Wednesday, November 20, 2013

Is infinity a quality or quantity: A series from notes about infinity, II

As previously noted, I'm writing a series of blog posts that are adapted from notes I made as preparation to talk with philosopher and author Peter Boghossian's Atheism class at Portland State on November 19, 2013. This is the second post in this series, which I anticipate will span four posts. The visit to his class was to address infinity and God, following from the theme presented in my new book, Dot, Dot, Dot: Infinity Plus God Equals Folly.

In this post, I hope to answer questions 3, 4, and 5 from my original list of five prepared background questions for his class to ponder.

3. How can infinity be thought of as a quality and as a quantity?
4. Can the quantitative aspect of infinity be removed from the concept of infinity?
5. How is William Lane Craig's thinking muddled on the idea of "potential" infinities on a future timeline if God is supposed to be separate from the universe and eternal? Particularly, how is Craig treating God in a local frame when his God hypothesis requires a global one? (See Craig's Q&A #325 for context.)

As a note, Evangelical apologist William Lane Craig's Q&A #325, "Infinity and God," was to be background reading for the discussion, and my notes assume that and make use of it.


3. How can infinity be thought of as a quality and as a quantity?

Infinity is both a quality--being inexhaustible, being able to remove elements without changing the size--and a quantity--a sense of the “number of things” present in infinite collections like the set of (all) natural numbers, {1,2,3,...}. Because infinity answers the question “how many?” at a very intrinsic level, it is immediately and inextricably tied to the notion of being a quantity as well as a quality.

Admittedly, this question seems a little weird, but it is key if we are to untangle the thoughts of apologists like William Lane Craig who wish to think of the infinite only as a quality. Of course, it's important to note that apologists and theologians, Craig in particular, are often unabashed about letting their theology lead their philosophy, even on topics like the philosophy of science or the foundations of mathematics. In this case, Craig's theology, or really his defense of the multiply-failed Kalam Cosmological Argument for the existence of an "uncaused first cause" demands that he reject the infinite as a quantity and consider it only qualitatively, as an abstraction that captures the notion of being inexhaustible.

Potential and actual infinities (and physical ones too)

Craig talks about this typically in the context of "potential" and "actual" infinities. A potential infinity is seen when we have some kind of a process that should go on interminably such that if it were completed, it would result in an infinite collection. An "actual" infinity is the completed infinite collection itself--which may still only be abstract in nature. We see a potential infinity when we think of counting: one, then two, then three, and so on, knowing that we need never stop. The set of numbers that would be generated by "completing" that task, called the natural numbers, is an example of an "actually" infinite set.

These should be distinguished with physical infinities, like if we were to have good reason to believe that the universe itself is infinite in scope and containing an infinite quantity of stars or other materials. A physical infinity would imply an actual infinity, but this implication does not go the other way around because we can have actual infinities of abstract objects like numbers, including many that do not count anything physical at all. Potential infinities merely suggest an actual infinity may be meaningful but need not imply one, and they certainly do not imply a physical infinity.

Infinity as abstraction

Craig takes the position that actual infinities (yet another lamentable name for something) are abstract at best and essentially non-existent. He is in some company in doing so. I take the tack personally that infinite cardinalities, unless proved otherwise by finding out more about our universe, are abstractions and "exist" as mental stuff, not to be confused with Platonic idealism which gives more real ontology to abstract objects than I'm willing to confer.

A small group of mathematicians called finitists (who might be right about this) agree with Craig's stronger position that actual infinities do not exist even as abstractions. I feel that this position may go too far, as I am comfortable with letting abstractions be abstractions, even if that renders some or all of them "useful fictions" and nothing more.

For my part, I'd argue that like numbers, infinite cardinals are abstractions that we use to make models (also abstractions) with which we try to understand the world. Indeed, I even say that infinite collections themselves are abstractions as well since we don't have any reasons to believe that they enumerate anything physical. And, even if they did count a real, physical infinity of things--because of the fact uncovered by Georg Cantor in the 19th century that given any collection, we can use it to define a larger collection--we'd soon be able to get to larger infinite cardinals that, for all intents and purposes, could only be seen as abstractions.

