An Argument Against Newtonian ‘Absolute’ Time From the Identity of Indiscernibles

An interesting thought occurred to me recently while I was reading through the early pages of Bas C. van Fraassen’s An Introduction to the Philosophy of Time and Space. I would not be surprised if this thought is unoriginal (indeed, I might even be slightly surprised if Leibniz himself hadn’t already thought it), but, for what it’s worth, the idea did genuinely occur to me, so, for all I know, it might be original. In any case, I think it may be of some interest, so I’m going to try to briefly flesh it out.

In order to do so, I will have to set the stage by very briefly explaining some of the basics of an Aristotelian view of time (at least, insofar as they are pertinent), and juxtaposing that with a Newtonian view of time as absolute. I will come around, near the end, to a brief reflection on what this argument might tell us, if anything, about the philosophical status of the generic A-theory, or the generic B-theory.

Aristotle is well known for championing a view of time on which time is dependent upon motion. Granted, what Aristotle means by motion bears only mild resemblance to our modern (much more mechanistic) notion. Motion, for Aristotle, is analyzed in terms of potentiality and actuality (which are, for Aristotle, fundamental conceptual categories). Roughly speaking (perhaps very, very roughly speaking), for any property P and being B, (assuming that having property P is compatible with being a B), B either has P actually, or else B has P potentially. For B to have property P actually is just for it to be the case that B has the property P. For B to have property P potentially is just for it to be the case that B could (possibly) have, but does not (now) have, the property P. In other words, potentiality represents non-actualized possibilities. A bowling ball is potentially moving if it is at rest, just as it is potentially moving at 65 mph if it is actually moving at 80 mph. A phrase like ‘the reduction of a thing from potentiality to actuality,’ common coin for medieval metaphysicians, translates roughly to ‘causing a thing to have a property it did not have before.’ This account may be too superficial to make die-hard Aristotelians happy, but I maintain that it will suffice for my purposes here. Aristotle, then, wants to say that in the absence of any reduction from potentiality to actuality, time does not exist. Time, in other words, supervenes upon motion in this broad sense – what we might, in other contexts, simply call change. Without any change of any sort, without the shifting from one set of properties to another, without the reduction of anything from potentiality to actuality, time does not exist.

Newton is well known for postulating absolute time as a constant which depends, in no way, upon motion (either in the mechanical/corpuscularian sense, popular among empiricists of his time, or in the broader Aristotelian sense).[1] In this he was, there is little doubt, infected by the teachings of his mentor, Isaac Barrow, who overtly rejected the Aristotelian view;

“But does time not imply motion? Not at all, I reply, as far as its absolute, intrinsic nature is concerned; no more than rest; the quality of time depends on neither essentially; whether things run or stand still, whether we sleep or wake, time flows in its even tenor. Imagine all the stars to have remained fixed from their birth; nothing would have been lost to time; as long would that stillness have endured as has continued the flow of this motion.”[2]

Newton’s view of time was such that time was absolute in that its passage was entirely independent of motion. It is true, of course, that Newton fell short of thinking that time was absolute per se; indeed, he viewed time as well as space as being non absoluta per se,[3] but, rather, as emanations of the divine nature of God. However, since God was absolute per se, as well as necessary per se (i.e., because existing a se), time flowed equably irregardless of motion, just as space existed irregardless of bodies.

To illustrate the difference, imagine a world in which everything is moving along at its current pace (one imagines cars bustling along the streets of London, a school of whales swimming at 2500 meters below sealevel, planes reddying for landing in Brazil, light being trapped beyond the event horizon in the vicinity of a black hole in the recesses of space, etc.), and, suddenly, everything grinds to a halt. It is as though everything in the world has been paused – there are no moving bodies, the wind does not blow, there are no conscious experiences, light does not propagate, electromagnetic radiation has no effects. Does time pass? On the Newtonian view, it certainly does. This sudden and inexplicable quiescent state might persist for a short amount of time, or a very long time, or it may perdure infinitely. On the Aristotelian view, this is all nonsense; instead, we are simply imagining the world at a time. To imagine that this world persists in this state from one time to another is just to be conceptually confused about the nature of time; time doesn’t merely track change, its relationship to change is logically indissoluble. So, for Aristotle, time cannot flow independently of motion (i.e., of change), while, for Newton, time flows regardless of what, or whether, changes were wrought in the world.

Now, I want to try to construct an argument for thinking that this Newtonian view may be logically impossible. I will start with an appeal to no lesser an authority than Gottfried Leibniz, who was easily Newton’s intellectual superior. He famously championed a principle which has come to be called the identity of indiscernibles (though, McTaggart tried, unsuccessfully, to relabel it as the dissimilarity of the diverse).[4] As Leibniz puts it, “it is never true that two substances are entirely alike, differing only in being two rather than one.”[5] To put it in relatively updated language: “if, for every property F, object x has F if and only if object y has F, then x is identical to y. Or in the notation of symbolic logic:

∀F(Fx ↔ Fy) → x=y.”[6]

The suggestion was that not only were identicals indiscernible (which is indubitable), but that absolutely indiscernible things must be identical. In other words, if there is not a single level of analysis on which two things can be differentiated, then the two things are really one and the same thing.

‘What is the difference,’ you might ask ‘between this ball here and that ostensibly identical ball over there?’ Well, for one thing, their locations in space (one is here, and the other is there – and this difference suffices to make them logically discernible), to say nothing of which of them is closer to me at this present time, or which one I thought about first when formulating my question (Cambridge properties suffice to make things discernible in the relevant sense). If two things do not differ with respect to their essential properties, they must (if they are genuinely distinct) differ at least in their relational properties, and if not in real relations, at least in some conceptual relations (or, what Aquinas would have called relations of reason).[7] This principle is a corollary, for Leibniz, of the principle of sufficient reason – for, the reason two indiscernible things must be identical is that, if they are truly indiscernible, then there is no sufficient reason for their being distinct. For any set of things you can think of, if they share all and only the very same properties (and, thus, are absolutely indiscernible), then they are identical – they are not a plurality of things at all, but merely all one and the same thing.

Assume that this principle is true (in a few moments, I will explore a powerful challenge to this, but spot me this assumption for the time being). Now, suppose there are two times t1 and t2, such that these two times are absolutely indiscernible. We can help ourselves here to the previous thought experiment of a world grinding to a halt; this perfectly still world is the world at t1, and it is the world at t2. No change of any kind differentiates t1 and t2. There is no discernible difference between them at all. But then, by the identity of indiscernibles, t1 and t2 are identical. To put it formally;

  1. For any two objects of predication x and y, and any property P: ∀P(Px ≡ Py) ⊃ x=y
  2. Times are objects of predication.
  3. Times t1 and t2 share all and only the same properties.
  4. Therefore t1 = t2.

This argument is so straightforward as to require little by way of clarification. I assume that times are objects of predication not to reify them, but simply to justify talking as though times have properties.

