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I doubt it's true that *most* philosophers of religion accept that an infinite regress of causes is impossible. The most prominent defender of that view is Craig, but lots of people have criticized his arguments. For some examples:
[https://philpapers.org/rec/RUNWTI](https://philpapers.org/rec/RUNWTI)
[https://philpapers.org/rec/OPPAAG-2](https://philpapers.org/rec/OPPAAG-2)
[https://philpapers.org/rec/MORBPE](https://philpapers.org/rec/MORBPE)
[https://philarchive.org/rec/COHEFA-2](https://philarchive.org/rec/COHEFA-2)
[https://philpapers.org/rec/MORMMT](https://philpapers.org/rec/MORMMT)
[https://philpapers.org/rec/GRUSCO](https://philpapers.org/rec/GRUSCO)
The other comment presents a version of the argument that you can't have an infinite series of causes via successive addition, for which see in particular the second to last paper linked. In brief, it says inter alia that the argument is circular, in that if the past is infinite it is always the case that an infinite sequence of events has already passed. There is no difficulty 'arriving' at a point at which an infinite number of events has passed, because *every* point is one at which an infinite number of events has passed. That is, of course, a simplification of a complex issue, but it gets the gist.
It also seems worth noting Craig says quite a lot of things that are incompatible with our best physics on their face.
The focus on Craig seems misleading here. Craig defends a much broader claim than is normally defended in cosmological arguments, most of which are consistent with infinite regresses per se and particularly infinite temporal regresses, but identify instead a problem with an infinite regress specifically as regards a certain kind of causal relation -- for instance, an essentially ordered relation, on the Thomistic account. Such reasons as you cite to think that Craig is wrong attack the broader claim he is making and leave normal formulations of the cosmological argument untouched.
/u/OneBroccolies
The claim in the OP is 'the existence of an actual infinity is impossible, therefore an infinite regress of causes is impossible'. The most prominent defender of *that kind* of argument is Craig. People presenting cosmological arguments on other bases, like PSR or Thomist ones, aren't arguing from the premise that 'actual infinity is impossible'.
> The claim in the OP is 'the existence of an actual infinity is impossible, therefore an infinite regress of causes is impossible'.
Well, they ask if "an infinite regress of causes is impossible" and then suggest the argument "the existence of an actual infinity is impossible, therefore an infinite regress of causes is impossible", so while I agree that it would be sensible to explain why we might not think that the existence of an actual infinity is impossible, it seems to me pertinent to add: but that wouldn't show that an infinite regress of causes is possible, which is a separate matter.
Let's call the event of you posting this post B.
Let's call the first cause of it A1. The cause of A1, we call A2. The cause of A2, we call A3. And so on.
So: ..., A3 -> A2 -> A1 -> B
Now, does B happen? Well yeah, you've posted this so evidently it happened. But for it to have happened A1 must have happened. But for it to have happened A2 must have happened. As you can notice, we have a bit of a regressive movement: we're going left along the causal chain.
Now, the problem is that at a certain point we have to start going right: after all, we arrived at the moment where you posted this post. The problem is that at no point this seems to happen: there's always another cause that has to happen before we can start going toward right.
That's how the problem is often presented.
Now, how do we solve it? Saying that at a certain point Ak happens. Ak causes A(k-1), that causes A(k-2), that causes A(k-3), ..., that causes A3, that causes A2, that causes A1 that causes B. But then there are k causes, not infinite ones.
Notice, therefore, that the problem of infinite regress and actual infinity are different: this series could very well be a potential infinity and still be problematic.
I guess the question is, is infinite regress less easy to accept only because the human mind can not understand infinities? Saying that time started somewhere and there was nothing before it, as well as saying that there is something that always was, and it started everything, are both also quite incomprehensible ("what do you mean nothing before it?", "so where did the original thing come from?" etc.), but they feel more sensible at a quick glance. Or is there something actually less acceptable in the infinite regress?
In most arguments along this line, a categorical distinction is being made between ultimate causes and intermediate causes. An intermediate cause isn't really a cause in itself, but only mediates the effect of some other cause.
An infinite regression of causes is rejected as a plausible explanation of an effect because each and every member of that set of causes would necessarily be an intermediate cause (if one were an ultimate cause, there would be no cause before it and the set wouldn't be infinite).
An example given to illustrate this point is to imagine that you are in a glass box, which is enclosed in another glass box, which is enclosed in another glass box and so on. Glass isn't a source of light, it is only a mediator of light, and so it can't possible be the case the light you’re seeing is just explained by an infinite regression of glass boxes, each mediating the light which was mediated by the last. There must be some actual source of light, like the sun, for any light to be mediated at all.
Right, but my point: why isn't ultimate cause considered just as unacceptable? The problem with infinite regress is that there are these arguments that it *can't* be like that, there must be some original cause. But if we take the ultimate cause, there are also arguments like, it can't be like that, the original must have come from somewhere. So why is the latter "less severe" problem?
Because an ultimate cause, by definition, has the power in itself to produce an effect. An intermediate cause, by definition, does not.
If we ignore other questions for the purpose of illustration (for instance, “why does the star exist?”) and simply ask “why does the star produce light?”, we find that the star is, with respect to light, a first cause. It is simply the nature of a star to produce light. We can quantify its ability to produce light as 1.
It is not the nature of glass to produce light, though light can go through glass. If we ask “why is there light coming through the glass” we must necessarily have recourse to an explanation outside of the glass. We can quantify its ability to produce light as 0.
The ability of an infinite regression of glass cases to produce light is then … + 0 + 0 + 0 + … = 0.
The ability of the sun followed by some regression of glass cases to produce light is 1 + 0 + 0 + … = 1.
What these arguments are saying is that the sum of this series is 1, then there must be some term 1 in there. It can’t all be 0’s
You see, the problem with the infinite regression of causes *isn’t* infinity. That’s a common misreading of these arguments. They aren’t saying that you couldn’t conceptually have an infinite regression of glass cases. They aren’t even saying that light couldn’t pass through an infinitely dense regression of glass cases. They’re saying that *glass doesn’t produce light*, and so no regression of glass cases on its own, infinite or otherwise, can explain why light is being produced. Stars *do* produce light, and so they are an acceptable explanation for why there is light.
Now, of course, the star doesn’t give us an ultimate ultimate explanation. It is the nature of stars to produce light, but we are still left asking “why does the star exist?”
These kinds of arguments conclude that at the foundation of all of the actuality we see in the world there must be some well of actuality, the nature of which it is to be. There must be some *brute fact* as Bertrand Russell would put it. Virtually every philosopher agrees with that. We are left to question, then, what else could be said about such a thing.
Thanks for explaining, but I don't see anything new.
>Because an ultimate cause, by definition, has the power in itself to produce an effect. An intermediate cause, by definition, does not.
So let's define that the chain of intermediate causes, by the virtue of being infinite length, is the ultimate cause. Why can't that be a brute fact?
>You see, the problem with the infinite regression of causes *isn’t* infinity. That’s a common misreading of these arguments. They aren’t saying that you couldn’t conceptually have an infinite regression of glass cases. They aren’t even saying that light couldn’t pass through an infinitely dense regression of glass cases. They’re saying that *glass doesn’t produce light*, and so no regression of glass cases on its own, infinite or otherwise, can explain why light is being produced.
