Thomson's Lamp and change

Start with two Thomson's Lamps. They each have toggle switches and are on at 10 a.m. The switches are toggled at 10:30, 10:45, 10:52.5 and so on. Now suppose, as is surely possible, that regardless of what, if any, state the lamps would have had at 11 a.m., aliens come and instantaneously force the first lamp to be on at 11, and force the second to be off (say, by breaking the bulb!).

Now some but not all causal interactions are changes. Which lamp's on/off state did the aliens change? There are four possible answers:

1. The first but not the second
2. The second but not the first
3. Both
4. Neither.

Symmetry considerations rule (1) and (2) out of court. Can we say that in both cases, the aliens changed the on/off state of the lamp? Surely not. For if something can have only two states, it can't be that each of the two possible state inductions counts as a change. Moreover, if the lamp were to have changed state, what state did it change from? For inducing an on state only counts as a change if the induction starts with the lamp in an off state, and vice versa. But the induction didn't start with the lamp on, nor did it start with the lamp off. That leaves only last option: Neither.

But it seems that if an object has been persisting, and a causal interaction induced a state in that objection, that causal interaction either was a state-change or a state-maintenance. So if in neither case did the aliens change the state of the lamp, then it seems that in both cases they maintained the state. But we get analogues of (1)-(4), and analogues of the above arguments also lead to the conclusion that in neither case did the aliens maintain the state.

So the Thomson's Lamp story forces us to reject the dichotomy between state-change and state-maintenance.

Here's another curious thing. It seems that the following is true:

5. If an object has state A at t₁ and non-A at t₂, then the object's having state non-A at t₂ is the result of a change.

But applying (5) shows that both lamps' final states are the results of a change. But that change must have thus been from the opposite state. And yet the final state doesn't follow right after the opposite state.

One could use this as an argument against the possibility of infinitely subdivided time. Alternately, one could use this as an argument against principles like (5) and the idea that the concepts of change and maintenance are as widely applicable as we thought them to be.

IRToWebThingy

For quite some time, my older daughter has wanted me to make her Great Wolf Lodge Magiquest wand do something at home. So I made a simple IRToWebThingy (the link gives build instructions). It takes infrared signals in many different infrared remote controls and makes them available over WiFi. As a result, we can now watch Netflix with a laptop and a projector and the ceiling and play/pause with the Magiquest wand (and adjust volume with our DVD remote) with a pretty simple python script. My son (with some help) made a 3D etch-a-sketch script that lets him draw in Minecraft with the DVD remote. I made a fun script that lets you fly a pig in Minecraft with a Syma S107 helicopter remote (see photo). You can even control rooted Android devices with IR remotes and shell scripts.

Knowledge and scepticism

Some years back I learned that when epistemologists talk about the "problem of scepticism", they are thinking about theses that deny that we have much knowledge about, say, the external world. When I found out this, I was disappointed. I thought that the problem that epistemologists were dealing with was the problem whether we can be reasonable in believing things about the external world. I care about that, not about whether we know. Suppose it were to turn out that knowledge has to be infallible, and so indeed we know nothing empirical about the external world. This would hardly bother me. It's too bad that I couldn't say that I know that I have two hands, but my rational confidence that I have two hands would be just as high as it ever was.

We sell the scepticism at the beginning of Descartes' Meditations and throughout Sextus Empiricus far short if we think about it as simply a denial of knowledge. If we take seriously Descartes' invocation of the possibility of dreaming or of a deceitful demon, that not only has a tendency to undercut the claim that we know, but even the claim that we are rational in being confident in ordinary empirical beliefs, and maybe even the claim that it is rational to hold such beliefs to be more likely true than not. This is the more radical and more interesting scepticism: it questions whether we have any non-circular reason to think we aren't being deceived. (And in the end I think Descartes' answer to it is pretty good.)

The problem of induction in mathematics

Let's say I have some algorithm that generates the sequence of numbers, and I run ten iterations on the computer and get

• 1
• 1.5
• 1.41666666667
• 1.41421568627
• 1.41421356237
• 1.41421356237
• 1.41421356237
• 1.41421356237
• 1.41421356237
• 1.41421356237
• 1.41421356237

I will now be very confident that the sequence of numbers converges, and indeed that it converges to the square root of two. But why? Convergence is a property that the sequence has at infinity. The first 10 items in the sequence are an infinitely short proportion of infinity. Moreover, why do I assume that the sequence converges to the square root of two. Maybe it converges to the square root of two plus e⁻¹⁰⁰. Such possibilities ensure that my credence that the limiting value is the square root of two is strictly less than one. But the credence stays high.

In other words, the standard problems of induction come up not in just in science, but in mathematics. We should, thus, hope that whatever solutions we adopt to the problems as they come up in science will apply in the mathematical cases as well.

