Comparing the resurrection rate of humans to the resurrection mendacity rate

Hume argues against miracles by means of his balancing principle:

• (HBP) You should believe p on the basis of testimony only if p is at least as probable as the falsity of the testimony.

There are two interpretations of HBP, depending on whether “probable” refers to the prior probabilities (the probabilities before the evidence of the testimony is accounted for) or posterior ones (the probabilities after the evidence has been weighed). On the posterior interpretation, HBP is almost completely obvious (at least if the “should” is that of epistemic normativity). On the prior interpretation, HBP is well-known to be false: the standard counterexample is that it’s reasonable to believe that you won the lottery on the basis of a newspaper report of the winner even if the chance of a newspaper error exceeds your chance of winning the lottery.

I think the prior interpretation fits Hume’s text better, even if it’s bad epistemology.

In this post I want to suggest that there could be reasonable assignments of priors for a theist on which the prior probability of the falsity of the testimony is less than the prior probability of the miracle.

Assume we are theists. Take the resurrection of Jesus. First, let’s say something about the prior probability of the resurrection of a human. Given theism, there is a good God, and it wouldn’t be surprising at all if there were resurrections. In fact, we might expect it from a loving God. But how often would they happen? What is the resurrection rate in human beings? Well, here we need to turn to empirical data. Let’s grant Hume that apart from the case under examination, there are no resurrections. There have been approximately a hundred billion human deaths, so we have an upper bound on the resurrection rate of less than one in 10¹¹. It’s not unreasonable, I think, given the moderate prior probability that someone would be resurrected, and the lack of resurrections in 10¹¹ cases, to suppose the probability of a particular person getting resurrected would be something like (1/2) ⋅ 10⁻¹¹.

But what is the probability of false testimony? Well, as an initial back of the envelope calculation, suppose we have 11 witnesses, and each has an independent 1/20 chance of lying or being mistaken that Jesus was resurrected. So, the chance that they all lied or were mistaken would be (1/20)¹¹ or (1/2048) ⋅ 10⁻¹¹.

With these numbers, the prior probability of Jesus getting resurrected is about 100 times bigger than the prior probability of the 11 witnesses lying that he was resurrected. And so even in its prior probability formulation, HBP doesn’t destroy the testimony to the miracle.

Of course the numbers are made up. Probably the main problem has to do with the assumption of the independence of the witnesses. But that problem is to some degree balanced by the fact that 1/20 is way too high for a probability of lying or being mistaken that they witnessed a resurrection. (What percentage of the people you know testified to witnessing a resurrection?)

In any case, I think the above shows that it is far from clear that, assuming theism, a reasonable estimate of the resurrection rate of humans would be lower than a reasonable estimate of the resurrection mendacity rate for groups of 11 people.

Now what if we don’t assume theism, but assume, say, a 1/10 chance of theism? Well, that approximately cuts our estimate of the resurrection rate of humans by a factor of 10. But that’s still not enough to make it clear that the resurrection rate of humans is less than the resurrection mendacity rate for groups of 11.

God's timelessness, the A-theory of time, and two kinds of Cambridge change

Classical theism holds that God is timeless and knows all objective truths. According to A-theories of time, objective truths change (e.g., what exists simpliciter changes on presentism, and on other A-theories at least what time is objectively present changes). There is a prima facie conflict here, which leads some classical theists to reject the A-theory of time.

But there is also a widely accepted reply. Classical theism also holds that God is simple. One of the consequences of divine simplicity is that if God had created a different world, he wouldn’t have been any different intrinsically—and yet he would know something different, namely that he created that world rather than this one. Seemingly the only good solution to this problem is to suppose that God’s knowledge is in part extrinsically constituted—that facts about what God knows about contingent things are partly constituted by these contingent things.

But the same move seems to save timelessness and the A-theory. For if God’s knowledge is partly extrinsically constituted, then as the created world objectively changes, as the A-theory holds, God’s knowledge can change without any intrinsic change in God. Basically, the change of God’s knowledge is only a Cambridge change in God—a purely relational change.

I have always been pulled two ways here. Since I accepted divine simplicity, the response seemed right. But it also seemed right to think there is a tension between God’s timelessness and the A-theory of time, thereby yielding an argument against the A-theory.

I haven’t settled this entirely to my satisfaction, but I now think there may well be an argument from classical theism against the A-theory.