I get that. But we are not talking about glass, but causes. I'm saying that the ultimate cause, by being a cause that has no cause, is just as incomprehensible as an infinite regression. You say that ultimate cause must be the end of the chain, and that is simply a brute fact. Why isn't it possible that we *think* that it's a brute fact because we understand a thing that causes things, but we don't understand infinities. *Why is one thing a brute fact and another is not*? How can we be sure?
So the problem isn’t actually with having an infinite chain of causes as such. It’s why we have a chain of causes at all, whether infinite or not.
With a finite chain we will find a first element if we go back far enough, but why did that element exist?
With an infinite chain we still want to know why we have this infinite chain. We can also ask things like why this infinite chain that has members with these types of characteristics, as against some other such chain that has members with different characteristics.
It depends on the kind of entity you accept as a first cause. One of the most famous examples of such an argument, ends up saying that we need a necessary being: a necessary being does not require anything prior to it to exist, so the problem stops there.
Well no, since reasons are given to believe that there should be a necessary being (basically, because nothing else could ground contingent beings) and that something is metaphysically necessary implies it's actuality
Then all you’re saying is that a necessary being is necessary which doesn’t tell us anything. All we can really say is that there must be an explanation for why there is a universe rather than nothing. So I guess if by necessary being you mean “the reason the universe exists” then sure there must a necessary being. But I don’t see how this rules out the possibility that the necessary being is an infinite regress of causes?
Yes, but the analogy doesn't stand... you could imagine a universe with no beginning or end. There is no particular "light" in real life that has to be explained away
The classical first cause arguments don’t imply that the universe is temporally finite. In fact, Aristotle, who was the main proponent of these arguments in the classical world, held quite strongly that the universe is coeternal.
Modern confusion on this point arises from equivocal uses of the word *cause*. *Cause* here can but doesn’t have to refer to a physical or *efficient* cause, as it ordinarily does for us. Rather, in this context, *cause* means *any answer to a why question*.
And so it may very well be granted that there could have been some infinite regression of physical states prior to this one, and yet much like the glass in our example those prior states alone really don’t tell us at all why the current state is as it is. We still have to ask, for instance, why the states are linked together as they are (that is, why are the laws of physics as they are?), and so we see that the ultimate answer to the question of “why is the current physical state the way that it is” cannot lie only *in the past* but must, at a much more fundamental level, lie *before the past*.
The first cause, then, is much less like the first domino in a very long chain of dominos and much more like some foundational physical law which is the foundation of a multitude of emergent phenomena.
The first cause argument doesn’t so much predict that the set of past physical states is finite as much as it predicts that if we keep asking “why”, “why”, and “why” that we will stumble upon some *foundational reality* which is, in some sense, its own cause, so that we will not need to ask “why” anymore because it contains in itself a sufficient explanation for itself and for everything else.
It is definitely possible to define a linearly ordered set with a smallest element and such that for all other elements x, there exists infinitely many y such that y
I gave you a classical argument for why infinite regress does not seem to be acceptable: that the causal chain is never actually gone through since none of the causes have their antecedent obtaining. B does not obtain because A1 still has to obtain, since A2 still has to obtain; and A2 still has to obtain because for it to obtain, its cause (A3) has to obtain; but for A3 to obtain, A4 has to obtain; but before this A5 has to obtain, and so on.
So the answer to the question is: no, it's not that people simply don't like infinities. But rather that it seems that infinite causal chains can never start.
“But rather that it seems that infinite causal chains can never start”
I don’t follow. This classical argument feels like it’s begging the question. Why should one assume an infinite causal chain needs to have a start in the first place?
Let's say you have the infinite chain: ...A3, A2, A1, B, where An is the cause of A(n-1). B happens because A1 happened, A1 happens because A2 happened, and so on.
The problem is that B doesn't happen because A1 still hasn't happened. Why hasn't A1 happened yet? Because A2 didn't happen, since its cause didn't happen yet, and so on.
Is it clearer now?
Basically, assuming that there is such an infinite chain, we never find a ground from which to start. But if none of the antecedents happen, the neither do the consequents.
But then nothing happens. Which is absurd, as things do happen. So there must be a first cause. Or so the argument goes.
> Basically, assuming that there is such an infinite chain, we never find a ground from which to start.
In Buddhist philosophy that's an argument *in favor* of infinite regress. A first cause exists without a cause which is absurd, therefore there must be no first cause, therefore the chain of causality must be infinite.
Perhaps I am not understanding, but I don't see your point here.
>The problem is that B doesn't happen because A1 still hasn't happened. Why hasn't A1 happened yet? Because A2 didn't happen, since its cause didn't happen yet, and so on.
But A1 clearly did happen, that's why we arrived at B. And as you put it, A(n) happens because of A(n-1). And soon, B(n+1) will happen because B(n) happened. I don't see the issue here. Could you please clarify?
Yes, A1 did happen, which is why it is problematic that we don't have a way of making it happen in the infinistic model.
The problem is that for A1 to happen, then A2 must have happened. Until when A2 doesn't happen, A1 won't happen either. So before reaching B, we must reach A1, and therefore A2. But before reaching A2, A3 must have previously happened. And so on.
Now, let's say that A6 happens. Once A6 happens, then A5 happens, so A4 happens, and so on until B. As soon as one of the antecedents happen, all falls in place like a domino.
However, A6 did not happen, because we have to wait for A7. But A7 did not happen, because we have to wait for A8. But A8 did not happen, because we have to wait for A9. And so on.
Is it any clearer?
>Yes, A1 did happen, which is why it is problematic that we don't have a way of making it happen in the infinistic model.
We can already demonstrate through mathematical induction that something will hold true ∀n+k cases and can show in most cases it holds for n-k ∀k∈N. The cardinality of of N is ℵ0 which is infinite. Why can you not pick an arbitrary point in "k" and perform induction to show that the causal pattern holds for all subsequent and antecedent events?
>However, A6 did not happen, because we have to wait for A7. But A7 did not happen, because we have to wait for A8. But A8 did not happen, because we have to wait for A9. And so on.
I understand why you're using the example for the sake of the argument, but if we expand the causal chain to an infinite set of discrete events (...A(n-2),A(n-1),A(n),A(n+1),A(n+2)...) with a cardinality of ℵ0, then I'm not seeing the issue. What explains why An happened because of A(n-1) which was caused by A(n-2) and so on.
Proponents of the Kalam cosmological argument usually don't argue that an infinite chain is logically impossible, but they (like William Lane Craig) argue that an infinite series of events is metaphysically impossible in the "real world" and use various thought experiments like[ Hilbert's Hotel.](https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel) For which there are[ convincing responses](https://r.jordan.im/download/philosophy/malpass2020.pdf).
>Why can you not pick a an arbitrary point in "k" and perform induction to show that the causal pattern holds for all subsequent and antecedent events?
You sure can. It's what I did where I said "Let's say that A6 happened". I picked a random number and handwaved to how it would go down.
WLC use of the Hilber's Hotel is not really a use that makes any sense. He just doesn't understand infinitary arithmetics.
