Friday, April 24, 2009

Van Inwagen on the Rate of Time’s Passage

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 It is based on an unpublished paper by the four of us that is available online here.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.


  1. At most van Inwagen's argument seems to imply that the rate of passage of time is a dimensionless quantity, but there is nothing wrong with dimensionless quantities. Maybe the idea is that rates (like velocities) usually are not dimensionless quantities. But the fact that most rates are not dimensionless, does not mean that all rates must have dimensions.

  2. I might agree with some of the points about division. But I take it that the main thrust of this problem is as follows. Loosely, to ask for a rate of change is to assess variation in one dimension against variation in another. To ask for the rate of change of time itself is to attempt to asses one dimension of variation against itself. So the only answers you can get are utterly uninformative.

    You reply "this may be uninformative, but that doesn't make it untrue".

    But if the answer to the question "what is the rate of passage?" is necessarily uninformative, how can you claim to have given any content to the idea that time passes? For example: Have you given any more sense to that idea, than to the rival claim that space passes at a rate of 1 metre per metre? If not, then what do you mean when you claim that time, unlike space, passes?

  3. I had a discussion with Joe Melia a few days ago that had me thinking about the van Inwagen (/Olson) argument --- and thinking it was wrong --- but I'm not sure how it squares with your response to the argument. Joe reminded me that physicists often like to use so-called "natural units", which have the result that certain physical constants end up getting the value "1" (with no further unit attached: not 1 cm or 1 ohm or whatever, but just 1). One system is used in relativity, in which the speed of light in a vacuum comes out as 1. So, for natural units L (length) and T (time), the speed of light = L/T = 1.

    So that's bad for van Inwagen's argument; if "how fast does light travel in a vacuum" can sensibly be answered with an unadorned "1", then "how fast does time pass" should be sensibly answered the same way. But the reasoning that lets the physicists get to the point of assigning light the value (1) seems suspiciously similar to the sort of reasoning that goes on in the (1)--(4) argument. (It's a little more complex, because the natural units for length and time aren't obviously the same units, but the result still goes via the thought that certain units can "cancel" each other.) So I'm not sure how to think about physicists' uses of natural units if your criticisms of (2) and (3) go through.

  4. You say, "One can, of course, say things like: 10kg divided by 5 kg is 2 kg." However that's not true - 10 kg divided by 5 kg is 2, not 2 kg. The sorts of things that can be divided are quantities, some of which have units and others don't. To see that it's the quantities and not the numbers that are being divided, note that 10 kg is the same as 10,000 g, so 10,000 g divided by 5 kg should be the same as 10 kg divided by 5 kg. If we were dividing the numbers, we would get 2,000 in one case and 2 in the other, which would be problematic.

    Admittedly, the mathematical operations applied to quantities don't seem to be quite the same as the operations applied to numbers, but I think they're generalizations of these operations. Quantities come in various types, and operations change the type of quantity involved, so a distance times a distance is an area, and a force divided by a mass is an acceleration. Notably, a mass divided by a mass is a number, and a number times a volume is a volume.

    The operations of addition and subtraction only work when the quantities being operated on are of the same type, while multiplication and division always make sense. Each type has various characteristic units that can be used, like meters, minutes, Newtons, kilograms per second, etc.

    But I'm not exactly sure where this leaves van Inwagen's argument. A "rate" isn't a single type - some rates (like speeds) have units of meters per second, others (like flows) have units of cubic meters per second, and you could have others (like frequencies) in numbers per second, and so on. I don't see why it isn't the case that the rate of time's passage is just 1.

    Unfortunately, that's not a very informative answer, but I'm not quite sure whether that means it isn't an answer.

  5. Oh, and I forgot to mention - I'm sure all this stuff about units and quantities and the like is discussed much better in the Luce, Krantz, Suppes, and Tversky Foundations of Measurement, though I haven't read it. I got these ideas from reading Field's Science Without Numbers and some idle thought about dimensionless quantities in physics.

  6. First, I think Jason, Kenny, and I are trying to make more or less the same point. (Am I wrong guys?)

    Second, the fact that the rate of passage of time is given by a dimensionless quantity is not particularly surprising. Those who think time passes plausibly think that the rate of its passage is a fundamental physical constant and such constants are often dimensionless quantities (their value is usually an artifact of our fundamental units of measurement). This last further suggest that the value of a physical constant is rarely informative (about the world).

    Third, what Kenny is saying about dividing 1m/1m is completely right and I think Ned et al. are wrong in saying that '/' is ambiguous between 'divided by' and 'per'. If it takes me 4 hours to travel 400km, my average speed (the rate at which I travel) is 400km/4h or, dividing both numerator and denominator by 4, 100km/(1)h or 400km/h. So, even in this context, '/' has the meaning 'divided by'.

