Anselm's Ontological Proof Does Not Add Up

Mike Arnautov, August 2007

One aspect of Anselm's so-called ontological proof of the existence of God appears to be largely unappreciated even by acknowledged experts: the proof is essentially mathematical in its nature, and is therefore open to criticism from the mathematical standpoint.

This always seemed to me an entirely obvious and uncontroversial observation, and I was quite taken aback when a mere mention of it provoked a rather hostile reaction from an otherwise eminently reasonable American philosophy lecturer. That experience caused me to pay attention to various discussions of Anselm's proof, and it gradually dawned on me that the mathematical angle was far from being widely known or appreciated. In fact, Richard Dawkins in his "The God Delusion" quotes Bertrand Russell as saying "It is easier to feel convinced that [the ontological argument] must be fallacious than it is to find out precisely where the fallacy lies." What is more, it would seem that Gödel himself took the ontological argument seriously enough to formalise it in (modal) symbolic logic. Nonetheless, it seems to me that there is a problem, apparently missed even by such great minds.

In a nutshell, the argument assumes that God, being perfect, must possess all "positive" attributes. "Existence" being (allegedly) a positive attribute, it must be also applicable to God, because without it God would be less perfect. I see a difficulty in that innocuous word "less". It inescapably means that one can compare the relative perfection of various sets of attributes. In other words it implicitly imposes some kind of measure on the space of all possible sets of positive attributes, which allows one to rank such sets according to their perfection. Hence my assertion that the ontological proof is mathematical in nature, because measure functions are very much a mathematical subject.

Proponents of the ontological proof do not tell us anything about the perfection measure they implicitly postulate. It could be a very simple one, e.g. assigning only two values corresponding to two levels of perfection (effectively classifying attribute sets into perfect and non-perfect), or it may permit a much more elaborate ranking. Anselm's proof and its variants do not concern themselves with such details. But the devil is, as is so often the case, in the detail.

It is asserted, without a supporting argument, that removing existence from the set of all positive attributes necessarily results in a less perfect set -- i.e. that it reduces the perfection measure of the set. Why might this be the case? There are two possible reasons. One is that the attribute of existence is so crucial that its removal by definition makes the set less perfect. But that would effectively reduce the argument to a circular proof by assertion: existence is a necessary attribute of God and therefore God exists. The other possibility is that the very removal of an attribute (any attribute, e.g. the "existence" one) necessarily reduces the perfection measure of the set.

It does indeed seem intuitively plausible that removing an element of a set would necessarily reduce the overall measure of the set for any reasonably defined measure function. But we have learned since Anselm's times that without knowing anything about the measure function being used, it is impossible to prove that such is indeed the case. In fact, following Cantor's exploration of cardinalities of infinite sets, we have explicit examples of sets and measures for which this is not the case. E.g. removing an end-point from a line segment does not reduce the length of the line segment. More generally, removing a finite number of elements from an infinite set, does not reduce the size of the set.

Reasoning which looked mathematically convincing in Anselm's time, can now be seen to be completely ineffective -- doubly so, given the theologians' fondness for associating God with infinities. Without some non-tautological definition of Anselm's perfection measure, it is impossible to assert that dropping some positive attributes necessarily decreases the perfection of a set of all positive attributes -- unless, of course, this is postulated axiomatically, in which case we are back to a circular proof by assertion.

In short, there is a gaping hole in the ontological proof, hiding in the words "less perfect".