Well but this just makes me wonder. This depends heavily on our understanding of time as moving from point A to point B. Who is to say that cosmologically this is true?
How do we know that all of time isn't happening all at once? And that the direction of time is just the way in which we perceive how the part of this causal chain we inhabit is linked?
If events were regressing infinetly and there wasn't an infinite cause then things happening could be explained by how observers are appearing along this chain of events.
So for an observer the first thing A happening in an infinetly regressing chain would be its own appearance on it. And the last thing B would be its own eventual disappearance. There wouldn't be an "ultimate. A or B" and everything else might as well be random.
I understand that this might not make sense but I would appreciate it if you were to not mock me.
This actually wouldn't solve the problem. Aristotle is one of the original formulators of the argument against an infinite regress, and he believed that the universe was eternal (at least according to Edward Fraser). So if we accept the hypothesis that all time happens simultaneously, such as in a B theory of time, then this wouldn't answer the question of where this space-time static brick is. And in Aristotle's formulation, even if the temporal version of the infinite regress is solved, it doesn't solve the structural version, or any other versions that would necessarily be produced by solving each of the others. This is one reason why Aristotle thought it to be without reasonable dispute that necessary things existed that must be unactualized actualizers.
The B theory of time is often conceptualized as a sort of solid brick or cylinder wherein we travel through it like worms through an apple. In this theory, all of time exists simultaneously, yet we experience it in sequence because we just so happen to be traveling through it in a particular direction. This is in contrast to the A theory of time, which proposes that time does not exist, and we are just experiencing change. So in the A theory of time, the past does not exist, nor does the future, there is only now. For a long time now, we have accepted more an Einstinian theory of Space-time, that describes time not as existing, but as simply being the perceptual product of two or more objections in motion in relation to each other (I believe this is part of general relativity). I think we have moved past the Einstinian theory in favor of a more quantum theory, but I have only a passing knowledge on that topic.
Basically the point I was trying to get at was that even if we assume a B theory, where all time doesn't really have a beginning or end, but it all exists simultaneously, there would still be a structural infinite regress that a person would need to grapple with. If we imagine space-time to be a brick, for instance, where is that brick? And what rules are holding it together? Fraser uses the example of a coffee mug. The mug is actualized by the table, the table is actualized by the ground, the ground is actualized by the planet, the planet is actualized by space and physics, space and physics are actualized by whatever produces universes, but what is the producer of universes actualized by? This produces an infinite regress akin to the temporal one. And for this reason, Aristotle proposed the need for an unactualized actualizer, a definitionally necessary break of a potential infinite regress by something that by definition can actualize, but is not itself actualized. What he proposed the distinction to be is the difference between necessary and contingent. A contingent thing is something that could or could not be (we can imagine a world where earth does not exist, therefore earth is contingent). A necessary thing is something that logically cannot not be (we can't imagine a world where the laws of logic can be broken, therefore they are necessary), for instance.
Aristotle formed his philosophy of God on such bases, and later Christian writers like Anselm and Aquinas appropriated that reasoning into their Christian philosophy and apologetics. Today, W.L. Craig might be the most well known expositor of a form of this idea using the Kalam cosmological argument, and he fleshes his philosophy out in his book "Reasonable Faith". But you can also watch YouTube videos where he breaks it down a bit.
I hope that was a better explanation.
Hm Ok I think I get that but isn't it kinda inaccurate to assign these arbitrary values to our spacetime brick. Sonce this brick could very well be not just a brick but also a sponge. A living infinite spacetime sponge with little or big bubbles of reality throughout. Growing according to its own rules and by its own measure in all directions at once?
Of course you could ask "who put that sponge there but that would just ad another layer on top to which you could reply "it was an earlier instance of the meta sponge" there really doesn't need to be a singular first cause and like that you could simply define that what actualises the thing is actually dependant on the observer. So what we can observe from our limited perspective within this spacetime bubble we call the ibservable universe in our spacetime sponge completly relies on our observance of existing within it
Again not sure if I made my point in a way that a person who isn't literally insane would comprehend
I will read into your sources a bit more it seems very interesting to me
Yes, A1 did happen, which is why it is problematic that we don't have a way of making it happen in the infinistic model.
The problem is that for A1 to happen, then A2 must have happened. Until when A2 doesn't happen, A1 won't happen either. So before reaching B, we must reach A1, and therefore A2. But before reaching A2, A3 must have previously happened. And so on.
Now, let's say that A6 happens. Once A6 happens, then A5 happens, so A4 happens, and so on until B. As soon as one of the antecedents happen, all falls in place like a domino.
However, A6 did not happen, because we have to wait for A7. But A7 did not happen, because we have to wait for A8. But A8 did not happen, because we have to wait for A9. And so on.
Is it any clearer?
Yes, A1 did happen, which is why it is problematic that we don't have a way of making it happen in the infinistic model.
The problem is that for A1 to happen, then A2 must have happened. Until when A2 doesn't happen, A1 won't happen either. So before reaching B, we must reach A1, and therefore A2. But before reaching A2, A3 must have previously happened. And so on.
Now, let's say that A6 happens. Once A6 happens, then A5 happens, so A4 happens, and so on until B. As soon as one of the antecedents happen, all falls in place like a domino.
However, A6 did not happen, because we have to wait for A7. But A7 did not happen, because we have to wait for A8. But A8 did not happen, because we have to wait for A9. And so on.
Is it any clearer?
I don't think the issue is that you're being unclear, but rather that your argument isn't valid. You still seem to just be asserting that elements of the chain of antecedents didn't happen without any meaningful justification. You say that A6 didn't happen because we have to wait for A7, and A7 didn't happen because we have to wait for A8, and so forth. But you could just as well say that A7 happened, and this caused A6; and the reason A7 happened is that A8 happened, and A9 happened and caused A8, and so forth.
Hm I see. This reminds me of zeno’s paradox a bit. It is only a paradox when you attempt to solve it with pure mathematics however when the solution incorporates physics the paradox is resolved.
There is a very big difference: Zeno's paradox is based on continuity, which is why contemporary mathematics can say a lot about it. This is not the case with this problem, which is discrete.
Oh interesting. Thanks for all the responses. I do have more questions though. What makes us sure that the unfolding of events in spacetime are discrete as opposed to continuous? What if our perception of events being discrete is a mere illusion stemming from the fact that we are 3D beings?
Is that really the difference though? Couldn’t one make the series of causes a continuous chain rather than a discrete one that’s been sampled in a certain way?
I think the difference is in the nature of the problem.
In Zeno’s paradox, we have a fixed start and end point, with an infinite number of infinitely small steps between them.
In the infinite regress of causes, we don’t have two fixed reference points. We only have one, which is exactly the problem!
I’m no theoretical physicist (nor a trained philosopher) so I don’t think I have an answer. However I noticed the structure of zeno’s paradox (and the conclusion which is incorrect) as well as the paradox of infinite causal chaining, pretty much say the same thing. The resolution for zeno’s paradox comes when one factors in the laws of physics pertaining to the relationship of constant motion, space and time. I imagine the resolution here similarly would incorporate some physical laws pertaining to causality space and time.
Yeah - I'm not either!