First, note that the extrinsic constitution move is aimed not specifically at a tension between God’s timelessness and the A-theory, but at a tension between God’s immutability and the A-theory. The move shows how an immutable being could have changing knowledge, because of extrinsic constitution. But while any timeless being is immutable, the other implication need not hold: timelessness is a stronger condition than immutability, and hence there could be a tension between divine timelessness and the A-theory even if there isn’t a tension between immutability and the A-theory.

Here is why I see a tension. The crucial concept here is of a merely relational change, a Cambridge change. The most common example of a Cambridge change is something like:

1. Bob became shorter than his daughter Alice.

Here, we’re not supposed to think that Bob changed intrinsically, but simply that Alice got taller!

But there is another kind of change that I used to lump in with (1):

2. Dinosaurs became beloved of children around the world.

Both are, I suppose, Cambridge changes. But they are crucially different. The difference comes from the fact that in (1), the change is between the slightly younger Bob being taller than Alice was then and the slightly older bob being sorter than Alice was then. While the change was due to Alice’s growth, rather than Bob’s shrinkage, nonetheless it is crucial to this kind of Cambridge change that we be comparing the subject at t₁, considered relationally, with the subject at t₂, again considered relationally. It is, say, the 2018 Bob who is taller than Alice, while it is the 2023 Bob who is shorter than Alice. I will call this kind of thing strong Cambridge change.

But when dinosaurs become beloved of children around the world, as they did over the course of the 20th century, this wasn’t a change between earlier and later dinosaurs. Indeed, the dinosaurs were no longer around when this Cambridge change happened. I will call this kind of thing weak Cambridge change.

Strong Cambridge change requires an object to at least persist through time: to be one way (relationally) at one time and another way (again, relationally) at another. Weak Cambridge change does not require even that. One can have weak Cambridge change of an object that exists only for an instant (think of an instantaneous event that becomes notorious).

A timeless being can “undergo” weak Cambridge change, but not strong Cambridge change. And I suspect that change in knowledge, even when the knowledge is extrinsically constituted, is strong Cambridge change.

Here is a piece of evidence for this thesis. Knowledge for us is partly extrinsically constituted—if only because (I am grateful to Christopher Tomaszewski for this decisive point) what we know has to be true, and truths is typically extrinsic to us! But now suppose that I have a case where the only thing lacking to knowledge is truth—I have a belief that is justified in the right way, but it just happens not to be true. Now suppose that at noon the thing I believe comes to be true (here we are assuming the A-theory). If we set up the case right, I come to know the thing at noon, though the change is a strong Cambridge change. But suppose that at noon I also cease to exist. Then I don’t come to know the thing! To come to know something, I would have to persist from not knowing to knowing. Prior to noon I was such that if the thing were true, I’d know it, but the thing isn’t true. After noon, I don’t know the thing, even though it isn’t true, because I don’t exist after noon. Change in extrinsically constituted knowledge seems to be at least strong Cambridge change.

Further, think about this. When God knows p in one world and not-p in another, this transworld difference is a difference between how God is in the one world and how God is in the other world, even if it is a relational difference. Similarly, we would expect that if God changes from knowing p at t₁ to knowing not-p at t₂, God exists at t₁ and also at t₂. And this does not seem to fit with God’s timelessness. (But don’t classical theists say God is omnipresent, and shouldn’t that include omnitemporal presence? Yes, but omnitemporal presence is not omnitemporal existence.)

In other words, I think for God to change in knowledge in lockstep with the objective facts changing, God has to exist in lockstep with these objective facts. To change from knowing to not knowing some fact due to the change in these facts, one needs to be a contemporary of these changing facts. And a timeless being is not (except should there be an Incarnation) a contemporary of anything.

In summary: A timeless being can only undergo weak Cambridge change, while it is strong Cambridge change that would be needed to maintain knowledge through a change in objective truths, even if that change is extrinsically constituted. One can uphold the A-theory with a changeless God, but not, I think, a timeless God.

Or so I suspect, but I am far from sure, because the distinction between weak and strong Cambridge change is still a bit vague for me.

And even if my specific arguments about God aren't right, I think the weak/strong Cambridge change distinction is worth thinking about.