There are now two things to consider; first, what implications (if any) this argument’s soundness would have for the generic A-theory of time, and, second, whether this is a powerful argument. With respect to the first, obviously Newton’s view of time was what we would today call A-theoretical. On the A-theory, there is a mind-independent fact about time’s flow – there is a fact about what time it is right now, et cetera. Time, on the A-theory, may continue to flow regardless of the state of affairs in the world. On the B-theory of time, by contrast, there is nothing which can distinguish times apart from change (in particular, change in the dyadic B-relations of earlier-than, simultaneous-with, and later-than between at least two events). It seems confused to imagine a B-series where the total-event E1 (where ‘total-event’ signifies the sum total of all events in a possible world, at a time) is both one minute earlier than total-event E*, and where the total-event E1 is also (simultaneously?) a year earlier than the total-event E*. Indeed, to use any metric conventions to talk about the amount of time E* remained unchanging might be confused (even if one opts for a counterfactual account of how much time would have been calculated to pass had a clock been running, there is still a problem – clearly, had a clock been running, it would have registered absolutely no passage of time for the duration of E*). So, there is just no rational way of speaking about the duration of a total-event E* by giving it some conventional measurement in the terms of some preferred metric.[8] If the B-relations of earlier-than, simultaneous with, and later-than, are not in any way altered from one time to another, then the times under consideration are strictly B-theoretically indiscernible, and, thus, identical. On the A-theory, by contrast, one can provisionally imagine an exhaustively descriptive state of affairs being both past and present.[9] One can imagine its beginning receding into the past while it (i.e., this total-event E*) remains present. I am not sure that every version of the A-theory will countenance this possibility, but it seems right to say that only the A-theory will countenance this possibility.[10] If my argument is right, and the reasoning in this paragraph hasn’t gone wrong, then the A-theory is less likely to be true than it otherwise would have been (we don’t even need to apply a principle of indifference to the different versions of the A-theory, so long as we accept that the epistemic probability of each version of the A-theory is neither zero nor infinitesimal).

In any case, the salient feature of what I’ve presented as the Newtonian view is that time may pass independently of any change in the world at all. I’ve suggested that there is a problem for the Newtonian view (whether or not such a view can be married to the B-theory) in the form of a violation of the principle of the identity of indiscernibles. The Newtonian might, of course, argue that God’s conscious awareness continues regardless of a quiescent world, so that God himself could act as a sort of clock for such a motionless universe. He, at least, would know how long it had been since anything was moving, or changed. In this case, however, the Newtonian is effectively conceding ground to the peripatetic; at least God, then, has to be reduced from potentiality to actuality (this suggestion will, of course, be repugnant, both to Aristotelians as well as to Catholics, but die-hard Newtonians typically aren’t either anyway).

Regardless, this argument may not be as strong as I initially hoped. After all, together with the principle of sufficient reason, the identity of indiscernibles has been the subject of sustained and impressive criticisms. While these criticisms may not present insuperable difficulties for defenders of the principle, they cannot be lightly dismissed. For a fair conceptual counter-example, one might think, in particular, about a perfectly symmetrical world in which there are only two physically identical spheres, neither of which has a single property that the other fails to have. Consider the following passage from Max Black’s ingenious paper, The Identity of Indiscernibles;

“Isn’t it logically possible that the universe should have contained nothing but two exactly similar spheres? We might suppose that each was made of chemically pure iron, had a diameter of one mile, that they had the same temperature, colour, and so on, and that nothing else existed. Then every quality and relational characteristic of the one would also be a property of the other. Now. if what I am describing is logically possible, it is not impossible for two things to have all their properties in common. This seems to me to refute the Principle.”[11]

There are no obvious and attractive ways out of this predicament for the rationalist, as far as I can see. One might be able to say that they have distinct potentialities (i.e., that to scratch or mutilate one would not be to scratch or mutilate the other, so that each one has a distinct potentiality of being scratched or somehow bent into a mere spheroid), but it isn’t clear how useful such a response is. One might argue that each one is identical with itself, and different from its peer, but it isn’t clear that self-identity is a bona-fide property. One may, out of desperation, ask whether God, at least, would know (in such a possible world) which was which, but it may be insisted, in response, that this is a pseudo-question, and that, while they are not identical, God could only know that there were two of them (and, of course, everything else about them), but not which one was which.

In passing, I want to recommend that people read through Black’s paper, which is written in the form of a very accessible and entertaining dialogue between two philosophers (simply named ‘A’ and ‘B’ – yes, yes, philosophers are admittedly terrible at naming things). Here is a small portion which, I feel, is particularly pertinent;

“A. How will this do for an argument? If two things, a and b, are given, the first has the property of being identical with a. Now b cannot have this property, for else b would be a, and we should have only one thing, not two as assumed. Hence a has at least one property, which b does not have, that is to say the property of being identical with a.

B. This is a roundabout way of saying nothing, for ” a has the property of being identical with a “means no more than ” a is a When you begin to say ” a is . . . ” I am supposed to know what thing you are referring to as ‘ a ‘and I expect to be told something about that thing. But when you end the sentence with the words ” . . . is a ” I am left still waiting. The sentence ” a is a ” is a useless tautology.

A. Are you as scornful about difference as about identity ? For a also has, and b does not have, the property of being different from b. This is a second property that the one thing has but not the other.

B. All you are saying is that b is different from a. I think the form of words ” a is different from b ” does have the advantage over ” a is a ” that it might be used to give information. I might learn from hearing it used that ‘ a ‘ and ‘ b ‘ were applied to different things. But this is not what you want to say, since you are trying to use the names, not mention them. When I already know what ‘ a’ and ‘ b ‘ stand for, ” a is different from b ” tells me nothing. It, too, is a useless tautology.

A. I wouldn’t have expected you to treat ‘ tautology’ as a term of abuse. Tautology or not, the sentence has a philosophical use. It expresses the necessary truth that different things have at least one property not in common. Thus different things must be discernible; and hence, by contraposition, indiscernible things must be identical. Q.E.D

[…]

B. No, I object to the triviality of the conclusion. If you want to have an interesting principle to defend, you must interpret ” property” more narrowly – enough so, at any rate, for “identity ” and “difference ” not to count as properties.

A. Your notion of an interesting principle seems to be one which I shall have difficulty in establishing.”[12]

And on it goes – but I digress.

Now, if such a world (with two identical spheres) is logically possible, it looks as though the spheres in it are indiscernibles even if they aren’t identical. No fact about their essential properties, or relations, will distinguish them in any way (and this needn’t be a case of bilocation either, for we are supposed to be imagining two different objects that just happen to have all and only the same properties and relations). If that’s correct, then (I take it) the identity of indiscernibles is provably false.

So, my argument will only have, at best, as much persuasive force as does the identity of indiscernibles. It persuades me entirely of the incoherence of imagining a quiescent world perduring in that state, but I doubt whether the argument will be able to persuade anyone who rejects the identity of indiscernibles.