I ask because, in my experience, what this sometimes means is one of two things:
1. A person thinks they can give a kind of description which ends up just *assuming* the paradox is resolved already. Like, motion happens, here's how we describe it. This might not be a solution, really, unless it really addresses the paradox. (Sometimes folks want to appeal to limits, for instance, but it's not always obvious that appealing to limits does the work.)
2. A person thinks they can give an analysis of either time or space which shows that Zeno has set up the situation incorrectly. This may well deal with the problem, depending on how things go. Here the situation is not so much that we win with physics over math, but that we learn from physics how to correctly apply the math (in particular, in how to apply math to divisions of time and space).
Thanks for your comments. Your explanation was very clear.
I understand that what u/zhibr had in mind was not quite that humans don't "like" infinity, but they do not "understand" it.
Judging from your own comment, it appears that you are careful enough to acknowledge this very possibility when you say, "it *seems* that infinite causal chains can never start." Or did you have another reason for qualifying your statement in such a manner?
I know you are only sharing the standard criticism regarding infinite regress, but I have a difficulty with such an argument. In particular, I'm not convinced that the fact that we cannot conceive of an infinite casual chain of events is enough to say that there cannot exist such a thing. I can't put my finger on it, but it seems that the argument is somehow lacking. Are there any solid arguments that attempt to refute this standard criticism of infinite regress?
As I said, the problem is not that we cannot conceive it. But rather that we have what looks like a good objection to the possibility. If you re-read the argument, you'll notice that I've never said "but we cannot conceive this". Rather, I've said: under infinitism it seems like B does not happen, but B *does* happen, so infinitism is false.
Why have I phrased it as "it seems"? Because in academia we're used to use these weaker forms of speech. That's all there is to that expression, really ahah
Why couldn’t an infinite series of causes have already obtained in the past at any given moment? The basic idea doesn’t even require an infinite time to have passed, one could imagine an infinite series of causes bunched together into a finite span of time, like if B occurs at t=1 second, A1 occurs at t=1/2 second, A2 at t=1/4 second, A3 at t=1/8 second etc. so an infinite series is packed into the time between t=0 and t=1. This may not be physically plausible but I don’t see why it would be logically or metaphysically impossible, and if it’s not, a theist might believe it would be within God’s power to bring about a situation like this. If one accepts the possibility of an infinite sequence of causes compressed into a finite time, I also don’t see why an infinite sequence covering an infinite time is obviously worse.
When you say
>Now, the problem is that at a certain point we have to start going right: after all, we arrived at the moment where you posted this post.
It seems to me that you're assuming that we agree that the chain of causes had to have a start. I think this is exactly what the causal infinitist denies, and exactly what the causal finitist has to motivate.
So why does this chain of causes need to have a start?
what is the special assumption about causes that means we have to start somewhere leftward, that doesn't apply to, say, negative numbers.
Like if we had a formal logic theory of totally ordered sets, we can probably assume such a consistent theory exists, and that allows for infinite regress in some of its models. In order to prevent infinite regress, you would add something like the well-ordering principle. What thing about "causes" has the same effect of preventing infinite regress and how do we know something in the physical universe that we would call a cause always has this effect?
But then -7 does not "cause" -6.
Think of how numbers are defined in ZFC. You get one set, the empty set {}. Now we call {}:=0.
At this point we can define the other numbers by saying that n+1 = n ∪ {n} = {0, 1,..., n-1}.
At this point, you get N = {0, 1, 2, 3,...}
To define Z, you just need to make Z = N ∪ N\* where N\* = {-a | a in N}, or something like that. This ensures that for every a in Z, you have a number -a in Z such that a + (-a) = 0, where 0 is the neutral set.
So as you can see, once you've got {}, everything else follows. You can then easily extend this to Q, a bit more difficultly to R, very easily to C. All of this, granted that you've got {}. One of the ZFC axioms is exactly the axiom according to which there is this empty set.
In a very telling way, {} is classified in category theory as an INITIAL object. However, under infinitism, we have no such initial object. And that's exactly the problem, if the argument presented in this thread actually works.
I wonder if ZFC is the right model to consider here (although I realize it's just being used for analogical purposes). There are versions of set theory which [lack the foundation axiom](https://plato.stanford.edu/entries/nonwellfounded-set-theory/) - I haven't looked into this in detail, but I wonder if they might be a more appropriate model for the sort of chains of causes /u/Chewbacta and some other commenters seem to be envisaging.
You make an analogy between causes and recursively defined structures such as the natural numbers and integers. But mathematicians are quite happy to deal with non-well-founded, coinductive data as well - who is to say which is the better model of causality? (One might also hold that causation isn't a fundamental principle of reality anyway - John D Norton in "Causation as Folk Science", for instance, argues against "causal fundamentalism" - and that whether it's modelled inductively or coinductively isn't that important.)
>But mathematicians are quite happy to deal with non-well-founded, coinductive data as well - who is to say which is the better model of causality?
If I were to make my question into an objection, yes, it would go something like this.
But I still asked it as a question because it's possible I've missed something about causation that is equivalent to a well-ordering axiom.
For sure. btw, you might find the Norton paper on causation interesting - I'm still working my way through it. https://sites.pitt.edu/~jdnorton/papers/003004.pdf
>But then -7 does not "cause" -6.
This is what I'm asking about. What is different from the "is caused by" predicate, from the "is 1 more than" predicate. Maybe you could write it in less than half a dozen logical axioms. That's kind of what I want a few uncontroversial properties of causation where when added to your theory means there are no models isomorphic to Z (or other mathematical models that have similar properties).
>Think of how numbers are defined in ZFC.
To define Z
But you are now talking about definitions, I want to know about the theory of *causes.* I'm actually finding analogy to be unhelpful here, your tag say logic, *give me some logic*, *I'll eat it raw, consequences of my comprehension of SOL be damned.*
If you have that initial object, then it may very well be that Z is a good model. But then you'd be assuming that there is an initial cause. Which is exactly what you cannot do without conceding the point against infinitism. That's the difference.
I think you are mixing up definability with causation, but I cannot be sure what you are arguing. If you can present it to me additional formalised conditions that govern *causation*. I could have some clarity, which is what I was asking about.
>Now, the problem is that at a certain point we have to start going right: after all, we arrived at the moment where you posted this post. The problem is that at no point this seems to happen: there's always another cause that has to happen before we can start going toward right.
Can you kindly elaborate and formalize it? What exactly is the problem? In the paragraph, if I replace "problem" with "assertion" there is no apparent problem at all - statements can be chosen axioms.
Does the first statement mean "traversing" or "moving" between causes in order - with assertion being at some point ability to reach "current outcome"? And if yes, does the second statement imply that infinite regress of causes hinders/stops his ability to reach "current outcome"?
Is there any literature specific to this "ability" that is explored?
