I think you'd be better to say that the absurdities aren't absurd, just a bit surprising. It's odd that someone who believes in Allah thinks that his own personal common sense determines what can and can't exist. Does he not think his deity can surprise him? For that matter, does he not think Allah exists and is actually infinite?

Anyway, I might read the discussion later if you'd like detailed comments (let me know if you do), but there's plenty good stuff in this thread to be going on with.

logical bob wrote:

Shrunk wrote:There's a form of maths called constructivism in which, for reasons of philosophical preference, there are no infinite sets. Constructivists don't say there are only finitely many numbers, they just say you can't capture them all in a set.

The idea that constructivism is about potential infinities rather than actual infinities seems to capture the feel of the mathematics pretty well, though I can't put a solid finger on it.

Another way to put it is to say that "infinite" means something different for the constructivist. Indeed, the meaning of "infinite" intuitionistically better reflects the etymology. In-finite. Not finite. Not finished. Unterminated. So the word describes non-terminating processes, not collections. You can count numbers without end. They are infinite.

However, I am quite happy to talk about infinite collections intuitionistically. The point about constructivism is that I could eliminate that talk and just think about processes instead. It is the shape of the reasoning that is therefore different. You can't pull the same logical tricks that you can in classical logic. Classical logicians assume that the tricks they use with finite collections works for infinite collections, but as an intuitionist, I would say this is not the case.

The hilarious irony is that Craig's arguments do not remotely reflect the subtle distinction I have in mind here, nor do they reflect anything about intuitionism versus classical logic, nor the point about extending reasoning from the finite to the infinite. Intuitionists still have Hilbert's Hotel, and everything said about the hotel makes perfect sense. Nothing in the conception of Hilbert's Hotel or the arguments about it rely on the axiom of choice of the law of excluded middle.

Craig really is full of shit on this one.

logical bob wrote:I think you'd be better to say that the absurdities aren't absurd, just a bit surprising.

I never found them surprising. The existence of uncountable infinities was a surprise. But the mere fact that extending a set changes its algebraic structure isn't exactly uncommon in mathematics. The number 1 has no additive inverse in the natural numbers. But if you add negative numbers, it suddenly does. How absurd. Negative numbers must be self-contradictory.

VazScep wrote:The idea that constructivism is about potential infinities rather than actual infinities seems to capture the feel of the mathematics pretty well, though I can't put a solid finger on it.

Another way to put it is to say that "infinite" means something different for the constructivist. Indeed, the meaning of "infinite" intuitionistically better reflects the etymology. In-finite. Not finite. Not finished. Unterminated. So the word describes non-terminating processes, not collections. You can count numbers without end. They are infinite.

I'm not familiar with the technicalities of construcivist systems so I guess I'm imaging ZF without the axiom of infinity. You can still produce numbers without end, you just don't have a single set that contains them all. Informally and intuitively, you can't accomodate a new guest in the hotel because you can't separately ask every one of infinitely many guests to move and without a completed set you can't give the collective command "everyone go one room number up". You need a set to be the domain of the function.

To look at it a different way, we accept that there's no set of all sets (because that leads to an actual paradox, not just a result that takes a moment to get your head around) but that doesn't entail any lack of an abundance of sets. In fact, it's the opposite. They don't go in a set because they're too abundant.

You're quite right to say that Craig isn't even on the page here. He practices total sophistry. The Kalam argument generally is full of fail on every level. If I was asked to teach an intoduction to philosophy for total beginners I'd probably start with Kalam because it's a sort of one stop shop for common fallacies and you'll learn quite a bit of basic philosophy by seeing it shredded. But if your none too critical audience knows no philosophy it sounds good to them and they come out thinking yeah, I understood that. Explaining why it fails just doesn't fit into soundbites precisely because it ties in so well with an introduction to philosophy. It's a handy argument for the format Craig operates in.

I'm read three posts in this thread last night that are gone this morning. What gives?

logical bob wrote:I'm not familiar with the technicalities of construcivist systems so I guess I'm imaging ZF without the axiom of infinity.

Ah okay. That would make constructivism pretty weak. The victims of intuitionistic mathematics are things such as excluded-middle, choice and extensionality, not the infinite. One way to state the axiom of infinity is to say there is a function which is one-one but not onto. Intuitionists certainly accept that: just take the successor function.

You can still produce numbers without end, you just don't have a single set that contains them all. Informally and intuitively, you can't accomodate a new guest in the hotel because you can't separately ask every one of infinitely many guests to move and without a completed set you can't give the collective command "everyone go one room number up". You need a set to be the domain of the function.

Which intuitionists definitely have, otherwise they would never be able to express functions on the natural numbers, or any other constructive domain. Hilbert's Hotel is safe: you can still talk about the operation of moving one along, and remark that the first room is not in the image of the operation. Once we get into the physical business of hotels, things are different. But then, it's not a matter of mathematics what these functions mean physically, while special relativity says that because of the finite limit on the propogation of signals, the operation of "moving one along" cannot be physically completed anyway. I know that metaphysics is supposed to occupy a middle ground here between the maths and reality, but all the evidence says that this middle ground occupies a rather dull space between Craig's ears.

You're quite right to say that Craig isn't even on the page here. He practices total sophistry. The Kalam argument generally is full of fail on every level. If I was asked to teach an intoduction to philosophy for total beginners I'd probably start with Kalam because it's a sort of one stop shop for common fallacies and you'll learn quite a bit of basic philosophy by seeing it shredded. But if your none too critical audience knows no philosophy it sounds good to them and they come out thinking yeah, I understood that. Explaining why it fails just doesn't fit into soundbites precisely because it ties in so well with an introduction to philosophy. It's a handy argument for the format Craig operates in.

Absolutely agreed.

logical bob wrote:I'm read three posts in this thread last night that are gone this morning. What gives?

Apart from one of Teuton's and one of Shrunk's posts, two of my posts are gone as well.
Maybe the mods can enlighten us.

Edit: I've just found out that the thread has been split. The missing (or split) posts can be found here.
.

Ah, thanks.

I study history, and would just like to say that much of what is being discussed in this thread is so alien to me it ought to pay the metoikion.