[1] Strictly speaking, I’m not entirely sure that Newton would have said that time can continue to flow independently of any change of any kind, but I do have that impression. Clearly, for Newton, time depends solely on God himself.  Below, I will consider one response a Newtonian could give which suggests that time flows precisely because God continues to change – however, to attribute this to Newton would be gratuitous and irresponsible. I am not a specialist with regards to Newton’s thinking, and I do not know enough about his theology to say whether, or to what extent, he would have been happy to concede that God changes.

[2] The Geometrical Lectures of Isaac Barrow, J.M. Child, Tr. (La Salle, III.: Open Court, 1916), pp. 35-37.

Reproduced in Bas C. van Fraassen An Introduction to the Philosophy of Time and Space, (New York: Columbia University Press, 1941) 22.

[3] William Lane Craig, Time and the Metaphysics of Relativity, Philosophical Studies Series Vol. 84. (Springer Science & Business Media, 2001), 114.

[4] See C.D. Broad, McTaggart’s Principle of the Dissimilarity of the Diverse, Proceedings of the Aristotelian Society, New Series Vol. 32 (1931-1932), pp. 41-52.

[5] G.W. Leibniz, Discourse on Metaphysics, Section 9; http://www.earlymoderntexts.com/assets/pdfs/leibniz1686d.pdf

[6] Peter Forrest, “The Identity of Indiscernibles,” in The Stanford Encyclopedia of Philosophy ed. Edward N. Zalta, (Winter 2016 Edition); https://plato.stanford.edu/entries/identity-indiscernible/

[7] See W. Matthews Grant “Must a cause be really related to its effect? The analogy between divine and libertarian agent causality,” in Religious Studies 43, no. 1 (2007): 1-23.

[8] I will not, here, explore the idea of non-metric duration.

[9] Interestingly, McTaggart would likely have begged to disagree. Indeed, one may be able to construct an argument along McTaggart’s lines for the impossibility of a world remaining totally quiescent over time by arguing that the A-properties of pastness and presentness were incompatible determinations.

[10] It is entirely possible, upon reflection, that I am dead wrong about this. Perhaps this is just my B-theoretic prejudice showing itself. Why, if the A-properties of Presentness and Pastness aren’t incompatible determinations of a total-event E*, think that the B-relations of being earlier-than and simultaneous-with are incompatible determinations of a total-event E*? I continue to persuade and dissuade myself that there’s a relevant difference, so I’m not settled on this matter.

[11] Max Black, “The identity of indiscernibles,” in Mind 61, no. 242 (1952): 156.

[12] Max Black, “The identity of indiscernibles,” in Mind 61, no. 242 (1952): 153-4,155. http://home.sandiego.edu/~baber/analytic/blacksballs.pdf

Banach-Tarski paradox, א Infinities, Infinitesimals, and the A-theory

I will offer an analysis of what is going wrong with the Banach-Tarski paradox suggesting that points, construed as infinitesimal surface areas, are nothing more than mathematically useful fictions. I will suggest that infinitesimals raise the same kinds of modally-prohibitive paradoxes in metaphysics as positing actually infinite quantities does (and for the same or similar reasons), and then consider an argument against the A-theory (in most of its forms) which can be purchased from these insights. I will then scout out some philosophical avenues available to the A-theorist.

The Banach-Tarski paradox is a famous mathematical paradox according to which it can be proved that if you divide the surface area of a sphere into little bits, and simply rearrange the bits appropriately, you can reconstruct two spheres each with the same surface area as the original sphere. In layman’s terms, you can prove (something just a shocking as) that 1=2.[1] To explain how it works, it may be worth calling to mind the various paradoxes associated with actual infinities.

Consider what it would be like to count upwards from -7 to infinity and stop only once you’ve arrived. Even if given an infinite amount of time you would never arrive, because no finite additions can sum up to a transfinite quantity. Subtract infinity from infinity, and what do you have? You have zero, but you also have infinity, and you also have 18.9801 (and every other real number); all of these are not just legitimate answers, they are mathematically correct answers. However, clearly 18.9801 is not equal to either zero, infinity, or anything else! Have a (Hilbert) hotel with an infinite number of rooms, all of which are occupied, and you want to check in an infinite number of new guests? No problem, just move every person from the room they are in (n) to the room with a room number equivalent to two times the original room’s room number (2n). Done; you’ve managed to move people around in such a way as to create an infinite number of vacant rooms without asking anyone to leave. Most of us (who are interested in this sort of thing) know the myriad paradoxes which arise from postulating even the possibility of an actual infinity. It seems relatively philosophically secure that there cannot be an א number of things (where א represents the first transfinite number, not to be confused with ∞ which symbolizes infinity taken as a limit rather than a quantity). If there are philosophically sophisticated caveats then so be it, but the point will remain that there are plenty of examples of things for which having an א number of them is clearly (broadly logically) impossible.

Let’s return, for a moment, to Hilbert’s Hotel, because it’s a particularly useful illustration. Suppose that the guest in room 3 checks out, while all the (infinitely many) other rooms remain occupied. The desk clerk decides that they want every room occupied, so they ask each person in room n (where n>3) to move one room over; that is, from room n to room n-1. That will fill up room 3, but the process will also leave no room empty because there is no room number n for which there is not an occupied room n+1. This works equally well for two dimensional shapes, such as circles; remove one ‘point’ from the circumference of a circle and you may have an infinitesimal gap, but simply move every other point along the circumference over (uniformly) by an infinitesimal amount and, voila, the gap is plugged and there will be no new gap. The trick in the case of the Banach-Tarski paradox is to apply the same reasoning to three-dimensional objects. For the best explanation of this paradox I’ve ever seen, (especially for readers who aren’t familiar with it, please make your life better and) check out Vsauce.

Alexander Pruss has noted on his blog that this result “is taken by some to be an argument against the Axiom of Choice.”[2] However, he argues that you can get the same paradoxical result in similar cases (and even in the same case) without the axiom of choice, so that the axiom of choice should be cleared of all suspicions. I agree (though I’m certainly no expert). Richard Feynman is purported to have said, upon being shown the proof, that “it’s fine you can do it with ‘continuous spheres’, since there’s no such thing. The important thing is you can’t do it with oranges, because oranges are made of a finite number of indivisible parts.” I think he is wrong about oranges (being actually comprised of indivisible finite parts, at least if the ‘parts’ are extended in three spatial dimensions), but his sentiment is appreciably insightful nonetheless.