The fleetingness of being

Imagine it’s the last moment of time. What’s next for you? Nothing! It’s a terrifying time, but it’s one that’s hard to describe well. Phrases like “You’re about to perish” don’t fit it logically, because they imply that you will perish, but at the last moment of time there is no “will”. You need awkward wide-scope negations like: “It is not the case that you will continue to exist.”

But I think the philosophical puzzles go beyond the choice of words.

Thing about a world where time begins and ends with t₁, where there is only one moment. That’s a world with no flux or flow or dynamism or change. It seems, then, that that’s a world where essentially temporal attitudes, like fear of ceasing to exist, are inappropriate. It doesn’t, it seems, to be a world where it’s right for you to feel the terror of facing nothingness. Indeed, it doesn’t seem like anything in this world is fleeting or lasting.

But the difference between the only-one-moment and last-moment scenarios is just with regard to the past. Now in the last-moment scenario you would have a reasonable (pace Epicurus) terror of impending nonexistence and a vivid feeling of the fleetingness of existence.

But taking away the past, and hence moving to the only-on-moment scenario, shouldn’t change any of that! It doesn’t make your existence last any longer. It makes you no more eternal. We have to be able to say that somehow in the only-one-moment world our existence would be tenuous and fleeting (indeed, it seems, maximally so).

This pulls us to a very deep conclusion here:

1. The phenomenon of fleetingness does not require the flow of time.

For in the only-one-moment world we have fleetingness but no flow.

So if we are to look at what grounds the fleetingness of our existence, it seems we must look away from the distinctive resources of the A-theory of time, and towards the B-theory.

One obvious thing to say is that there is an incompleteness to our existence when restricted to any finite compass. Eighty years is not enough for the kind of being we are, and a moment is much less. This is something an eternalist can say, whether or not they accept the A-theory or the B-theory of time. Though it’s not quite so clear that a presentist or Growing Blocker can say it, since on their views our future life is not a part of reality anyway, no matter whether it is finite or infinite.

But perhaps there is a resource available for the A-theorist, even the presentist. Instead of thinking that it is the present moment that is present, we can suppose that what is present is an interval between two succeeding times in a discrete account of time. If so, then neither the only-one-moment and last-moment scenarios work. Instead, one has only-one-interval and last-interval scenarios. And these are not so problematic. Even if there is only one interval of time, that’s enough for change and flow—things move from one state to another over an interval. The impending doom has to do with the fact that the later end of the present interval borders nothingness. And over that interval, we can say (if we have a flowy theory of time) that we are flowing—but not for long!

Of course, there are technical issues with the suggestion that what is present is an interval between two successive times. If there is flow during that interval, it sees can always ask: “How long before the interval is finished?” But any clear answer to that subdivides the interval and places us at a moment within it. So we must refuse to countenance any answer beyond: “I am flowing from t_n to t_n + 1.” (We might then say: We’re between 0 and t_n + 1 − t_n units of time before the next interval begins.)

I started thinking about an A-theory on which what is present is an interval just as an exercise in wacky theories of time. I am now thinking that perhaps this is the best version of presentism.

What has form?

On the question of what has a substantial form, I have tended to think something similar to van Inwagen’s answer to the question of what wholes there are. Namely, I assign form to:

1. organisms, and

2. fundamental objects in physics that are good candidates for being substances.

Regarding 2, if the correct physics is particle-based (which I doubt, in light of the apparent possibility of the world being in a superposition of states with different numbers of particles), these will be particles, or at least those particles that aren’t part of an organism. If the correct physics is field-based, the substances in physics will be fields (or maybe just one field-like object, namely “the global wavefunction”).

A lot of Aristotelians have substances, with forms, that are intermediate between (1) and (2), such as hydrogen atoms or water molecules or chunks of iron, and maybe astronomical objects like stars or galaxies. While I don’t have a knock-down argument against such substances, I also don’t see any reason to posit them.

My reasons for positing form for organisms and fundamental physical objects are quite different. For organisms, the reasons are largely normative. Parrots and oak trees can flourish or languish; they have ends and proper functions. In the case of humans, the normativity extends much further. Furthermore, we need well-defined boundaries for organisms for ethical reasons—there is reason not to harm an organism, especially but not only a human one—and there need to be well-defined persistence conditions for humans for moral responsibility. Something needs to ground all this. And the best candidate is form.