    Finally, if you are interested, the branch of mathematics that studies this kinds of problems is called dimensional analysis.

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  8. Gabrielle,

    I think we're all roughly making the same point, yes (but I wasn't tracking that at first, as I wasn't tracking "dimensions" as having to do with units). Only, I think I'm less confident than the two of you that it's on the money. For instance -- in the absence of natural units -- we might always think that multiplication or division of units is really two operations: division (which on this view is only defined between numbers) and a structurally similar unit-conversion-type operation.

    If we said this, we might think that in the "natural units" cases, the values aren't really 1; they're instead 1 something-or-other (maybe the "vel" is a fundamental unit of velocity, defined so that 1 vel = the speed of light in a vacuum; then (spatial and temporal) distance are defined from it in such a way that we can always just drop the "vel" from the unit names, leaving it as understood.

    I'm pretty sure these two ways of looking at things aren't just notational variants of each other, and that if the way I just mentioned is right, then Ned-et-al's argument hits the nail on the head. (They can say that the conversion "rule" for s/s leaves it as it is: that is, 1 s/s is different than just the value 1.) If the other way is right, then the argument is confused about units. But I don't have a good grip on which way is the (metaphysically) better way of thinking about units, so I'm not sure what to say about Net-et-al's argument.

  9. This argument against passage isn't new with van Inwagen. Price offers it in his (1996). But I suspect it's a v old chesnut which keeps getting rediscovered. As no doubt does the reply, my version of which (targetted at Olson) is forthcoming in this summer's Analysis here:

    I absolutely agree with Tim that this kind of reply doesn't answer deeper objections to the notion of passage. But I do think that it answers the objection as it's given.

    I also agree with Kenny's points about the division of quantities. But I don't see why we should think of rates as division operations. Only if we do that do we end up with a rate equal to 1.

    I'm no expert on natural units but it's not at all clear to me that they provide any reason to think that (e.g.) the speed of light is just one. No doubt the speed of light can be one _in some system of natural units_. But as the link Jason gives makes clear there is always a metrical value one can convert into. Whether the right metrical conversion is metres or coulombs isn't something that just magically appears out of thin air.

  10. Jason - I think I see the distinction you're making, but there may yet be arguments that there isn't a separate unit-conversion operation (or if there is, that it does the expected thing with s/s and cancels the units). For instance, consider something like the efficiency of an engine - it has as an input some amount of heat energy, and as output some amount of kinetic energy. If the machine takes in 10 J of energy for every 5 J of energy it puts out, then it seems right to say that it's efficiency is 50%. It doesn't seem right to ask of that 50% number what units it's expressed in. No matter what units the original question of engine efficiency was phrased in, the answer will be "50%". And it's clear that there is some natural sense in which the actual number .5 is expressed in this physical system, in a sense that 10 and 5 are only conventionally present.

    As for the case of natural units, I'm really not sure what to think. On the one hand, it seems that speeds and distances are just two different types of quantity, and they can't be added. But if natural units mean that these things are really dimensionless, then they should be the sorts of things that can be added. Therefore, natural units can't mean that things are dimensionless.

    On the other hand, from the little I understand, it sounds like general relativity says that mass and energy really are two aspects of the same thing. Observers from two different frames will disagree on the mass and energy of a given object, just as they'll disagree about its velocity. But they'll agree about its mass-energy, so therefore mass-energy seems to be one thing. But then Einstein's equation shows that e=mc^2, and if mass and energy are actually quantities with the same dimension, then the speed of light must be dimensionless.

    I don't know what to believe here. I suppose I just need to study dimensional analysis more. (I've heard from physicist friends that huge amounts of physics can be derived quite simply just from dimensional analysis, with the empirical work only required to fill in values of some constants, while the form of the equation is derived a priori.)

  11. The obvious analogue of "How fast is time?" would be "How big is space?" and this harmless game can be extended a bene placito. What is the duration of duration? What is the speed of speed?

    Well-educated readers, if any, might eventually recognize the "third man" under so many disguises... all' heteron ti tês metabolês aition. "Something else is the cause of this association," or maybe a more appropriate translation here would be "Something else is to blame for this hodge-podge."

    Special relativity provides the most facile solution for this puzzle, and there it makes sense to ask how much time passed, and how fast it passed, inside the inevitable example of a spaceship as it traveled from A to B, and then the annoying "third man" has apparently disappeared.

    But just as Alida Valli is eternally unimpressed by the Anglo-American earnestness of Joseph Cotten, and patiently awaits the return of her unruly lover Orson Welles, a "third time" intrudes to unite relatively compressed and uncompressed durations, and if we accordingly define "philosophical time" to measure progress with these mysteries since 420 BCE, no time has passed at all.