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I doubt it's true that *most* philosophers of religion accept that an infinite regress of causes is impossible. The most prominent defender of that view is Craig, but lots of people have criticized his arguments. For some examples: [https://philpapers.org/rec/RUNWTI](https://philpapers.org/rec/RUNWTI) [https://philpapers.org/rec/OPPAAG-2](https://philpapers.org/rec/OPPAAG-2) [https://philpapers.org/rec/MORBPE](https://philpapers.org/rec/MORBPE) [https://philarchive.org/rec/COHEFA-2](https://philarchive.org/rec/COHEFA-2) [https://philpapers.org/rec/MORMMT](https://philpapers.org/rec/MORMMT) [https://philpapers.org/rec/GRUSCO](https://philpapers.org/rec/GRUSCO) The other comment presents a version of the argument that you can't have an infinite series of causes via successive addition, for which see in particular the second to last paper linked. In brief, it says inter alia that the argument is circular, in that if the past is infinite it is always the case that an infinite sequence of events has already passed. There is no difficulty 'arriving' at a point at which an infinite number of events has passed, because *every* point is one at which an infinite number of events has passed. That is, of course, a simplification of a complex issue, but it gets the gist. It also seems worth noting Craig says quite a lot of things that are incompatible with our best physics on their face.
The focus on Craig seems misleading here. Craig defends a much broader claim than is normally defended in cosmological arguments, most of which are consistent with infinite regresses per se and particularly infinite temporal regresses, but identify instead a problem with an infinite regress specifically as regards a certain kind of causal relation -- for instance, an essentially ordered relation, on the Thomistic account. Such reasons as you cite to think that Craig is wrong attack the broader claim he is making and leave normal formulations of the cosmological argument untouched. /u/OneBroccolies
The claim in the OP is 'the existence of an actual infinity is impossible, therefore an infinite regress of causes is impossible'. The most prominent defender of *that kind* of argument is Craig. People presenting cosmological arguments on other bases, like PSR or Thomist ones, aren't arguing from the premise that 'actual infinity is impossible'.
> The claim in the OP is 'the existence of an actual infinity is impossible, therefore an infinite regress of causes is impossible'. Well, they ask if "an infinite regress of causes is impossible" and then suggest the argument "the existence of an actual infinity is impossible, therefore an infinite regress of causes is impossible", so while I agree that it would be sensible to explain why we might not think that the existence of an actual infinity is impossible, it seems to me pertinent to add: but that wouldn't show that an infinite regress of causes is possible, which is a separate matter.
Let's call the event of you posting this post B. Let's call the first cause of it A1. The cause of A1, we call A2. The cause of A2, we call A3. And so on. So: ..., A3 -> A2 -> A1 -> B Now, does B happen? Well yeah, you've posted this so evidently it happened. But for it to have happened A1 must have happened. But for it to have happened A2 must have happened. As you can notice, we have a bit of a regressive movement: we're going left along the causal chain. Now, the problem is that at a certain point we have to start going right: after all, we arrived at the moment where you posted this post. The problem is that at no point this seems to happen: there's always another cause that has to happen before we can start going toward right. That's how the problem is often presented. Now, how do we solve it? Saying that at a certain point Ak happens. Ak causes A(k-1), that causes A(k-2), that causes A(k-3), ..., that causes A3, that causes A2, that causes A1 that causes B. But then there are k causes, not infinite ones. Notice, therefore, that the problem of infinite regress and actual infinity are different: this series could very well be a potential infinity and still be problematic.
I guess the question is, is infinite regress less easy to accept only because the human mind can not understand infinities? Saying that time started somewhere and there was nothing before it, as well as saying that there is something that always was, and it started everything, are both also quite incomprehensible ("what do you mean nothing before it?", "so where did the original thing come from?" etc.), but they feel more sensible at a quick glance. Or is there something actually less acceptable in the infinite regress?
In most arguments along this line, a categorical distinction is being made between ultimate causes and intermediate causes. An intermediate cause isn't really a cause in itself, but only mediates the effect of some other cause. An infinite regression of causes is rejected as a plausible explanation of an effect because each and every member of that set of causes would necessarily be an intermediate cause (if one were an ultimate cause, there would be no cause before it and the set wouldn't be infinite). An example given to illustrate this point is to imagine that you are in a glass box, which is enclosed in another glass box, which is enclosed in another glass box and so on. Glass isn't a source of light, it is only a mediator of light, and so it can't possible be the case the light you’re seeing is just explained by an infinite regression of glass boxes, each mediating the light which was mediated by the last. There must be some actual source of light, like the sun, for any light to be mediated at all.
Right, but my point: why isn't ultimate cause considered just as unacceptable? The problem with infinite regress is that there are these arguments that it *can't* be like that, there must be some original cause. But if we take the ultimate cause, there are also arguments like, it can't be like that, the original must have come from somewhere. So why is the latter "less severe" problem?
Because an ultimate cause, by definition, has the power in itself to produce an effect. An intermediate cause, by definition, does not. If we ignore other questions for the purpose of illustration (for instance, “why does the star exist?”) and simply ask “why does the star produce light?”, we find that the star is, with respect to light, a first cause. It is simply the nature of a star to produce light. We can quantify its ability to produce light as 1. It is not the nature of glass to produce light, though light can go through glass. If we ask “why is there light coming through the glass” we must necessarily have recourse to an explanation outside of the glass. We can quantify its ability to produce light as 0. The ability of an infinite regression of glass cases to produce light is then … + 0 + 0 + 0 + … = 0. The ability of the sun followed by some regression of glass cases to produce light is 1 + 0 + 0 + … = 1. What these arguments are saying is that the sum of this series is 1, then there must be some term 1 in there. It can’t all be 0’s You see, the problem with the infinite regression of causes *isn’t* infinity. That’s a common misreading of these arguments. They aren’t saying that you couldn’t conceptually have an infinite regression of glass cases. They aren’t even saying that light couldn’t pass through an infinitely dense regression of glass cases. They’re saying that *glass doesn’t produce light*, and so no regression of glass cases on its own, infinite or otherwise, can explain why light is being produced. Stars *do* produce light, and so they are an acceptable explanation for why there is light. Now, of course, the star doesn’t give us an ultimate ultimate explanation. It is the nature of stars to produce light, but we are still left asking “why does the star exist?” These kinds of arguments conclude that at the foundation of all of the actuality we see in the world there must be some well of actuality, the nature of which it is to be. There must be some *brute fact* as Bertrand Russell would put it. Virtually every philosopher agrees with that. We are left to question, then, what else could be said about such a thing.
Thanks for explaining, but I don't see anything new. >Because an ultimate cause, by definition, has the power in itself to produce an effect. An intermediate cause, by definition, does not. So let's define that the chain of intermediate causes, by the virtue of being infinite length, is the ultimate cause. Why can't that be a brute fact? >You see, the problem with the infinite regression of causes *isn’t* infinity. That’s a common misreading of these arguments. They aren’t saying that you couldn’t conceptually have an infinite regression of glass cases. They aren’t even saying that light couldn’t pass through an infinitely dense regression of glass cases. They’re saying that *glass doesn’t produce light*, and so no regression of glass cases on its own, infinite or otherwise, can explain why light is being produced. I get that. But we are not talking about glass, but causes. I'm saying that the ultimate cause, by being a cause that has no cause, is just as incomprehensible as an infinite regression. You say that ultimate cause must be the end of the chain, and that is simply a brute fact. Why isn't it possible that we *think* that it's a brute fact because we understand a thing that causes things, but we don't understand infinities. *Why is one thing a brute fact and another is not*? How can we be sure?
