I thought I'd make this its own thread since the absolutism thread was getting a bit long. The question is when an infinite regress is a vicious infinite regress. Probably the best place to get situated on the problem (beyond the above thread) is Brandon's post from last fall. I meant to comment on it there, but never managed to. Allow me to quote from his analysis of Aquinas. (I changed the example from letters to subscripts to make my point clearer - Brandon used the alphabet in the original)
Consider the causal chain:
A0 <- A1 <- A2 <- A3 <- A4 <- A4...,
where A0 is moved by A1, and A1 by A2, and A2 by A3, and so on. In such a chain A1 mediates between A2 and A0, and so is an instrumental cause for A2; A2 mediates between A3 and A1, and so is an instrumental cause for A3. But, because we are dealing with moved, movers, one can say the same of cause-complexes: {A1, A2} is an instrumental cause mediating between A3 and A0, for instance; {A1, A2, A3} is an instrumental cause mediating between A4 and A0. So we have two options:
1. We can trace it back to some principal cause; and then everything between the principal mover and A0 will be a mover moved by the principal mover.
2. We can trace it back infinitely; and then everything that moves A0 must be an instrumental mover, but what moves A0 will not be an instrumental mover, because it will not be moved by anything. In other words, on the infinite regress, we have a mover (what moves A0) that is both moved and unmoved; and, as well, given that to be a moved mover requires that it be moved by something, the nonexistence of that thing means that it is both a mover and not a mover. This contradiction, I think, is the problem Aquinas sees with infinite regress.
Now as I mentioned, I'm anything but an expert in medieval philosophy. Brandon, anticipating my comments in the other thread, wrote, "it's common to accuse Aquinas of begging the question here, but I don't think he is--he's just being more concise than modern readers can usually follow." I guess I'm one of those modern readers as I just can't follow how it doesn't beg the question. I read through the above and thought about it for quite some time last night. But it just appears to me that he sneaks in his conclusions.
To me the "contradiction" that we both have and don't have an instrumental mover seems false. Let's look at the definitions (reading back into Brandon)
D1. Principle Cause. A cause that is uncaused
D2. Instrumental cause. A cause that mediates between two causes
D3. cause-complex. A set of instrumental causes acting that can be taken as a single instrumental cause. (i.e. {A1,...An} = C1)
Now if I follow Brandon's logic he argues that for any infinite series we can generate a cause-complex such that the cause-complex is made up of all instrumental causes we pick. Thus we both have an instrumental cause (the set of instrumental causes) but simultaneously don't because for this set, there won't be a cause prior. That is for any set of instrumental causes which are the instrumental causes there will be a missing cause prior to the instrumental causes.
Now that works for finite series, but doesn't for infinite series. So I think, if I am indeed reading him right, that Brandon's mistake is over the properties of infinite series or perhaps trying to treat the infinite as a finite series. I think the "contradiction" only works because he makes the cause-complex the infinite series prior to A0 and then asks, what is the cause prior to this cause-complex. But that's a no no for the same reason when we were kids someone would say "I dare you infinitity" and the next kid says "I dare you infinity plus one!" You simply can't treat infinite series like finite series. It seems to me that the appropriate answer is that even for an infinite series making up a cause-complex there is always an instrumental cause prior to it, thus allowing the cause-complex to truly be an instrumental cause.
The only way this would work is if for an infinite series cause-complex there was no prior cause. But that only works if we presuppose the necessity of a principle cause - the very point that Aquinas is trying to establish. Alternatively we might say that one simply can't make sense of infinite cause-complexes and we can only consider them as finite series.
Put more simply, I don't see how this isn't just an other manifestation of the same sort of error Craig made with respect to the Hilbert Hotel. There we had an infinite number of people in an infinite number of rooms and ask if there are any rooms left. Here we're saying we have an infinite number of instrumental causes in a cause-complex and ask if there are any instrumental causes left.
Id quod movet instrumentaliter, non potest movere nisi sit aliquid quod principaliter moveat. What moves instrumentally, cannot move unless there be something that principally moves. (Aquinas, SCG 1.13, quoted from Brandon's post)
This is a bad assumption that appears to be based on unnecessarily narrow semantics. There is no reason why an intermediate cause needs to be the instrument of some antecedent principal cause in the case of an infinite series of intermediate causes.
Mark, is that a conclusion or a premise though? (I don't have my Aquinas here at work and am too lazy to look it up on the net)
Aquinas actually didn't have a problem with infinite regress. Quoting from Copleston (Vol 2, pp 366): "Moreover, for St. Thomas as series can be infinite ex parte ante and finite ex parte post, and it can be added to at the end at which it is finite." In other words given a series as long as you find one end of it you can add things to the end you know about and there need not be a finite other end, so infinite regress is fine as long as it happens on one end of the series.