The problem with the paradox, in my submission, is that it divides the surface of the sphere up into points. However, points on a sphere, like points on a line segment, are infinitesimals. This is precisely why (Aristotelians) say that line segments are not composed of points the way walls are composed of bricks, but, instead, points act as the limits between which a line segment is continuously extended. An infinitesimal is a quantity which is infinitely small. It is non-zero, but it is also smaller than any finite quantity. Sure infinitesimals are useful for doing things like infinitesimal calculus, developed by one of my all time favorite philosophers Gottfried Wilhelm Leibniz, but they remain, I believe, nothing more than useful fictions. To borrow a phrase from W.L. Craig;

“They are akin to ideal gases, frictionless planes, points at infinity, and other useful fictions employed in scientific theories.”[3]

If we are to accept the possibility of infinitesimal quantities in reality, then we will quickly run into paradoxes like the Banach-Tarski paradox (which, quite apart from being obnoxious to the rational intellect, seems to violate the law of conservation of matter and energy). Positing infinitesimals is just as paradoxical as positing sets of actually infinitely many discrete things (where ‘things’ is an ontologically loaded term). I am suggesting that infinitesimals are just as paradoxical as actual infinities, and, at bottom, for the same reason(s). In fact, I have this intuition that every argument for thinking that there cannot be any actual infinities (as opposed to potential infinities, where ‘infinity’ merely acts as a limit), admits of a parody for an argument against the existence of infinitesimals. I’m not sure I can rigorously prove it, but I think it’s very plausible.

It seems to me that there’s something conceptually parasitic about infinitesimals relative to infinities. They each conceptually supervene on each other symmetrically. To visualize this symmetry, consider plotting the function ƒ(x)=  1/x which will look like this:

[http://mathworld.wolfram.com/images/eps-gif/AsymptotesOneOverX_1000.gif]

The distance between the curved line and the x-axis (i.e., y=0) as x approaches (positive or negative) infinity is shrinking (or, at least, its absolute value is shrinking), and approaching an infinitely small non-zero measure. When X is infinite, the absolute value of the y-axis coordinate of the curved line (i.e., the distance between the curved line and it’s asymptote, here being the x axis) is infinitesimally small. This example helps to illustrate the point that the concept of an infinitesimal is bound up with the concept of infinity, so that in the absence of one the other would be inconceivable. That at least motivates the suspicion that if one turns out to be metaphysically impossible, so will the other.

What relevance does this have for the philosophy of time? Well, consider that on the A-theory there is such a time as the present. How long does the present last? What, precisely, is its magnitude, its duration? Let’s consider the following argument:

  1. If the A-theory is true, then the present is either infinitesimal in duration, or it is finite in duration.
  2. The present cannot be infinitesimal in duration.
  3. The present cannot be finite in duration.
  4. Therefore, the A-theory is false

Premise 3 can be established with Leibniz’ argument against the (logical) possibility of a physically indivisible element, or ‘atom’ (in the etymologically literal sense). For anything extended in three-dimensional space, however small, it will always be logically possible for me to divine it in two, even if I am physically incapable of doing so (due to some constraint, such as not having the appropriate equipment for the job, or maybe not even being able to develop any tool which could do the job). Physical impossibilities are not (all) logical impossibilities, and logically there is no constraint on how many times I could divide an object extended in space. To say that there is an object extended in space which is not logically possibly divided up into smaller constituent pieces is, according to Leibniz, incoherent. The exact same argument, mutatis mutandis, works against there being chronons (i.e., atomic chunks of time).

The denial of premise 2 is absurd given our observations that positing infinitesimals leads to modally unconscionable paradoxes like Banach-Tarski.

Ways out: I see four ways, not all of them equally viable, for an A-theorist to escape the conclusion of this argument.

First, they could challenge premise 3 on the grounds that, if there are chronons, then by definition they are entities which cannot be physically divided. The suggestion would be that the prima facie absurdity of a Chronon de dicto doesn’t entail the impossibility of a chronon de re. This dangerously dislocates rational intuition from epistemic reliability, but I can imagine extreme empiricists embracing this response.

Second, they could challenge premise 2 by arguing that positing any more than one real infinitesimal of any kind might be problematic, but that there’s no way to derive similar paradoxes from positing a maximum of one infinitesimal. In other words, perhaps paradoxes involving infinitesimals only arise when there are n infinitesimals, where n ∈ ℕ, and n>1. Multiply an infinitesimal by any natural number, or even a transfinite number, and you will still get an infinitesimal result, so it seems harder to show that from one infinitesimal you could derive some kind of contradiction.

Quick thought: Perhaps if there are rules/axioms such as (i) no infinitesimal can be larger or smaller than any other infinitesimal, but (ii) anything (other than 1) to the power of itself is larger than itself, you could derive a contradiction by taking an infinitesimal X, running it through Xx=Y, and then asking whether Y is larger than X, or the same size (it appears to be both). However, I don’t have the kind of facility in mathematics to be able to produce a rigorous proof that even a single infinitesimal would lead to some kind of contradiction or unconscionable paradox. Moreover, it isn’t entirely clear to me what relevance that kind of mathematical paradox would have for the metaphysical consideration at hand. In any case, the second challenge to premise 2 cannot be lightly dismissed.

Third, one could adopt a really wild philosophy of time, such as the Apresentism I wrote about in the last post (thus denying the first premise).

Fourth, one could deny the first premise by adopting what has been called a non-metric view of the present. This is the view preferred by William Lane Craig.[4] I have more than expended my allotted time for blogging and casual writing today, so I will leave this post here for now. I may return to the idea of non-metric present in the (near) future in another post.

[Ha, I don’t presently have time to write more. Get it?]

[1] For fun, check out and try to find the mistake in the following mathematical proof that 1=2 here: https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html

[2] http://alexanderpruss.blogspot.ca/2013/06/the-banach-tarski-paradox-and-axiom-of.html

[3] William Lane Craig, “Response to Greg Welty,” in Beyond the Control of God: Six Views on the Problem of God and Abstract Objects, ed. Paul Gould (A&C Black, 2014), 102.

[4] See: Craig, William. “The extent of the present.” International Studies in the Philosophy of Science 14, no. 2 (2000): 165-185.

Two (new?) Versions of the A-theory

I’m not sure if this is truly original, but I have never encountered anyone defending either of the following two versions of the A-theory of time. Possibly because they are so wildly esoteric as to stand on the periphery of intelligibility. To explain them, let’s begin by thinking about the properties of Presentism, and C.D. Broad’s growing-block theory of time. On Presentism the set of things which exist is identical to the set of things which exist right now. Presentists think that there are no real future events (the future is not ‘waiting for us to get there’), and past events have literally gone out of existence. To become present is to become, full stop. On such a view not all the A-properties distinguished by J.M.E. Mctaggart in his famous essay The Unreality of Time are instantiated. It seems that a prerequisite for an event to literally have a property, such as the property of being ‘future,’ is that the event has to literally exist. If a thing literally fails to exist then it cannot literally have properties of any kind. Similarly, on Broad’s growing-block theory of time, things which are present come into existence at the present, and they continue to exist even once they have inherited the property of being ‘past,’ but future events do not literally exist (or have A-properties) at all.

Keeping these in mind, here are two other views. Since I’ve never encountered them in the literature before, I will take the liberty of giving them names. Let’s call them Apresentism and deteriorating-block theory, respectively.

