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.

Aren’t worlds comparably better or worse?

Si c’est ici le meilleur des mondes possibles, que sont donc les autres?
~Voltaire

I here mean to explore a paradox about comparing worlds with each other insofar as they are supposedly comparably better or worse. On the one hand I will maintain the alethic truism that there is no such thing as a best of all possible worlds, and conversely that there is no such thing as a worst of all possible worlds. To see why, consider that a best of all possible worlds is a world than which no better world could be conceived. However, the concept of a world than which no better could be conceived seems to be incoherent. Stephen T. Davis explains:

Take the notion of the tallest conceivable human. This notion is incoherent because, no matter how tall we conceive a tall human to be, we can always conceptually add another inch and thus prove that this person was not, after all, the tallest conceivable human. Just so, it may be argued, the notion of the best of all possible worlds is incoherent. For any possible world, no matter how much pleasure and happiness it contains, we can always think of a better one, i.e., a world with slightly more pleasure and happiness.[1]

Alvin Plantinga offers a more amusing illustration:

Just as there is no greatest prime number, so perhaps there is no best of all possible worlds. Perhaps for any world you mention, replete with dancing girls and deliriously happy sentient creatures, there is an even better world, containing even more dancing girls and deliriously happy sentient creatures. If so, it seems reasonable to think that the second possible world is better than the first. But then it follows that for any possible world W there is a better world W’, in which case there just isn’t any such thing as the best of all possible worlds.[2]

This truth seems indubitable once it has made its first impression on the mind, but it also leads to a conspicuous problem. To obviate the problem, we should turn first to St. Thomas Aquinas:

“The terms ‘more’ or ‘less’ only make sense if something is the maximum in a genus.”[3]

If there is neither a best nor a worst of all possible worlds, and if Thomas is right, then what sense can we make of calling some worlds better than others? Solutions do not abound. What I intend to do in this article is to survey some candidate solutions, and then share what has become my preferred solution to this problem inspired by St. Thomas Aquinas’ Quinque viæ (five ways), in particular from the fourth argument for God’s existence.

Gottfried Leibniz was the philosopher who introduced logically possible world semantics in the first place, and though he was clearly exceptionally brilliant, contemporary philosophers like Alvin Plantinga have raised their eyebrows high at Leibniz’ suggestion not only that there is a best of all logically possible worlds, but that this is it! Plantinga has, in fact, taken to calling this ‘Leibniz’ lapse.’[4] Although amusing, this charge has been criticized by those attempting to protect Leibniz’ good name. Thus, for example, Dr. George Gale, one of my philosophy professors at Concordia, has attempted an answer to the following effect: he has argued that, at least for Gottfried Leibniz, “this most perfect, best of all possible worlds is so only in accordance with a mathematical formula [relating simplicity of laws to abundance and variety of phenomena], and not in accord with our normal, everyday, Candidate-like notions of perfection, i.e., moral ones.”[5]

Thus, Plantinga’s criticism is guilty of an equivocation, since Plantinga must have something other than Leibniz’ notion of better-making properties in mind. As an aside, it seems to me that Leibniz’ solution is open to a deeper objection from William Lane Craig and possibly also from Alexander Pruss,[6] who have both concluded that it is absurd to posit an actually infinite number of contingent beings. Leibniz needs for the aggregate of all contingent beings, his ‘monads,’ to be actually infinite in number. Leibniz’ eclectic notion of better-making properties notwithstanding, however, unless one is inclined to think that Leibniz is right about better-making properties, it seems that Plantinga, Davis and others like them have posed an indissoluble difficulty. One, at least, for which a plausible and satisfying answer will not be found in Leibniz’ work.

One could, of course, simply bite the bullet and admit that what makes one world better than another is merely a matter of taste, and that the ascriptions ‘better’ or ‘worse’ express nothing more profound than preference. This solution is likely to be only as satisfying as moral nihilism is, and for the same reasons. A world in which a thousand more pregnant women get kicked in the stomach than another world seems, ceteris paribus, a much worse world relative to the other, and its being so isn’t simply a matter of taste or convention, like the way a world with a thousand fewer key-lime pies than ours seems worse than ours to me.

One could object to Plantinga that, though Leibniz does not have the right conception of what properties make worlds better, neither does Plantinga (at least, not as reflected in his thought experiment). Thus one can argue that Utilitarian standards of better-making properties are simply the wrong standards in the same way as Leibniz’ mathematical standards are; a world with more deliriously happy sentient creatures may be no better for it. Perhaps there are some other standards (known or unknown, discernible or indiscernible) which, like Leibniz’ standards, admit of a maximally good (or bad) world. This solution also strains credulity, however, as one need not be a Utilitarian to concede that, all things being equal, a world with more deliriously happy creatures really is better. Moreover, some Utilitarians could argue that two worlds in which the same average happiness obtains are really just as good as each other, so that the addition of more deliriously happy sentient creatures makes no calculable difference to how good a world is. So, Plantinga’s suggestion is about as far from being Utilitarian as he is. Moreover, for just about any standards it seems that one can simply run a parody of the kinds of arguments presented by Plantinga and Davis – perhaps even Leibniz’, if there really are different sizes of infinity and no greatest size of infinity. I am familiar enough with set-theory to know that there are different sizes of infinity, since some infinite sets cannot be bijected into others, but I am not familiar enough with set-theory to know if there is a species among these different ‘sizes’ of infinity than which no greater size exists. Either way, perhaps somebody opting for this Leibnizian avenue could argue, in a fashion similar to our hypothetical Utilitarian above, that once a world has the quality of instantiating an actually infinite number of desideratum it can no longer be meaningfully called ‘worse’ than any other world, even if that other world has ‘more’ desiderata.