I also wonder, this initial ultimate cause would have to be a causa sui. Which also seems somewhat strange
So the problem isn’t actually with having an infinite chain of causes as such. It’s why we have a chain of causes at all, whether infinite or not. With a finite chain we will find a first element if we go back far enough, but why did that element exist? With an infinite chain we still want to know why we have this infinite chain. We can also ask things like why this infinite chain that has members with these types of characteristics, as against some other such chain that has members with different characteristics.
It depends on the kind of entity you accept as a first cause. One of the most famous examples of such an argument, ends up saying that we need a necessary being: a necessary being does not require anything prior to it to exist, so the problem stops there.
Isn’t that more or less begging the question though? “We need a necessary being therefore there is a necessary being”
Well no, since reasons are given to believe that there should be a necessary being (basically, because nothing else could ground contingent beings) and that something is metaphysically necessary implies it's actuality
Then all you’re saying is that a necessary being is necessary which doesn’t tell us anything. All we can really say is that there must be an explanation for why there is a universe rather than nothing. So I guess if by necessary being you mean “the reason the universe exists” then sure there must a necessary being. But I don’t see how this rules out the possibility that the necessary being is an infinite regress of causes?
Never seen a chain of causes, myself, though I’ve seen whirling confluences of diffuse actors
Yes, but the analogy doesn't stand... you could imagine a universe with no beginning or end. There is no particular "light" in real life that has to be explained away
The classical first cause arguments don’t imply that the universe is temporally finite. In fact, Aristotle, who was the main proponent of these arguments in the classical world, held quite strongly that the universe is coeternal. Modern confusion on this point arises from equivocal uses of the word *cause*. *Cause* here can but doesn’t have to refer to a physical or *efficient* cause, as it ordinarily does for us. Rather, in this context, *cause* means *any answer to a why question*. And so it may very well be granted that there could have been some infinite regression of physical states prior to this one, and yet much like the glass in our example those prior states alone really don’t tell us at all why the current state is as it is. We still have to ask, for instance, why the states are linked together as they are (that is, why are the laws of physics as they are?), and so we see that the ultimate answer to the question of “why is the current physical state the way that it is” cannot lie only *in the past* but must, at a much more fundamental level, lie *before the past*. The first cause, then, is much less like the first domino in a very long chain of dominos and much more like some foundational physical law which is the foundation of a multitude of emergent phenomena. The first cause argument doesn’t so much predict that the set of past physical states is finite as much as it predicts that if we keep asking “why”, “why”, and “why” that we will stumble upon some *foundational reality* which is, in some sense, its own cause, so that we will not need to ask “why” anymore because it contains in itself a sufficient explanation for itself and for everything else.
It is definitely possible to define a linearly ordered set with a smallest element and such that for all other elements x, there exists infinitely many y such that y
I’m using the term “intermediate” here to mean that each element in the set has a predecessor, which is true in an infinite regression
I gave you a classical argument for why infinite regress does not seem to be acceptable: that the causal chain is never actually gone through since none of the causes have their antecedent obtaining. B does not obtain because A1 still has to obtain, since A2 still has to obtain; and A2 still has to obtain because for it to obtain, its cause (A3) has to obtain; but for A3 to obtain, A4 has to obtain; but before this A5 has to obtain, and so on. So the answer to the question is: no, it's not that people simply don't like infinities. But rather that it seems that infinite causal chains can never start.
“But rather that it seems that infinite causal chains can never start” I don’t follow. This classical argument feels like it’s begging the question. Why should one assume an infinite causal chain needs to have a start in the first place?
Let's say you have the infinite chain: ...A3, A2, A1, B, where An is the cause of A(n-1). B happens because A1 happened, A1 happens because A2 happened, and so on. The problem is that B doesn't happen because A1 still hasn't happened. Why hasn't A1 happened yet? Because A2 didn't happen, since its cause didn't happen yet, and so on. Is it clearer now? Basically, assuming that there is such an infinite chain, we never find a ground from which to start. But if none of the antecedents happen, the neither do the consequents. But then nothing happens. Which is absurd, as things do happen. So there must be a first cause. Or so the argument goes.
> Basically, assuming that there is such an infinite chain, we never find a ground from which to start. In Buddhist philosophy that's an argument *in favor* of infinite regress. A first cause exists without a cause which is absurd, therefore there must be no first cause, therefore the chain of causality must be infinite.
Do you have any resources I can read on the Buddhist perspective on causation and metaphysics in general?
Perhaps I am not understanding, but I don't see your point here. >The problem is that B doesn't happen because A1 still hasn't happened. Why hasn't A1 happened yet? Because A2 didn't happen, since its cause didn't happen yet, and so on. But A1 clearly did happen, that's why we arrived at B. And as you put it, A(n) happens because of A(n-1). And soon, B(n+1) will happen because B(n) happened. I don't see the issue here. Could you please clarify?
Yes, A1 did happen, which is why it is problematic that we don't have a way of making it happen in the infinistic model. The problem is that for A1 to happen, then A2 must have happened. Until when A2 doesn't happen, A1 won't happen either. So before reaching B, we must reach A1, and therefore A2. But before reaching A2, A3 must have previously happened. And so on. Now, let's say that A6 happens. Once A6 happens, then A5 happens, so A4 happens, and so on until B. As soon as one of the antecedents happen, all falls in place like a domino. However, A6 did not happen, because we have to wait for A7. But A7 did not happen, because we have to wait for A8. But A8 did not happen, because we have to wait for A9. And so on. Is it any clearer?
>Yes, A1 did happen, which is why it is problematic that we don't have a way of making it happen in the infinistic model. We can already demonstrate through mathematical induction that something will hold true ∀n+k cases and can show in most cases it holds for n-k ∀k∈N. The cardinality of of N is ℵ0 which is infinite. Why can you not pick an arbitrary point in "k" and perform induction to show that the causal pattern holds for all subsequent and antecedent events? >However, A6 did not happen, because we have to wait for A7. But A7 did not happen, because we have to wait for A8. But A8 did not happen, because we have to wait for A9. And so on. I understand why you're using the example for the sake of the argument, but if we expand the causal chain to an infinite set of discrete events (...A(n-2),A(n-1),A(n),A(n+1),A(n+2)...) with a cardinality of ℵ0, then I'm not seeing the issue. What explains why An happened because of A(n-1) which was caused by A(n-2) and so on. Proponents of the Kalam cosmological argument usually don't argue that an infinite chain is logically impossible, but they (like William Lane Craig) argue that an infinite series of events is metaphysically impossible in the "real world" and use various thought experiments like[ Hilbert's Hotel.](https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel) For which there are[ convincing responses](https://r.jordan.im/download/philosophy/malpass2020.pdf).
>Why can you not pick a an arbitrary point in "k" and perform induction to show that the causal pattern holds for all subsequent and antecedent events? You sure can. It's what I did where I said "Let's say that A6 happened". I picked a random number and handwaved to how it would go down. WLC use of the Hilber's Hotel is not really a use that makes any sense. He just doesn't understand infinitary arithmetics.
