Though, I should point out, Voyager was right to say “This says nothing about infinite chains of regress.” By which I mean, whether or not we choose to apply or withhold the mere words “number” or “quantity” or “magnitude” or so on to particular objects can’t actually change the non-terminological facts about those objects; it only affects how we choose to categorize them. Purely linguistic conventions can only have purely linguistic consequences.
Anyway… pardon the hijack.
That’s not true. You can have an Euler series that equals 1/2, for example.
What of it? You can have an Euler series that sums to an infinite quantity just as well (set x to -1).
What I was pointing out was that if you reasoned in the same manner as Voyager did in the quoted excerpt, and forced the definition of “number” to mean specifically “natural number”, then of course you would conclude that infinity “isn’t a number”; because you’ve taken specific steps to gerrymander it out! But this is no more enlightening or illuminative than would be a corresponding proof that 1/2 is not a number on the grounds that 1/2 is not a natural number.
I said natural numbers (ie integers). There is another definition for reals. Since he was trying to add 1, the definition of integers, which is simpler, should work just fine.
I’m pretty sure I can show that infinity isn’t a real number either, but looking at the definition informs me that I’d have to pull the definition of a ring out from where it has been hiding for over 35 years.
If someone is trying to prove or disprove the existence of god based on things like infinities, you need to be very precise about having your proof correspond to mathematics. Otherwise the hole you think you have found may not exist. You might as well prove god by dividing by zero as by adding one to infinity.
But the number of steps is a number, so you can’t have an infinite number of steps unless you assume your premise - that infinity is a number. If you are going to do that you don’t need to worry about Euler - an infinite number of successor functions will give an infinite number just as well.
However, by induction you can easily prove that given a finite number of steps, adding another will not give you infinity. Similarly, 1/2 is not a natural number because a finite number of steps in the expansion won’t get you there.
It is up to you to show how you get infinity from any axiomatic definition of a number without using infinity already. Infinities are just different.
I agree that infinity is neither an integer nor a real number. But that doesn’t mean infinity can’t be construed as a number, and it doesn’t mean one can’t add 1 to infinity in some other system of numbers besides the integers and the reals (e.g., one can do so in the projectively extended reals).
None of which will prove the existence of God, of course. But while we’re being precise, we might as well not make overbroad claims like “infinity is not a number”; at least, not in contexts where we’re liable to give these claims more import than they deserve.
My point is that there isn’t a single, one-size-fits-all axiomatic definition of number. There are different conceptions for different purposes. Many of these do accomodate entities which would be considered “infinity” or “infinite”. E.g., we can, if we like, define numbers as lines passing through the origin in the Cartesian plane (conceptually, we are identifying these lines with their slope) and make a system of arithmetic to match; then a vertical line is naturally thought of as an unsigned infinity, so on this formalization of “number”, “infinity” is most definitely a “number”. On other formalizations, it won’t be. There isn’t just one system to choose from.
I had no intention of bringing up Euler series myself, incidentally. I was just responding to the mistaken claim that Euler series could justify taking 1/2 as a number in a way in which they could not justify taking an infinite quantity to be a number.
Don’t bother; we all know you can’t increment up to/down to/up from/down from infinity. Proponents of infinite regress just assume the existence of an infinite block of prior steps, presuming that since they already have it, they can create it by adding one to it in succession. Remember, this is not a circular argument. (Keep repeating that.)
But regardless, all Aquinas really demonstrated is that, either infinte regresses are allowed, or uncaused causes* are allowed. (Once you clean up his erroneous argument formation into something sensible, that is.) That proves God…not at all. Even if you don’t accept infinite regresses.
Nobody is a proponent, here. YOU are the one assuming that infinite regress is logically impossible; it is up to you to provide justification for that assumption. All we are saying is that you can’t rule it out without providing a reason to do so.
Forgot to address this:
He didn’t even demonstrate that. It is possible to have an unbroken chain without a beginning or end; it’s called a circle. Aquinas didn’t rule out the possibility that the chain of causes curves back on itself. I know, it’s out there, but nonetheless it remains a possibility.
That’s a good observation, DrCube, though I suppose it doesn’t change much, since probably the intended use of the term “infinite regress” is one which covers that situation just as well; i.e., were it to be more formalized, probably the intended use of the term “an infinite regress” in this context is as any situation in there is an unending (thus, infinite) sequence of the form “A was caused by B was caused by C was caused by…”, whether or not that sequence contains repetitions (i.e., circles around).
