Some time ago I wrote a response to an article by evangelical theologian/apolologist William Lane Craig. Craig's article argued that Wolfhart Pannenberg, Phillip Clayton, and F. LeRon Shults hold a kind of pantheism despite their efforts not to. Craig's argument rested on his interpretation of the theologians as saying something like, "Given X, we must conclude Y", when in fact the theologians are saying something more like, "If we think in terms of X, we must then conclude Y. However, we need not think in terms of X." More specifically, the theologians were arguing that (by my interpretation), "If infinity relates to finite reality the same way that finite things relate to other finite things, then we must conclude pantheism. However, Infinity does not relate to finite objects they way that finite objects relate to eachother."
Two things struck me as curious about the article. (1) Craig does not address the problem the theologians themselves are addressing in making their arguments- perhaps they are off his radar and this is why he misunderstands them; (2) Craig does not seem conversant with the subject.
Below is an excerpt from the article I wrote, pointing out Craig's confusion.
Craig's Confusion of the Mathematical Infinite and the Metaphysical Infinite
There are two approaches to describing infinity: the mathematical and the metaphysical. The mathematical infinite is quantitative and has to do with extension. The metaphysical is qualitative and has to do with such concepts as ultimacy, perfection, completeness, ground of being, etc. Historically, these two accounts of infinity have interacted with and informed each other in different ways. One way to speak of the infinite is to say that it is that which is always beyond. In this sense, insofar as mathematical analysis always presupposes a point of reference (from a finite perspective), a mathematical account of infinity will always be confined and limited; there always will yet be a beyond. In other words, a mathematical or quantitative description of infinity is a domesticated infinity. Many theologians are concerned about construing God's infinity in a quantitative way. This brings us to the next section of Craig's paper.
Craig's paper goes on to argue that the vague metaphysical definition of infinity put forth by these theologians have been supplanted by a clear and concise mathematical definition of infinity derived from Georg Cantor. Craig seems to think the two definitions represent opposing approaches to understanding Infinity. He then compares the metaphysical descriptions he takes from our theologians to the mathematical account of Cantor in a way that only confuses the issues. Those who are familiar with the topic understand that the mathematical approaches to infinity are a part of the broader philosophical discussion regarding metaphysical infinity. Much of what Craig says can be accounted for by his confusion..
According to Craig, Cantor's definition of infinity is a feature of a set of objects whereby each object in that set can be “paired off” with objects in one of it's proper subsets. Craig then says that,
Cantor’s definitions completely subvert the Hegelian argument. For it is not true, as Pannenberg’s version of Hegel’s argument affirms, that
2. The infinite is defined as that which is not finite.
According to Craig, Cantor's work on set theory subverts the notion of infinity in Hegelian metaphysics and hence the theology of Pannenberg, Clayton, and Shults. However, it is not that simple. I would argue that a proper discussion on the interaction between the work of Cantor and Hegel (and our theologians) is quite involved and requires a degree of elucidation.
Craig says:
Two questions then present themselves. First, why think that the metaphysical infinite is privileged over the mathematical infinite as the concept of the “true infinite”? Why not think that the true infinite is the mathematical concept, and the qualitative idea just an analogical notion? Indeed, given the rigor and fecundity of Cantor’s analysis in contrast to the imprecise, subjective, and poorly understood metaphysical concept, do we not have good grounds for elevating the mathematical concept to the status of the true infinite? At least there is no reason to make it play second fiddle to its metaphysical cousin.