The real question is does Aquinas believe this causal chain to be finite or infinite? Again quoting from Copleston (Vol 2 p 367) where Copleston quotes Aquinas as saying, "We must hold firmly, as the Catholic faith teaches, that the world has not always existed". If he holds this belief the causal chain is finite and there is no problem with this argument
St. Bonaventure held that it could be proved that the world has a beginning in time. However, Aquinas held that it could not be proved that it had a beginning. So on this argument the world/universe might be infinite ex parte ante, and the causal chain would be infinite. If this is what he held then his argument will have the same errors (like the infinite hotel argument) that any argument has that treats infinity naively and not rigorously with something like set theory.
But David, the whole argument about the unmoved mover is explicitly an argument about an infinite series needing two ends. I think Brandon mentioned in the other thread disagreeing with Craig whether temporal regress was vicious. So certainly he feels Aquinas doesn't take regress to be always vicious. The question is this particular kind of regress of cause.
I'm a bit busy at the moment, so I'll be back in a bit with further thoughts for you Clark. With regard to Mark Butler's remark: I don't understand what you mean at all; you seem to be saying that the law of noncontradiction is just a matter of semantics, which I don't think is what you mean. If it doesn't have a principal cause, it's not an instrumental cause; whatever else it may be. We can't just go around arbitrarily mixing and matching concepts and still consider ourselves rational. But I'm assuming that I just am missing your point.
Thanks Brandon, I always appreciate your thoughts. And no rush. I know how life can get busy. Blogging always comes last.
Clark, just a few brief fringe comments until I can read your post a bit more carefully.
-> A cause that principally moves is not the same as an uncaused cause; a principal cause can also be an instrumental cause. The unmoved mover(s) come(s) in only at the conclusion, when we have considered all instrumental causes, even those that are also principal causes in their own order.
-> An instrumental cause is a moved mover, i.e., a subordinate cause mediating between the principal cause and a more distant effect insofar as it (the instrumental cause) is moved to have the effect. So I can legitimately say that I am making the marks on a piece of paper because the pencil that is directly making the marks is operating in virtue of my causation. The pencil, in other words, is a distinct cause; but its causing of the marks is not a distinct causing from my causing the marks. The pencil causes the marks in virtue of my causal ability to move the pencil to make the marks. Instrumental causation, then, has the following structure:
The instrumental cause I has effect E in virtue of being caused to cause E (or, in some cases, being caused to be able to cause E) by some other cause (which for convenience we call the principal cause).
The cause-complex comes in because we can embed instrumental causes within instrumental causes (e.g., I can use a stick to press the key to send the signal to make the letter appear on the screen). The whole system of instrumental causes or moved movers in such an embedded complex is itself a moved mover or instrumental cause. If we are regressing in moving causes, we are regressing in instrumental causes or moved movers.
-> We're not asking whether there are any instrumental causes left; we're asking whether the whole system of movers prior to the effect is to be regarded as an instrumental cause. If yes, we have a contradiction; if no, we have an unmoved first mover.
As far as I can tell it is a premise to an independent argument:
Tertia probatio in idem redit, nisi quod est ordine transmutato, incipiendo scilicet a superiori. Et est talis. Id quod movet instrumentaliter, non potest movere nisi sit aliquid quod principaliter moveat. Sed si in infinitum procedatur in moventibus et motis, omnia erunt quasi instrumentaliter moventia, quia ponentur sicut moventia mota, nihil autem erit sicut principale movens. Ergo nihil movebitur.
** [This is the way:] What moves instrumentally, cannot move unless there be something that principally moves. But if one were to proceed infinitely in moving and moved, all would be like instrumental movers, because they would be posited as moved movers, and nothing would be a principal mover. Therefore nothing would be moved.
As translated / abridged by Rickaby, Aquinas' other arguments appear to contain the similar assumptions:
WE will put first the reasons by which Aristotle proceeds to prove the existence of God from the consideration of motion as follows.
Everything that is in motion is put and kept in motion by some other thing. It is evident to sense that there are beings in motion. A thing is in motion because something else puts and keeps it in motion. That mover therefore either is itself in motion or not. If it is not in motion, our point is gained which we proposed to prove, namely, that we must posit something which moves other things without being itself in motion, and this we call God. But if the mover is itself in motion, then it is moved by some other mover. Either then we have to go on to infinity, or we must come to some mover which is motionless; but it is impossible to go on to infinity, therefore we must posit some motionless prime mover. In this argument there are two propositions to be proved: that everything which is in motion is put and kept in motion by something else; and that in the series of movers and things moved it is impossible to go on to infinity.(SCG 1.13, Rickaby abridged translation)
The idea that something in motion requires something else to either put or keep it in motion is unsupported and incorrect.
The Philosopher also goes about in another way to show that it is impossible to proceed to infinity in the series of efficient causes, but we must come to one first cause, and this we call God. The way is more or less as follows. In every series of efficient causes, the first term is cause of the intermediate, and the intermediate is cause of the last. But if in efficient causes there is a process to infinity, none of the causes will be the first: therefore all the others will be taken away which are intermediate. But that is manifestly not the case; therefore we must posit the existence of some first efficient cause, which is God. (ibid)
Same assumption here as before.