On Apresentism events bear either the A-property of being future, the A-property of being past, or both, but no events bear neither. In particular, no events bear the A-property of being present. This is because the present, on this view, literally does not exist. The category of the ‘present’ is simply a heuristic tool, a useful fiction, but what it picks out is events which have both the A-properties of being past, and the A-properties of being future. All that ever really happens, on this view, is that events go from having exclusively future A-properties, to having both future and past A-properties, and then finally to having exclusively past A-properties. Is this view logically possible? I’m not inclined to think that it is, but showing what, precisely, has to be wrong with it remains a tall order. If one can show what is wrong with it, then I suspect one will be able to show what is wrong with presentism, but this is just a philosophical hunch.[1]

The deteriorating-block theory is exactly what it sounds like. In place of events having the A-property of being present, and then inheriting the A-property of being past, but never literally having had the A-property of being future (so that layers and layers are added on to the world as the present rolls on), this view suggests that events have the A-property of being future, or the A-property of being present, but to become past is to fall out of existence entirely. Thus, the future is real, the present is real, and the present represents a sort of ontological precipice – events which cease to be present cease to be.

[1] Here’s a quick thought – on presentism construed in the most ontologically conservative way, God would, even if Sempiternal, not literally have a past or a history, anymore than any of us would.

Foreseeing Problems with Engineering Optimized Academic Languages

I want to share some thoughts I recently had about the possibility of literally engineering languages optimized for academic research, and what consequences we might expect to follow from implementing the use of such languages as academically standard. The thought came to me in an example packaged with the intellectual accouterments of my own area of expertise (or at least area of academic focus), the philosophy of time, but the point generalizes (I think). Philosophy of time, perhaps especially as it exists within the Anglo-American or ‘Analytic’ tradition, requires key distinctions like the distinction between ‘time’ and ‘tense’ in order to get off the ground. Not all natural languages have developed in a way that allows one to make this distinction, however, and as a consequence some languages turn out to be more optimal for the study of such niche philosophical areas than others. Recently William Lane Craig pointed out[1] that French, for example, allows no room for the distinction between time and tense, since both ‘time’ and ‘tense’ are represented by one single word in French: ‘temps.’ He recalls that a French Graduate student in philosophy struggled to communicate his ideas to his academic peers:

“As a French speaker, he found it next to impossible to communicate to his colleagues his interest in tensed versus tenseless theories of time. Since in French the word for time and the word for tense is the same, namely, temps, he found himself quite a loss to how to communicate something like tenseless time. People didn’t even know what he was talking about!”[2]

This is (presumably) precisely why philosophy of time has been so stagnant in the French-speaking academic world.

As I was reflecting on this it occurred to me that some language might be maximally optimal for the study of philosophy in general, or for epistemology, or metaphysics. This will not sound unfamiliar to those who are well read in the continental tradition of philosophy (think of Heidegger and Hegel and their ilk who treat the German language itself as indispensable not just for their own philosophies, but for the world’s most pristine philosophy). However, what I have in mind is more radical than this. Given that some natural languages are better suited to certain intellectual pursuits, it seems entirely plausible (theoretically) that we could optimize a language for a discipline. I mean that we would literally engineer a language, from the ground up, to be optimized for a species of academic/intellectual pursuit. Presuming indefinitely many more advances to come in the worlds of cognitive science and linguistics I can see no reason why this suggestion is infeasible in principle (in fact, we need only presume a few more leaps and bounds forward in these areas, so we could even dispense with the ‘indefinitely many more advances to come’ optimism). Imagine having an artificial language which we specially constructed to be optimized for the study of philosophy, or for the study of psychology, or chemistry, or even linguistics itself. What would the academic world look like if we were to do this? If people in these fields all used a highly specialized language (not just technical vocabulary) then how good would it be for academia in general? Would there be such a thing as a maximally optimized language for a discipline (is there a ceiling to how optimized these languages might be, or would some disciplines have ceilings while others did not – and presuming that at least a few had ‘ceilings,’ would some be lower than others, and could we infer anything interesting from that)? Interesting questions.

I can, it is true, imagine such a project being an academic boon in certain respects, but I can just easily (perhaps more easily) imagine it being academically detrimental in other respects. For instance, if somebody who studies chemistry is (potentially) a brilliant chemist, but is no good at all with linguistics, wouldn’t they be disadvantaged if it became academically necessary for them to learn a new language to study in their chosen field? How many gifted prodigies would we disadvantage compared to the polymaths we would be advantaging? Moreover, we already have a problem with academic overspecialization and intellectual insulation; academics aren’t always good at keeping up with what is going on around them in other fields (political science, physics, social science, philosophy, musicology etc.), and it seems as though encouraging academics to speak in literally idiosyncratic languages optimized for, but peculiar to, their own fields, would only exacerbate this problem.

I can imagine specially engineered academic languages causing even deeper academic divisions. Imagine, for instance, that a political science student writes a thesis on how the specialized language of biology – say – were subliminally charged with a left/right-wing political ideology, or imagine that a gender-studies major writes a thesis on how the specialized language of physics is inherently and structurally sexist.[3] Such criticisms, which would be sure to come, would create not academic rapprochement but alienation and perhaps, in the worst case, even antagonism. In the end it isn’t clear to me just how helpful developing and implementing such specialized artificial languages would really be, and I suspect (for many of the reasons I’ve alluded to) that the payoff wouldn’t be worth the cost.

[1] William Lane Craig, “The Passage of Time,” http://www.reasonablefaith.org/the-passage-of-time

[2] William Lane Craig, “The Passage of Time,” http://www.reasonablefaith.org/the-passage-of-time

[3] For fun, see: https://www.youtube.com/watch?v=mg0CZZUFyeo

An Argument for Analogy

I have heard some atheists and skeptics about God’s existence claim that the Thomistic doctrine of analogy (i.e., that we can only speak about God by analogy, since we form our concept about God under the influence of empirical impressions of creatures, and so cannot possibly form a univocal concept of God’s essence as such), cannot be right because there is nothing about which we can only speak by analogy. In other words, if we can speak about anything by analogy, then we can speak about that thing univocally as well (followers of Duns Scotus will often press this point).[1] In this post I want to explore a way in which I think the multiverse hypothesis, which is popular among naturalists today, implies that there are some things, after all, about which we can speak, write and think, only by way of analogy, or at least by way of analogy alone.

There are a few different definitions of the multiverse hypothesis, but I will here take the multiverse hypothesis to be the thesis that there is an ensemble of universes, including our own, all of which have their own entirely separate spaces and times. On this hypothesis other space-times exist, but, it turns out, their spaces and times are incommensurable with our own. Everything may seem fine so far, but an interesting thing happens when we reflect more deeply on this (hypothetical) situation. It turns out that, in a rather straightforward way, the space and time of any alternative universe isn’t really what we refer to as space, or what we refer to as time. It makes no sense to ask ‘how far away’ an object in the space of another universe is, or ‘how long ago’ an event in another universe was, precisely because those universes do not share our time, or our space. It makes no sense to ask how old another universe is compared to ours, or how large another universe is compared to ours, for they cannot stand in such relations to each other. Those comparisons break down at this level because they become semantically vacuous. Such relations simply do not obtain between different universes.