So, it's both quantitative and qualitative

So, we can think of the infinite as a quality of being beyond any real-world reckoning, true inexhaustibility, or we can think of the infinite as being a quantity, the "number of" objects in some set with infinite cardinality. The contemporary understanding of infinity uses both ideas.

In the context of the set theory that underlies modern mathematics, the infinities act more like quantities that enumerate different kinds of infinite sets, and in context of its use in fields like calculus, it is probably more accurate to think of infinity as a quality: say, for instance, going out sufficiently far in a certain kind of process to get arbitrarily close to something nice to work with. Still, even in this last sense, infinity acts like a quantity in important ways, e.g. having enough iterations of the process to get arbitrarily close or, for "divergent" processes, a comment that the quantity obtained by repeating the process indefinitely returns an indefinitely large return.

4. Can the quantitative aspect of infinity be removed from the concept of infinity?

As hopefully is implied above, I do not think so, at least not meaningfully. To do so requires us, at the least, to reject what is called the Axiom of Infinity, which guarantees as a consequence that at least one infinite set exists. Mathematics can be done without it, but the set of axioms that serve as the backbone of most of the mathematics done in the last century include that axiom.

As noted, removing the quantitative aspect of infinity binds us to a philosophical position in mathematics known as “finitism,” which asserts that the infinite is only a quality of inexhaustibility (sometimes replacing the term “infinite” with “indefinite”). I haven’t deeply explored finitist mathematics, but finitists claim that any math that can be done with infinity (even calculus) can be done without it. Of course, this comes at a cost. From what I have seen, a great deal more mathematical machinery is needed to do things this way. In that regard, infinity may be a useful fiction in the sense that it greatly simplifies some conceptual matters in highly useful mathematics.

Committing ourselves to finitism also raises uncomfortable and unanswerable questions about large numbers, particularly like “when do larger numbers stop possessing meaning? And why?” As noted in the previous post in this series, it also renders the idea of a perfect circle nonexistent, which may well be how things are but feels very uncomfortable. It could be that these questions are themselves meaningless, but they are intuitive. Further, transfinite mathematics has proved to be a rich, interesting field that has shown usefulness in laying theoretical foundations under other practical fields like differential equations.

Rolling back the clock

For fairness, the finitists might be right. That said, to reject the quantitative notion of the infinite would be to roll back the clock about 120 years on our understanding of set theory and other mathematical objects, particularly those related to infinity. I feel like this is a common theme with Evangelical apologists like William Lane Craig--it seems they want to do it with certain results from cosmological physics as well, at the least.

Of course, more fundamentalist apologists than Craig want to do it with biology as well, seeking to undo the core theorem of that field: evolution by natural selection. Since we know that these apologists and theologians are led by their theology, including on their philosophy and even their science, it is my opinion that we are not presented with good reasons to roll back the clock on any of our thinking, to a time when God was easier to defend or otherwise. The right theology could lead us to conclude just about anything, so it is not a reliable method for determining how we should view or think about the world. 

5. Can you see how Craig's thinking is muddled on the idea of "potential" infinities on a future timeline if God is supposed to be separate from the universe and eternal? Particularly, how is Craig treating God in a local frame when his God hypothesis requires a global one?

This question will be addressed more fully as we explore the text of Craig’s Q&A response in more detail in future posts, but as I laid it out separately here, I'll take a moment to comment on it.

A general note about Craig and mathematics

On the whole, although some of his mathematics is a little sloppy, I do not really disagree with much of what Craig thinks about numbers and about infinity, though I do disagree with his uses of it and his conclusions. It's worth noting that for a lay person in mathematics, some of his use of mathematical topics is quite savvy, though the rough edges are plainly visible when one knows where to look. Particularly, his use of arguments about infinity in order to employ the infamous Kalam argument is not only weak but quite poor (committing both a category error and circularity).

Further, as noted, Craig appears to be a finitist--though largely because it seems to be the only way he can maintain a defense of some of the philosophical arguments he wishes to make for his God. This is important to note because it appears to reveal yet again that his theology leads his philosophy, even his philosophy of mathematics. This reeks of a kind of circularity that shouldn't be allowed to pass unexamined. Of course, assuming his God in the first place is a major problem all apologists appear to be guilty of.