This leads to another solution which I find not altogether unattractive. Perhaps instead of talking about a single ‘best’ or ‘worst’ of all possible worlds, one can identify a class of worlds than which no greater world can be conceived, and then speak of this class of worlds as the standard against which the goodness of worlds is measured. Any world with an actually infinite number of desideratum would surely be a world than which no greater could be conceived (I put no stock in any distinction, here, between conceivability and ‘possibility’ simpliciter).

I see two problems with this recommendation. First, due to the looseness of the definition, the set of all possible worlds would qualify as a set of worlds than which there could be no better world. Second, however, in order to isolate a class of worlds than which no better world could be conceived, and other than which every world is worse, seems to require positing an actual infinity of some sort, and we are led straight back to the problem posed by Dr. Craig.

Given that the impossibility of an actually infinite number of beings seems to pose such a problem for talk of best/worst possible worlds, perhaps one could run the following response in the form of an argument:

  1. If there is a best/worst of all possible worlds, then it includes an actually infinite number of beings and/or (better/worse-making) properties.
  2. If there is no best/worst of all possible worlds, then no world is better or worse than any other(s).
  3. But, at least some worlds are better or worse than some other(s).
  4. Therefore, there is a best/worst of all possible worlds.
  5. Therefore, there is at least one world which includes an actually infinite number of beings and/or (better/worse-making) properties.

I don’t find that satisfying myself, but that’s because W.L. Craig has sold me on thinking that it is logically impossible (indeed clearly incoherent) to talk about any actually infinite aggregate of contingent beings (I note in passing that he has not sold me on the idea that there couldn’t be an actually infinite number of events, anymore than there couldn’t be an actually infinite number of true propositions – to reify either of these into quasi-beings is a mistake, and though I would agree that if time were tensed there could not possibly be an actually infinite number of past events, I am adamantly a B-theorist).

A philosopher named Jean David Robert brought one solution to my attention which attempts to show that the objection to there being better or worse possible worlds is based on the presumption that there are better and worse possible worlds, and thus the objection cannot go through. Scilicet, the objection only works if the objection fails. He explains:

If it’s true that “there is no such thing as the best of all possible worlds because one can always conceive of a better world,” then it’s false that “one can never conceive of a better world because there is no such thing as the best of all possible worlds.” […]
Consider the following counterexample: on my desk, I have a potentially infinite number of rulers of different lengths. In other words, I have one potentially infinite ruler. I also have two rulers of different finite lengths. I compare the length of these two rulers using the potentially infinite ruler, and determine that one of the finite rulers is 1 cm longer than the other. Now imagine that the rulers are in fact possible words and the length of the rulers correspond to the objective value of these possible worlds. We can see that it does make sense to speak of one possible world being objectively better than another.

I like this answer; it has an almost Moorean quality to it. An actually Moorean answer may also be provided; perhaps we all know that some worlds are better than others, and we are surer of this truth than we are or can be sure of all the clever arguments against it. However, the trouble with the Moorean answer, and in a subtler way the trouble with J.D. Robert’s answer, is that it doesn’t actually help us make good sense of ‘better’ or ‘worse’ possible worlds. It may help us sleep at night, but it doesn’t get us anywhere. J.D. Robert’s doesn’t precisely because his potentially infinite ruler consists of the indefinite put-together of differently sized rulers, but for any ruler to have a size relative to any other it must be in principle comprised of commensurable units of length. However, to say that there are in principle commensurable units of length is just to say, pace the metaphor, that there is some standard against which these worlds can be compared so as to make one ‘better’ than another.

All of the aforementioned solutions seem to leave me high and dry. None of them seem to me to represent a plausible answer to the question ‘how can we meaningfully say of one world that it is better or worse than any other?’ I have come to think, however, that there may be a natural theological solution to the problem. Once again, I turn to St. Thomas Aquinas:

Among beings there are some more and some less good, true, noble and the like. But “more” and “less” are predicated of different things, according as they resemble in their different ways something which is the maximum, as a thing is said to be hotter according as it more nearly resembles that which is hottest; so that there is something which is truest, something best, something noblest and, consequently, something which is uttermost being; for those things that are greatest in truth are greatest in being, as it is written in Metaph. ii. Now the maximum in any genus is the cause of all in that genus; as fire, which is the maximum heat, is the cause of all hot things. Therefore there must also be something which is to all beings the cause of their being, goodness, and every other perfection; and this we call God.[7]

This may sound appealing to Theists like me, but presumably we shouldn’t be satisfied with merely gesturing in the direction of Theism, as though saying ‘God!’ loudly enough solves every problem. How can one cash-out this solution?