It is a central commitment of Aristotelianism that all of physical reality is grounded in physical substances and their accidents. But it is false that all of physical reality is grounded in organisms. There was a time when the physical universe had no organisms. So we need other substances. The fundamental objects of physics are the best candidates. They are active and have very clear kind-boundaries. The electromagnetic field is a different kind of thing from the gravitational field (which is just spacetime, according to Einstein). Photons are clearly different from electrons. (Though if it turns out that particle number is indeterminate, then particles won’t be the fundamental objects of physics.)

Granted, it is not obvious (and somewhat counterintuitive) that organisms have well-defined kind-boundaries and identity conditions. And it is not obvious (and somewhat counterintuitive) that fundamental physical objects have norms. But here I just take these to be consequences of the theory. Organisms have well-defined kind-boundaries and identity conditions, but we don’t know where they lie. Fundamental physical objects have normative properties, but I suspect they are perfect instances of their kind, and always do exactly what they should (C. S. Lewis says something like that in Mere Christianity).

Neither of my two reasons applies much to objects like atoms, molecules, chunks of stuff, or astronomical objects. There is no strong independent reason to suppose that they have normative properties in their own right, and their boundaries are, if not quite as fuzzy as those of organisms, pretty fuzzy. How far apart do I get to move a hydrogen atom from two oxygen atoms before I destroy a water molecule? How many sodium and chloride ions do I add to water to change it from water with impurities to a salt solution? (I suppose the concept of impurity pulls in the direction of thinking there are normative properties. But here is a reason to think this is mistaken. If impure water is languishing, then we have reason to distill water independently of any practical benefit to any organism, just for the sake of the water itself. That seems absurd.)

That the reasons don’t apply doesn’t show that there aren’t other reasons to posit substantial forms for these other candidates. But I don’t see such reasons. And so we can apply Ockham’s razor.

Wacky reality dynamics

The dynamics of reality over time vary between theories of time. On growing block, at each moment a new present slice is added to reality. On presentism, at each moment the formerly present slice is subtracted from reality and a new present slice is added to replace it.

But once we admit that there are at least these two such “reality dynamics”, other possibilities show up.

Centisecondism: At each moment, the slice one millisecond (i.e., 0.01 seconds) in the past (if there is one) is subtracted from reality, while a new present slice is added. As a result, except during an initial one centisecond warmup when reality is just growing, reality is always a one-centisecond thick chunk. Centisecondism is superior to presentism in multiple ways. First, it seems hard to fit consciousness in an infinitesimally long reality, as in presentism, but a centisecond is good enough. Second, a centisecond is long enough for diachronic causal relations to be unproblematic. Third, presentism suffers from the problem that on presentism we never see reality. Because of light-travel times, we always see the past, and the past is not real! On centisecondism, we have a chance to see reality as it is.

Of course, a centisecond is arbitrary. The actual slice thickness could be bigger or smaller. It may seem ad hoc what it is. But it’s no more ad hoc than, say, the fine structure constant or any other constant in the laws of nature. If there is a God, he can decide on the slice thickness, in his wisdom, just as he decides on the fine structure constant. If there is no God, the thickness constant can be brute.

Eschatological growing block: Presentism is true right now: reality is one-moment thick. But then comes an eschaton. At the eschaton, suddenly all those past slices that had disappeared due to presentism pop back into reality, and we stop subtracting from reality, and begin to just add. Now we have growing block. This could give us a kind of transcendent outlook on the past in the eschaton. The eschatological growing block has the interesting consequence that being-real-at is not a symmetric relation. For instance, the time of the eschaton is not real at 2023, but 2023 is real at the time of the eschaton. This may seem strange, but in fact is true on any growing block theory.

Eschatological eternalism: Presentism is true right now. But eternalism starts to be true at the eschaton—at the eschaton not just one moment, and not just the past, but the whole past, present and future pop into reality. This provides a kind of temporalized version of Leftow’s model of a timeless God’s relation to a presentism time—the beings in the eschaton have an eternalist relation to our presentist time.

One might think that these theories require hypertime. That is not true for centisecondism or eschatological growing block, because centisecondism and eschatological growing block have room for defining the present moment without moving to hypertime. The present moment on both theories is just the leading the edge of reality. On eschatological eternalism, if we could get in a moving spotlight, then we could define a present moment. (Or could we have an eschatological eternalism on which “at the eschaton” all the “past, present and future” are actually present?)

