More on the potential / actual infinite part 1.2

0. Introduction

This is just a short post, as I am currently in the middle of working on the second (and hopefully third) longer posts in this series. I just want to get a point down on paper (as it were) for reference’s sake.

In a footnote to his book on the Kalam, Craig considers what I called ‘Cantor’s intuition’ (also known as ‘Cantor’s thesis’ or the ‘domain principle’), which is the thesis that the potential infinite entails an actual infinite. Craig claims that this thesis is refuted in a paper by W D Hart. That paper is called ‘The Potential Infinite‘ (Proceedings of the Aristotelian Society, New Series, Vol. 76 (1975 – 1976), pp. 247-264). It is true that Hart takes the hierarchy of sets to be a potential infinity without being an actual infinity, and that this is a rebuttal of the thesis. He says:

“We can take this as evidence that the existence of an actual infinity is not implied by there being potentially infinitely many F’s. This is a strong rebuttal of Cantor’s thesis” (p. 263)

This is more or less how I argued in the previous post. Not every potential infinite presupposes an actual infinite.

However, our question was less general (and more specific) than whether Cantor’s thesis is true in its widest scope. We were primarily interested in time, not sets. We wanted to know if the potentially infinite future presupposes an actually infinite future. And it is worth noting that Hart does touch on this in the paper – the paper that Craig cites as support for rebutting Cantor’s thesis, which is crucial to his defence against Morriston’s attack. What Hart says about this is interesting, and I just want to explain that here.

  1. Lack of clarity of potential infinite

First, note that part of the point of Hart’s paper is to clarify the notion of the potential infinite, which he thinks is far less clear than that of the actual infinite. As he notes:

“Cantor’s achievement was to bring the actual infinite out of the philosophical shadows into the scientific light. Can we do for the potential infinite what Dedekind and Cantor did for the actual infinite? That is my topic.” (p. 248)

Clearly, for Hart, the notion of the potential infinite is not settled mathematical cannon, unlike the notion of the actual infinite. It is an open question, one which is ‘his topic’, as to how it is to be understood in a formal sense. And although it is his project to look at this question, he does not settle it in this paper. He goes on:

“I do not claim to have analysed the potential infinite adequately. Instead, I shall explore two natural approaches that have been mentioned in the literature. I reach no decisive conclusion on the merits of either, but perhaps the explorations can turn up intuitions which are at least candidates for the eventual material adequacy conditions in terms of which a genuine analysis of the potential infinite should be judged. Such, at any rate, is my hope.” (ibid)

So the notion is problematic for Hart. There is no non-controversial definition of it which can be supposed that all mathematicians agree on. This is the problematic area he is working on, and he doesn’t claim to have settled the question. I just want to make that clear. When Craig appeals to the notion of the potential infinite, he is appealing to something that is not settled within the mathematical and philosophical literature. Of course, the paper was written almost 40 years ago, but it is contemporaneous with Craig’s book, and there is still considerable discussion of this topic today (see, for example, Dahl (2017)).

The simple point is just that the distinction between the potential and actual infinite is contentious in the academic literature, and the notion of the potential infinite is seen as problematic in particular by Hart.

Anyway, let’s move on to when Hart addresses an idea similar to Craig’s, and see what he says about it.

2. The temporal model

Immediately after the passage quoted above, Hart goes on to touch on an idea very similar to Craig’s (he says he wants to mention it “if only to get it out of the way”):

“For all I know, the best theory of the potential infinite identifies it with a process in time conceived of as a series of moments isomorphic to the natural numbers.” (ibid)

This does seem to be like Craig’s view. Consider this from Taking Tense Seriously:

“…virtually all philosophers who espouse a tensed, or A-theory of time, hold that the series of successively ordered, isochronous events later than some denominated event is potentially infinite.”

The ‘series of successively ordered isochronous events’ fits this bill pretty closely. Remember that Craig distinguishes between the past and present, which are ‘real’, and the future which is not. Hart seems to encode this intuition in the following considerations:

“Such a process might (1) have one input given at a moment zero prior to any operation of the process; (2) for any output the process has actually already yielded at a moment t, the process can take that and only that output as an input at the next moment t+1, and; (3) the process never yields the same output at two different moments and never destroys its input (so that what it once yields exists ever after). For such a process, there is no moment at which it can have produced an infinity of outputs, but no matter how many outputs it has yielded at a given time, at some later time it can always yield more.” (Hart, The Potential Infinite, p. 248)

So this is like counting up from 0 starting now, and writing down each number you have counted on a bit of paper. As you do so, the process can always go on further, but at no point will you have written down an actual infinity of digits. Writing the numbers down on a bit of paper is an analogue of Craig’s idea that once something has happened – once it has gone from being future, to being present / past – it is ‘real’. So, this seems to be substantially like Craig’s idea of the potential infinite.

But, what does Hart say about such a proposal? He says the following:

“The trouble with such a sketch is that we have no settled theory of processes in which to imbed it, so we have no sharp way to establish whether it satisfies reasonable desiderata for potential infinities” (ibid, p. 249)

What he is saying is that this is too messy and vague to know how to evaluate it. It presupposes too much of which is unclear, about the nature of time and how processes work, for it to be a proposal from which we can apply any meaningful considerations. He goes on:

“For example, does it presuppose a completed actual infinity of moments? This question is central to an issue raised by a thesis of Cantor’s to be stated below” (ibid)

The “issue raised by a thesis of Cantor’s” is exactly that which we considered in the previous post, namely the thesis is that a potential infinite presupposes an actual infinite.

Hart is saying that one of the problems with a proposal such as the temporal one described here (which looks just like Craig’s) is that it is not clear whether Cantor’s thesis holds of it or not. That is to say, it is unclear whether a potential infinite conceived of in that way presupposes a corresponding actual infinite or not. Thus, Hart is decidedly unhelpful for Craig in what he says which is directly relevant to the point in dispute.

3. Conclusion

To sum up the point here, Craig wants to say that the endless future is merely potentially infinite, and not actually infinite. He addresses ‘Cantor’s thesis’, which is that a potential infinite presupposes an actual infinite. When he does so, he references Hart’s paper, saying that Hart rebuts the claim. True, Hart rebuts Cantor’s thesis by arguing that the hierarchy of sets is a counterexample, i.e. it is potentially but not actually infinite. But, crucially, when he addresses the temporal model that Craig endorses, he refuses to treat them the same way. He suggests that it is unclear whether such a temporal account of the potential infinite presupposes an actual infinite or not. And the issue is that such an account is too vague for clear formal considerations to be applied to it productively. So while the paper is an example of someone arguing that Cantor’s thesis is false, it is only the most general form of the thesis that is rejected. Whether it can be applied to the temporal case is definitely not rebutted in this paper.