Cardinality paradoxes

Some people think it is absurd to say, as Cantorian mathematics does, that there are no more real numbers from 0 to 100 than from 0 to 1.

But there is a neat argument for this:

1. If the number of points on a line segment that is 100 cm long equals the number of points on a line segment that is 1 cm long, then the number of real numbers from 0 to 100 equals the number of real numbers from 0 to 1.
2. The number of points on a line that is 100 cm long equals the number of distances in centimeters between 0 and 100 cm.
3. The number of points on a line that is 1 meter long equals the number of distances in meters between 0 and 1 meter.
4. The number of distances in centimeters between 0 and 100 cm equals the number of real numbers between 0 and 100.
5. The number of distances in meters between 0 and 1 meters equals the number of real numbers between 0 and 1.
6. A line is 100 cm if and only if it is 1 meter long.
7. Equality in number is transitive.
8. So, the number of points on a line that is 100 cm is equals the number of points on a line that is 1 meter long.
9. So, the number of distances in centimeters between 0 and 100 cm equals the number of distances in meters between 0 and 1 meters.
10. So, the number of real numbers between 0 and 100 equals the number of real numbers between 0 and 1.

Amusing probability case

Write down a decimal point. Then choose a digit at random, with equal probability 1/10 of each possible digit. Repeat ad infinitum, with all the digits chosen independently. Let X be the number you've written down the infinite decimal expansion of.

Suppose you find out that X is going to be either 1/4 or 1/3. Which of the two is more likely? Answer: 1/4. For there are two ways of getting 1/4: 0.250000... and 0.249999.... But there is only one way of getting 1/3: 0.333333..., and each infinite sequence is equally likely. Thus, intuitively P(X=1/4 | X=1/3 or X=1/4)=2/3. Surprised?

Another interesting fact here. In the technical probability-theory sense, X is uniformly distributed on the interval [0,1]. But in the intuitive sense, it's not. So the technical probability-theory sense does not capture the notion of uniform distribution.

Similarly, the technical probability-theory sense of independence does not capture the intuitive notion of independence. Suppose that a random process uniformly picks out a number Y in the interval [0,1], and suppose you get a dollar if and only if the number is 1/2. Let A be the event that the number picked out is 1/2 and let B be the event that you get a dollar. Then P(A&B)=P(A)=0=P(A)P(B), and hence in the probability-theoretic sense A and B are independent. But intuitively they are far from independent: B is entirely determined by A.

Maybe a better definition of independence for philosophical (though maybe not mathematical) purposes is that both P(A|B)=P(A) and P(B|A)=P(B). And then conditional probabilities should not be defined by ratios of unconditional probabilities.

Real numbers

For a long time I've been puzzled—and I still am—by this. Our physics is based on the real numbers (complex numbers, vectors, Banach spaces—all that is built out of real numbers). After all, there are non-standard numbers that can do everything real numbers can. So what reason do we have to think that "the" real numbers are what the world's physics is in fact based on?

I think one can use this to make a nice little argument against the possibility of us coming up with a complete physics—we have no way of telling which of the number fields is the one our world is based on.

Thomson's lamp

Thomson's lamp has an on-off switch. It begins in the "off" position. At noon the switch is toggled, and the lamp comes on. Half a minute later, the switch is toggled, and the light goes off. A quarter of a minute later, the switch is toggled again, and the light comes on. And so on. There are no other switch flippings than these, and the switch survives at least until 12:01 pm. At 12:01 pm, is the switch on or off?

As paradoxes go, this one seems really flimsy. As best I can see, the argument to a paradox is something like this:

1. Time is actually infinitely subdivided.
2. If time is actually infinitely subdivided, the story of Thomson's lamp is possible.
3. Necessarily, if the story is true, then the switch is either on or off at 12:01.
4. Necessarily, if the story is true, then the switch is not on at 12:01.
5. Necessarily, if the story is true, then the switch is not off at 12:01.

The argument for (4) is, presumably, that after every time the switch is on, there is a next time when it is off, and the argument for (5) is similar.

There are a couple of ways of showing what's wrong with the argument. Here is one. In order to argue for (4) and (5), it needs to be a part of the story that

6. At each time t after noon, the position of the switch is the result of the last switch-flipping event prior to t.

For suppose that we deny this. Then we can allow that the switch is on at 12:01, but not due to any switch-flipping event. Or we can allow that the switch is off at 12:01, but not due to any switch-flipping event. After all, perhaps, the switch, instead of being flipped, just undergoes a quantum leap from one position to another.