This makes clear that our time is what we refer to when we talk about time, and our space is what we refer to when we talk about space. We are, when conceiving of other universes, taking our concepts of time and space and saying of a reality actually incommensurable with our own that it is ‘like this.’ That, however, is just to say that we are using our concepts of space and time analogously, and this is precisely the way in which the Thomist thinks we can, and must, speak about God. These other universes do not literally or univocally have any space, or any time, where these words are understood in their literal senses. You can satisfy this for yourself by thinking through some obvious considerations; for instance, consider that anything extended in time is, by logical necessity, earlier, later, or simultaneous with all other events in time. In the case of another universe this is not so, for anything extended in the ‘time’ of another universe is not earlier, later, or simultaneous with all other events in time. The same can be said of space, since any two (non-identical, non-overlapping) things extended in space are, necessarily, some distance apart from each other, but objects extended in different spaces are no distance apart from each other. The only way to make sense of talk of spaces and times incommensurable with our own is by analogy; we can speak about other space-times only by adopting a propositional attitude according to which we recognize our statements to be predicated by way of analogy. Our terms are inherited from the world with which we are familiar, and we are using them to speak about realities which we otherwise (than by analogy) cannot speak or think about at all. Nevertheless, the multiverse hypothesis can be both coherent and even true.

If I am right, then what this shows is that analogous predication is coherent and legitimate after all, at least if the multiverse hypothesis is a coherent hypothesis (it may not be, of course, but at least the naturalist/skeptic who takes it to be a coherent hypothesis will not be able to turn around and say that the Thomistic doctrine of analogy must be wrong because there isn’t anything about which we can speak only by analogy). Perhaps the naturalist will recoil at this point and argue that even if different universes have incommensurable times and spaces, that doesn’t mean that there is no way to predicate anything univocally about these different universes. For instance, perhaps two universes can stand in some real relation (for instance of similarity) to one another, or perhaps we can say that both exist in exactly the same sense of ‘exist.’ In response, I want to say that I am doubtful that any two universes can stand in any real relation to each other at all (I think this is ultimately a linguistic confusion), and even if many different universes could be said to exist in a univocal sense, their spaces and times considered as such could only be described and conceived of by analogy with our own. Perhaps it is not inappropriate to point out, as well, that existence is not a first-order predicate anyway (a point with which the naturalist will almost certainly agree), so that the fact that it can be applied apparently univocally shouldn’t worry us precisely because it isn’t a property. As such, it contributes absolutely nothing to the idea of the thing in question, and the doctrine of analogy maintains that it is our idea of the thing which can be formed exclusively by analogy.

Another objection may be that space and time are complex ideas which are conceptually formed by putting together combinations of simpler ideas, each of which can, as it turns out, be used univocally as applied to our universe and to others. For instance, somebody could suggest that time is nothing other than the direction of increasing entropy, and that ‘entropy,’ ‘direction’ and ‘increasing’ are concepts which can be applied univocally across different universes.[2] I think that this is wrong for a few reasons. First, ‘direction’ doesn’t seem to be univocally applied across different space-times (maybe it is, but it isn’t clear to me that it is). Second, I see no reason to think that time is defined by the direction in which entropy increases. In fact, the only reason we think of entropy increasing over time is because as time passes we observe an increase in entropy, but had it been the other way around we would have defined time as the direction in which entropy decreases, and, indeed, there are presumably some (at least possible) universes in the multiverse in which, as time goes on, entropy does decrease – and if this is even possible, given the multiverse hypothesis, then time cannot mean merely the direction in which entropy increases. If time simply meant the direction in which entropy increases then a universe in which entropy decreases over time would be physically impossible, but that, as far as I know, is not the case (perhaps someone could raise a quibble here about the second law of thermodynamics, but that is articulated precisely with the presumption that it is about our universe). Moreover, if one simply defines time as the direction in which entropy increases then I think it follows trivially that in no universe is it physically possible for entropy to decrease over time, but there is no good reason at all to accept this definition of time, and there are some very deep philosophical reasons for rejecting such a definition.[3] In any case I think our concept of time is more primitive and basic than our concept of entropy; we discovered that entropy increases as time passes, but we did not and could not have discovered the reverse.

In conclusion then, it seems to me that the naturalist faces a dialectical dilemma here. Either analogous predication is coherent and legitimate, in which case we can countenance both the doctrine of analogy and the multiverse hypothesis, or else it isn’t, in which case we cannot. If the naturalist wants to appeal to the multiverse hypothesis, even as a merely coherent hypothesis (for instance, as a possible explanation of the appearance of fine-tuning), then they will have to concede to the Thomist that we can, in principle, speak about God by analogy alone (not to be confused with the concession that we can only speak about God by analogy).

 

[1] See: Thomas Williams, “John Duns Scotus,” in The Stanford Encyclopedia of Philosophy (Summer 2015 Edition), Edited by Edward N. Zalta, http://plato.stanford.edu/entries/duns-scotus/#NatThe

[2] My thanks to a friend for bringing this point to my attention.

[3] For more on this topic please visit: http://www.reasonablefaith.org/questions-on-the-arrow-of-time

 

A Scientistic argument for Determinism, and some related thoughts

I would like to write a little bit, today, about Determinism. First, I want to try to give another argument for determinism which occurred to me recently (though I think it is a very poor argument, but may be worth mentioning if for no other reason than that it is some kind of argument for determinism). Following this I wish to draw on a thought experiment presented by Alexander Pruss to show that libertarian free will can be consistently combined with physical determinism.

What arguments are there for determinism? Let us take determinism to be the thesis that for any even E, E either follows of causal necessity from some prior (or posterior) event(s), or else from E every event follows of causal necessity. To avoid trouble, let us stipulate that no two events are both simultaneous and non-identical (i.e., events are complete states of affairs at a moment). Obviously the reason I used ‘causal’ necessity in the definition, as opposed to logical necessity, is that at any time t1, plausibly there is a future-tense (or past-tense) fact about any time t1+n (where n can be negative), so that at least one proposition at any time (and thus for any E) will logically entail every other proposition at every other time. Even the libertarian accepts that, so we should be careful not to conflate that with determinism.

I have said previously that I can think of one argument for a modest kind of determinism which would still be strong enough to rule out libertarian free will; 1) that human beings are entirely material entities, 2) that all material entities are governed entirely by deterministic physical laws, and therefore 3) human beings are determined to act and think exactly as they do act and think. I mentioned that this argument seems implausible to me for two reasons; first, that human beings are not plausibly entirely material entities,[1] and second that the laws of physics are not actually deterministic.[2] However, notice that this argument, even if it were sound, would not go as far as to entail that determinism per se is true (but only that physical determinism is true), nor would it give us any justificatory reason(s) for believing that determinism is true. Additionally, the restriction to physical determinism may actually undermine determinism per se. On determinism per se, even the universe is deterministically caused to begin to exist (assuming it does so), but on physical determinism there is no physical determinant responsible for the beginning of space, time, energy and matter. Physical determinism would, then, imply that materialism (and anything like it) is false, or that determinism per se is false. That’s a hard bullet for the champion of scientism to bite.

