I don't know if Wittgenstein would have changed his mind if he were alive today--nor do I think it matters. I'm not interested in him as an authority or guru figure, but as someone who came up with some interesting ideas. Insofar as he rejects those ideas in favor of the status quo or conventional wisdom he is actually less interesting.

However, Subfallen is correct in stating that Wittgenstein's views here are not very clear. He is never a very clear writer to start with and since most of what we are working from is unpublished work and because his views on meaning change, there is significant disagreement about his actual views on math. I think the main use is to think of his position as an alternative, if minority, understanding of the nature and meaning of mathematics. For instance, if you feel inclined to reject the actual infinite, then his philosophy of maths provides a more fully-worked out way of doing so.

As for your point more generally, I think most modern philosophers would tend to agree with you. For example, Quine and Hilary Putnam are both naturalists, but argue that because of the indispensability of these kinds of maths to modern science we should accept mathematical objects as real.

Let's say that I take a box and put 2 apples in it. Now, I can say that it is physically impossible for there to be 3 apples in it right now given that there are only 2 apples in it right now. In some sense that is true. However, there is no general physical principle that says it is impossible to put 3 apples in the box.

What I'm saying is, if I were god and creating a universe and I create a universe with 2 apples in it, then in some sense it is physically impossible for there to be an infinite amount of apples in the universe right then given that I only created 2 apples. However, that doesn't mean that there is some general physical principle that prevents me from creating a universe with an infinite amount of apples in it.

What I'm saying is that in order for Craig's premise to be justified, the "impossibility" claim has to be at least strong enough for us to say that if I were god and creating a universe then I couldn't created one with an infinite amount of objects in it. Any justification that rests solely on the fact that the actual universe doesn't have an infinite amount of objects is insufficient, because there are other ways that the universe could have been.

I just want to emphasize this paragraph, because this is mainly what I've been arguing. My claim is that of course for any mathematical platonist actual infinites exist. What I've been saying is that in order for Craig's claim to be consistent he has to accept an understanding of math that rejects not only the existence of abstract objects, but also is revolutionary about accepting the validity of certain commonly accepted mathematical fields (such as set theory), or give a much richer account of physical impossibility.

Hmm...I guess I'll take your word for it. Hopefully someday I can re-read Philosophical Grammar and try to bridge the gap between my interpretation and what you've said here.

too use another analogy about infinite regress of time in the past. If the past is infinite then there are an infinite amount of events. Today is real. In order for todays events to occur, you must complete an inifinite of past events in order for today to then come about. But completing an actual infinite amount of events in order for today to happens is absurd. But of course, today is actually here. So how is this possible. Unless you hold a B Theory of time (All instances of time are real at every moment) this isn't possible on the standard A theory of time which is commonly held today. The burden of proof is apon the person who says that an actually infinite amount of events can occur in the past in order for todays events to then take place. When Craig lays this burden of proof, its not a bad tactic on his part. He has the evidence of todays events actually existing! But of course, this cant be possible unless the past is finite and not inifinite in the past. This isn't the exact line of reasoning he takes in his arguments but it might help to think of it this way.

kyleLP -

But it's not "absurd" in any obvious sense to posit a set T of moments and a t₀ ∈ T which has infinitely many predecessors. For example, using ZFC: Let T be an uncountable well-ordered set with the property that every initial segment of T is countable. Let t₀ be the smallest element whose initial segment is infinite.

Ta-da! We have a moment with infinitely many preceding moments. Why is that absurd?

(I guess I should note, as Original Position did in conversation with duffe, that if you can show an "absurdity" in the construction above, you have a paradigm-shattering publication in your future. So you needn't really post it here; just publish and we'll all hear about it soon enough.)

the problem with craig's arguments against infinite regress is they use language in one way or another intended to misdirect you to think of the problem as if you must traverse infinity from a beginning point and/or in a finite amount of time ("must complete", "in order to reach today" etc). in your case you are thinking -

1. we've reached today, so treat today at as "finish"
2. look back to the "beginning"
3. move in reverse direction
4. it would be impossible to reach the "beginning"
5. therefor the past can't be infinite

obviously if the past is infinite there was no start, no beginning, and this line of reasoning doesn't apply at all.

in itself there is nothing absurd or illogical about an infinite amount of days before today if the past had no beginning. that is very simple if you think about it in the right direction without complications.

shifting the burden of proof is never a bad tactic for an apologist with unsupportable claims.

I'm not sure if you're still following, NotReady but I've started on that book on time by Craig which you recommended (only the very early sections so far) and it provoked the following question.

