This post is co-authored by Hud Hudson, Ned Markosian, Ryan Wasserman, and Dennis Whitcomb. It is based on an unpublished paper by the four of us that is available online here.
In the 2nd edition of his book, Metaphysics (Boulder, CO: Westview Press, 2002), Peter van Inwagen offers a new argument against the passage of time. In the 3rd edition of the book (Westview Press, 2009) the same argument appears, and it also appears in a recent Analysis paper by Eric Olson (“The Rate of Time’s Passage,” Analysis 61: pp. 3-9). Here’s a quote from van Inwagen.
Does the apparent “movement” of time… raise a problem? Yes, indeed… the problem is raised by a simple question. If time is moving (or if the present is moving, or if we are moving in time) how fast is whatever it is that is moving moving? No answer to this question is possible. “Sixty seconds per minute” is not an answer to this question, for sixty seconds is one minute, and – if x is not 0 – x/x is always equal to 1 (and ‘per’ is simply a special way of writing a division sign). And ‘1’ is not, and cannot ever be, an answer to a question of the form, ‘How fast is such-and-such moving?’ – no matter what “such-and-such” may be… ‘One’, ‘one’ “all by itself,” ‘one’ period, ‘one’ full stop, can be an answer only to a question that asks for a number; typically these will be questions that start ‘How many…’… ‘one’ can never be an answer, not even a wrong one, to any other sort of question – including those questions that ask ‘how fast?’ or ‘at what rate?’. Therefore, if time is moving, it is not moving at any rate or speed. And isn’t it essential to the idea of motion that anything moving be moving at some speed…? (2002: 59)
Here’s the gist of van Inwagen’s argument. If time passes, then it has to pass at some rate. And even if that rate is expressible in a number of different ways (e.g., 60 minutes per hour, 24 hours per day, etc.), it must also be true (if time passes at all) that time passes at a rate of one minute per minute. But one minute per minute is equivalent to one minute divided by one minute. And when you divide one minute by one minute, you get one (since, van Inwagen says, “if x is not 0 – x/x is always equal to 1”). But ‘one’ (not ‘one’ of anything, but just plain old ‘one’) is the wrong kind of answer to any question of the form “How fast…?” So it must be that time does not pass after all. QED.
We can put the reductio part of van Inwagen’s argument a bit more carefully as follows.
(1) The rate of time’s passage = 1 minute per minute.
(2) 1 minute per minute = 1 minute ÷ 1 minute.
(3) 1 minute ÷ 1 minute = 1.
(4) The rate of time’s passage = 1.
We have several problems with this argument, but will discuss only two of them here. (We discuss some other problems, and the two problems raised here in more detail, in the paper linked to above.)
First problem: It’s not true that for any x distinct from 0, x ÷ x = 1. Take for example the Eiffel Towel. If you divide the Eiffel Tower by itself, you don’t get 1. You don’t get anything, because division is not defined for national landmarks. Division is an operation on numbers, and a minute – like a meter or a tower or a car – is not a number. So 1 minute ÷ 1 minute is undefined, and thus (3) is false.
(One can, of course, say things like: 10kg divided by 5 kg is 2 kg. But we take this to be loose talk – it is the numbers, not the quantities, that are being divided. Similarly, one can show that a rate of one kilometer per minute is equal to sixty kilometers per hour by multiplying fractions and canceling out units: 1k/1m x 60m/1hour = 60k/1hour. Once again, we take this to be a loose way of speaking – it is the fractions, not the rates, that are being multiplied.)
Second problem: (2) is also false. Van Inwagen supports it by saying that “…‘per’ is simply a special way of writing the division sign.” (2002: 59) We disagree. The forward-slash (‘/’) can be used to abbreviate both ‘per’ (i.e., ‘for every’) and ‘divided by’, but it is a mistake to treat ‘per’ as synonymous with ‘divided by’. To see this, consider the claim that time passes at a rate of one minute per minute. This may be uninformative, but that doesn’t make it untrue. A minute does pass every time a minute passes, just as a car passes every time a car passes. So ‘1 minute per minute’ expresses a genuine rate. But now consider the claim that time passes at a rate of 1 minute ÷ 1 minute. This is worse than uninformative – it is nonsensical. That is because 1 minute ÷ 1 minute is a division problem (without a defined answer) and a division problem is not a rate of change. One might as well say that time passes at a rate of orange x banana. So ‘1 minute ÷ 1 minute’, unlike ‘1 minute per minute’, does not express a rate.
We conclude that van Inwagen’s anti-passage argument fails, for (2) and (3) are both false.