I think centisecondism and eschatological growing block are both coherent if standard growing block is coherent. If I were a growing blocker, I think I would think that God could make a world where presentism or centisecondism or eschatological growing block are true, or almost true (by that I mean that in those worlds there wouldn’t be time, but time^*).

But I am B-theorist eternalist, and I am just giving all these stories for fun. I suspect that they are all ultimately impossible, as are presentism (of a standard sort) and growing block.

Two implications of Aristotle's theory of time and locality

Aristotle thinks that time is infinitely subdivisible but only finitely subdivided. Thus, there will be moments t₁ < t₂ such that there is actually no moment between t₁ and t₂, but there could have been. What would make there have been a moment between t₁ and t₂? Presumably, this would be if something happened strictly between those times. Time is the measure of change, so if, say, some object started or finished changing at a time between t₁ and t₂, then there would have been a time between them, say t_1.5.

But here is a curious consequence. Suppose that in the actual world, w₀, I am living from t₁ to t₂, which are so close together in time that there are no time between them. But in another world, t₁, where everything in our galaxy was the same, in some other galaxy indeterministically an event happened between t₁ and t₂, namely at t_1.5. Then:

• In w₀, it is not true that I exist at t_1.5 (because there is no t_1.5).

• In w₁, it is true that I exist at t_1.5.

And what is responsible for that difference is that indeterministic event in another galaxy. So it seems that something in another galaxy is responsible, in a faster-than-light way, for whether I exist at t_1.5. In other words, the Aristotelian theory seems to imply highly non-local influences.

There is perhaps a way out. Perhaps fundamentally time sequences are internal to substances. Thus, I have a time sequence internal to me, you have one internal to you, and things in that other galaxy have time sequences internal to them. There are, additionally, connections (probably causal ones) between objects that allow one to form a global time sequence. That global time sequence will include moments that don’t correspond to any moments internal to me. For instance, it will include moments earlier than my conception, but more interestingly, it could be that for me t₂ immediately succeeds t₁, but something else has a time that fits between t₁ and t₂, and so global time could have times corresponding to t₁ and t₂, but also some intermediate time between them.

The difference between w₀ and w₁, then, would not be a difference in my internal time sequence. What happened in that other galaxy wouldn’t affect my internal time except perhaps once the light from that galaxy could reach me.

On this account, while it is true that I exist at t_1.5, my existing at t_1.5 is not an intrinsic feature of me. The difference between my existing at t_1.5 in w₁ and my not existing at t_1.5 in w₀ is a merely Cambridge difference.

I think it is hard to make this story fit with presentism. When t_1.5 is present, then it had better be intrinsic to me that I exist presently, i.e., at t_1.5. A similar point applies to growing block.

Maybe, though, there is a way of making this story fit with a moving spotlight A-theory. We could suppose that at global time t_1.5, what is “lit up” by the spotlight is the time t_1.5 for the thing in the other galaxy that has something happening to it then, but for me what is lit up is the entire interval between t₁ and t₂.

If I am right, then

1. Locality, and

2. Aristotle’s theory of time

seem to imply:

3. Internal time is primary

4. Eternalism is true.

Skepticism and a causeless beginning to the universe

Suppose that the universe began with the Big Bang, and the Big Bang had no cause, so the universe came into existence ex nihilo. It then went through many (infinitely many if time is continuous) stages—the one-second-old universe, the ten-second-old universe, the five-minute-old universe, the one-year-old universe, the 7-million-year-old universe, before arriving at the present 13.7-billion-year-old universe, and then (presumably) going onward.

Question: Why did the universe start in the Big Bang state rather than in one of these many other states?

Assuming the universe has no cause, there does not seem to be any compelling reason to think the Big Bang state is somehow “more likely” as a starting state of the universe. (If anything, it seems a less likely state, because it has lower entropy than the later states.)

Now, granted, if the universe started in a sufficiently “late” (as compared to our state) stage, there would be no observers in that universe other than perhaps Boltzmann brains, due to the universe having expanded too much. So we have an interesting bit of fine-tuning: the universe started in a sufficiently early stage that there was still time for life.