Thanks for the clarification, although I'm not sure that my mistake about principal movers changes the logic.
To say "whether the whole system of movers prior to the effect is to be regarded as an instrumental cause" really seems like the child saying, "infinity plus one."
My point is that it is wrong to exclude infinite causal chains simply because they do not fit into a improperly narrow definitional mold. Aquinas repeatedly assumes that all causal chains must have a first term.
I think the question is, Mark, whether he ever argues for why they are vicious and thus why they must have a first term. Also, as I think Brandon said in the other thread, Aquinas didn't appear to make this assumption with regard to the temporal series where he appears to have considered it a matter of faith.
Frederick Copleston has the following explanation:
When Aquinas talks about an "order" of efficient causes he is not talking of a series stretching back into the past, but of a hierarchy of causes, in which a subordinate member is here and now dependent on the causal activity of a higher member ... We have to imagine, not a lineal or horizontal series, so to speak, but a vertical hierarchy, in which a lower member depends here and now on the present causal activity of the member above it. It is the latter type of series, if prolonged to infinity, which Aquinas rejects. And he rejects it on the ground that unless there is a "first" member, ... a cause which does not depend on the causal activity of a higher cause, it is not possible to explain the ... causal activity of the lowest member. His point of view is this ... Suppress the first efficient cause and there is no causal activity here and now. If therefore we find that ... there are efficient causes in the world there must be a first efficient, and completely non-dependent cause. The word "first" does not mean first in the temporal order but supreme or first in the ontological order.(Copleston quoted in Hierarchical causes in the cosmological argument by Stephen Davis)
A requirement for a first cause in a simultaneous chain of efficient causes (though unproven) seems reasonable to me.
That's more how I was taking it Mark. I'd add that Leibniz has an infinite number of efficient causes but also a first cause. He's probably somewhat unique in that though. For the record I'm not sure there has to be a first cause. Indeed I strongly suspect there doesn't or if there is it is unprovable what it is or that it is.
I'm having difficulty following what arguments are being given here, so I don't think I'm understanding either of you correctly.
(Clark) To say "whether the whole system of movers prior to the effect is to be regarded as an instrumental cause" really seems like the child saying, "infinity plus one."
I'm not sure what you mean here. Where is the equivalent to "plus one" in the argument? The claim of the argument is not that you need another cause behind all this infinite series; the claim is that the characterization of the series itself becomes self-contradictory on the failure of a particular condition. This is not a childish claim to make; in any sort of discussion of an infinite, mathematical or otherwise, a characterization that implies contradictory statements can't just be dismissed by waving one's hands vaguely and saying, "Oh, but we are talking about infinites, so anything goes". (And assimilating the issue to mathematical infinity is misleading, in any case, since what generates the contradiction is not infinity but the particular type of causal dependence. The mistake, which appears to be clearly expressed in the infinity-plus-one analogy, seems to be in thinking that Aquinas's argument is that the infinity is somehow generating the problem on its own; when it actually is that the particular kind of causal dependence -- which for the purposes of the sub-argument against infinite regress is assumed to exist, having been argued for in a previous premise -- is interfering with the supposition of infinity.)
Either the whole series of prior causes is taken as a first mover or as an instrumental mover; if the former, there is a first mover; if the latter, there is a contradiction on the supposition that the series is infinite, because on such a supposition the whole series of prior causes will, given the nature of instrumental causation, be an unmoved moved mover. The only possible third option is to deny that you can take a whole series of prior causes; but this is ruled out by the nature of this particular causal dependence.
I find the last sentence of your second-to-last paragraph in the post to be very obscure; is the group of causes being taken the group of all prior causes, or is it not? If it isn't, your objection doesn't deal with the argument. Aquinas's point is not that you can't in principle have a cause prior to any particular cause (he accepts that you can), but the above dilemma when we take the whole series of prior causes, which we should be able to do with instrumental causation.
(Mark) My point is that it is wrong to exclude infinite causal chains simply because they do not fit into a improperly narrow definitional mold. Aquinas repeatedly assumes that all causal chains must have a first term.
As I said before, this second sentence is false. E.g., none of your three bolded premises are equivalent to each other, and none of them are taken by Aquinas in such a sense that they entail on their own that there must be a simply first term to any causal chain. The only one that can plausibly be read that way is the third, and in context it is clearly just indicating that particular sorts of causal chains require a relative first term. (BTW, the "A thing is in motion because something else puts and keeps it in motion" premise is very misleadingly translated by Rickaby; the premise is more literally, "Whatever is moved is moved by another." The Aristotelian meaning of the term 'motion' makes this a necessary truth: denial of it is logically equivalent to claiming that you can have uncaused effects.) I would need to see the actual argument for the claim that the definition is improperly narrow before I comment further; but just going on what your objection sounds like it would be, i.e., if it is really an improperly narrow definition that is the problem, this wouldn't affect the argument against infinite regress of this sort of cause at all -- it would just deny that any such causes actually exist, which would be an entirely different part of the First Way.
