Anselm's Ontological Proof Does Not Add Up

Mike Arnautov, August 2007-December 2008

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. 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, Gödel apparently 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 ontological proof (further just the proof) 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. There is, however, a problem with 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 proof is mathematical in nature, because measure functions and measure theory in general are very much a mathematical subject.

Proponents of the 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 whole argument to a circular proof by assertion: existence is a crucial attribute of our notion of God and therefore God exists. The more interesting possibility is that the very removal of any attribute (e.g. the one of existence) necessarily reduces the perfection measure of the set.

It does indeed seem plausible that removing an element of a set would reduce the overall measure of the set for any reasonably defined measure function. But we have learned since Anselm's times that this is not necessarily the case. In order for the proof to work, it has to specify the measure function, and the set on which it is being defined. Further, it has to demonstrate that this is in some way a reasonable (or better still an obvious) definition. All of this must be done without an appeal (explicit or implicit) to the statement intended to be proved. Unless these conditions are met, the proof cannot be considered valid – with no prejudice to the existence or non-existence of God.

In the absence of any other indication (and assuming that Anselm's argument is subtler than a proof by assertion), it would seem that the proof as offered may be using as its measure function of a set of attributes, the simple cardinality of that set – i.e. the number of elements in the set. No other measure being stated or even implied, if this measure definition cannot be shown to work, the proof cannot be considered valid.

At first glance, this may appear to do the trick. Intuitively, removing an element from a set, necessarily reduces its cardinality. However, after Cantor's revolutionary exploration of set cardinalities, a mathematician is bound to ask whether the original set is finite or infinite, because removing an element from an infinite set does not actually reduce the set's cardinality. Hence without a proof that the set of all "positive" attributes is in fact finite, the ontological argument fails in proving its intended conclusion.

Are there infinitely or finitely many such attributes? I am not aware of any version of the proof that even considers this basic question, let alone attempts to answer it.

But let us suppose that it can be somehow established that the set in question is finite. Would that mean that Anselm is home and dry? I do not think so.

Let me be quite clear: I am not suggesting that the issue of infinity is necessarily a stumbling block for the proof. I am merely using it as an example which shows that non-trivial assertions about set measures require that a clear definition is given of sets being measured, and that the appropriateness of the measure function to be used is justified within the context of that use. Even if we accept that all potential sets involved (directly or indirectly) in the proof are finite, it does not follow that set cardinality is the appropriate measure for ranking these sets in their "order of perfection".

As already noted, Anselm's argument restricts its consideration to "positive" attributes only, but is it really as simple as that? There are reasons to doubt this. As random examples, redness or sphericity would seem to be a positive attributes. Is the measure of God's attribute set smaller than the measure of all "positive" attributes if it does not include redness or sphericity? A proponent of the proof may retort that clearly enough, only attributes applicable to a person are in fact being ascribed to God. Leaving aside the small matter of the attributes of omnipotence and omniscience not being ascribed to any person other than God, does the set of Godly attributes include also the ones of, e.g., humility and modesty? On the basis of the existing written record, this also appears doubtful.

It would seem, therefore, that some "positive" attributes are considered to have the measure of zero in the context of the argument, or at least a measure smaller than that of some other attributes (in particular, smaller than that of the attribute of existence). This is in fact sometimes explicitly acknowledged in some statements of the proof, which state that "existence" is such a crucial attribute of perfection, that it must be ascribed to God. This necessarily means that some other attributes may be considered "less crucial".

So on closer inspection, it would appear that we cannot use the simple cardinality of attribute sets cardinality for the purposes of the proof. In the absence of any other explicit definition, I cannot see how the proof can possibly work except as a piece of blatantly circular reasoning: God must exist therefore God exists.

But let's once again assume that the ontological proof can be somehow rescued from this predicament as well. Let us suppose that some suitable measure function can be reasonably postulated for the purpose. Is it obvious that "existence" must be ascribed a non-zero measure by such a function? In other words, is it reasonable to assume that existence is a necessary attribute of perfection? I suggest that this is far from obvious.

Consider the notion of a perfect sphere. Basic physics dictates that any actual sphere will be seen to depart from perfect sphericity under a sufficient magnification, even before fuzzing out completely due to quantum effects. Is the perfection of the notion of a perfect sphere in any way diminished by the fact that it necessarily lacks the attribute of actual existence? Certainly not.1 Hence unless the existence or non-existence of God is deemed to be qualitatively different in some way (in which case all bets are off, and the proofs argument is reduced to a kind of a Humpty-Dumpty hand waving), this strongly suggests that in the case of God the attribute of existence could arguably be assigned zero measure.2 So yet again, we see that the verdict should be at best "not proven".

To summarise, Anselm's proof implicitly relies on postulating a "perfection" measure function defined on subsets of a set of attributes. In doing so, it fails to give any clear definition of the function in question or of the domain on which it is defined. Consequently, it fails to substantiate its central claim that attribute sets lacking the attribute of existence are necessarily "less perfect". In short, the ontological proof fails in proving God's existence.

- o O o -

If you feel like leaving a comment, please feel free to do so!

1 The notion of a perfect sphere undeniably exists, but this must be distinguished from the existence or otherwise of an actual instance of the notion, which in turn must be distinguished from the possibility of such an instance actually existing, the latter being a property of our universe rather than that of a sphere. Of course, we are free to include the attribute of existence in the notion of a perfect sphere, but such an act of mental gymnastics does not make the actual existence of an instance of a perfect sphere any more feasible

2 Interestingly enough, I understand that Leibniz was of the opinion that the ontological proof could be made to work, provided one could first establish the possibility of the existence of God.

To the polemical contributions page
To the Mipmip home page
Feel free to leave a comment!
Mike Arnautov (23 December 2016)