## Tuesday, May 6, 2014

### Infinite regress explanations

Consider Thomson's toggle lamp—each time the button is pressed, the lamp toggles between on and off—but suppose it existed from eternity and every January 1 the switch has been pressed once, and only then. Why is the lamp on now? Consider the regress explanation: It's on in 2014 because it was off in 2013 and toggled on January 1, 2014. And it was off in 2013 because it was on in 2012 and toggled on January 1, 2013. And so on.

Hume will say that this is a complete explanation. But surely not. Surely the whole story does not explain why the lamp is on in even numbered years and off in odd numbered years.

Notice an interesting thing. The following are perfectly fine explanations:

1. The lamp is on in 2014 because it was off in 2013 and toggled at the beginning of 2014.
2. The lamp is on in 2014 because it was on in 2012 and toggled at the beginnings of 2013 and 2014.
3. The lamp is on in 2014 because it was off in 2011 and toggled at the beginnings of 2012, 2013 and 2014.
And as we go down this list of explanations, our explanations get more and more ultimate. However, we can't take this to infinity. Each of the explanations in the list has wo conjuncts: a fact about the state of the lamp in year n, and then facts about the lamp being toggled in successive years. The facts about the lamp being toggled in successive years can be taken to infinity, but aren't enough to explain it. The following clearly isn't enough to give us an ultimate explanation of why the lamp was on in 2014:
1. The lamp was toggled at the beginnings of ..., 2010, 2011, 2012, 2013 and 2014.
Can we take the first conjunct in explanations (1)-(3) to infinity? Well, we certainly can't in general say that the lamp was on, or that it was off, in year −∞, since even if such a year existed, dubious as that is, the lamp need not have existed then—it need only be supposed to exist in all finite-numbered years. So what can we say? Well, we could let the lamp state in year n be L(n)—0 being off and 1 being on—and then say:
1. The limit of L(2n) is 1 as n→−∞ and the limit of L(2n+1) is 0 as n→−∞ (both limits over the integers only).
So if we think about how to complete our regressive explanation, it seems that it will need to be something like this:
1. The lamp is on in 2014 because of (4) and (5).
Very good. But even if (4) were to be ulitimately explained (maybe there is some mechanism where each toggling is caused by the preceding, which according to Hume would give an ultimate explanation of (4)), it is clear that (5) calls out for an explanation as well, and so the regressive explanation just isn't ultimate explanation.

So infinite regresses aren't enough for ultimate explanations, pace Hume.