Strictly speaking, Aquinas's argument is consistent with Leibniz's claim; if you posit a first cause, you could have an infinite series of causes between it and an effect, because the relevant contradiction wouldn't arise. Aquinas has other reasons for rejecting this possibility, but they don't come into play in this argument, precisely because it's the introduction of the first cause that avoids the contradiction.
Of course a natural "hierarchical" causal chain composed of finite elements has a non-zero propagation delay. An infinite natural finite element HCC would have infinite propagation delay, placing it clearly out of the realm that Aquinas apparently intended here.
As a modern apologetic, the cosmological argument seems rather lacking, presupposing a hierarchical chain of being for which there is no empirical evidence. Interestingly, Mormonism has a similar concept in regard to the "light of Christ", which proceeds forth from the presence of God to fill the immensity of space, and which is the law which governeth all things (paraphrasing D&C 88).
Brandon, I'm afraid you're going to have to explain how the contradiction follows then. I just don't see how it does from your reading or presentation. It doesn't seem to just be an issue of causal dependence, but the relationship of the properties of infinite sets and causal dependence. It still seems to me like your contradiction can only appear if you treat infinite sets like finite sets.
The problem is that when you take the whole series you still have room left over. Thus the contradiction never happens. It's just one of the weird properties of infinite sets.
Just before I go to bed, I was reading some Scotus that might clarify things somewhat. (Always a dangerous situation though - sleepy and reading Scotus. Last time I did that. . .)
Scotus' approach to this problem can be taken like the following. An infinite series ends up being like an infinite regress of ifs. So take the following.
Case 1: If A then B
Case 2: If ( if A ) then B
Case 3: If ( if ( if A ) ) then B
. .
Case n: If ( if ( if ( ... [n ifs] ... ( if A ) )...) then B
Now if I have Scotus right, his objection is that it doesn't matter how many ifs you have. Even if you have an infinite number you still eventually need that A to be able to say B. And this is why I think Brandon is saying we are having a contradiction and not seeing how it is related to infinity. He sees it not a problem of infinite sets but a problem of causality. To be able to say if ... then ... you have to have something to put after the if.
The problem is that all this presupposes that the structure is an "if A then B." That is that assumption of an A is hidden in our belief about the nature of the "ifs." Thus it ends up being a sneaky way of merely saying that we have to have to have "if A then B." But supposedly, that was what we were setting out to prove. So it ends up begging the question.
Clark, show me where in the argument this supposed instance of treating an infinite set like a finite set is supposed to occur.
I don't know what you mean by "when you take the whole series you still have room left over", unless it is just that there is nothing in the series as such that does not allow for yet one more prior cause. That would be true; as you say, that's just a property of infinite sets (and not even, really, a weird one, since we have to presuppose it in quite a few different kinds of reasoning). But it's not the issue; the issue is the nature of the whole series itself, namely, that (1) either it is the whole series or it is not, and if it is not, we would be engaged in an ignoratio elenchi; (2) if it is the whole series there is a principal cause (whether an initiating cause in the series or the whole series itself, doesn't matter) or there is not; (3) if there is not then, given that we are considering instrumental causes, and series of instrumental causes are necessarily instrumental causes, the whole series is an instrumental cause; (4) if it is an instrumental cause, since instrumental causes are by definition moved movers, and it is not the whole series. You keep reading it as if it were supposed to be a contradiction about the elements of the series ('infinity plus one' is always put forward as a claim about the elements of the series, namely, that they can't be infinite) when the actual claim is that the nature of the particular kind of series generates a contradiction if we suppose that there is no first element, of whatever sort it may be. The issue of whether or not the series is actually infinite does not arise at all; what arises is that it is not logically possible to consider the whole series of prior causes as not either involving or being a first mover.
(A side note. As I said, it's not relevant here, but it seems to me that even if you were arguing against, say, someone like Craig, you're not being sufficiently demanding on yourself to make a telling point. By 'demanding' I mean this: you keep coming back to the problem of treating infinite sets as finite; but you've done nothing whatsoever to show that the nature of the series itself doesn't introduce a property that can only apply to finite sets. So long as you don't do this, you haven't even come close to making your argument, and this would be true even if you were arguing with someone like Craig, who is clearly vulnerable to this sort of objection. If the sort of collection which Craig is discussing, namely, the series of moments in time, were such that it necessarily had a property conflicting with its creating an infinite set, like, e.g., the collection of air molecules in this room, he would be entirely right. He's not, since the collection is not like that; but if he were, all the insistence on not treating infinite sets as finite would be utterly futile, even though, for all that you've actually said here, you would have exactly the same reasons for saying it. It is impossible that there be an infinite set of actual air molecules in this room; this is entailed by the nature of the collected elements and the collection itself. So even if you were arguing against Craig's argument rather than Aquinas's, you wouldn't have made your point.)
