Against falsifiability

What if time suddenly stopped everywhere for 5 seconds?

Your first instinct might be to laugh at the question and write it off as meaningless, given that such questions are by their nature unfalsifiable. I think this is a mistaken impulse, and that we can in general have justified beliefs about such questions. Doing so requires moving beyond outdated philosophies of science, and exploring the nature of evidence and probability. Let me present two thought experiments.

The Cyclic Universe

imagine that the universe evolves forward in time in such a way that at one time t1 its state is exactly identical to an earlier state at time t0. I mean exactly identical – the wave function of the universe at time t1 is quantitatively identical to the wave function at time t0.

By construction, we have two states of the universe that cannot be distinguished in any way whatsoever – no observation or measurement that you could make of the one will distinguish it from the other. And yet we still want to say that they are different from one another, in that one was earlier than the other.

But then we are allowing the universe to have a quantity (the ‘time-position’ of events) that is completely undetectable and makes no measurable difference in the universe. This should certainly make anybody that’s read a little Popper uneasy, and should call into question the notion that a question is meaningless if it refers to unfalsifiable events. But let’s leave this there for the moment and consider a stronger reason to take such questions seriously.

The Freezing Rooms

The point of this next thought experiment will be that we can be justified in our beliefs about unobservable and undetectable events. It’s a little subtler, but here we go.

Let’s imagine a bizarre building in which we have three rooms with an unusual property: each room seems to completely freeze at regular intervals. By everything I mean everything – a complete cessation of change in every part of the room, as if time has halted within.

Let’s further imagine that you are inside the building and can freely pass from one room to the other. From your observations, you conclude that Room 1 freezes every other day, Room 2 every fourth day, and Room 3 every third day. You also notice that when you are in any of the rooms, the other two rooms occasionally seem to suddenly “jump forward” in time by a day, exactly when you expect that your room would be frozen.

Room 1

Room 2 Room 3

So you construct this model of how these bizarre rooms work, and suddenly you come to a frightening conclusion – once every twelve days, all three rooms will be frozen at the same time! So no matter what room you are in, there will be a full day that passes without anybody noticing it in the building, and with no observable consequences in any of the rooms.

Sure, you can just step outside the building and observe it for yourself. But let’s expand our thought experiment: instead of a building with three rooms, let’s imagine that the entire universe is partitioned into three regions of space, in which the same strange temporal features exist. You can go from one region of the universe to another, allowing you to construct an equivalent model of how things work. And you will come to a justified belief that there are periods of time in which absolutely NOTHING is changing in the universe, and yet time is still passing.

Let’s just go a tiny bit further with this line of thought – imagine that suddenly somehow the other two rooms are destroyed (or the other two regions of space become causally disconnected in the extended case). Now the beings in one region will truly have no ability to do the experiments that allowed them to conclude that time is frozen on occasion in their own universe – and yet they are still justified in this belief. They are justified in the same way that somebody that observed a beam of light heading towards the event horizon of the universe is justified in continuing to believe in the existence of the beam of light, even thought it is entirely impossible to ‘catch up’ to the light and do an experiment that verifies that no, it hasn’t gone out of existence.

This thought experiment demonstrates that questions that refer to empirically indistinguishable states of the universe can be meaningful. This is a case that is not easy for Popperian falsifiability or old logical positivists to handle, but can be analyzed through the lens of modern epistemology.

Compare the following two theories of the time patterns of the building, where the brackets indicate a repeating pattern:

Theory 1
Room 1: [ ✓,  ]
Room 2: [ ✓, ✓, ✓, ]
Room 3: [ ✓, ✓, ]

Theory 2
Room 1: [ ✓, , ✓, , ✓, , ✓, , ✓, , ✓ ]
Room 2:  [ ✓, ✓, ✓, , ✓, ✓, ✓, ✗, ✓, ✓, ✓ ]
Room 3: [ ✓, ✓, ✗, ✓, ✓, ✗, ✓, ✓, ✗, ✓, ✓ ]

Notice that these two theories make all the same predictions about what everybody in each room will observe. But Theory 2 denies the existence of the the total freeze every 12 days, while Theory 1 accepts it.

Notice also that Theory 2 requires a much more complicated description to describe the pattern that it postulates. In Theory 1, you only need 9 bits to specify the pattern, and the days of total freeze are entailed as natural consequences of the pattern.

In Theory 2, you need 33 bits to be able to match the predictions of Theory 1 while also removing the total freeze!

Since observational evidence does not distinguish between these theories, this difference in complexity must be accounted for in the prior probabilities for Theory 1 and Theory 2, and would give us a rational reason to prefer Theory 1, even given the impossibility of falsification of Theory 2. This preference wouldn’t go away even in the limit of infinite evidence, and could in fact become stronger.

For instance, suppose that the difference in priors is proportional to the ratio of information required to specify the theory. In addition, suppose that all other theories of the universe that are empirically distinguishable from Theory 1 and Theory 2 starts with a total prior of 50%. If in the limit of infinite evidence we find that all other theories have been empirically ruled out, then we’ll see:

Initially
P(Theory 1) = 39.29%
P(Theory 2) = 10.71%
P(All else) = 50%

Infinite evidence limit
P(Theory 1) = 78.57%
P(Theory 2) =21.43%
P(All else) = 0%

The initial epistemic tax levied on Theory 2 due to its complexity has functionally doubled, as it is now two times less likely that Theory 1! Notice how careful probabilistic thinking does a great job of dealing with philosophical subtleties that are too much for obsolete frameworks of philosophy of science based on the concept of falsifiability. The powers of Bayesian reasoning are on full display here.

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