Part of the problem with Craig's question is that it does not recognize the present philosophical discussions concerning infinity. Current philosophical reflection holds the mathematical infinite is an incomplete, inappropriate, and finite infinity as it is rooted in conceptions of extension and quantification; the mathematical infinite is something of a poor parody of the metaphysical infinite. It should be pointed out that Cantor himself would never have regarded the mathematical infinite as superior to, or even alternative to, the metaphysical infinite. Indeed, he understood his mathematical work as reflective of the greater metaphysical infinity that is beyond mathematical description. Cantor wrote:
The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent other-worldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third, when the mind grasps it in abstracto as a mathematical magnitude, number, or order type. I wish to make a sharp contrast between the Absolute and what I call the Transfinite, that is, the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite.3
Cantor makes a sharp contrast between the metaphysical infinite (which he describes as God) and the (actual) mathematical infinite in his work. He understands his mathematical infinite as not really infinite, but itself finite. Philosophers recognize the difference and mathematicians understand that theoretical study of transfinite mathematics touches on the metaphysical.
To better understand this, it may be helpful to review the more interesting and more novel aspects of Cantor's discussion of infinite sets that Craig does not mention. Remember, a set is infinite if its elements can be “paired off” with the elements of one of it's proper subsets. For instance, we can pair off each element of the set of natural numbers {1, 2, 3, 4, …, n, ....} with each element from the set of all squares {1, 4, 9, 16, …, n2, …}. The set of all squares is a proper subset of the set of natural numbers; that is, all of the elements in the set of squares are also in the set of natural numbers. Yet each element in one set can be paired off with an element in the other set (with none leftover). This is a feature not enjoyed by finite sets. For instance, we cannot pair off all of the elements in {1, 2, 3, 4, 5} with the elements in one of it's proper subsets. If we try to pair off each element in this set with the subset {1, 2, 3, 4}, there will be an element left over.
Cantor showed that one can “pair off” each of the elements of the natural numbers with all of the integers as well as with all the rational numbers. However, one cannot evenly pair off all of the natural numbers with all of the real numbers. He demonstrated that no matter how one paired off the elements of the set of natural numbers with the set of real numbers, there will always be found a real number left over. In other words, given the definition of infinity outlined by Cantor (i.e., the feature of a set whereby each element can be paired off with each element in one of it's proper subset), some infinities are “larger” than others. Because the set of real numbers have “more” elements in it than the set of natural numbers, we say that the set of real numbers have a larger “cardinality” than the set of natural numbers. That's not all. No matter what cardinality an infinite set may have, there will always be a set with a larger cardinality. The power set of a set, that is, the set of all subsets of a given set, will always have a larger cardinality.4 The point is that while one would think that Cantor's set theory successfully subjects infinity to mathematical analysis, the truth is that such an attempt is futile. There are always larger infinities.
Furthermore, the attempt to capture “true infinity” in terms of set theory leads to paradoxes. Consider, for example, the paradox of the set of all sets: If there existed the set of all sets (that is, a set whose elements would be all sets, including itself) then, according to set theory, it's power set would have a higher cardinality! (Hence, it would not be the set of all sets). Paradoxes such as these are what Cantor called “inconsistent totalities”, which he believed to be (at least alluding to) the metaphysical infinite, or the 'true infinite.'
On this aspect of Cantor's research, philosopher A.W. Moore says:
This relates back to a view of Cantor's, namely that whatever is subjected to mathematical investigation, even if it is infinite in a suitably technical sense, is thereby seen to enjoy a kind of finitude: the truly infinite is that which resists mathematical investigation.5
Moore then comments on this result:
….the concerns and preoccupations that dominated the history of thought about the (mathematically) infinite until the time of Cantor had at least as much to do with inconsistent totalities as with infinite sets. In fact the history serves as a useful corrective against those inclined to say that, through the combined efforts of such people as Bolzano, Dedekind, and Cantor, we have at last been brought to see what infinitude really is: it is a property enjoyed by any set whose members can be paired off with the members of one of it's proper subsets – as though this were the last word on everything that exercised Aristotle, Kant, Hegel, and all the rest.6
Craig does not seem to be familiar with this part of the discussion. His conception of the mathematical account of infinity being in opposition to a metaphysical account of infinity deviates from mainstream philosophical discussion and is indeed problematic. For the reasons just outlined, we can respond Craig's question – Do we not have good grounds for elevating the mathematical concept to the status of the true infinite? – No, we do not.