Brandon, that's for expanding the argument a tad. That's very helpful. The key area of misunderstanding appears to be the meaning of "the whole series" and how that is meaningful in the context of the earlier definitions.
Taking "the whole series" as a single entity then
a. that series is the principal cause or
b. there is no principal cause
c. If (b) then the series is an instrumental cause.
d. if (c) then there must be an other element. (by definition of instrumental cause)
e. but (d) contradicts our claim that we are taking the whole series.
Is that way of rewriting it accurate? If so, then clearly the flaw is in assuming we are taking the whole series or can take the whole series. That is we presuppose in our definition of instrumental cause that a true infinite series is impossible. Thus it still sounds like begging the question.
With regards to Craig, I think I critiqued his view in the link I provided. It was a contradiction by a demonstrable example with infinite sets. Craig, recall, is the one who brings up Hilbert's Hotel.
Put perhaps a little better, what you call ignoratio elenchi might not be a straightforward as you say. Let me change the topic slightly so as to present an analogy that perhaps will clarify things somewhat.
Let I be the set of integers
We wish to find out if there is an element e in I such that e is the highest integer.
Now either e is in I or it isn't.
If it is, the I is the set with the highest integer.
If it isn't then clearly there is an integer higher than the highest element in I.
But if there is an other element e that isn't in I, then we didn't have the full set of integers after all.
Therefore by contradiction we didn't really have the set of integers.
Therefore in I is a highest integer
Now I'll admit to being obtuse far more times than I care to admit. But I don't quite see the difference between what you say and the above.
Clark, I discussed or mentioned the issue about whether we can take the whole series somewhere in a comment above; so when I have more time I'll have to go back and see whether I need to expand on the point; and I'd need to see the argument that it is a flaw. Your reconstruction is very close; with regard to (a) it would be more accurate to say that either the whole series has a principal cause or is itself a principal cause. In the case of the unmoved mover, strictly speaking Aquinas doesn't rule out the possibility of the latter until he argues that the unmoved mover has to be simple. With regard to (c) and (d) I would prefer not to bring in talk of elements at all; the primary contradiction in the argument is not whether it's the whole series or not but: given that this is the whole series, it would have to be both unmoved (because it is the whole series) and moved (because it is an instrumental cause). But I think we've finally clarified the basic point.
Even if wrong, it can't be begging the question: the definition of instrumental causation is not ad hoc or arbitrary but the result of a rather extensive analysis of motion. This is why I keep saying a much more plausible criticism of Aquinas's First Way is that there isn't actually any instrumental causation in the real world. A Humean, for instance, would have to say this. The argument against the infinite regress would still stand, but it would be completely otiose.
On Craig, fair enough; my primary point was just that we need to consider the nature of the thing to which infinity is being attributed before we can say anything on the matter.
Clark, where is the analogue in the integer argument to the notion of instrumental causation? And where is the analogue to the notion that a series of instrumental causes is an instrumental cause?
The structural analogy seems kind of straightforward to me. But I'll relook at it tonight to see if I made some obvious error. (Hardly unusual)
With regards to your comment, "we need to consider the nature of the thing to which infinity is being attributed before we can say anything on the matter," we may be agreeing. Put an other way, if our concepts presuppose finitude and we try to argue for some semblance of finitude because infinity doesn't work with our definitions, the problem can rest either with our ascribing infinitude or with our definitions. I think I'm suggesting the problem is with the definitions while you're suggesting it is with infinitude.
Perhaps this question might also help clarify things: what do you mean by a concept presupposing something? One way in which people sometimes colloquially use 'presupposition' makes it non-question-begging, because on that usage, any rigorous inference, say, from A to B, will be such that A 'presupposes' B; that is, the criterion for identification of a 'presupposition' is that A implies B. Now, this is clearly not question-begging because it would apply to any rigorous inference from A to B, however we came by A; but an inference is only question-begging if we originally got A from B (in some way). For another way in which people use the term 'presupposition' is genealogical: If you originally got B from A, B presupposes A. And when this happens, A -> B is question-begging. It is absolutely essential to distinguish these two to avoid saying that all of mathematics, and all application of mathematics to science, and all use of deductive logic, is question-begging. But another use of 'presupposition' is to take it as domain-identifying: (A presupposes B) iff (A is applicable only if B obtains). This sense of presupposition would give a different meaning; it also requires a different pattern of argument, since one would have to have independent reasons to think that there is a condition on the applicability of A that makes it so that it applies only if B obtains. So there are several different things you might mean by 'presupposition'. I took you as meaning the genealogical version, and so pointed out that Aquinas develops his definitions through a considerable amount of analysis; but your latest comment sounds more domain-identifying. There is, indeed, domain-identifying presupposition here; but it's not surprising, because, in essence, that's what the argument is setting out to determine: is there a domain-identifying presupposition in the case of instrumental causation? In fact, this is just the question, "Is this infinite regress vicious?" put in different terms; when you ask whether a infinite regress is vicious, you are simply asking whether the sort of regress involved presupposes finitude. But coming to the conclusion that there is a domain-identifying presupposition isn't question-begging, either; it's just good straightforward analysis and inference.
