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A puzzle about rates of change

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Most of our best scientific descriptions of the world employ rates of change of some continuous quantity with respect to some other continuous quantity. For instance, in classical physics we arrive at a particle’s velocity by taking the time-derivative of its position, and we arrive at a particle’s acceleration by taking the time-derivative of its velocity. Because rates of change are defined in terms of other continuous quantities, most think that facts about some rate of change obtain in virtue of facts about those other continuous quantities. For example, on this view facts about a particle’s velocity at a time obtain in virtue of facts about how that particle’s position is changing at that time. In this paper we raise a puzzle for this orthodox reductionist account of rate of change quantities and evaluate some possible replies. We don’t decisively come down in favour of one reply over the others, though we say some things to support taking our puzzle to cast doubt on the standard view that spacetime is continuous.

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  1. For a classic defense of the reductionist at–at theory of velocity, see Russell (1937). See Tooley (1988), Arntzenius (2000, 2012, ch. 2), Carroll (2002), Lange (2005), and Easwaran (2014) for further discussion on both sides of the debate.

  2. For the classic papers on grounding, see Schaffer (2009), Rosen (2010), and Fine (2012). For more discussion, see the papers collected in Correia and Schnieder (2012).

  3. We don’t intend anything metaphysically significant by our ‘fact’ talk throughout. Indeed everything we say is neutral on whether grounding is an operator or a relation, so we’ll just ignore this issue here.

  4. Of course this points to a quite radical reaction to our puzzle; namely, reject the continuity of spacetime in favour of a gunky or discrete view. We’ll discuss this reaction in Sect. 7, but until then we’ll take the standard view for granted.

  5. In fact there are two other remaining answers: partially ground [o has velocity v at t] in o’s position facts only before t, or alternatively in o’s position facts only after t. However, the issues we’ll raise for the Egalitarian View carry over straightforwardly to these two alternatives, so we’ll set them aside in the main text.

  6. This relevance criterion is explicit in both Rosen (2010) and Fine (2012). See Raven (2013) and Dasgupta (2014) for further discussion of the relevance condition. It’s also worth noting that a similar relevance condition would hold if we were running the puzzle using the framework of metaphysical semantics or truthmaking.

  7. One of us (TT) is sympathetic to the at-at theory even in the case of temporal derivatives like velocity. The other (DB) is more sympathetic to intrinsic temporal derivatives for independent reasons (see Builes (MS)). Nevertheless, we both agree that our puzzle isn’t a good reason to reject the at-at theory for any rate of change, even those with respect to time.

  8. You might worry that the standard definition of a derivative predicts that facts about real numbers are amongst the partial grounds for facts about objects’ velocities at times. We’ll continue working with the standard definition in the main text, but we want to emphasize that every point we’ll make has an analogue in terms of nominalist-friendly definitions of a derivative, which don’t quantify over real numbers (in particular, we have in mind the definition offered by Field (1980)).

  9. For example we may set δ0 = |t0t| + 1 and ε0 = max{|(p’  −  p)/(t’  −  t)  −  v| + 1: ∀t’(|t’ − t| < δ0)}.

  10. In order to draw our conclusion that [o has position p0 at t0] is a partial ground for [the time-derivative of o’s position at t is v] we are tacitly appealing to the transitivity of partial ground. While transitivity is widely endorsed, there has been some push-back (e.g. Schaffer (2012)). We’re inclined to side with the orthodoxy in endorsing the transitivity of partial ground, but we think that even those who are skeptical of it in full generality should endorse it in this case. However, see Raven (2013) for a full defense of transitivity.

  11. Although we reject the Egalitarian View for velocity, we remain open to it being the correct account of any derivative-like quantities in pure mathematics that are stipulatively defined using the standard epsilon-delta definition. Moreover, we suspect that the relevance intuitions we used to argue against the Egalitarian View in the case of velocity will be less strong as concerns these purely mathematical quantities. We'll discuss a further disanalogy between the cases in footnote 16. Thanks to two anonymous referees for encouraging us to say more about this issue.

  12. Notice that this claim requires reading the initial universal quantifier in (5) as a binary quantifier ranging only over reals in the open interval (0,2), rather than as a unary quantifier ranging over all objects whatsoever, which is then further restricted by some conditional antecedent. We intend a similar reading of the initial quantifiers in (1), though we’ll generally elide the difference in the main text.

  13. There are several variants of indeterminacy-based reactions. For instance, perhaps while position facts sufficiently far away from the relevant time are determinately not partial grounds, no position facts are determinately partial grounds. Instead it simply becomes indeterminate what the partial grounds are once you get close enough to the relevant time. We’ll discuss only certain paradigm examples of appeals to indeterminacy in the main text, but the points we make apply generally to indeterminacy-based replies to our puzzle.

  14. In the main text we’re taking for granted that there isn’t any genuinely metaphysical indeterminacy (following, e.g., Lewis (1986)). While some philosophers accept the possibility of metaphysical indeterminacy, we count ourselves amongst those who find the notion mysterious. Nevertheless, many of the points we make could also be phrased in terms of metaphysical indeterminacy, given the main candidate accounts of metaphysical indeterminacy on offer. For a survey of some arguments against metaphysical indeterminacy together with possible responses, see Barnes (2010). For some of the main options in the area, see Barnes and Williams (2011) and Wilson (2013).

  15. One response to this objection would be to adopt a Humean conception of laws, according to which laws are certain especially simple and informative summaries of the facts at a world. Humeans may embrace indeterminacy in what the fundamental laws are, since it can be explained in terms of there being “ties” amongst which laws are the simplest and most informative. Neither of us is particularly sympathetic to the Humean conception, so we leave it to Humeans to adjudicate whether to simply accept this widespread indeterminacy in what the fundamental laws are. We’d regard it as an interesting result in its own right if our puzzle shows that the metaphysics of rates of change has this striking implication for the Humean view.

  16. Note that our final objection to the indeterminacy-based approach applies only to rates of change that appear in the laws of our best fundamental physical theories. We thus remain more open to the indeterminacy-based approach as applied to other rates of change (see also footnote 11). Thanks to both anonymous referees here.

  17. The details of the criticisms differ, but for a sampling see Hofweber (2009), Daly (2012), Wilson (2014), Koslicki (2015), and Miller and Norton (2017).

  18. An anonymous referee wonders whether proponents of the continuity of spacetime can reply by adopting whatever fundamental physical properties proponents of gunk embrace to replace standard rate of change quantities (which presuppose continuity). In fact there is nothing inconsistent about this package: the main replacements proffered by proponents of gunk, such as fundamental average values, are also well-defined in standard continuous spacetimes (see Arntzenius and Hawthorne (2005) for a survey of the options available to gunk-theorists here). Nevertheless, we think the combination is an unhappy one: abandoning the standard calculus treatment of rates of change undermines a central motivation for believing in the continuity of spacetime in the first place. Given the various problems that arise for the orthodox continuum view, we doubt anyone willing to concede the gunk theorist's conception of the fundamental physical properties (which carries costs of its own) would nevertheless still cling to orthodoxy.


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For helpful comments and discussion, we’d like to thank David Albert, Cian Dorr, Daniel Hoek, Boris Kment, Tim Maudlin, Miriam Schoenfield, Bradford Skow, Jack Spencer, Stephen Yablo, and two anonymous referees.

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Builes, D., Teitel, T. A puzzle about rates of change. Philos Stud 177, 3155–3169 (2020).

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