Fine, then, says the paradoxer: Add (6) to the story.

However, now (2) becomes false. The defender of actually infinitely subdivided time can simply deny (2), since the story is plainly inconsistent: the position of the switch at 12:01 is determined by the last flip before 12:01, but there is no last flip before 12:01. It is a story as plainly inconsistent as this one: "Whether there is an obligatory side of the street to drive on is determined by the content of the will of the king of France. And France is a monarchy." The question of the position of the switch is rather like asking: "If atheism were true, would God want us to be atheists?"

Perhaps the paradoxer will say that that was her whole point, but nonetheless the defender of actually infinitely subdivided time has to affirm that this inconsistent story is possible. But why? It is an easy game to construct inconsistent stories by including a stipulation that something is sufficient to determine something, and then adding to the story something that denies the existence of the determiner. In addition to my present king of France story, consider this one:"A lamp with an on-off switch that can only have two positions, on and off, is produced ex nihilo by God at t₀. The position of the switch at any time is fully determined by how it has last been flipped." Then ask: What is the position of the switch at t₀? Obviously, we have an inconsistency in the story—if the lamp came into existence ex nihilo at t₀ it came into existence with the switch in a particular position, but that position was not determined by a flipping.

But does not the defender of actually infinitely subdivided time think that a lamp switch's being flipped in the supertask way is possible? Certainly. But she has to hold that this is only possible in those worlds in which either something other than the last flip determines the position of the switch at 12:01 or the Principle of Sufficient Reason is violated (I don't think there are any such) or both.

Is time a continuum?

The following argument is valid:

1. (Premise) If one compressed all the events of an infinitely long happy life into a minute, by living a year of events in the first half minute, then another year of events in the next quarter minute, and so on, then one would be exactly as well off as living the finite life as the infinite one.
2. (Premise) If supertasks are possible, then the antecedent of (1) is possible for any infinitely long happy life.
3. (Premise) If time is an actual continuum, supertasks are possible.
4. (Premise) There is a possible an infinitely long happy life that would make for full human well-being.
5. (Premise) A finitely long life could not make for full human well-being.
6. (Premise) If a life makes for full human well-being, then so does any life that makes one exactly as well off.
7. Therefore, if supertasks are possible, there is a finitely long life that would make for full human well-being (1, 2, 4, 6).
8. Therefore, supertasks are impossible. (5, 7)
9. Therefore, time is not an actual continuum. (3 and 8)

Sharp cutoffs in the moral life

Ted Sider apparently has an argument (I reporting second-hand) that there is a continuum in the degree of sinfulness, but there is no continuum in the heaven-hell welfare spectrum, since there is a sharp jump in welfare as one moves from an eternity of suffering to an eternity of joy. Therefore, he concludes, divine judgment cannot be just if the outcomes are heaven or hell.

Now one way to answer this is to say that there really are sharp cutoffs in the moral life, such as that between those in a state of mortal sin and those not in a state of mortal sin. The cutoffs would not be defined by some kind of a moral arithmetic[note 1], but by a qualitative fact about the state of the person's will. Thus, Aquinas defines the state of mortal sin in terms of the lack of charity. Now, charity is a fairly sharply defined state of friendship with God (which state is always the fruit of grace). The mortal sinner lacks charity entirely, though the charity will be restored in repentance and forgiveness. Now, there might be a continuum in the degree of charity, say from zero to a hundred, but the difference between zero charity and even the tiniest bit of charity is deeply significant. Even a tiny bit of charity makes one fit for eternity with God (but the more charity there is, the more blissful that eternity will be). But a complete lack of charity makes one fit for damnation.

Is it plausible that there should be such sharp cutoffs in the moral life? Well, what led me to this reflection was watching the excellent 1953 film Pickup on South Street. The central character, Candy (Jean Peters), is a woman who has lived somewhat on the wrong side of the law, and is now trying to leave that life behind, but has one last task of greyish legality. However, she finds that she is enmeshed in a situation of Soviet espionage. And then it becomes clear that she sees a yawning gulf between mere crime and treason, and she assumes, perhaps wrongly, that other people living on the wrong side of the law see it this way, too. It is one thing, in her mind to be a pickpocket (though she is not one herself), and quite a different to work for the Reds. The film makes it plausible that there is indeed a sharp cutoff between other crimes and treason. It's almost as if treason were an allegory for mortal sin. See the film—it is really good. (If you have Netflix, it's available from their Watch Instantly section—that's how I watched it.)