Given the Christian belief that people will ultimately live with God for eternity, why doesn't their experience of the afterlife constitute an actual infinity?

He distinguishes between potential and actual infinity which was first shown by Aristotle. The natural numbers, for instance, is a potential infinity. It's unending, or boundless, in one direction and will never be reached.

I've thought that if Cantor had just said potential instead of actual there would have been a lot less dispute over this issue.

So the afterlife is a potential infinite? Nobody will actually spend an eternity with God?
I think he was positing a fundamental shift from the Aristotelian understanding of the infinite (which was known to cause problems, I vaguely recall).

It is a potential infinite. We will never finish spending the afterlife so it won't be a completed eternity - generally when the phrase is used - "Spend eternity with God" - what is meant is that it will never be the case that we are not with God, or that we will always be with Him.

Cheers. I don't really see why a past 'one-sided' infinite should give rise to Hilbert Hotel problems but that a future 'one-sided' infinite doesn't. Nonetheless, I'm not looking to rehash this thread (it's kind of spent, hey?) it just occurred to me when I started reading some of Craig's opening remarks.

is god not an example of an actual infinite?

Question 106

Subject: Is God Actually Infinite?

http://www.reasonablefaith.org/site/...rticle&id=7087

I disagree with you - i think these 3 possibilities are completely obvious and beyond question.

Start with a mutually exclusive and mutually exhaustive set of possibilities:
1. Universe was designed
2. Universe was not designed

Then split #2 into 2 subchoices that are similarly mutually exclusive/exhaustive
2a. randomness was involved
2b. randomness was not involved

Slightly reword each of the 3 choices, same meanings, just more distinctive lingo:
1. necessity (ie, no design, no randomness)
2. chance (ie no design, with randomness)
3. design (ie design ldo)

Note that you don't have to believe in a fine-tuned universe for this argument. You can skip that premise and just point to "how did the universe come to its current state" - and you have 3 mutually exclusive/exhaustive choices, design, neccessity, chance.

I do agree with you that his conclusion is rubbish (which answer he chose and why). But narrowing his question to 1 of these 3 possibilities is logically valid.

I agree that these are the three most plausible choices, but I cannot accept that they are necessarily exhaustive. The two sub-options for 2 seem kind of vague to me (what do you mean exactly by "randomness was involved?). Also, I think that a combination of 2 out of 3 is possible as well. Here are two possibilities:

- The universe was designed, but still some of the variables were left to evolve randomly.
- The values of the constants were determined by chance, but the options were narrowed down by necessity (i.e., the degrees of freedom were not as many as Craig assumes). This can happen, for example, if there are some hidden variables which we currently don't know about, which bound the possible values of the constants within certain limits.

But you are right about saying that for him to assume that the universe was fine-tuned is flawed in the first place (he assumes his conclusion that life is special and qualitatively different than non-life, whereas as far as the universe is concerned, life does not hold a special status to non-life).

Your instinct is quite correct. In reality, there can be a frame in which a proper time of infinity has passed but only a finite amount of time has passed in another frame so it does not seem one can be allowed while the other is impossible. Surprisingly, these frames can be causally connected.

http://en.wikipedia.org/wiki/Malamen...arth_spacetime

So the notion that the universe could not have always existed because of some common sense argument about how only a finite amount of time must have passed to get us to "now" is also certainly wrong, not that I expect bad philosophers to stop using it .

how can they not be exhaustive? A + ~A = 1. right? therefore either there was some randomness involved in the creation/evolution of the universe or there wasn't. A + notA = 1.

- your first example (design with some randomness) fits solely in the category i labelled as "design". The description of "chance" had a clear description "not designed, with randomness).
- your second example (some chance, some deterministic) fits solely in the "chance" category since some chance was involved.

you can break these categories down into more specific sub categories if you want. But A + ~A = 1. substitute B + ~B = ~A, and you get A + B + ~B = 1 = a mutually exclusive/exhaustive set of choices.

Am I right to be astonished by this? I find this quite incredible - thanks for posting it.


I think this is a very powerful counter to Craig's approach - although I also think he rejects relativity (at least the totality of the theory) so it may not have much power for him. I know he covers it in the book of his I'm reading now.

Yeah, it is very surprising imo that that is a feature of relativity.

The way you define the categories, they are exhaustive. But I don't think Craig defines them like that, since he thinks that the "chance" explanation is so ridiculously unlikely. But if my second example is correct, chance is not that unlikely at all. So, this is different than his chance option.

Just to clarify: what I mean to say is that the options Craig provided are not exhaustive because his categories (1, 2, and 3) are narrower than the ones you presented.