It seems to me that if your problem is with the definitions, then strictly speaking you aren't finding a flaw in the argument against infinite regress; you are just denying that the relevant sort of regress (regress of moved movers) exists, or that, if it exists, Aquinas has erred in his characterization of it. And that doesn't actually affect the argument against infinite regress; it just affects whether it is relevant to anything.
Perhaps you're right Brandon. I'll have to think about that for a while. Unfortunately right now I'm exhausted and my mind is mush. So I'll have to rest on it a bit.
Very quick and dirty comparison. I may rework this more tomorrow. (Sorry about the formatting - I was in a hurry)
| Let A be any set where either A or a proper subset of A does not contain the highest integer | Let A be the set of causes where either A or a proper subset of A mediates between two causes. (i.e. does not contain the first cause) |
| Let Sn be the set composed of adding integers to A such that Sn > Sn-1 | Let Sn be the set composed of adding causes to A such that Sn causes Sn-1 |
| Let F be the full ordered series of integers | Let F be the full ordered series of causes |
| Either F is in A or F is not in A | Either F is in A or F is not in A |
| If F is in A: then we have don't have a highest integer and thus we must have an other integer outside of F by definition of highest integer | If F is in A then we must have a cause outside of F by definition of mediating |
| If we have an other element then F is not the fiull ordered set of integer | If we have an other element then F is not the full ordered set of causes |
| But this is a contradiction | But this is a contradition |
| Therefore F is not in A and F contains the highest integer | Therefore F is not in A and F contains the first cause |
I don't want to pile more on your busy schedule, but I thought I'd at least say briefly why I think the parallel here is only superficial. Actually, they're very much what I've already mentioned. Note that steps (5) of your reconstruction are actually very different; both use definitions, but 'highest integer' and 'mediating cause' function very, very differently in the two arguments (for the parallel to be close, the analogue of 'highest integer' would have to be 'first cause'; but that's not the case).
Second, mediating causes are not related to their causes and effects in the way integers are related to their predecessors and successors. This is the whole point of the distinction between the infinite regress of accidental causes (which are integer-like in their relations to each other because they don't cause each other to be causing) and the infinite regress of essential causes -- in this case, of instrumental causes, where one cause's causality is instrumentally linked to another cause's causality. A crude example: my hand moving a stick to write in the dust; the stick causes the marks in the dust as a part of my causing marks in the dust.
Third, this may be my lack of a strong background in mathematics, but it isn't really clear to me that the move from step (3) to step (4) on the left makes any sense; is a series of integers really an integer? But a series of instrumental causes, because of the way they depend on each other, is an instrumental cause; and that's what motivates the relevant move in the case of causes.
I'm slowly getting caught up.
For step 3 - 4, I tried to make it in terms of ordered sets. So the issue isn't an integer but whether the property "the highest integer is a member" is true. So we have one set and then an other set. We're trying to find out if the second set is a subset of the former. I probably should have tightened up the language slightly. Rather than say, "is in," I should have said, "is a subset." But I was used to saying "is in" as a shortcut for that.
I'm afraid I don't quite understand what you're saying about how instrumental causes function so differently formally. I understand in terms of content, mind you. I just don't see how that affects the form of the argument. That's why rather than just talking about integers I talked about ordered sets of integers so I could capture the formal structure of instrumental causes.
That's not to say I've not made a mistake. I've honestly not looked at the above since I wrote it Sunday night. But while I think your criticism of "first cause" is perhaps apt, I'm not sure the rest is.
Well, it could very well just be that I don't understand the particular parallel you are drawing. Instrumental causes aren't related to effects like integers at all; a series of instrumental causes is not like a series of integers because an instrumental cause's being a cause at all is part of its principal's being a cause. So these sorts of causes aren't merely ordered in terms of places; they are linked. This is why a series of instrumental causes is also a single instrumental cause. I don't see how anything like this is being appealed to in the integer case; but it is this that actually is incompatible with infinite regress. Without you don't get a contradiction. This is why I pointed out the difference between the types of infinite regress. One way to put the reason why Aquinas doesn't think a temporal infinite regress is vicious is in precisely these terms: since Aquinas holds that time is simply a measure of duration, moments of time are just parts of a measurement, and so are related to each other entirely as integers in an ordered series are related to each other. But this is not the case with instrumental causation; the causes are not merely ordered but are ordered according to dependence. The dependence is what is incompatible with infinite regress. I don't see how a parallel would arise in the case of integers.
Perhaps we should clarify what the formal properties of an instrumental cause are? Also note that I didn't just have a series of integers but rather sets of ordered integers. The properties are different. Thus I could easily deal with the example in your initial post is an isomorphic way. Now perhaps there is a way the two aren't isomorphic. And perhaps that's where I'm going astray. But it seems that leads us to the formal meaning of instrumental causes.
