There's an argument for a beginning of time from the grim reaper paradox that Alexander Pruss made popular several years ago.1 I don't think he came up with it but he made a blog post about it, and William Lane Craig started talking about it, and that made it popular. Today, I want to explain the paradox, how it shows that time had a beginning, and answer an objection to it.
THE ARGUMENT
Let's suppose that time is infinite in the past. That means there are an infinite number of equal intervals of time. I say, "equal intervals of time" because it's mathematically possible to have an infinite number of unequal intervals of time, and yet they sum up to a finite number. For example, if you had. . .
0.1 minutes + 0.01 minutes + 0.001 minutes + . . .
. . .it would sum up to a finite number of minutes even if you were adding an infinite number of them. But if you were adding equal intervals of time, and if there were an infinite number of them, then you'd have an infinite amount of time.
If the past is infinite, then there are an infinite number of equal intervals of time in the past. So let's divide the past up into hours. Let's suppose that during each hour in the past, a grim reaper was created and set to go off at some time in the future.
A grim reaper is kind of like an alarm clock except that when it goes off, it kills you instantly unless you're dead already. Let's say you set a grim reaper to go off at noon, and another to go off at 1 pm. Once it gets to be noon, the grim reaper will kill you instantly, but when it's 1 pm, the grim reaper set to go off then doesn't do anything to you because you're dead already. It doesn't do anything if you're dead.
Now, let's suppose that the grim reapers are set in a particular way. Between 10 and 11 am, grim reaper #1 is set to kill you at 1/1 hours past noon, i.e. at 1 pm. Between 9 and 10 am, grim reaper #2 was set to kill you at 1/2 hours past noon, i.e. at 12:30 pm. Between 8 and 9 am, grim reaper #3 was set to kill you at 1/3 hours past noon, i.e. at 12:20 pm. In fact, during each hour in the past, a grim reaper is set to kill you at 1/n hours past noon, with 'n' being the number of the grim reaper.
If the past is infinite, then there will be an infinite number of grim reapers set to kill you between noon and 1 pm because there are an infinite number of hours in the past. This creates a paradox. Lemme explain.
If you are alive when you meet a grim reaper, that grim reaper will kill you instantly. If you are already dead, it will do nothing. So what happens between 11 am and 2 pm? You will have to pass through that dreaded hour when all those grim reapers are set to kill you. On the one hand, you can't possibly survive between noon and 1 pm without a grim reaper killing you because any grim reaper that goes off during that time will kill you if you're not dead already. But which one? You see, there is no first grim reaper. The grim reapers are set to go off in the same order that they were created. If the past is infinite, then there is no first grim reaper created and set in the past. Every grim reaper has a grim reaper that came before it. That means there’s no first grim reaper set to go off between noon and 1 pm. Every grim reaper that goes off will have a grim reaper that already went off before it. So no matter what grim reaper you reach, it can't kill you since there will have been one that came before it that already killed you.
That's the paradox. On the one hand, any grim reaper you meet will kill you if you're still alive, but on the other hand, there is no grim reaper that can kill you. Still, time goes on. The time between noon and 1 pm will happen, and if you're alive when it does, you will either be killed or you won't be killed. This paradox is actually a contradiction because it would follow from the stipulations of the scenario that (1) you must be killed, and (2) you cannot be killed. The whole scenario, then, is a logical impossibility.
Here's the argument for the beginning of time.
1. If the past is infinite, then the grim reaper scenario is logically possible.
2. But the grim reaper scenario is not logically possible.
3. Therefore, the past is not infinite.
THE OBJECTION
One objection I've heard to this argument is that it is physically impossible to actualize the grim reaper scenario even if the past is infinite. The reason is because as you go backward in time from 1 pm to noon, the grim reapers will get closer and closer together in time. Eventually, the time between two grim reapers will get smaller than the Planck time of 5.391 x 10-44 seconds. (A lot of sources round it to 10-43 seconds, so I’ll use that for the rest of this post for simplicity.) The Planck time is the smallest unit of time and time can’t be divided any smaller than that. With that being the case, you cannot fit an infinite number of sequential grim reapers within a finite interval of time with one coming after the other. I addressed this objection in another post,2 but I want to come at it from a different angle this time.
First response
It may not be true that the Planck time is the smallest unit of time and that time can’t be divided any further. I’ll explain that later (see the fourth response below), but for the sake of this response, I’ll take it a as a given that the Planck length is the smallest unit of time. My first response to this objection is to say that Planck time is irrelevant to the argument. The grim reaper paradox is a thought experiment designed to show that an infinite past is logically impossible. A thought experiment doesn’t have to be physically possible to serve as a useful tool that has some bearing on reality.