First, it is clear that if God exists, then his nature is identical to ‘the Good.’ In fact, if God exists then it seems like the only way to intelligibly predicate anything of him will have to avoid univocal predication just as much as equivocal (indeed, in God’s case, univocal predication is equivocal). Thus, we can predicate things of God in two ways: either by analogy, or metaphorically. We can say that God is our king metaphorically, while we can say that God exists or is good analogously. I will spare myself the trouble of having, here, to explain St. Thomas’ whole philosophy of language. I will, instead, take the liberty of presuming that the reader is at least relatively familiar with Thomistic philosophy of language.[8] God, ex hypothesi, is clearly the bearer of superlative attributes which serve as the paradigms of those attributes insofar as they are identified as instantiated in the world. In other words, for any predicate P, if P is an intrinsic superlative attribute of God, then God’s nature serves as the paradigm according to which P is predicated of contingent beings. Thus, if a being is good, it is good to the extent that it imitates (or intimates) the nature of God. If a thing is beautiful, it is beautiful to the extent that it intimates the pleasure of ‘seeing God’ (note that beauty is defined by Aquinas as that which, upon being seen, pleases).

I suspect, therefore, that when we say one logically possible world is better or worse than another, we mean that it is better or worse in the very same (or, at least, similar enough) sense as one person may be better or worse than another. Clearly, though, the Theist (at least of the Thomistic variety) will say that one person is good to the extent that they intimate God.[9] They are virtuous to the extent that their character intimates the character of God. A possible world, therefore, is good to the extent that it intimates the nature of God (i.e., to the extent that God’s nature is intimated in that world). This may mean that it reflects God’s moral goodness as well as his justice, his wrath as well as his mercy. Thus, the suggestion is that one possible world W is better than some other world W’ just in case it better intimates the nature of God.

I am convinced that this answer is not only appealing, but exactly right. In fact, I am tempted to make an argument of it for Theism. I will end this article with a brief sketch of how this argument is likely to go:

  1. If some possible worlds are better/worse than others, then either (i) there is a best/worst of all possible worlds which acts as the standard against which the goodness of worlds is measured, or (ii) there is a class of best/worst of all possible worlds which acts as the standard against which the goodness of worlds is measured, or (iii) God’s nature serves as the paradigmatic standard against which the goodness of worlds is measured.
  2. Some possible worlds are better/worse than others.
  3. There is no best/worst of all possible worlds which acts as the standard against which the goodness of worlds is measured.
  4. There is no class of best/worst of all possible worlds to act as the standard against which the goodness of worlds is measured.
  5. Therefore, God’s nature serves as the paradigmatic standard against which the goodness of worlds is measured.
  6. If God’s nature serves as the paradigmatic standard against which the goodness of worlds is measured then God’s nature exists.
  7. Therefore, God’s nature exists.
  8. If God’s nature exists, then God exists.
  9. Therefore, God exists.

Somebody may wish to wiggle out of this argument by splitting the horns of the trilemma in the Major premise, for instance by suggesting that moral Platonism may be a fourth alternative. However, the argument could be appropriately amended by changing the first premise to include the supposed alternative, and then we could insert a ‘premise 4.1’ which denied that moral Platonism is a solution.


[1] Stephen T Davis, ed. Encountering Evil [New Ed]: Live Options in Theodicy. (Cokesbury.com, 2001): 75.

[2] Alvin Plantinga, God, Freedom, and Evil. (Wm. B. Eerdmans Publishing, 1974): 61.

[3] Apologies to the reader: I can’t seem to find this quote in St. Thomas’ works, and I’m not sure where it came from either – it may not have come from the Summa Theologiae. It’s in one of his writings, somewhere.

[4] Plantinga, Alvin C. “Which worlds could God have created?.” The Journal of Philosophy 70, no. 17 (1973): 548.

[5] Roger Stuart Woolhouse, ed. Gottfried Wilhelm Leibniz: critical assessments. Philosophy of mind, freewill, political philosophy, influences. Vol. 4. (Taylor & Francis, 1994): 453.

[6] Alexander Pruss, “Probability on Infinite Sets and the Kalaam Argument” http://alexanderpruss.blogspot.ca/2010/03/probability-on-infinite-sets-and-kalaam.html

[9] Interesting thought: if God had not incarnated setting the paradigmatic standard of a best of all possible persons, could people still be (have been) meaningfully said to be better than others? The answer is, obviously, bound up with the suggestion I am here in the business of elaborating. It could, I think, make sense, even without a best of all possible men, just in case the measure of a man’s goodness is the degree to which he intimates the divine nature.

[10] Alexander Pruss, “One Thing I have Learnt from Hume” http://alexanderpruss.blogspot.ca/2007/12/one-thing-i-have-learned-from-hume.html