Here’s a more ambitious argument for determinism:

  1. Determinism is a necessary presupposition of the scientific method.
  2. The scientific method is the only, or in any case the best, avenue to genuine discovery (i.e., the finding of truth, since a discovery of something false is not a genuine discovery).
  3. Therefore, the presupposition of determinism is a necessary condition of the only, or in any case the best, avenue to genuine discovery (i.e., to the truth).
  4. For any P, if P is a presupposition necessary for the only, or in any case the best, avenue to genuine discovery, then P ought to be believed.
  5. Therefore, determinism ought to be believed.
  6. For any P, if P ought to be believed then P is true (i.e., nothing untrue ought to be believed).
  7. Therefore, determinism is true.

We might call this a presuppositionalist argument for determinism. If it were sound then it would provide us with a good reason to believe that determinism is true.[3]

Is this argument any good? Unsurprisingly, I think not. To start off, the first premise seems dubious, especially in light of the same points I made in response to the last argument for determinism which I examined – namely that quantum mechanics may not be deterministic (and yet clearly indeterministic theories of quantum mechanics are scientific, whether or not they are possibly true), and even Newtonian mechanics is certainly not deterministic (and yet, again, is clearly scientific, regardless of whether it is true, or even possibly true – scientific theories can suggest metaphysical impossibilities without ceasing to be scientific). The second premise is also problematic in my view, since it seems to me to simply enunciate the prejudice of scientism, which we have no good reasons for accepting, along with very good reasons for rejecting. So, I outright reject both of the first two premises of this argument.

I also think there are significant problems with the sixth premise which, even though I accept it, seems dubious on the assumptions of determinism and scientism. If determinism is correct, that seriously threatens the possibility of genuine ethics, including the ethics of belief, and if scientism is true then we have no good reason for believing that there are no false beliefs which we ought to adopt (for instance, if scientism and determinism are true, maybe I ought to believe that I am free in a morally relevant sense, even though, in fact, I am not and cannot be – or, paradoxically, if determinism/scientism are true, then, possibly, I ought not to believe that they are true). The whole reason for thinking that a belief ought to be believed if and only if it is true is based on a kind of metaphysical conception of truth on which truth, beauty and goodness are, we might say, ‘natural siblings.’ This makes perfect sense on the Christian way of seeing things, as well as many (perhaps most) other worldviews, but it does not make much sense on materialism or naturalism (which scientism enjoins on us). I’m not even sure it makes much sense on any non-materialist, but yet deterministic, view of the world (like the Calvinist worldview).[4]

Returning to the topic of physical determinism, I would now like to talk about an illustration I found in Pruss’ writing which helps to show that physical determinism is logically compatible with libertarian free will. Pruss uses the image of a cannonball flying through the air to clarify the difference between the Principle of Sufficient Reason (PSR) and what he calls the “Hume-Edwards-Campbell Principle” (HECP). According to the HECP, if each member of an infinite set could be explained in terms of the preceding member(s) then (i) every member of the set would be explained, and (ii) the set itself would stand in need of no additional explanation. The HECP is sometimes used as a response to cosmological arguments from contingency, for obvious reasons. Hume, for instance, writes:

Add to this that in tracing an eternal succession of objects it seems absurd to inquire for a general cause or first author. How can anything that exists from eternity have a cause[?]… In such a chain, too, or succession of objects, each part is caused by that which preceded it and causes that which succeeds it. Where then is the difficulty? But the whole, you say, wants a cause. I answer that the uniting of these parts into a whole, like the uniting of several distinct countries into one kingdom or several distinct members into one body, is performed merely by an arbitrary act of the mind and has no influence on the nature of things. Did I show you the particular causes of each individual in a collection of twenty particles of matter, I should think it very unreasonable, should you afterwards ask me, what was the cause of the whole twenty. This is sufficiently explained in explaining the cause of the parts”[5]

This principle is, I believe, demonstrably wrong for several reasons, though my favorite demonstration is provided by Pruss who proves that an infinite series of successive explanations is logically equivalent to one great big viciously circular explanation. However, my interest here is not to find out whether the HECP is correct, but how thinking through the HECP can help make clear how physical determinism is compatible with libertarian free will.

To illustrate the difference, let’s imagine that there were a cannonball flying through the air in a logically possible world where there was no time at which the cannonball was not flying through the air. Every point in time at which the cannonball was in a certain place, going a certain speed in a particular direction, those facts could all be explained by pointing to facts about where a cannonball was (how fast it was going, and in what direction it was moving) at a preceding point in time (at least presuming the regularity of its motion, which is to say that its motion is governed by certain laws). For Hume (et al) it would make no sense to ask for an explanation over and above this for the fact that there is a cannonball flying through the air; we can explain why the cannonball exists, why it is moving as fast as it is, and why it is going in this direction rather than that direction, all by referring to facts about the laws governing its movement along with the fact that it existed, where it was, how fast it was moving, and in what direction it was going at some previous time. Where the HECP makes any further explanation unnecessary, the PSR demands that there be an explanation for why there is a cannonball at all, for the PSR demands that there be an explanation of any contingent fact.

Keeping this distinction in mind, let us imagine a logically possible world W which had no beginning, but was just stretched out temporally infinitely in its past, and in which physical determinism is true. At any point in time in W, W could be said to have existed for an infinite number of some unit of temporal length (hours, days, milliseconds, etc.), call this unit T, so that it had no beginning in the sense that there is no first T in W.[6] Now, for any state of affairs in W picked out by any time tn, all the facts about that state of affairs can be explained by the facts which obtain in W at a slightly earlier time tn-1. So, for any state of affairs at any time in W, there is an adequate explanation for that state of affairs in terms of some other state of affairs at another time which deterministically brings it about. In W, the HECP is satisfied by the facts we have laid out, whereas the PSR requires a deeper explanation for the existence of W, and for contingent facts obtaining in W. The PSR reminds us that such explanations are possible, and this will help us to see that libertarian free will possibly coincides with physical determinism.

Bear in mind that all we need to do in order to demonstrate the compossibility of two propositions is to show that there is a logically possible world out there which satisfies both propositions. In W, physical determinism is satisfied. If in/at W there is at least one libertarian-free act (or, technically, even just one libertarian-free agent), then the compossibility of libertarian free will and physical determinism will have been logically demonstrated. Clearly, however, it is logically possible that the existence of W is explained by the voluntary election of a libertarian-free divine agent (i.e., God). If God, in a libertarian-free capacity, chose to create such a world, then the world and all of its happenings would ultimately be explained in terms of God’s acting freely to create it. Thus, physical determinism is clearly logically compatible with libertarian free will. This is because God is, ex hypothesi, not a material entity. Suppose, further, that people are not merely material entities (i.e., the mind is immaterial), but that epiphenomenalism is true of all embodied people, and the mind, which persists after bodily death, becomes libertarian free once freed of the body. So long as this is logically possible its very possibility goes to show that physical determinism is demonstrably logically compatible with libertarian freedom.