Never mind physical reality for a moment, and consider a number line with a zero in the middle, the positive numbers to the right, and the negative numbers to the left. For each negative integer (n = -1, -2, -3, . . .) there is a corresponding point on the positive side equal to 1/-n. So, -1 corresponds to 1, -2 corresponds to ½, -3 corresponds to 1/3, and so on. Notice that the negative integers on the negative side are in the same order as the positive fractions between zero and 1. The most distant negative integers corresponds to the smallest fractions closest to zero. Since there is no first integer going left to right on the negative side, there is no first fraction going left to right on the positive side. Now imagine traveling along the number line from -1 to +1. Somewhere between zero and +1, you’re bound to run into one of those points, but which point will you run into first? Herein lies the contradiction. No matter what point you run into, you will have already run into another point since each point has a point that came before it. It’s impossible to run into a first point since there is no first point, yet there must be a first point if you are to run into any points at all. This illustrates that an infinite regress of any sort is logically impossible.
A physical constraint, such as quantized time, may prevent you from carrying out an experiment like the grim reaper paradox in real life. But if you remove that physical constraint, the logical constraint will still be standing in the way. All a physical constraint does is add an additional obstacle. On top of the logical impossibility, you’d also have a physical impossibility. The logical constraint doesn’t vanish just because the physical constraint stops you first. The logical constraint is still there in spite of the physical constraint.
You could point to all sorts of physical reasons for why you couldn't perform the grim reaper scenario in reality. To create an infinite number of grim reapers, you’d need an infinite amount of matter. It’s impossible to kill somebody instantaneously. It’s also impossible to build a machine that you could set to go off with the precision required of grim reapers that were generated trillions and trillions of years ago.
But the fact that you couldn't actually build grim reapers in physical reality has no bearing on whether the grim reaper paradox, as a thought experiment, shows a logical impossibility. The reality of Planck time and the physical constraints of building clocks with infinite precision may prevent you from physically making and setting grim reapers to go off in the above scenario, but those physical impossibilities do not turn a logical impossibility into a logical possibility.
Think about it. How could our physical inability today to put an infinite number of sequential grim reapers into a finite interval of time somehow render it logically possible for the past to be infinite? The grim reaper paradox shows that an infinite past generates a contradiction. The contradiction is entailed by an infinite past. The grim reaper scenario merely illustrates why, and it’s entirely hypothetical.
Second response
Let’s suppose I’m wrong, though. Let's grant that the laws of physics are such that the smallest unit of time is 10-43 seconds, and that this prevents the grim reaper paradox from making an infinite past impossible. Now, consider this thought experiment. Imagine that God came down and changed the laws of physics, doing away with Planck time, and making time smooth and continuous. And let’s say he caused this change to take effect from noon until 1 pm. Now, Planck time cannot prevent the grim reaper paradox from making an infinite past impossible. If there were an interval of time in which the grim reaper paradox could be generated, then the past would have to be finite.
One might be tempted to say that whereas it is possible for the laws of physics to change in the future, the reality of the matter is that such a change will probably never happen. As long as it never happens, the grim reaper paradox can't be generated, and the past can still be infinite.
That’s an inadequate response, though. As long as it's even possible for the laws of physics to change in such a way as to generate the grim reaper paradox, the past cannot be infinite. Let me explain.
The past is fixed because it has already happened. It’s either infinite or it’s finite. If it’s infinite, it can’t become finite. That means that if it’s possible for the laws of physics to be changed in such a way as to generate the grim reaper paradox, then the past has been finite all along. If the past were infinite, then nothing could be done in the future to cause the past to be finite. So if the past is infinite, then it’s logically impossible for the laws of physics to ever change in such a way as to generate the grim reaper paradox since that would make an infinite past impossible. To summarize this argument in a syllogism:
* If the past is infinite, then it is impossible for the laws of physics to ever change in such a way as to generate the grim reaper paradox.
* It is not impossible for the laws of physics to ever change in such a way to generate the grim reaper paradox.
* Therefore, the past is not infinite.
So the mere possibility of the laws of physics changing generates the grim reaper paradox and makes an infinite past impossible.
Third response
For the purposes of my third response, it doesn’t even matter whether it’s possible for the laws of physics to ever change. As long as it’s possible for the laws of physics to have been otherwise all along, the grim reaper paradox makes an infinite past impossible. Unless you think Planck time is logically or metaphysically necessary, you have to allow for the logical or metaphysical possibility that it could have been otherwise.
The grim reaper paradox doesn’t just prevent the past from being infinite. It demonstrates that an infinite past is logically impossible. Logical impossibility is a formidable thing. It isn’t like physical impossibility. Physical impossibility can be overcome by changing the physical circumstances. The laws of logic can’t be changed, though, because they are necessary. It’s impossible for them to be otherwise. If something is logically impossible, then it can’t be actualized under any circumstances. If there is a possible circumstance under which an infinite past is logically impossible, then an infinite past is logically impossible under any other circumstances as well.