  12. Dimensionless numbers are all over physics---in fluid mechanics the ratio of inertial forces to viscous forces is the reynolds number, a dimensionless quantity which tells you a lot about the behavior of a fluid. So it certainly makes sense to divide a force by a force. You can divide my velocity by the velocity of light, to get a useful (dimensionless) measure of my velocity. It doesn't seem to me like a good response to the price/van inwagen/olson argument to say that you cannot meaningfully divide forces, or velocities.

    I suspect the problem with their argument is somewhere else. Each quantity has an associated dimension: speed has dimension (length)/(time), energy has dimension (mass)(length)^2/(time)^2, the reynolds number of the Hudson river is dimensionless. Seems like their argument turns on the claim that:

    A quantity is not a RATE (or a rate of change) unless it has dimension (OTHER DIMENSIONS)^n/(time).

    The "rate" at which time passes, though, is the ratio of two quantities which each have dimension (time); so the ratio is dimensionless. So this claim entails that this ratio is not a rate (or not a rate of change).

    But I don't see why the claim is at all appealing. I would have thought the correct claim was:

    A quantity is not a rate unless it is the quotient of two quantities, the second of which has dimension (time).

    Then the rate of time's passage can be a rate.

  13. Brad -- you invoke the principle that if a ratio is a ratio of two quantities each with the same unit/dimension, then the ratio is dimensionless. But why accept that? In the time case, why not think that the ratio is a ratio of time to time.

    I'm not suggesting that you can't divide times (forces, velocities etc.). But why assume that a ratio is simply a division operation?

  14. Tim: It sounds like you are saying that you agree with us about PVI’s argument, but think that there are two deeper problems for the passage view: (1) the problem of explaining what it means to say that time passes, and (2) an argument about the view’s inability to give a coherent and informative answer to the question ‘How fast does time pass?’ I agree that these are deeper problems for the passage view. (For what it’s worth, my 1993 paper, “How Fast Does Time Pass?” is an attempt to deal with exactly these two problems.)

    Gabriele, Kenny, Jason, and Brad: My view is that division is an operation on numbers, and not defined for any other entities. (I’m pretty sure that my co-authors have the same view.) So I agree with the position spelled out by Jason on our behalf in his post at 3:38AM on April 26th. (Perhaps it wasn’t 3:38AM where Jason was when he made the post.)

    Notice, though, that if this is wrong, and if you guys are right that 1 is a legitimate rate for the passage of time, then PVI’s argument still fails (but for a different reason).

    Ian: I am on your side when you ask, “why assume that a ratio is simply a division operation?” In fact, I would go further, and say that there are excellent reasons to deny that a ratio (like 10km per hour, for example) is simply a division problem. Here is one. If 10km/hour is a division problem, then so is 1km/hour. But 1km/hour = 1km/1hour. And if that were a division problem, then the 1’s would cancel out, which means that 1km/hour would be equal to km/hour, i.e., kilometers divided by hours. But kilometers cannot be divided by hours. (“How many hours are there in a kilometer?” sounds like something Chico Marx would say. And not in a good way.)

    Here’s another reason to deny that a ratio like 1 minute per minute is a division problem. If it is, then we have to say that the rate of time’s passage = 1, that the rate of time’s passage = 5 - 4, that the rate of time’s passage = the speed of light, and that the rate of time’s passage + 1 = an even prime number.

    Here’s a third reason. When we say that Abibe Bikila’s rate over the course of the marathon is 12 miles per hour, we are saying that his position changes by 12 miles for every one hour change in the passage of time. (And that’s all we’re saying.) But there is no temptation to think of this last claim as any kind of division operation. It is simply a comparison of one change to another.

  15. Ian---

    I may agree with both you and Ned. Here is how I was thinking. First, there are quantities, like the mass of this computer, or the volume of water in Bellingham Bay. Each quantity has a dimension, and each dimension can be written as a product of powers of the fundamental dimensions (length, time, mass). We use numbers to measure magnitudes of quantities---though the number used to measure the magnitude depends on a choice of a system of units of measurement.

    The ratio of any two quantities is itself a quantity. The dimension of this new quantity is got by dividing the dimension of the first quantity by the dimension of the second, and then canceling. The magnitude of this new quantity is also measured by a number, relative to a choice of a system of units.

    Some quantities are dimensionless. Then the number that measures their magnitude is the same in all systems of units of measurement. But the number is still being used to measure the magnitude of the quantity.

    The Reynolds number of the water in the hudson river is a dimensionless quantity. Maybe it is 33. But it remains true that the Reynolds number is the ratio of two forces. That fact doesn't disappear when you say that the quantity is dimensionless.