My favorite story about induction, the theistic story that God would have good reason--and hence be not unlikely to--create a well-ordered universe does not apply to mathematics, since pace Descartes, God doesn't choose the truths of mathematics. A relative of the story does apply, however. The truths of mathematics are grounded in the necessary nature of the mind of God, as Augustine held, and it is to be expected that the necessary nature of the mind of God will exhibit beauty and elegance. And where there is beauty and elegance, there is pattern and some purchase for induction.

Emotivism about art

Being largely tone-deaf, I don't appreciate music much at all. In fact, the older I get, the more music tends to annoy me, particularly when I meet it in the background of daily activities. I would much prefer listening to 4'33'' than to Mozart or Beethoven.

Yet I know that Mozart and Beethoven are much more beautiful music. Hence I embody a counterexample to emotivism about art. It is fun to be a counterexample (but I realize that it would be more fun to appreciate music, and I hope to do so in heaven).

Teaching programming with Python and Minecraft

Last summer, I taught programming to gifted kids with Python and Minecraft. Here's my Instructable giving a curriculum for a course like that.

Teaching programming with AgentCubes

I made an Instructable giving the curriculum I used for a mini-course for gifted middle- and high-school kids teaching programming by making 3D games with AgentCubes.

RaspberryJamMod for Minecraft/Forge 1.10

My RaspberryJamMod, which enables the Minecraft PI API for Python programming, has been updated to work with Minecraft/Forge 1.10. Still alpha-quality, but everything I've tried seems to work.

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.

Finitism and mathematics

Finitists say that it is impossible to actually have an infinite number of things. Now, either it is logically contradictory that there is an infinite number of things or not. If it is not logically contradictory, then the finitist's absurdity-based arguments are seriously weakened. If it is logically contradictory, on the other hand, then standard mathematics is contradictory, since standard mathematics can prove that there are infinitely many things (e.g., primes). But from the contradictory everything follows ("explosion", logicians call it). So in standard mathematics everything is true, and standard mathematics breaks down.

I suppose the finitist's best bet is to say that an infinite number of things is not logically contradictory, but metaphysically impossible. That requires a careful toning down of some of the arguments for finitism, though.

Measure of confirmation

Suppose I played a lottery that involved my picking ten digits. The digits I picked are: 7509994361. At this point, my probability that I won, assuming I had to get every digit right to win, is 10⁻¹⁰. Suppose now that I am listening to the radio and the winning number's digits are announced: 7, 5, 0, 9, 9, 9, 4, 3, 6 and 1. My sequence of probabilities that I am the winner after hearing the successive digits will approximately be: 10⁻⁹, 10⁻⁸, 10⁻⁷, 10⁻⁶, 10⁻⁵, 10⁻⁴, 0.001, 0.01, 0.1 and approximately 1.

If we think that to get a significant amount of evidential confirmation for a hypothesis we need to give a significant absolute increment to the probability, only the final two or three digits gave significant confirmation to the hypothesis that I won, as the last digit increased my probability by 0.9, the one before by 0.09, and the one before by 0.009. But it seems quite wrong to me to say that as I am listening to the digits being announced one by one, I get no significant evidence until the last two or three digits. In fact, I think that each digit I hear gives me an equal amount of evidence for the hypothesis that I won.

Here's an argument for why it's wrong to say that a significant absolute increment of probability is needed to get significant confirmation. Let A be the collective evidence of learning digits 1 through 7 and let B be the evidence of learning digits 8 through 10. Then on the absolute increment view, A provides me with no significant confirmation that I won, but when I learn B after learning A, then B provides me with significant confirmation. However, had I simply learned B, without learning A first, that wouldn't have afforded me significant confirmation--my probability of winning would still have been 10⁻⁷. So the combination of two pieces of evidence, both insignificant on their own, gives me conclusive evidence.

There is nothing absurd about this on its own. Sometimes two insignificant pieces of evidence combine to conclusive evidence. Learning (C) that the winning number is written on a piece of paper is insignificant on its own; learning (D) that 7509994361 is written on the piece of paper is insignificant on its own; but combined they are conclusive. Yes: but that is a special case, where the two pieces of evidence fit together in a special way. We can say something about how they fit together: they fail to be statistically independent conditionally on the hypothesis I am considering. Conditionally on 7509994361 being the winning number, the event that 7509994361 is what is written on the paper and the event that what is written on the paper is the winning number are far from independent. But in my previous example, A and B are independent conditionally on the hypothesis I am considering, as well as being independent conditionally on the negation of that hypothesis. They aren't pieces of evidence that interlock like C and D are.

Temporal nonlocality of consciousness

We are incapable of having a pain that lasts only a nanosecond (or a picosecond or, to be really on the safe side, a Planck time). Anything that is that short simply wouldn't consciously register, and an unconscious pain is a contradiction in terms. But now suppose that I have a constant pain for a minute. And then imagine a human being who lives only for a nanosecond, and whose states during that nanosecond are exactly the same as my states during some nanosecond during my minute of pain. Such a person wouldn't ever be conscious of pain (or anything else), because events that are that short don't consciously register.