After conceding that it may be appropriate to speak of God in terms of a metaphysically infinite, Craig later says:
But then we come to the second question occasioned by the distinction between the mathematical and metaphysical infinite, namely, why think that the Hegelian concept of the metaphysical infinite is correct? Why think that Hegel has correctly understood the notion of the metaphysically infinite?
I think that it is fair to say that the theologians in question do not completely agree with Hegel's account of the metaphysical infinite, but do try to understand Hegel's place in the discussion well enough to know what aspects of his work are worthy of interacting with and what questions of Hegel are worthy of a response. Hegel, like Cantor and Kierkegaard, has a place in the overall discussion as philosophers and theologians critically assess his unique contributions and questions. It does not do to simply say, as Craig does, that the theologians in question are under the bad influence of Hegel. Rather, we must assess how these theologians have critically appropriated Hegel, which involves a more in-depth exploration of issues. In any case, what is at issue here is not simply Hegel's influence but the broader conversation of the metaphysically infinite (in which Hegel plays one role), and I do not believe that Craig has successfully evaluated these theologians' engagements with that overall topic. For instance, Shults has argued for a retrieval of divine infinity from within the Christian tradition, where the same philosophical issues are found without relying on Hegel’s contributions.7 Discussion of Hegel is reserved for section that outlines modern philosophical movements.
Craig's confusion of the relationship between the metaphysical and mathematical infinite leads him to some problematic comparisons as he tries to show how Cantor's notion of an infinite set displaces the metaphysical definitions put forth by these theologians. Consider, for instance, the following:
Neither, on Cantor’s account, is it true, as Pannenberg suggests, that
2′. The infinite is that which includes the finite.
For (2′), on Cantor’s definitions, is clearly false, for one can have infinite collections which have no members in common. So, for example, -2 is not included in the natural number series, despite the fact that that series is infinite.
Cantor’s definitions also make it clear that Clayton’s premiss
- If something is infinite, it is absolutely unlimited.
is false. The collection of natural numbers have a lower bound 0 but is nonetheless infinite. The series of fractions between 1 and 2 has both an upper and lower bound, namely, 2/1 and 1/1, but is for all that infinite. Thus, given Cantor’s definitions, the crucial premises in the monistic arguments are false.
Craig is evaluating the statements he takes from Pannenberg and Clayton in terms of the properties of sets. However, there simply are no grounds that allow him to evaluate these statements within the criteria of set theory. To show that the statements do not correspond to sets (the way that Craig suggests) does absolutely nothing to invalidate them. The above argument suggests that Craig interprets infinity primarily in a mathematical or quantitative way. It is precisely this tendency that the theological notion of true infinity is meant to avert. I would suggest that much of Craig's analysis in this section is rooted in this confusion.
Craig's dependance on quantitative categories are revealed in his last sentence:
Rather God’s metaphysical infinity should be understood in terms of His superlative attributes which make Him a maximally great being.
To say that God has “superlative” attributes that makes God a “maximally great being” is to put God at the top of a quantitative scale. The terms “superlative” and “maximally” are concepts that involve a kind of comparative relationship that only make sense in terms of something relatively 'less than.' They are extensive concepts and depicts God as finite. Again, this is precisely the kind of relationship that the criteria of 'true infinity' is meant to avert.
In sum, Craig believes that the Hegelian definition of infinity is “subverted” by Cantor's set-theoretical definition. This obscures the understanding of current philosophical discussion on infinity. Craig supposes there is no reason the mathematical infinity cannot be elevated to the status of the 'true infinity' discussed in theology. Cantor himself and thinkers thereafter have shown that the mathematical infinity enjoys a kind of finitude that precludes this option. Craig's preference for mathematical descriptions lead him to problematic theological conceptions that his theologians critique in the first place.
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