Still, it’s a good catch in terms of what the logical possibilities are (or at least, start out as before someone decided to impose an ordering requirement upon causality).
To be fair, begbert2 covered that with his mention of time travel in the footnote.
Actually, the God argument has its own regress problem, far worse than the non-God one. Say we assume God exists from infinity. Say we have a bi-omni god. If he decides to take some action, the decision must be before he knows he will take it, or the decision is constrained, which means he is not omnipotent, and thus is not god. However, given any moment, if there is a time before that moment when god does not know the future, then he is not omniscient, and is thus not god. I’m sure some will say god is outside of time, but this knowledge and these decisions are intimately connected with time, and thus must be mapped onto our time line, however far back that is.
BTW, this thread shows the intimate connection between science and religion. Aquinas is justifying god through the science of his day. The statement that every action has a cause is a scientific one. The major thrust I see here is noting that since this science has been shown to be incorrect, so is the argument.
He seems to be pretty explicity excluding prior causes (and self-causes) to me, by asserting them to be “impossible”. That would kibosh the ‘loop’ theory (which is basically what a time-travel theory is.)
Well, yeah. There are probably half-a-dozen arguments by which the omnimax god can be driven to self-contradiction. But we’re having enough fun attacking Aquinas’s arguments in this thread without bringing others in, eh?
Intimate connection, heh. I see it as showing that the less you pay attention to the religious ‘science’, the better religion looks.
I agree. It either makes religion looks foolish because of who much it contradicts reality and logic; or it makes God look evil or uncaring.
I was just thinking last night about the arguments I hear from believers trying to defend the idea of a benevolent God. About how he’s unknowable, how we can’t understand his plans or motives and we can’t attribute terms like malice or other human terms to what looks evil in our eyes. And it suddenly struck me just how Lovecraftian that makes God sound. As something so alien that we can’t understand it’s thoughts or motives and which may crush us without us ever understanding why.
Yeah, I didn’t think the loop theory was a very good one (and begbert, I’m sorry for not catching your footnote preempting it). Nevertheless, Aquinas didn’t eliminate the possibility. He can’t just assert something is impossible; he has to show it with reasoning and/or evidence.
Also, I don’t think time travel is the only way to achieve a circular chain of causes; perhaps time itself is circular. I don’t think this jibes well with our current understanding of cosmology, but it seems we’re mostly discussing what is logically possible, not what fits the evidence best.
Anyway, are there any Aquinas defenders left? I notice the OP bailed on the first page and Caclulon (great name ;)) has been gone awhile…
Looking at his posting history in this thread I notice he tends to post later in the day. He may well be back.
What I find aggravating about this religious attitude, when I find it (and I often find it in my students), is its attitude toward reason. Okay, fine, maybe if we understood God from the perspective of Divine Reason we would understand why (say) He just slew 100,000 people. But all we have is *human *reason, and if you cannot show me according to *human *reason why it is rational to believe, then the possibility that it might be rational according to some other reason (which might not exist, and whose existence I am supposed to take on faith) is meaningless to me.
You are just in awe of this unholy ACTING TALENT!!! 
I only have time for a few points, because I have other stuff to do.
On infinite regress
I still maintain that an infinite causal series of events cannot exist. The central problem is that iterating over the elements of an infinite set is an impossible task. Since infinity - n = infinity, no matter how many elements you go through, you still never get any closer to the end.
And so it is logically impossible for the universe to contain an infinite amount of events before this one. If we define ourselves at the infinith step in the chain, (which is not the first) then there is no way the universe could progress through all of the events that separate this event to that infinith event.
The objection is not solved by some sort of ring set-up. Again it is required that to arrive at any point the universe must first traverse through an infinite number of pother points, which is impossible. Creating a ring only adds the extra fun that in order to exist the loop must first exist. It only adds to the problem, not solve it.
On causeless events and proofs
One point is that Aquinas, in order to make the argument, does not need to prove his premises. No-one does, that is not how logical arguments work. To prove the premises of the arguement you would need another arguement, which again has more premises which need to be proved and so on. That would lead to an infinite regress of proofs. Premises are taken as axiomatic, that is the point of them. That someone does not prove the premises is not a meaningful argument against their position. To invalidate an argument it is necessary to show that the premises are not true in every case, or that the premises do not connect to the conclusion.
Calculon