THE LEIBNIZ COMET

Having in the previous entry agonized over the fact that our dear Leibniz was not a nice man by any stretch of imagination, I still have this entry left to give him his due as one of the greatest geniuses of all time. In the words of the English essayist Thomas De Quincey (1785-1859), which have inspired the title of this entry: “There are such people as Leibnizes on this earth, and their office seems not to be that of planets to revolve within the limits of one system, but that of comets: to connect different systems together.” Having studied a long time ago Frege’s mathematical logic, this becomes a personal thing for me to try to reach into the depths of Leibniz’s genius (as deep as I can), to find out how this great intellect was capable of overturning the millennia of Aristotelian unassailable logic (not that the latter was to be proven faulty in any measurable way, but by the same token as Lobachevsky, and, perhaps, even Gauss before him, would be much later capable of reaching beyond the Euclidian flat-earth truth) and produce the science of mathematical logic, which, most ironically, was never published in his lifetime, but was kept virtually under lock and key until the twentieth century when, to everybody’s great astonishment, it was accidentally discovered. But this thrilling incursion into Leibniz’s logic will take place much later, if at all, under the customary condition, by now, that I shall have the luxury of time in the future to engage myself in this adventure.

Leibniz’s philosophy (of the published kind) is most famous for the eerie fantastic concept of “windowless monads.” Here again we are turning to our Bertrand Russell to provide us with the stock summary (I have compressed and rearranged this summary, to the point that it no longer looks like his quotation):

Like Dèscartes and Spinoza, Leibniz based his philosophy on the notion of substance, but he differed from them radically as regards the relation of mind and matter and the number of substances. Dèscartes allowed three substances: God, mind, and matter; Spinoza admitted God alone. In Dèscartes’ view extension is the essence of matter; for Spinoza both extension and matter are attributes of God. Leibniz held that extension cannot be an attribute of a substance. He believed in an infinite number of substances, called monads. Each can be roughly likened to a physical point, but in fact each monad is a soul. This follows naturally from the rejection of extension, leaving thought as the only remaining possible essential attribute. Thus, Leibniz is led to deny the reality of matter, and to substitute an infinite family of souls… The Cartesian doctrine that substances cannot interact was retained by Leibniz. No two monads, he declared, can ever have any causal relation to each other. “Monads are windowless,” in his words. The semblance of interaction is produced by a “pre-established harmony” between the changes in one monad and another. This parallels the concept of two clocks which strike time simultaneously not because of their interaction, but because of their accuracy in keeping the correct time.

Monads form a hierarchy and thus in the human body, which consists of monads entirely, there is one that is dominant, which is what is called the soul of the person to whom the body belongs.

Of the infinite number of monads, no two are exactly alike. (This is one of his famous principles called the “identity of indiscernibles,” also known as Leibniz’s Law.)

Talking about Leibniz’s fundamental philosophical Principles, here they are:

Identity. If a proposition is true, then its negation is false, and vice versa.

Identity of indiscernibles. Two things are identical if and only if they share the same properties.

Sufficient reason. There must be a sufficient reason often known only to God for anything to exist, for any event to occur, for any truth to obtain.

Pre-established harmony. The appropriate nature of each substance brings it about that what happens to one, corresponds to what happens to the others, without, however, their acting upon one another directly. (Discourse on Metaphysics, XIV) A dropped glass shatters because it knows it has hit the ground, and not because the impact with the ground compels the glass to split.

Continuity. Natura non saltum facit. A mathematical analog to this principle would proceed as follows. If a function describes a transformation of something to which continuity applies, then its domain and range are both dense sets.

Optimism. God assuredly always chooses the best.

Plenitude. He believed that the best of all possible worlds actualizes every genuine possibility, and argued, in Théodicée, that this best of all possible worlds would contain all possibilities, with our finite experience of eternity giving no reason to dispute nature’s perfection.

A much more questionable endeavor on Leibniz’s part was to finalize the corpus of metaphysical proofs of God’s existence. (The reader knows of my objection, as a matter of principle, to any effort to prove matters of faith, under the slogan: if the proof were available, who would need the faith?) Still, Leibniz made this effort obviously in vain, but for us it has the indisputable value of giving us these grounds for arguing with him. His arguments are (following Russell’s order):

(1) the ontological argument; (2) the cosmological argument; (3) the argument from the eternal truths; (4) the argument from the pre-established harmony, which may be generalized into the argument from design, or the physico-theological argument, as Kant calls it.

The ontological argument is terribly arcane, and at the end of a successful demonstration the proof is most probably achieved through the listeners complete bafflement and resignation to the prover, rather than by a genuinely successful proof of anything whatsoever. The argument strives to prove the existence of God by virtue of the existence of the idea of God. Its silliness becomes instantly exposed when we consider any of the universally known purely fictitious characters, like, say, Zeus of the Greeks, or Goethe’s Mephistopheles. Many of the heroes of literary fiction and folklore have very elaborate ideas spun around their existence but that kind of existence by virtue of an idea by no means proves their existence in reality. Leibniz sees the basic flaw of the ontological argument, but instead of rejecting it altogether, he decides to supplement it by formally suggesting that God is the ideal sum of all perfections, and existence being among those necessary perfections, it means that God exists. The way Leibniz goes about it, reminds me of Zeno the Trickster and his signature “proofs” of the realistically impossible, yet mathematically comprehensible, and of course the value of Leibniz’s ontology of God is about the same as Zeno’s argument (actually, quite valuable for science!) that Achilles will never catch up with the tortoise.

The cosmological argument is once again based on a shaky foundation. Everything has a cause for being, but for some reason it is believed that the backtrack of the causal chain must be finite, and thus there has to be the First Cause, itself uncaused, of everything, and that First Cause must be God. This argument reminds me of the backtracking puzzle of the chicken and the egg. In this latter case we are resigned to the fact that in this case the first cause (either the chicken or the egg) does not exist, but still we allow the First Cause of everything being possible, and call it God. Leibniz obviously tweaks this argument, to make it more plausible, and he succeeds in formulating a challenging logical puzzle, which however does not result in solving the problem of the objective existence of God.

The other two arguments follow the same pattern. Leibniz sees the deficiencies of the arguers before him, and he considerably improves on each argument in so far as its logical sophistication is concerned, but he obviously fails to perfect what is impossible to prove to make it possible.

All of these arguments are of interest to me in terms of their formal logic, but not in terms of their pragmatic value in proving what they set out to prove. In the future I may probably write a lengthy analysis of this whole effort, say, in the Acorn section. But for now, I will only refer the curious reader to Russell’s chapter on Leibniz, where he gives his own take on Leibniz’s vanity project. In the meantime, we are now about to move on from here.

And next we are coming to the most interesting part of Leibniz’s philosophy: the unpublished lot, and to do it justice, we might just as well start a new entry, which is the one that now follows, to be posted tomorrow.