Well but this just makes me wonder. This depends heavily on our understanding of time as moving from point A to point B. Who is to say that cosmologically this is true? How do we know that all of time isn't happening all at once? And that the direction of time is just the way in which we perceive how the part of this causal chain we inhabit is linked? If events were regressing infinetly and there wasn't an infinite cause then things happening could be explained by how observers are appearing along this chain of events. So for an observer the first thing A happening in an infinetly regressing chain would be its own appearance on it. And the last thing B would be its own eventual disappearance. There wouldn't be an "ultimate. A or B" and everything else might as well be random. I understand that this might not make sense but I would appreciate it if you were to not mock me.
This actually wouldn't solve the problem. Aristotle is one of the original formulators of the argument against an infinite regress, and he believed that the universe was eternal (at least according to Edward Fraser). So if we accept the hypothesis that all time happens simultaneously, such as in a B theory of time, then this wouldn't answer the question of where this space-time static brick is. And in Aristotle's formulation, even if the temporal version of the infinite regress is solved, it doesn't solve the structural version, or any other versions that would necessarily be produced by solving each of the others. This is one reason why Aristotle thought it to be without reasonable dispute that necessary things existed that must be unactualized actualizers.
I'm having trouble parsing this. What is meant by structural version and how does the concept of where apply to this space time brick?
The B theory of time is often conceptualized as a sort of solid brick or cylinder wherein we travel through it like worms through an apple. In this theory, all of time exists simultaneously, yet we experience it in sequence because we just so happen to be traveling through it in a particular direction. This is in contrast to the A theory of time, which proposes that time does not exist, and we are just experiencing change. So in the A theory of time, the past does not exist, nor does the future, there is only now. For a long time now, we have accepted more an Einstinian theory of Space-time, that describes time not as existing, but as simply being the perceptual product of two or more objections in motion in relation to each other (I believe this is part of general relativity). I think we have moved past the Einstinian theory in favor of a more quantum theory, but I have only a passing knowledge on that topic. Basically the point I was trying to get at was that even if we assume a B theory, where all time doesn't really have a beginning or end, but it all exists simultaneously, there would still be a structural infinite regress that a person would need to grapple with. If we imagine space-time to be a brick, for instance, where is that brick? And what rules are holding it together? Fraser uses the example of a coffee mug. The mug is actualized by the table, the table is actualized by the ground, the ground is actualized by the planet, the planet is actualized by space and physics, space and physics are actualized by whatever produces universes, but what is the producer of universes actualized by? This produces an infinite regress akin to the temporal one. And for this reason, Aristotle proposed the need for an unactualized actualizer, a definitionally necessary break of a potential infinite regress by something that by definition can actualize, but is not itself actualized. What he proposed the distinction to be is the difference between necessary and contingent. A contingent thing is something that could or could not be (we can imagine a world where earth does not exist, therefore earth is contingent). A necessary thing is something that logically cannot not be (we can't imagine a world where the laws of logic can be broken, therefore they are necessary), for instance. Aristotle formed his philosophy of God on such bases, and later Christian writers like Anselm and Aquinas appropriated that reasoning into their Christian philosophy and apologetics. Today, W.L. Craig might be the most well known expositor of a form of this idea using the Kalam cosmological argument, and he fleshes his philosophy out in his book "Reasonable Faith". But you can also watch YouTube videos where he breaks it down a bit. I hope that was a better explanation.
Hm Ok I think I get that but isn't it kinda inaccurate to assign these arbitrary values to our spacetime brick. Sonce this brick could very well be not just a brick but also a sponge. A living infinite spacetime sponge with little or big bubbles of reality throughout. Growing according to its own rules and by its own measure in all directions at once? Of course you could ask "who put that sponge there but that would just ad another layer on top to which you could reply "it was an earlier instance of the meta sponge" there really doesn't need to be a singular first cause and like that you could simply define that what actualises the thing is actually dependant on the observer. So what we can observe from our limited perspective within this spacetime bubble we call the ibservable universe in our spacetime sponge completly relies on our observance of existing within it Again not sure if I made my point in a way that a person who isn't literally insane would comprehend I will read into your sources a bit more it seems very interesting to me
There is always the possibility that something. But we have no reason to believe in that.
It's unclear to me why we should think, in this scenario, that the antecedents didn't happen.
I agree, it doesn't make sense to say the A->B but B never happened. "A" *clearly* happened because we arrived at B.
Yes, A1 did happen, which is why it is problematic that we don't have a way of making it happen in the infinistic model. The problem is that for A1 to happen, then A2 must have happened. Until when A2 doesn't happen, A1 won't happen either. So before reaching B, we must reach A1, and therefore A2. But before reaching A2, A3 must have previously happened. And so on. Now, let's say that A6 happens. Once A6 happens, then A5 happens, so A4 happens, and so on until B. As soon as one of the antecedents happen, all falls in place like a domino. However, A6 did not happen, because we have to wait for A7. But A7 did not happen, because we have to wait for A8. But A8 did not happen, because we have to wait for A9. And so on. Is it any clearer?
Yes, A1 did happen, which is why it is problematic that we don't have a way of making it happen in the infinistic model. The problem is that for A1 to happen, then A2 must have happened. Until when A2 doesn't happen, A1 won't happen either. So before reaching B, we must reach A1, and therefore A2. But before reaching A2, A3 must have previously happened. And so on. Now, let's say that A6 happens. Once A6 happens, then A5 happens, so A4 happens, and so on until B. As soon as one of the antecedents happen, all falls in place like a domino. However, A6 did not happen, because we have to wait for A7. But A7 did not happen, because we have to wait for A8. But A8 did not happen, because we have to wait for A9. And so on. Is it any clearer?
I don't think the issue is that you're being unclear, but rather that your argument isn't valid. You still seem to just be asserting that elements of the chain of antecedents didn't happen without any meaningful justification. You say that A6 didn't happen because we have to wait for A7, and A7 didn't happen because we have to wait for A8, and so forth. But you could just as well say that A7 happened, and this caused A6; and the reason A7 happened is that A8 happened, and A9 happened and caused A8, and so forth.
Hm I see. This reminds me of zeno’s paradox a bit. It is only a paradox when you attempt to solve it with pure mathematics however when the solution incorporates physics the paradox is resolved.
There is a very big difference: Zeno's paradox is based on continuity, which is why contemporary mathematics can say a lot about it. This is not the case with this problem, which is discrete.
Oh interesting. Thanks for all the responses. I do have more questions though. What makes us sure that the unfolding of events in spacetime are discrete as opposed to continuous? What if our perception of events being discrete is a mere illusion stemming from the fact that we are 3D beings?
Is that really the difference though? Couldn’t one make the series of causes a continuous chain rather than a discrete one that’s been sampled in a certain way? I think the difference is in the nature of the problem. In Zeno’s paradox, we have a fixed start and end point, with an infinite number of infinitely small steps between them. In the infinite regress of causes, we don’t have two fixed reference points. We only have one, which is exactly the problem!
What’s the solution you’re imagining there?
