark: I believe that solving the optimization problem will lead to many local maxima and minima.

    For decades the dispute between the optimistic view of Leibniz and the sceptic view of Bayle went on. The term “optimism” by the way, goes back to the Jesuits. They applied this term as a sarcastic characterization of Leibniz’s theses. (3)

    Alexander Pope published in 1734 his “Essay on Man”. There he wrote: Man’s destination is not to investigate, or scan as he said, God. Instead man should understand himself, his mediocrity and his limited capabilities. He talks about a sceptic side and a stoic’s pride and meant: man should be able to suppress his emotions and become a pure intellectual being. However, where does this end ? In chaos of thought, in endless error hurled.

    Right in the middle of the dispute between philosophers Lisbon’s earthquake took 1755 place.
    Voltaire’s first reaction he made public was the “poeme sur le désastre de Lisbonne…” In his poem he rejects the optimism and makes fun of the principle “tout est bien”. He puts the philosophers into their place: …will ye reply:”you do but illustrate the iron laws that chain the will of God?”..What crime, what sin, had those young hearts conceived.. With this poem Voltaire wanted Europe to make this disaster her prominent cause.

    Emmanuel Kant wrote 3 scientific texts about philosophy of nature and natural science. He was fascinated by earthquakes. By explaining why earthquakes happen Kant applied the knowledge of chemistry and physics at that time. In the case of the Lisbon earthquake he said: “we have the cause below our feet”. He mentioned a sudden pressure release, which shakes the water and makes “the water propagating like a rigid body”. “the soil below us is hollow”. He mentioned “subterranean caverns” and “explosive gases propagate along these caverns”. Kant goes on writing: ”the shaking of countries, the raging of the ocean, the fire spewing mountains are not less implanted in nature by God as a just consequence of constant laws. … the contemplation of such dreadful events is edifying.”

    The optimistic thinking to understand earthquakes by searching the causes took 100 years, until the mechanism was found, which could explain these many natural effects. Today we see the cause of earthquakes in a mechanism called plate tectonics.

    One day we will also be able to predict earthquakes, that is, we will be able to predict the location and the time, when the earthquake will take place.

    Neither Bayle’s sceptic view and Pope’s fear, that we will end in the chaos of thoughts, nor Schopenhauer’s pessimistic view, that we live in the worst of all possible worlds does help in dealing with disasters.

    (1) Pierre Bayle: Dictionaire Historique et Critique
    (2) Odo Marquard: Die Krise des Optimismus und die Geburt der Geschichstphilosophie
    (3) G.Langer, T.Unger: Das Erdbeben von Lissabon und der Katastrophendiskurs im 18. Jahrhundert, 2005 Annual Conference oft he DGEJ

  8. Pingback: What matters to me, and why? | Quantum Frontiers

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