Infinity, Time, and Successive Addition

ABSTRACT

According to an influential line of argument, the past must be finite because no infinite series can be formed by successive addition. The present paper pinpoints the non sequitur at the heart of this argument, disentangles the ambiguities that disguise it, and dismantles the misleading picture of ‘traversing the infinite’ that gives the argument so much of its allure. Finally, the paper critically explores the related argument that a beginningless series of past events is impossible because there could be no explanation of its having been ‘completed’ at one time rather than another.

KEYWORDS:

• successive addition
• traversing the infinite
• actual infinite
• potential infinite
• relationism
• substantivalism

Disclosure Statement

No potential conflict of interest was reported by the author.

Notes

1 One thinks immediately of the thesis argument of Kant’s First Antinomy. But, in one form or another, the argument goes back at least to John Philoponus (c. 490–570 CE). In our own time, it has been championed by William Lane Craig and many others.

2 But see Dretske [1965], Oppy [2006: 61ff], and Malpass and Morriston [2020].

3 Craig sometimes says that the number of future events is ‘zero’, since none of them yet exist [Craig and Sinclair 2012: 116]. This won’t do. Even given presentism, events that will occur are in principle as numerable as those that have occurred. So, even if the events in E don’t (yet) exist, the fact remains that they will; and, since each will be followed by another, their number is ℵ₀.

4 Puryear’s own worry about the SA-argument is different. He argues that if time is continuous then a beginningless past need not be composed of an actually infinite sequence of successive steps.

5 Andrew Loke takes up the challenge in a way that doesn’t fall neatly into either of these categories [2017: 67–75]. His argument is difficult to interpret, but the heart of it seems to go like this: (1) any series of events that occur ‘one-after-another’ is ‘constituted’ by repeated additions of a finite number of events (‘one’) to a finite number of them (‘another’) [ibid.: 72]; but (2) the result of adding a finite quantity to a finite quantity is always finite; and so (3) any such series must be finite. However, premise (1) is ambiguous. Does it say only that each successor event is added to its (finite) predecessor? Or does it also imply that each event is added to a finite running total? On the first reading, (1) is true but trivial. ‘1 + 1’ is always equal to ‘2’, but this does nothing to rule out the possibility that the series as a whole is infinite. On the second reading, the problem is that nothing has been done to show that the running total to which fresh events are added must be finite. To see this, imagine that every year BCE was preceded by another. I am not asserting that this is so, but I take it to be obvious that if it were, then each year BCE would have been ‘added to’ a running total of years that was already infinite. To assume without argument that any series of events that occur ‘one-after-another’ must be formed by repeatedly adding a finite quantity to a finite total is either to misunderstand what’s involved in a beginningless series or else to beg the question against the possibility of one.

6 I wish to thank Landon Hedrick, Alex Malpass, and an anonymous referee for their sharp eyes and wise counsel.