# Posts Tagged actual infinite

### Contra Anselm’s Ontological Argument

Posted by rambleandrant in Philosophy on April 21, 2011

Refer to this if you wish to get acquainted with the basics of the argument.

A formulation of the argument by William Matson goes as follows:

1. Whatever is understood exists in the understanding

2. When the fool hears of the GCB [greatest conceivable being], he understand what he hears

3. Therefore something exists in the understanding, at least, than which nothing greater can be conceived.

4. Suppose the GCB exists in the understanding alone: then it can be conceived to exist in reality;

5. which is greater.

6. Therefore, if that than which nothing greater can be conceived exists in the understanding alone, the very Being than which nothing greater can be conceived is one than which a greater can be conceived.

7. Obviously this is impossible.

8. Assuredly that than which nothing greater can be conceived cannot exist in the understanding alone.

9. There is no doubt that there exists a Being than which nothing greater can be conceived, and it exists both in the understanding and in reality. (This is God.)

Essentially, the argument works in 3 general steps:

1. Anyone can conceive of a GCB

2. This GCB would have the property of existing beyond mere thought if it is to be a GCB

3. Therefore, this GCB exists beyond conception.

I’m going to specifically address the premises (4 and 5 are dependent premises and are the target at hand) that state that a thing/being that exists is a greater than one that exists only in conception.

Let us suppose that I conceive of a circle. If my conception is to reflect that of the abstract definition, I would conceive the circle as having an infinite number of points to be able to reflect its infinite number of sides, which would allow it to be a perfect representation of the abstract definition. While the circle is the greatest circle I can conceive of, it would be greater yet if it were to exist outside of conception. Am I warranted to jump to the conclusion that such a circle must exist outside of conception? I think not. An actual infinite could possibly lead to absurdities (such as Hilbert’s Hotel). Moreover, I would argue that in this case, the conception of a circle is greater than any possible representation of a circle in our world. What’s greater about a circle drawn on paper, for instance, if it fails to perfectly represent a circle?

The possibilities are: (1) the circle would be comprised of a finite amount of matter, in which case it would not be a perfect representation, (2) the circle would be comprised of an infinite amount of matter, in which case absurdities and paradoxes may come about due to the existence of an actual infinite, or (3), no such circle could exist outside of our conception. The circle cannot be perfect if it doesn’t have an infinite number of points to allow it to have an infinite number of sides. It would be perfect if it had access to an infinite amount of matter, but an actual infinite, let alone an infinite amount of matter, makes no sense given our current understanding of the world and the theoretical problems of actual infinites. I conclude that Anselm’s argument fails on the account of premise 4 and 5.

If you’re interested in reading further, here are some standard objections to Anselm’s argument.