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The Non-Cognitive Nature Of Infinity

by Francois Tremblay



Infinity has a somewhat paradoxical role in theology. On the one hand, infinity is a qualifier of a god’s attributes: infinite power, infinite knowledge, infinite benevolence, infinite presence, and so on. On the other hand, the Kalam argument rejects the possibility of an actual infinite regress in order to support the assertion that the universe had a beginning.

William Craig argues for the impossibility of infinite regress because of the contradiction involved in traversing an infinite number of moments.

One cannot form an actually infinite collection of things by successively adding one member after another. Since one can always add one more before arriving at infinity, it is impossible to reach actual infinity. Sometimes this is called the impossibility of “counting to infinity” or “traversing the infinite.” It is important to understand that this impossibility has nothing to do with the amount of time available: it belongs to the nature of infinity that it cannot be so formed. Source

The truth of this is immediately obvious. If there is infinite regress, then there must be an infinitely distant beginning, from which no actual point in time can be reached. Thus infinite regress of time implies that there can be no present time. This is obviously contradictory.

Objections have been raised against this reasoning, although they all seem to rely on misdirection. For instance, some argue that time is itself infinitely divisible. While this is not incongruous, it is besides the point since the problem with infinite regress is the amount of time passed, not the nature of the passage of that time. We experience finite movements in time, that we routinely add together.

In a debate with Craig, Michael Tooley has tried to use another analogy to prove that infinities can be actual. He argued in his first rebuttal that unbounded space admits an infinite number of regions of equal length. But the case of unbounded space is an apparent infinity, not an actual one: we know very well that there is a qualitative limit to such space even if we can move around freely within it, just like we know that the surface of a sphere is limited even though it is unbounded.

In short, the atheological arguments offered to disprove the impossibility of infinite regress are very unconvincing. But I must continue to stymie atheologians for a bit more, since I also contend that there is no necessary relation between the impossibility of infinite regress, and infinite sets. While it is true that Craig uses the argument that an actual infinity is impossible, it is not necessary to uphold this argument as valid in order for infinite regress to be invalid. the argument from traversing the infinity is sufficient.

In this perspective, we must view claims like Dan Barker’s with skepticism:

lthough Kalam deals with temporal succession, the same logic applies to non-temporal antecedent events, if such things are a part of reality. If the series were infinite, then God never could have traversed the totality of his own antecedent mental causes to arrive at his decision to say “Let there be light.” Therefore, sticking with Kalam, there must have been a “first antecedent” in the mind of an actual God, which means that God “began” to exist. Source

It is unclear why God would traverse an infinite number of mental causes before making a decision. While we must acknowledge that a god’s mental capacities and knowledge are infinite, it is an infinite set. Having an infinite set, unlike an infinite sequence such as a sequence of time, does not demand traversing every element. When we make decisions, we likewise do not have to traverse all our knowledge.

This is where I must now part ways with William Craig. While his argument against infinite regress is reasonable, we also have to contend with his belief in an infinite god. And we have to ask, what does it mean for a god’s knowledge, power, benevolence, and presence to be qualified by “infinite”?

The word “infinity” is defined negatively. An infinite set is a set having a cardinality greater than any finite number, to which no finite number can be added. An infinite set can also have a one-to-one correspondence with one of its subsets. The second premise is a corollary of the first. In short, infinity is what which is not finite and does not partake of the properties of finite numbers.

This is satisfying in mathematics: indeed, we use infinity for many useful tasks. But to posit an actual infinity is to place ourselves outside of the realm of unitary existence, indeed to deny it. Just as the hypothesis of infinite regress demands us to reject moments of time, the existence of an actual infinite cardinality demands us to reject unitary existence altogether.

But if this is the case, then the concept’s specificity is automatically nil. There is nothing that it could possibly mean for us for something to be infinite, or to have infinite cardinality.

This argument is limited in terms of application, since most theologians try to rationalize incoherency arguments by limiting God’s power. We can admit that a god’s attributes need not be infinite in the mathematical sense. It is probable that only unsophisticated accounts of theism fall prey to this problem. Nevertheless, it remains important to restrict theistic arguments to finiteness, if they are to have any sense at all.

Last updated: August 2, 2004