Let’s ignore that otherwise intriguing observation, however, and push ahead. Let S_t be the t-units-of-time-past-Big-Bang stage of a universe like ours. And let’s ask:

• Conditioning on all our observations, and given that the universe has no cause, is it more likely that the universe starts with S₀ (i.e., starts with the Big Bang) than with, say, S_t1, where t₁ > 0?

If the answer is negative, then the supposition of a causeless universe leads to a skeptical hypothesis: we have no more reason to think the universe started with the Big Bang than that it started in the t₁ stage.

The answer may be positive in some cases. As we already saw, if t₁ is past the time of there being any observers—or just past the time of there being any observers making observations like ours—then we can be sure the universe didn’t start in S_t1.

If it turns out that evolutionary is logically necessary for consciousness, as some materialists think (because, basically, evolutionary history is necessary for defining proper function for making functionalism run), we might get a positive answer if t₁ is so close to the present that there was insufficient evolutionary history for our consciousness if the universe started at t₁.

But if t₁ is any time between 0 and about 3.7 billion years ago, our observations do not differentiate between the hypothesis that the universe started with S₀ or that it started with S_t1. Thus, the answer to the bulleted question above is negative. Moreover, it’s negative for each of the many t₁ in the relevant range (more than 0 but less than about 10 billion years). Thus, on a no-cause hypothesis we should not prefer the Big Bang hypothesis over a whole bunch of skeptical hypotheses holding that the universe started in a more developed form.

This argument will please theists who think the universe had a cause. Of course, the theist still has to find a way to give a positive answer to the bulleted question. I think there are two ways of doing so. First, likely a perfect being would have a preference for greater creaturely participation in creation, and creating the universe in, say, the 10 billion year stage, or indeed in any post-Big Bang stage, would pointlessly cut short creaturely participation in creation (e.g., in the 10 billion year stage, the Solar System would be created ex nihilo, rather than being generated from a star-forming nebula). Second, such a being would be likely to create a world that doesn’t make true what is intuitively to us a “skeptical hypothesis”.

The argument will also please the now-less-common atheist who thinks the universe is eternal.

Losing track of time

Suppose that the full saturated truthbearers are tensed propositions (which I think is essentially what the A-theory of time comes to). Now consider an atom with a half-life of a week. I observe the atom exactly at noon on Monday, and it hasn’t decayed yet. I thereby acquire the belief that the atom has not decayed yet. Now suppose that for the next week I stop changing in any relevant respect, and maintain belief in the same truthbearers, and the atom doesn’t decay. In particular I continue to have the tensed belief that the atom hasn’t decayed yet.

But an odd thing happens. While my belief is reliable enough for knowledge initially—it has a probability 0.9999 of remaining true for the first minute after observation—eventually the reliability goes down. After a day, the probability of truth is down to 0.91, after two days it’s 0.82, and after a week, of course, it’s 0.5. So gradually I lose reliability, and (assuming I had it) knowledge, even though nothing relevant has changed in the world in me or around me.

Well, that’s not quite true. For something seems to have changed: my observation has “gotten older”. But it’s still kind of odd—the time slice of the world is relevantly the same right after the observation as a week after.

Partial eclipse

Handheld, through welder's glass

Hanheld, through eclipse glasses, with a whiff of sunspots (I'll have to get proper solar film for total eclipse photography this spring)

Consciousness and AI

Here are three interesting intuitions (or maybe evan data points) worth chewing on:

1. Large language models are smarter than squirrels.

2. Large language models are not conscious.

3. Squirrels are conscious.

I feel—without having formulate a rigorous argument—that these three data points rather nicely support Ben Page’s thesis that computation-without-consciousness is what we would expect non-theistic evolution to yield.

Reducing binary distance relations to unary properties

Some philosophers say that space is fundamentally constituted by points. Others that it is fundamentally constituted by regions, and points are logical constructions out of regions. Here is an interesting advantage of an approach base on regions. Relations are more mysterious that properties. A point-based account is likely to involve distance relations: x and y are α units apart.

But a region-based account need not suppose a distance relation, but a diameter property. Intuitively, the diameter of a region is the largest distance between two points in the region, and hence is defined in terms of a distance relation (to account for regions that are not compact, we need to say that the diameter is the supremum of the distances between points in the region). But we could also suppose that the diameter property is more fundamental than distance, and just as we might define points as constructions out of a region-based ontology, we might define distances as constructions out of diameters plus region mereology.