Consider some possible world in which time is continuous, and the Planck time doesn’t apply. In that world, there is no temporal constraint from the Planck time that would prevent the grim reaper paradox from being generated in physical reality. Therefore, in that world, Planck time cannot be invoked to refute the argument from the grim reaper paradox. Therefore, the grim reaper paradox shows that an infinite past is logically impossible in that world.
If an infinite past is logically impossible in one possible world, then it’s logically impossible in all possible worlds. The reason is because that’s what it means to be logically impossible. If a situation is logically impossible, that means it can’t exist in any possible world. If there were any possible world that included some state of affairs, that would mean the state of affairs is not logically impossible. The laws of logic are the same in all possible worlds. In fact, logic is what determines whether a world is possible or not. Possible worlds are defined by what is logically possible. If something is contradictory in one possible world, then it’s contradictory in all possible worlds. So a logical impossibility in one possible world carries over into all possible worlds, including the actual one. So if an infinite past is logically impossible in one possible world, then it’s logically impossible in the actual world.
That means if an infinite past is logically impossible in a world without Planck time where time is smooth and continuous, then an infinite past is also impossible in a world with Planck time where time is not smooth and continuous. So Planck time doesn’t refute the argument for a finite past from the grim reaper paradox.
The only way to evade this argument is to insist that Planck time (or something like it) is logically or metaphysically necessary and smooth continuous time is impossible. That still wouldn't refute the grim reaper paradox, though, because the grim reaper paradox doesn’t need to be carried out in physical reality. It’s a thought experiment, as I explained in my first response, and it shows the past to be finite whether Planck time is real or not.
Fourth response
The objection is based on the idea that the Planck time is the shortest unit of time, and time cannot be divided into any shorter length. But this may not be true (and I doubt that it is). Planck time is the point at which general relativity breaks down because Planck time is a function of the gravitational constant (G) and the speed of light (c) in a vacuum.
tp = (ℏG/c5)1/2
Any duration shorter than that cannot be described by general relativity because quantum gravitational effects start taking over.3 You would need a quantum theory of gravity, which we don’t have. Some unknown laws of physics would have to be invoked to describe any duration shorter than the Planck time, but that doesn’t mean the Planck time is the shortest possible duration. The Wiki page on Planck time puts it like this:
Because the Planck time comes from dimensional analysis, which ignores constant factors, there is no reason to believe that exactly one unit of Planck time has any special physical significance. Rather, the Planck time represents a rough time scale at which quantum gravitational effects are likely to become important. This essentially means that while smaller units of time can exist, they are so small their effect on our existence is negligible [unless a grim reaper is set to kill you!]. The nature of those effects, and the exact time scale at which they would occur, would need to be derived from an actual theory of quantum gravity.4
There is experimental evidence suggesting that time and space are not quantized. There were two studies that looked at images of distant galaxies from the Hubble Space Telescope. The images were sharper than we should expect them to be on the assumption that space and time are quantized.5 However, physicists don’t know for sure whether time is quantized or not.6
Planck time is actually just a unit of time defined as how long it takes a photon to travel the Planck length.7 There can be 1, 2, or 1/2 or 1/4 Planck time. It’s just a unit, like a centimeter or a second. So Planck time doesn’t put any limitations on your ability to run the grim reaper paradox and therefore doesn’t serves as an adequate refutation of the argument for the finitude of the past from the grim reaper paradox.
EDIT: In 2009, there was a study aimed at testing Loop Quantum Gravity (LQG). Because of the graininess of space in LQG, some theorists predict that higher energy photons should travel slower than low energy photons. Observations from a gamma ray burst disconfirmed this prediction, which casts doubt on the view that spacetime is grainy/quantized, which in turn casts doubt on LQG. Matt O'Dowd who hosts the PBS Spacetime channel on YouTube talked about this study in two videos:
"Loop Quantum Gravity Explained" (beginning at the 12:35 point) https://www.youtube.com/watch?v=L2suMPiuog4
"We Might Be Alone In the Universe" (beginning at the 13:39 point) https://www.youtube.com/watch?v=8wa1l7M5gU8
NOTES
1. "The Grim Reaper Paradox” by Alexander Pruss
"From the Grim Reaper Paradox to the Kalaam Argument” by Alexander Pruss
2. "Sibling rivalry and the beginning of time” by Me
3. http://astronomy.swin.edu.au/cosmos/P/Planck+Time
4. https://en.wikipedia.org/wiki/Planck_time
5. http://en.minghui.org/emh/articles/2003/4/8/34268.html
6. https://www.scientificamerican.com/article/is-time-quantized-in-othe/
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