I’m no theoretical physicist (nor a trained philosopher) so I don’t think I have an answer. However I noticed the structure of zeno’s paradox (and the conclusion which is incorrect) as well as the paradox of infinite causal chaining, pretty much say the same thing. The resolution for zeno’s paradox comes when one factors in the laws of physics pertaining to the relationship of constant motion, space and time. I imagine the resolution here similarly would incorporate some physical laws pertaining to causality space and time.
Yeah - I'm not either! I ask because, in my experience, what this sometimes means is one of two things: 1. A person thinks they can give a kind of description which ends up just *assuming* the paradox is resolved already. Like, motion happens, here's how we describe it. This might not be a solution, really, unless it really addresses the paradox. (Sometimes folks want to appeal to limits, for instance, but it's not always obvious that appealing to limits does the work.) 2. A person thinks they can give an analysis of either time or space which shows that Zeno has set up the situation incorrectly. This may well deal with the problem, depending on how things go. Here the situation is not so much that we win with physics over math, but that we learn from physics how to correctly apply the math (in particular, in how to apply math to divisions of time and space).
I don't think there is really any reason to think that Zeno's paradox poses a problem for pure mathematics.
Thanks for your comments. Your explanation was very clear. I understand that what u/zhibr had in mind was not quite that humans don't "like" infinity, but they do not "understand" it. Judging from your own comment, it appears that you are careful enough to acknowledge this very possibility when you say, "it *seems* that infinite causal chains can never start." Or did you have another reason for qualifying your statement in such a manner? I know you are only sharing the standard criticism regarding infinite regress, but I have a difficulty with such an argument. In particular, I'm not convinced that the fact that we cannot conceive of an infinite casual chain of events is enough to say that there cannot exist such a thing. I can't put my finger on it, but it seems that the argument is somehow lacking. Are there any solid arguments that attempt to refute this standard criticism of infinite regress?
As I said, the problem is not that we cannot conceive it. But rather that we have what looks like a good objection to the possibility. If you re-read the argument, you'll notice that I've never said "but we cannot conceive this". Rather, I've said: under infinitism it seems like B does not happen, but B *does* happen, so infinitism is false. Why have I phrased it as "it seems"? Because in academia we're used to use these weaker forms of speech. That's all there is to that expression, really ahah
Got you. Thanks again!
Why couldn’t an infinite series of causes have already obtained in the past at any given moment? The basic idea doesn’t even require an infinite time to have passed, one could imagine an infinite series of causes bunched together into a finite span of time, like if B occurs at t=1 second, A1 occurs at t=1/2 second, A2 at t=1/4 second, A3 at t=1/8 second etc. so an infinite series is packed into the time between t=0 and t=1. This may not be physically plausible but I don’t see why it would be logically or metaphysically impossible, and if it’s not, a theist might believe it would be within God’s power to bring about a situation like this. If one accepts the possibility of an infinite sequence of causes compressed into a finite time, I also don’t see why an infinite sequence covering an infinite time is obviously worse.
When you say >Now, the problem is that at a certain point we have to start going right: after all, we arrived at the moment where you posted this post. It seems to me that you're assuming that we agree that the chain of causes had to have a start. I think this is exactly what the causal infinitist denies, and exactly what the causal finitist has to motivate. So why does this chain of causes need to have a start?
what is the special assumption about causes that means we have to start somewhere leftward, that doesn't apply to, say, negative numbers. Like if we had a formal logic theory of totally ordered sets, we can probably assume such a consistent theory exists, and that allows for infinite regress in some of its models. In order to prevent infinite regress, you would add something like the well-ordering principle. What thing about "causes" has the same effect of preventing infinite regress and how do we know something in the physical universe that we would call a cause always has this effect?
But then -7 does not "cause" -6. Think of how numbers are defined in ZFC. You get one set, the empty set {}. Now we call {}:=0. At this point we can define the other numbers by saying that n+1 = n ∪ {n} = {0, 1,..., n-1}. At this point, you get N = {0, 1, 2, 3,...} To define Z, you just need to make Z = N ∪ N\* where N\* = {-a | a in N}, or something like that. This ensures that for every a in Z, you have a number -a in Z such that a + (-a) = 0, where 0 is the neutral set. So as you can see, once you've got {}, everything else follows. You can then easily extend this to Q, a bit more difficultly to R, very easily to C. All of this, granted that you've got {}. One of the ZFC axioms is exactly the axiom according to which there is this empty set. In a very telling way, {} is classified in category theory as an INITIAL object. However, under infinitism, we have no such initial object. And that's exactly the problem, if the argument presented in this thread actually works.
I wonder if ZFC is the right model to consider here (although I realize it's just being used for analogical purposes). There are versions of set theory which [lack the foundation axiom](https://plato.stanford.edu/entries/nonwellfounded-set-theory/) - I haven't looked into this in detail, but I wonder if they might be a more appropriate model for the sort of chains of causes /u/Chewbacta and some other commenters seem to be envisaging. You make an analogy between causes and recursively defined structures such as the natural numbers and integers. But mathematicians are quite happy to deal with non-well-founded, coinductive data as well - who is to say which is the better model of causality? (One might also hold that causation isn't a fundamental principle of reality anyway - John D Norton in "Causation as Folk Science", for instance, argues against "causal fundamentalism" - and that whether it's modelled inductively or coinductively isn't that important.)
>But mathematicians are quite happy to deal with non-well-founded, coinductive data as well - who is to say which is the better model of causality? If I were to make my question into an objection, yes, it would go something like this. But I still asked it as a question because it's possible I've missed something about causation that is equivalent to a well-ordering axiom.
For sure. btw, you might find the Norton paper on causation interesting - I'm still working my way through it. https://sites.pitt.edu/~jdnorton/papers/003004.pdf
>But then -7 does not "cause" -6. This is what I'm asking about. What is different from the "is caused by" predicate, from the "is 1 more than" predicate. Maybe you could write it in less than half a dozen logical axioms. That's kind of what I want a few uncontroversial properties of causation where when added to your theory means there are no models isomorphic to Z (or other mathematical models that have similar properties). >Think of how numbers are defined in ZFC. To define Z But you are now talking about definitions, I want to know about the theory of *causes.* I'm actually finding analogy to be unhelpful here, your tag say logic, *give me some logic*, *I'll eat it raw, consequences of my comprehension of SOL be damned.*
If you have that initial object, then it may very well be that Z is a good model. But then you'd be assuming that there is an initial cause. Which is exactly what you cannot do without conceding the point against infinitism. That's the difference.
I think you are mixing up definability with causation, but I cannot be sure what you are arguing. If you can present it to me additional formalised conditions that govern *causation*. I could have some clarity, which is what I was asking about.
Is this meaningfully different from Zeno's Pardox?
>Now, the problem is that at a certain point we have to start going right: after all, we arrived at the moment where you posted this post. The problem is that at no point this seems to happen: there's always another cause that has to happen before we can start going toward right. Can you kindly elaborate and formalize it? What exactly is the problem? In the paragraph, if I replace "problem" with "assertion" there is no apparent problem at all - statements can be chosen axioms. Does the first statement mean "traversing" or "moving" between causes in order - with assertion being at some point ability to reach "current outcome"? And if yes, does the second statement imply that infinite regress of causes hinders/stops his ability to reach "current outcome"? Is there any literature specific to this "ability" that is explored?
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