How this would work depends on the details of the point construction. One kind of point construction identifies points with (equivalence classes of) sequences of regions that get smaller and smaller. Some have done this with special concentric regions like balls, but one can also do it with more general regions making use of the diameter D(A) of a region A. Specifically, we can let a point be (an equivalence class of) a sequence of A₁, A₂, ... of regions, where we requires that later regions in the sequence always being subregions of the earlier ones, and that the limit of D(A_n) is zero.(The equivalence relation can be defined by stipulating that the sequences A₁, A₂, ... and B₁, B₂, ... are equivalent just in case D(A_n+B_n) converges to zero where A_n + B_n is the fusion of A_n and B_n.) We can then stipulate the distance between the points defined by the sequences A₁, A₂, ... and B₁, B₂, ... is equal to the limit of D(A_n+B_n). We’re going to need some axioms concerning diameters and regions for all this to be well-defined and for the distance to be a metric.

Or we can take a version of Lewis’s construction where points are just identified with balls of a specific diameter δ₀, with the intuition that we identify a point with the ball of diameter δ₀ "centered on it". And we can again define distances in terms of diameters: d(A,B) = D(A+B) − δ₀.

This does not rid us of all relations. After all, we are supposing the mereological parthood relation (in its "subregion" special case). However, one might think that parthood is more of a fundamental binary predicate than a relation. And at least it’s not a determinable relation, in the way that distance is.

I am not myself fond of mereology. So the above is not something I am going to push. But it would be fun to work out the needed axioms if nobody’s done it (quite likely someone has—maybe Lewis, as I haven’t actually read his stuff on this, but am going on hearsay). It would make a nice paper for a grad student who likes technical stuff.

A variant of Thomson's Lamp

In the classic Thomson’s Lamp paradox, the lamp has a switch such that each time you press it, it toggles between on and off. The lamp starts turned off, say, before 10:00, and then the switch is pressed at 10:00, 10:30, 10:45, 10:52.5, 10:56.25, and so on ad infinitum. And the puzzle is: Is it on or off at 11? It’s a puzzle, but not obviously a paradox.

But here’s an interesting variant. Instead of a switch that toggles on or off each time you press, you have a standard slider switch, with an off position and an on position. Before 10:00, the lamp is off. At 10:00, 10:45, 10:56.25, and so on, the switch is pushed forcefully all the way to the on side. At 10:30, 10:52.5, and so on, the switch is pushed forcefully all the way to the off side.

The difference between the slider and toggle versions is this. Intuitively, in the toggle version, each switch press is relevant to the outcome—intuitively, it reverses what the outcome would be. In the slider variant, however, each slider movement becomes irrelevant as soon as the next time happens. At 10:45, the switch is pushed to the on side, and at 10:52.5, it is pushed to the off side. But if you skipped the 10:45 push, it doesn’t matter—the 10:52.5 push ensures that the switch is off, regardless of what happened at 10:45 or earlier.

Thus, on the slider version, each of the switch slides is causally irrelevant to the outcome at 11. But now we have a plausible principle:

1. If between t₀ and t₁ a sequence of actions each of which is causally irrelevant to the state at t₁ takes place, and nothing else relevant to the state takes place, the state does not change between t₀ and t₁.

Letting t₀ be 9:59 and t₁ be 11:00, it follows from (1) that the lamp is off at 11:00 since it’s off at 10:00, since in between the lamp is subjected to a sequence of caually irrelevant actions.

Letting t₀ be 10:01 and t₁ still be 11:00, it follows from (1) that the lamp is on at 11:00, since it’s on at 10:01 and is subjected to a sequence of causally irrelevant actions.

So it’s on and off at 11:00. Now that’s a paradox!

Another weird discrete theory of time

Suppose time is discrete. The usual story then is that we are always at some point of time. But what if, instead, we are always between times? I.e., the times themselves are something like imaginary points—we don’t occupy them on their own. It is only the interval between two successive times that we occupy, and we occupy the interval as a whole. Such an interval is a “now”.