The issue of dependence doesn't seem to avoid the parallel though. The issue isn't whether causes are merely ordered but rather if they are ordered in any way. But by making the example use sets rather than just integers, I think I did that. So the dependence ordering can easily be mapped into sets of integers.
Fair enough; but I'm not sure if anything changes when we think of sets, either. But, again, that may just be my not having sufficient mathematical background. It's more than that causes are ordered in any way; it's that they are ordered in a particular way, so that (as I said before) one is a cause by being part of another's being cause. That's what it is to be a moved mover, and it's that relation that generates the contradiction. Is something analogous to that really operative in the integer case? I don't see what it would be, so if it's there you'll have to spell it out for me.
In other words, instrumental causation is transitive, irreflexive, and asymmetric, and in some ways analogous to proper parthood. However, it is different in that causation is not actually mereological; being 'part' of X's causation means dependence on X. Since integers are not linked causally, I'm not seeing why there's supposed to be such an analogy between the two.
Looking at my argument, I think I need to bring in the notion of power sets rather than just proper subsets. I'll try and rework it slightly tonight.
As to your argument, the question is what formally being linked causally entails. That is, what is the formal sense of causation in this sense. I'm taking it of the form "if X then Y" which can be considered isomorphic to { {An-1}, {An}} with order maintained.
Clark, I'm not really sure what you're looking for. Show me the formal apparatus you have in mind that has causal dependence as an ordering relation, and I'll look into it to see whether it fits Aquinas's notion.
My best answer on the issue of formal property, allowing for the fact that I'm not sure what you are asking: What is relevant to the original argument is the cause's being the sort of thing it is, namely, a moved mover. In other words, maintaining order in the sense that this usually applies to sets doesn't suffice to characterize the dependence; even partial ordering isn't accurate, since it doesn't capture the relevant dependence. If two integers were linked in this way, say A was dependent on B, A's being an integer would depend entirely on its being related to B's being an integer in such a way that if B were suppressed, A would not be an integer. I'm not sure what that would even mean; things are not integers incidentally, such that whether a number is an integer just depends on what ordered set it's in. This is why I don't understand why you keep thinking it would be so easy to formulate a causal relation in the same way one would formulate an ordering relation between integers. That seems to me to be close to begging the question; I don't know the reasons behind your attempt to treat causal relations as if integers could possess something precisely analogous to them. You can take ordered sets of elements arbitrarily representing causes all you wish; it isn't clear why this would tell you anything about them as causes. And likewise with sets of sets. All that such a representation will describe (as far as my limited knowledge suggests) are those formal properties of causes that allow them to be represented by integers. But there's nothing about the ability of causes to be represented this way, nor even how the integers representing them can be related to each other in an ordered set, that makes the infinite regress even an issue. The relation is not obviously something that would obtain between integers at all. Now, I wouldn't know if there is even anything that has been proposed in set theory as an asymmetric, transitive, irreflexive relation between one integer and another that makes the one integer an integer, whatever that would be; but such an analogue would be necessary to make the parallel.
But again, it's possible that I'm just not sufficiently clear on the sort of argument you're making; what little I've studied of set theory was quite a while ago. (There are lots of question where I don't know if I'm onto something or just asking about something that's completely obvious to you, e.g., to name a minor one, it isn't clear to me why you are making the parallel to the highest integer, rather than (say) to the lowest natural number, beyond wanting to find a mapping that fits your point. Is there something that makes the integers especially suited to represent causes? Is there something about a highest integer that makes it especially suitable as an analogue to first cause? If there were one would assume the conclusion of the left-hand argument to be, in effect, that if there is no highest integer it would be impossible for anything to qualify as an integer. But that doesn't seem what it's getting at.)
Brandon, I'm not neglecting the thread. I want to be able to write something cogent up. But that means I'll not be able to respond until probably tonight.
Are we still live here?
I'm reading your post and comments, and Brandon's original post, and I'm siding with Clark. In Brandon's original, in section 9, Aquinas' argument (as translated) is just "But you have to have a first cause", which is (as I understand it) the question before the floor.
Also, Brandon, your objection to the analogy between integers (or sets of integers, either will work) and causes doesn't hold. Math is all about the pattern of relationship, whether it be 'is caused by' or 'is greater than' or 'stands to the left of' or what have you. The key insight about infinite sets and series, from Cantor, is that they don't behave like finite sets of the same things. Thus, it's possible that you can have a rule like "a finite series of intermediate causes is an intermediate cause" without being able to be sure that "an infinite series of intermediate causes is an intermediate cause".
For this argument, I for one have no problem calling { A1, A2, A3, ... } a prime mover, an uncaused cause. And I don't begin to claim that it's God. It's just an infinite regress of causes, and I'm OK with that.
p.s. I was heartbroken to find a "Craig" who messed up Hilbert's Hotel so badly.
I've been so busy this weekend and week that I've not answered Brandon, despite promising to.