    So I think it's correct to say that the rate of time's passage is 1. I don't think you need to add "seconds per second," though you can; seconds is a particular unit for measuring time, and the rate of time's passage is 1 no matter what the choice of unit. I also think that, even though the rate of time's passage of 1, this quantity is still the ratio of two quantities, the second of which has dimension (time), so I still think it qualifies as a rate.

    But, like Ned (I think), I do not think that the rate of time's passage is a number. The rate of time's passage is a quantity that is measured by a number. (So "The rate of time's passage is 1" is not an identity statement; it asserts that a certain relation of measurement holds between the rate (a quantity) and the number 1.)

  16. There's an analogy between asking how fast time passes and asking how dense space is. Other things' rates can be given in units of quantity per unit time interval, and other things' densities can be given in units per unit volume, e.g. kg/m3. If someone tells you that space has a density of 1 m3 per m3 surely you'd think they were talking some kind of nonsense. Both cases are unlike those dimensionless quantities like Reynolds number which result from the cancelling out of dimensions. Reynolds number quantifies a ratio of two different kinds of forces, inertial force and viscous force, the ratio of which is physically significant although they have the same dimensions and so the dimensions disappear in the ratio. Even angle, which some authorities say is dimensionless, because it is a ratio of turns, is meaningful in a way which neither time's rate nor space's density is.

  17. I think now that the argument about the absurdity (or otherwise) of the rate of passage of time is no more problematic for the idea of time passing than for a B-theorist. Suppose a tap drips at a constant rate of 5 drips per minute. A B-theorist doesn't have to think of the drips and their rate as passing from future to past via the present. In any temporal interval there are so many events of dripping, in a minute, five. (Drips, it seems, don't have their own physical quantity dimension, but still they are different from other kinds of event). A rate is just a temporal density. So it is just as absurd to ask how much time there is per unit time as it is to ask how much space there is per unit space. Time no more passes than space is spread out. If space were spread out like honey on toast, it would make sense to wonder whether it is denser in some places than others.

  18. How about relativity? It may be uninformative to say "The flow of time is one minute per minute", but it may be informative to say about a traveller who went faster than light that he got older 1 minute of his time in, say, 1 hour of our time. This seems to be a difference to the question "how big is space?"
    If one allows for different rates of change in different places one could also allow for changing velocity of the passage of time. This only makes sense (to me at least) if there is a rate of passage. Hhm. Needs further thinking

  19. I agree about the need for more thought but relativistic differences apply to space too: suppose two fast spaceships a mile long pass each other in opposite directions. From the reference frame of either one, the other is shorter than it is.

  20. "However that's not true - 10 kg divided by 5 kg is 2, not 2 kg."

    Technically, isn't it 2 kg/kg? After all, there's a difference between "two items" and "two kilograms of salt per kilogram of sand."

    "Even angle, which some authorities say is dimensionless, because it is a ratio of turns"

    Isn't angle dimensionless because it is a ratio of lengths? Subtended arc length to radius length for a fixed sector of a circle centered at the angle's vertex?

  21. Saying 2kg/kg is different then two on the basis of "two kilograms of salt per kilogram of sand" is just to sneak in a different unit of measurement. The ratio indicated is 2 (kg salt)/(kg sand).

    Regardless, I think there's reason to believe there's more to scalar measurements than the units. Consider torque (the scalar).
    T = (distance)X(Force)X(angle) the units of which is (distance)X(distance)X(mass)/(time)X(time)
    Consider kinetic energy:
    E = .5X(mass)X(velocity)X(velocity) the units of which is:
    Perhaps there's some special relationship between torque and energy that I'm unaware of, but it seems to me that a measurement is more than its units.

  22. VS Bandaneer here,

    Well how do you know time is passing? The sun
    rising passing and setting is one way---also
    thoughts coming and going, actions, events. I submit that there is no time---,sans events, and
    that indeed events are what time consists of. Time is abstracted from the coming and going of
    events--in my view.
    How could you--or why would you--have such a notion as time if nothing new showed up? How would the notion come about?
    We trace events with a clock---a clock is cycles--actions--ticks--
    that we compare with other actions or events;
    the travel of the sun--the length of the day-----is what a clock tracks---the sun's travel is so many actions--clicks of a clock.
    Whether there is time as an independent entity
    or no is an ontological question----Is there some entity apart from the comparison of actions--clock actions compared to
    non-clock actions--we call time --or is time just a notion absracted from this set up and there is no actual entity.
    WHether time is an independent entity involves
    down to the issue of medeival scholasticism--the prolem of universals---(Abelard pronounced on this,what progress philosophy has made!)------is time just a name or is it a thing?
    And ultimately the question arises from the
    mind/body split.
    How you gonna decide that?
    Whether abstraction or entity--material thing--whatever that means) ---seems more an assumption.
    You all are considering time as a material thing
    apparently----why should I assume as you do?
    I'd love to read your argument.