Confusion on potential and actual infinities

Now, to address the question directly, I must briefly summarize Craig's position. Craig is quick to talk about potential infinities in the future timeline of the universe, say when asked (as in his Q&A #325). He is also quick to dismiss a potential infinity in the past timeline of the universe (again as in Q&A #325) by arguing about an actual infinity there.

This is important. For him to accept a potential infinity in the future but not in the past reveals a bias. A potential infinity in the future means that however far forward in time we look (or wait), we eventually will be able to look (or wait) a little further, this property existing without a boundary--one might say that "time keeps on ticking, ticking, ticking into the future," for instance. On the other hand, a potential infinity in the past should mean just the same: however far back we look, we should conceivably be able to look back a little further. A potential infinity in the past does not assume that we must measure time from infinitely long ago.

Craig dismisses this by saying that if the timeline in the past were infinite, then an infinite number of moments must have passed to get to the present moment. Here, though, we see him confusing a potential infinite in the past, described above, with an actual infinite in the past. In essence, what Craig is saying with regard to the past is that we either have a beginning of time finitely long ago or infinitely long ago, and then he dismisses the infinite case as nonsense (which it actually is--if we have an infinite past timeline, then there was no beginning). But this jumps to an actual infinity in the past and assumes a beginning, which is what Craig wants to establish via this argument in the first place. I raise this point in Dot, Dot, Dot.

God's independence from time

Now, to get to the meat of my study question, note that God is supposed to be both eternal and outside of the universe, including both time and space. Thus, God should be able to take a global view of the timeline of the entire universe. To the best I can tell, this idea is the God that Craig defends, and it seems to be integral to the general understanding of the deity, among a few other things. If God is outside of time and eternal, omniscience should imply that our timeline (inside of time) should be simultaneously and entirely visible to him all at once. This is a global view, available only from outside, instead of the local view we have on the inside.

The key point here is that from God's perspective, the same argument Craig gives to dismiss an infinite past timeline should also prohibit an infinite future timeline. It seems easy to sidestep this point by saying that the universe must therefore also have an end, but it isn't so. Christian theology binds him there. Even if the universe as we know it comes to an end, Christian theology proposes an eternal (infinite) hereafter. Presumably, God would have full knowledge or awareness of this infinite timeline in the afterlife, and thus God is committed to an actual infinity in the future time direction. That removes, from the global perspective, any qualified reason to reject an infinite timeline in the past time direction.

Some rebuttals rebutted

The rebuttal, of course, is to say that the hereafter is a different realm, and that the universe has actually ended, so time stopped. The problem with this claim is that every description of heaven that I am aware of ever having heard about contains the idea that there will at least be sentient minds/souls present that are capable of experiencing things, presumably in time. People will "get to spend eternity with Jesus." People will "get to live forever with their friends and relatives." Etc. Those kinds of statements only make sense in the context of a future timeline, either in our universe or in some (imaginary) meta-universe where God and heaven are, and that timeline is always described as being eternal, i.e. infinite in scope. To me, that suggests the only escape from this issue is either to admit actual infinities, which breaks Craig's avenue to the Kalam, or to revise Christian theology with a finite future timeline.

A final point to defend here is the case where we could have a finite past and infinite future. That's technically true: it is a possibility. That's not the point, though. The point is that Craig defends that the past must be finite because infinity doesn't exist. He's employing a double-standard on the past and future timelines based upon a question-begging assumption that the universe had a beginning (either finitely long or infinitely long ago)--and all this to establish an uncaused first cause, which is less even than a Deistic creator!

There's still more to come! In the next post in this series, I plan to start addressing Craig's Q&A #325, "Infinity and God". I don't think I'll cover all of it in a single post because it is fairly long and requires a diversion into another of Craig's Q&A's (#323, "The Concept of God") since it was the basis for the question posed in #325.

If you like how this reads or think this kind of subject matter is interesting, it's very likely that you'll enjoy reading Dot, Dot, Dot, so please give it some consideration.

Edit: Part III in this series is now available here.

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