So here's the deal. Like Craig mentioned, by formal analysis, I just mean trying to find the way to represent the argument without appeal to the definitions as such. That is, to find the formal or logical structure. That was the point of the integers. To find a mathematical argument that Aquinas' argument is isomorphic.
The heart of the issue is the unstated premise that the instrumental cause can have an infinite number of elements. In my formal analogy, I attempted to represent this as a set of sets that can contain set and so forth. The only restriction was that ultimately any set can not be empty or contain an empty set. Any set must thus either be a set of sets or else consist of ordered integers or a combination of the two.
This is the attempt to capture the formal structure of instrumental causes as possibly consisting of further instrumental causes.
The ultimate formal comparison is then between a set of all possible sets according to our definition which compares to the single instrumental cause that consists of all the infinite number of instrumental causes that make it up.
Now the key complaint is some property of these sets which the two structures share. For instrumetnal causes it is related to their intermediate nature. That is they mediate and thus there must be something before them. The isomorphic situation in the set of sets or ordered integers is the question of the highest integer. Now we recognize there is no highest integer, and thus even the set of all integers can't have that property. Yet with the definition of instrumental causes we have the definition requiring something like that.
So the question really becomes, does this definition of instrumental causes render it incompatible with an infinite series or does it render an initial cause (or something equivalent) necessary? I favor the former while Brandon favors the later. But it seems to me that ultimately the problem is in the application of the set of infinite instrumental causes making up the instrumental causes.
A better way of looking at it is by considering the process of trying to generate the single instrumental cause that is the whole series. Can we say we are done, such that we've actually formed this instrumental cause? I don't think so, because at any point there is an instrumental cause left out. Yet to generate the contradiction in Brandon's argument, we have to have reached the point when we've generated this infinite series and can ask, what is next.
Craig, whether math is about the pattern of relationships really isn't relevant; what is relevant is whether Clark's proposed parallel is adequately parallel. I don't think it is (and I think the difference in the fifth step noted above suffices to show it, independently of this other issue of whether one can adequately capture the relations between causes by relations between integers). And as I noted above, Aquinas's argument is not, even as translated, "But there is a first cause"; the closest it comes to saying such a thing is pointing out (rightly, or the whole point would be moot) that on the infinite regress there is no first cause. He then goes on immediately to look at second movers, however. If he were making the move you are suggesting, this would be otiose; but Aquinas in this context is being very concise (the First Way is not the argument from motion but a summary of it), so it's implausible that he would just throw in something that he didn't think important to the argument.
Clark, the mediation of instrumental causes is not successive mediation (in A, B, C, B happens to be in the middle) but causal mediation (in A -> B -> C, B mediates A's causation of C). This is quite explicit in the scholastics, because it's a major reason why they allow for infinite regresses of accidental causes (which mediate only successively; cf. Aquinas's recognition that an infinite past is not absurd). Between integers, however, we only have successive mediation; integers do not mediate integrality to each other. This is why I can't make any sense of your proposed isomorphism. We seem to be back to my original complaint, which is that your mathematical models keep ignoring the nature of the causal links, which is the whole point of the argument. It's the infinite regress of this sort of link that generates the contradiction, not the infinity of the infinite regress itself. Yes, if causes were related as integers can be related to each other (as accidental causes and temporal points are), there would be no problem; but that's precisely what's denied in these contexts.
I don't understand the argument in your final paragraph.
Brandon, I think the way to consider it is that the use of "instrumental cause" seems to entail a use of infinity some might disagree with. That unstated but apparently implicit relationship to infinity is what generates the contradiction. It also isn't obvious from the pure definition that one can make the instrumental cause "the whole series." The isomorphism is really just to illustrate how we can say things that seem alright, but once we start applying it to infinite sets we find our language assumptions fail. Thus the appeal to highest integer which makes sense for finite sets but not infinite sets.
As you mentioned earlier, we can simply disagree with the definition. (As I do) If the definition includes this feature, then the argument holds. But that sort of avoids the fundamental issue of whether, given an infinite set, we can conceive of one more member.
Sorry, Clark, it just might be that I didn't get enough sleep last night, but I'm still notunderstanding you. What use of infinity is being disagreed with? And it sounds like your first paragraph is saying that the point of running the parallel was just to say that people can be mislead by infinites -- which would be true, but doesn't really seem to be an objection.
I take it that, given an infinite set (simply speaking), we can conceive of one more member. But we don't have an infinite set (simply speaking) here; we have a series of causes mediating the causation of other causes, and this causal mediation is an important element. Or is that what you meant, i.e., the series of causes mediating the causation of other causes? But the issue here (as I said above) isn't whether you can have one more member (as far as I can see it makes no difference to the argument whether you can or can't) but whether you can have the whole series at all without something acting as the principal cause for it -- which is not the same question as whether an infinite set, or even an infinite series of instrumental causes, can have one more member. So I don't see what you mean by calling it a fundamental issue; it's not an issue at all. Or perhaps I'm just missing something.
I put my response in a new thread.
I've closed comments in order to avoid spam since I don't check this older blog as much anymore.
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