If t_n and t_n + 1 are successive times, then we say that at (t_n,t_n + 1) (think of this as an ordered pair or an interval—your choice of mathematical representation!):

• x is F iff x is F at t_n and at t_n + 1

• x is non-F iff x is non-F at t_n and at t_n + 1

• x exists iff x exists at t_n and at t_n + 1

• x non-exists iff x is does not exist at t_n or at t_n + 1

• x is changing from F to non-F iff x is F at t_n but not at t_n + 1

• x is changing from non-F to F iff x is non-F at t_n and at t_n + 1

• x is coming into existence iff x exists at t_n + 1 but not at t_n

• x is ceasing to exist iff x exists at t_n but not at t_n + 1

• x is coming to be F iff x is F at t_n + 1 and either does not exist at t_n or exists at t_n but is not F then

• x is ceasing to be F iff x is F at t_n and either does not exist at t_n + 1 or exists at t_n + 1 but is not F then.

Here is a plausible thesis:

1. x fails to exist or x is F or x is non-F.

On the theory we are exploring, this is false in a now. Instead:

2. x non-exists or is coming into existence or is ceasing to exist or is F or is non-F or is changing from F to non-F or is changing from non-F to F.

This theory is a variant of one I tried out in an earlier post, minus the possibility of the now being a point.

Love and obedience

“This is the love of God: that we keep his commandments” (1 John 5:3)

But what is the connection between love and commands? Indeed, why would a loving God issue us commands?

Many things can be said about this. But here is one more that has occurred to me. God is unchanging and has complete beatitude. Yet love seems to fit particularly well with vulnerability. And by commanding one becomes vulnerable to the person one has commanded. For it detracts from one’s “extrinsic wellbeing” if one’s commands are broken.

Thus while one might think of the issuance of commands as the mark of dignity and greatness, and it is that, it also turns the tables, by making the commander be at the mercy of the commanded, at least with respect to the fulfillment of this particular aspect of the commander’s will.

Aristotle thought that love between gods and humans was impossible because of the inequality, since love involves a kind of equality. Kierkegaard wrote much about the difficulty of a love relationship between the infinite and the finite. But commanding us, paradoxically, is a way of introducing a kind of equality.

Of course, our disobedience does not change God, or impact his intrinsic beatitude. But it does impact his external wellbeing, and detracts his extrinsic honor.

And this is a different kind of vulnerability from that which the second person of the Trinity acquires by the Incarnation. It is a vulnerability of God as God.

Complexity and skeptical hypotheses

Suppose a strong epistemic preference for simpler theories of the world. One might then think that a simulation hypothesis is automatically more complex than the best physical story about our world, because in addition to all the complexity of our simulated cosmos, it includes the complexity of whatever physical cosmos houses the hardware running the simulation.

But this need not be the case. The best physical story about our world makes our world include vast amounts of information that would not need to be included in the simulation. To simulate the history of the human race, we at most need information on the particles wihtin a sphere of radius about a hundred thousand light-years, so basically just the Milky Way Galaxy, a very small fraction of the particles in the world. And even that is a vast overstatement. One can surely have a low simulation resolution for a lot of stuff, simulating things only on a macroscopic level, and only including particle-level information when the simulated humans peer into scientific instruments. So the information content of the simulation software could be much, much lower than the information content of the physical world that our best theories say we live in.

But what about the simulation hardware itself? Wouldn’t that need to live in a complex physical universe? Maybe, but that universe need not be as complex as our physical theories claim ours to be. It could be a universe that has a level of physical complexity optimized for running the computing hardware. The granularity of that universe could be much coarser than ours. For instance, instead of that universe being made of tiny subatomic particles like ours, requiring many (but fewer and fewer with progress in miniaturization) particles per logic gate, we could suppose a universe optimized for computing whose fundamental building blocks are logic gates, memory cells, etc.

I am dubious, thus, whether we can rule out simulation hypotheses by an epistemic preference for simpler theories. The same goes for Berkeleian skeptical hypotheses on which there is no physical world, but we are disembodied minds being presented with qualia.

And of course the “local five minute hypothesis”, on which the universe is five minutes old and has a radius of five light-minutes, posits a world with intuitively much less complexity than the world of our best theories, a world with vastly fewer particles.

But if we cannot avoid skeptical hypotheses on grounds of complexity, how can we avoid them?

My current view is that we simply have to suppose that our priors are normatively constrained by human nature (which on my Aristotelian view is a form, a real entity), and human nature requires us to have non-skeptical priors. This is a very anthropocentric account.