However, there may be another way in which libertarian free will is compatible with physical determinism, at least on the B-theory of time. Suppose that there is a set of physical states of affairs P, consisting of {P1, P2, P3… Pn}. Now, suppose that any Pn+1 follows from Pn of causal necessity (for closed physical systems).  Every physical state of affairs in P is causally explained by some other physical state of affairs in P. Nevertheless, it is logically possible that the sufficient reason for a state of affairs in P involves the fact that a libertarian-free agent in that world makes a libertarian-free decision Fn at some time tn. Here, we might schematize this relationship as follows:

P1 → P2 → P3 → P4
↑↑↑↑       ↑↑↑↑
F1           F2

So, although P2 is physically-causally explained (i.e., HECP explained) by P1, P1 and P2 may only be sufficiently explained (i.e., PSR explained) by appeal to F1 (which itself is sufficiently explained just in case it is logically possible that facts about libertarian-free acts can be sufficiently explained).[7] It may seem strange to talk about libertarian-free acts which occur, in some sense, independently of their space-time context (for, if they occurred within that context, then physics, as we’re imagining it, would provide the context and impetus for the decision, along with determining the decision), but certainly that’s no stranger than thinking of God’s choices as libertarian free even though they are independent of any space-time context.

There is also, perhaps, a stranger way in which we can conceive of this relationship of free choices in a space-time context and a physically deterministic world. I should note that I’m not entirely sure whether this is coherent (it may run into unforeseen problems which more extensive analysis could tease out), but my suspicion is that it is coherent (and, therefore, logically possible). We might imagine that the context in which a libertarian free choice is made is physically under-determinative, but that, once a free decision is made, the result is that the world is supplied physical properties which make that decision appear physically determined. Here we have to imagine that free decisions occur with a limited space-time context (an under-determinative one), and that backwards causation is possible (i.e., events from the future can cause things in the past). Then, we might imagine that even though a temporally antecedent state of affairs P1 causally determines that P2 occurs next, a person’s free choice after the time at which P1 is the case, and before the time at which P2 is the case, is the sufficient reason P1 has the causally determinative features it has for bringing P2 about. On this view, a libertarian free agent makes a decision in light of an under-determinative slice of P1, and their making a decision has temporally backwards-reaching effects which supply P1 with all the physical features necessary for it to deterministically bring P2 about. On this view, a libertarian free decision can be the sufficient reason why P1 deterministically brings P2 about, even though P2 is HECP explained adequately in terms of P1 alone. This view is strange only because we generally think of causal sequences as parallel with temporal sequences, but, at least on the B-theory of time, there is no reason causal antecedence and temporal antecedence need to go hand-in-hand; my free decision may (atemporally) cause features of the past, and maybe those features physically-deterministically cause events in the future.

The Temporal Sequence: P1 → Fn → P2

The (atemporal) Causal Sequence: P1* → Fn → P1 → P2

In conclusion then, we still have no good arguments for believing either in determinism per se, nor in physical determinism. Moreover, even if physical determinism were true, we would have, it seems, no good reasons to doubt the fact that we are libertarian free, at least if we accept the possibility of temporally backwards causation (and, therefore, the B-theory). This can more easily be seen when we distinguish the HECP from the PSR, and note the two different levels of explanations which satisfy them. The PSR needn’t be true, but explanations of the kind it demands, if even possible, carve out a space for libertarian-free decisions even in a physically deterministic world.

[1] For further reading on this point, see Koons, Robert C., and George Bealer, eds. The Waning of Materialism. Oxford University Press, 2010.

[2] I cited the Copenhagen theory of quantum mechanics, as well as John Norton’s now famous example of a ball on a dome (in the comments section, in response to a reader), which illustrates that even Newtonian mechanics is not entirely deterministic. I could easily have added (though I did not think to) that Newton’s laws were all stipulated for closed systems anyway, and it is no part of those laws as such to stipulate that the physical universe as a whole is a closed system, so that his laws cannot imply physical determinism. Newtonian physics did not preclude God’s intervention in the world, for instance, and this is precisely why Newton was not being inconsistent when he maintained both that his laws were true, and that God occasionally intervened in the physical world (for instance by providing the planets with an extra ‘push’ every now and again). This demonstrates clearly that Newton’s laws, even if they were deterministic for closed systems (which the ball-on-dome example disproves), wouldn’t come anywhere near to entailing physical determinism.

[3] Not all sound arguments are good arguments, for the soundness of an argument is neither a necessary, nor sufficient, condition of the goodness of an argument (just as the goodness of an argument is neither necessary nor sufficient for soundness). I will discuss this distinction in more detail in an upcoming post. For now, however, observe that if this argument were sound, then it would give us good reasons for accepting its conclusion, or at least for accepting premise 5.

[4] Calvinism requires a compatibilist view of free will and determinism in order to allow normative statements about what one ought or ought not to believe, but I’m not convinced that such accounts are even coherent. In fact, I am convinced they are not.

[5] Pruss quoted a passage from Hume, but I have provided a more extended excerpt of the same passage from Hume. David Hume, “Dialogues Concerning Natural Religion” in Modern Philosophy: An Anthology (Second Edition), Edited by Roger Ariew and Eric Watkins (Indianapolis: Hackett Publishing Company, 2009), 622.

[6] At first I thought, in passing, that if somebody had trouble with the idea of an infinite past I could just say that there was a world W* in which temporally backward causation (i.e., causation from future events to past ones) is the only kind of causal relation which obtains, and where every event is deterministically caused by some posterior event, and although the world has a temporal beginning, it has no temporal end. However, one should only be concerned with actually infinite regresses of past events if one is i) an A-theorist, or ii) worried about infinite chains of causes. If one is an A-theorist, they will not likely accept the possibility of backward causation anyway, and if one is, like me, worried about infinite chains of causes, then they will have the same problem with W* as they had with W. If you, like me, do have a problem with accepting that W is logically possible then either suspend your modal suspicions here for the sake of argument, or just notice that any length of time can be infinitely subdivided, so that over any measurable length of time it is logically possible that an infinite number of causes are at play just in case it is true that there is no particular time tn at which no cause can logically possibly obtain.

[7] I strongly believe they can be sufficiently explained, and this is because I adamantly reject the assumption that all explanations can be reduced to, or expressed by, entailments. However, I will leave off giving an account of this highly contentious position for now; the reader who disagrees with me is invited to accept the weaker conditional claim that if facts about libertarian free actions could be sufficiently explained, then any combination of libertarian free acts might figure into a sufficient explanation for precisely why the physical states of the universe are precisely as they are. However, notice that, for the purposes of my argument, the PSR needn’t be true, it just needs to be possible that there be an underlying explanation which goes beyond the demands of the HECP.