Tuesday 10 February 2015

Possibility of the Actually Infinite

Mathematicians define two sets as being the same size whenever there is a bijective mapping from one to the other. That is to say, whenever each member of the one set can be paired up with exactly one member of the other. At face value this makes perfect sense. If you're at a dinner party and every guest has brought a significant other of the opposite sex, then clearly there are just as many men at this party as there are women.

Why does this matter? Because, if same size is understood as mathematicians define it, then it easily follows that a set can be the same size as its proper subset. In other words, the common objection to the possibility of actual infinities simply fails by definition.

It's all the more clearer with an example. The natural numbers {1, 2, 3, ... } and the even numbers {2, 4, 6, ... } are the same size because a bijective mapping exists between them: f(x) = 2x, which pairs 1 with 2, and pairs 2 with 4, and 3 with 6 and so on. And, of course, the even numbers are a proper subset of the natural numbers.

Now you might think, this sounds super abstract. Maybe this is some weird technical idea that mathematicians throw around, but surely all this mathematical mumbo jumbo isn't anything like my understanding of what it means for two sets to be the same size. But actually, it is.

When someone wants to know how big a set is, they count the members. They point to an object and say "one", they point at another and say "two", and so on. They don't know it, but they're proving the existence of a bijective mapping between the set in question and the subset of the naturals they're vocally describing. And because the size of the set they're counting out is identical to the value of the last member, they (all by intuition and not understanding the math) infer the size of the set in question. It's odd to think that something so simple as counting has a rigorous mathematical basis that utilizes the mathematicians very technical definition of same size, but it in fact does.

Of course we needn't proceed one member at a time when counting. We could just as well point to a pair and say "two", and another pair and say "four", and so on. Or we could go by groups of ten, or hundreds or, even, the collection as a whole. It follows from this that, contrary to popular opinion, we can count an infinite set after all, we just do it all at once instead of step by step.

Here's the catch, then. People who think that actual infinities are impossible (almost always because they think it's impossible for a set to be the same size as its proper subset) owe the rest of us an explanation of what 'same size' means, if not what mathematicians mean. Since they are unable to give any answer, their objections to actual infinities (usually to the possibility of an eternal past) fall flat.

Tuesday 3 February 2015

Evil as an Absence of Good?

It's very common for people to think that evil doesn't actually exist in of itself. Rather, like a hole in your shirt is just an absence of fabric, some would say evil is just an absence of good. Of course that doesn't mean there is no truth about evil, or that it wont affect your life. A large hole in your jacket, despite not being a thing in of itself, will still make you miserable on a cold, wet winters day.

But there is an obvious problem with this view, in that it would require all things (at least within the relevant domain) to be either good or evil. You can't have shirts that are neither whole nor have a hole, but you can have actions that are neither good nor evil—in fact most actions seem morally insignificant. What could be the moral value in cutting your grass, or eating a cheese burger? If you find some morally significant feature, one can always easily stipulate a scenario in which that feature isn't present.

And so defenders of this view must embrace the implausible and maintain that every action, no matter how seemingly insignificant, has moral value to some degree. This is why I favour theories of moral ontology on which good and evil are both real in a robust sense, neither being an absence of the other.

Monday 2 February 2015

Skeptical Theism and Divine Deception 2

I have become convinced that my argument outlined in Skeptical Theism and Divine Deception is not successful.

The problem is that I failed to distinguish between having justified belief, and being able to justify ones belief. The difference is that we can have justified belief without being aware of it. On externalism, the justification for a belief can be something that the subject might not even have access to, like the causal history that produced his belief. But justifying ones belief is an action rational people perform, a sort of giving of an explanation or an account of how one is justified in holding that belief.

It follows from externalism that, for all I know, the skeptical theist might be justified in believing God always tells the truth—my premise (3) is indefensible. And yet the spirit of my argument persists. If you think about the role divine revelation plays, it's always intended to account for the justification of religious belief. Religious folk would say "God has told us these things and therefore they are true," implicitly assuming that God is not lying.

But now it's clear that the implicit assumption is not plausible so long as we're committed to skeptical theism. After all, for all the skeptical theist knows, God could have a morally sufficient reason to lie about religious matters. (See the previous post for a more in depth analysis of this).

So while the skeptical theists' religious commitments might be justified, they are not something he can justify. And, since we should exercise a healthy skepticism about beliefs we cannot ourselves justify, there is still tension between skeptical theism and religious belief. It seems, at least, the skeptical theist should be no more confident about his religious beliefs than he thinks is appropriate for gratuitous evil.

We can re-formalize this argument as follows, where P is the sort of belief we can only justify by appealing to divine revelation (namely, religious belief):
  1. Without appealing to divine revelation, there is no way to justify the belief that P
  2. Skeptical theists cannot appeal to divine revelation to justify their beliefs
  3. Therefore, skeptical theists cannot justify their belief that P
  4. Doubt should be reserved for beliefs we cannot justify
  5. Therefore, skeptical theists should reserve doubt for their belief that P