Say that a property is temporally punctual provided that whether an object has that property at a given time depends at most on how the object is at that time. For instance, arguably, being curved is temporally punctual while being in motion is not. Say that a property is temporally nanoscale provided that whether an object has that property at a time t depends at most on how the object is on some interval of time extending a nanosecond before and after t. Any temporally punctual property is temporally nanoscale as well. An example of a non-nanoscale property is having been seated for an hour. The reflections of the first paragraph make it very plausible that our conscious experiences are neither temporally punctual nor temporally nanoscale. Whether I am now in pain at t depends on more than just the nanosecond before and after t. There is a temporal non-locality to our consciousness.

This I think makes very plausible a curious disjunction. At least one of these two claims is true:

1. My conscious states are non-fundamental.
2. The time sequence along which my conscious states occur is not the time sequence of physical reality.

For, plausibly, all fundamental states that happen along the time sequence of physical reality are nanoscale (and maybe even temporally punctual). Option (1) is friendly to naturalism. Option (2) is incompatible with naturalism.

Some non-naturalists think that conscious states are fundamental. If I am right, then they must accept (2). And accepting (2) is difficult given the A-theory of time on which there is a single pervasive time sequence in reality. So there is an argument here that if conscious states are fundamental, then probably the A-theory is false. There may be some other consequences. In any case, I find it very interesting that our conscious states are not nanoscale.

Naturalism and second-order experiences

My colleague Todd Buras has inspired this argument:

1. A veridical experience of an event is caused by the event.
2. Sometimes a human being is veridically experiencing that she has some experience E at all the times at which E is occurring.
3. If (1) and (2), then there is intra-mental simultaneous event causation.
4. If naturalism about the human mind is true, there is no intra-mental simultaneous causation.
5. So, naturalism about the human mind is not true.

In this argument, (4) is a posteriori: if naturalism is true, mental activity occurs at best at the speed of light. I am sceptical about (2) myself.

Conceiving people who will be mistreated

Alice and Bob are Elbonians, a despised genetic minority. It seems that unless the level of mistreatment that members of this minority suffer is extreme, it is permissible for Alice and Bob to have a child.

But now suppose that Carol and Dan are not Elbonian, but have a child through a procedure that ensures that the child is Elbonian. It seems that Alice and Bob's procreation is permissible, but Carol and Dan are doing something wrong. Yet in both cases they are producing a child that will be, we may suppose, the subject of the same mistreatment.

We understand, of course, Alice and Bob's intentions: they want to have a child, and as it happens their child will be Elbonian. But we have a harder time understanding what Carol and Dan are doing. Are they trying to make their child be the subject of discrimination? If so, then it's clear why they are acting wrongly. But we can suppose that both couples are motivated in the same way. Perhaps both couples really like the way that Elbonian eyes look, and that is why Alice and Bob do not seek out genetic treatment to root out the Elbonian genes while Carol and Dan seek treatment to impose these genes.

Thinking about this case makes me think that there is a significant difference between just letting nature take its course reproductively and deliberately modifying the course of reproduction. But there are a lot of hard questions here.

Some thoughts on the ethics of creating Artificial Intelligence

Suppose that it's possible for us to create genuinely intelligent computers. If we achieved genuine sapience in a computer, we would have the sorts of duties towards it that we have towards other persons. There are difficult questions about whether we would be likely to fulfill these duties. For instance, it would be wrong to permanently shut off such a computer for anything less than the sorts of very grave reasons to make it permissible to disconnect a human being from life support (I think here about the distinction between ordinary and extraordinary means in the Catholic tradition). Since keeping such a computer running is not likely to typically involve such reasons, it seems that we would likely have to keep such a computer running indefinitely. But would we be likely to do so? So that's part of one set of questions: Can we expect to treat such a computer with the respect due to a person, and, if not, do we have the moral right to create it?

Here's another line of thought. If we were going to make a computer that is a person, we would do so by a gradual series of steps that produce a series of systems that are more and more person-like. Along with this gradual series of steps would come a gradation in moral duties towards the system. It seems likely that progress along the road to intelligence would involve many failures. So we have a second set of questions: Do we have the moral right to create systems that are nearly persons but that are likely to suffer from a multitude of failures, and are we likely to treat these systems in the morally appropriate way?

On the other hand, we (except the small number of anti-natalists) think it is permissible to bring human beings into existence even though we know that any human being brought into the world will be mistreated by others on many occasions in her life, and will suffer from disease and disability. I feel, however, that the cases are not parallel, but I am not clear on exactly what is going on here. I think humans have something like a basic permission to engage in procreation, with some reasonable limitations.