Monday, August 15, 2011

Actual and potential infinities

I've spent much time on the ontological argument for God, but only ever invested one post, years ago, on the cosmological argument.  Looking back, my essay is a jumble of too many objections in too small a space.  I also pushed some of the most interesting objections aside because they were not particularly important.  Therefore, I am declaring a new blogging series: not "Why the cosmological argument is wrong," but "Here are a few things that are wrong about the cosmological argument."

I will place emphasis on math and physics, and deemphasize the cosmological argument itself.  I trust this post will demonstrate what I mean.

Actual and Potential Infinities meet Mathematics

In William Lane Craig's version of the cosmological argument, he makes a distinction between "potential" and "actual" infinities.  William Lane Craig (henceforth WLC) contends that potential infinities can exist in the real world, but actual infinities cannot.1

The thing is, in mathematics, there is no distinction between actual and potential infinities.  At least not one I've heard of.  Luckily, WLC explains.  He identifies actual infinity with the cardinality of natural numbers, ℵ0 ("Aleph-nought").  As for potential infinity...
Crudely put, a potential infinite is a collection which is increasing toward infinity as a limit, but never gets there. Such a collection is really indefinite, not infinite. The sign of this sort of infinity, which is used in calculus, is ∞.
From a mathematician's perspective, WLC's definition of actual infinity is perfectly well-defined, but his definition of potential infinity is poorly-defined.  In calculus, ∞ doesn't actually have any meaning on its own, but when inserted into mathematical expressions it gives the expressions new meaning.


These are two possible contexts in which ∞ can be used, and it technically has a different meaning in each context.2  WLC appears to be using ∞ in yet a third context where the meaning is unclear.  I can guess fairly well what he's trying to say, and the charitable thing to do would be to simply give potential infinity a precise and appropriate definition, even though WLC could only be bothered to give a crude definition.  But if I did that, WLC's supporters would likely claim that I've defined it incorrectly.

But let's try it anyway.
Any particular set of objects has an exact "size", called its cardinality.  The cardinality may either be a finite number (eg 0, 1, 2, 3), or an infinite cardinality (ℵ0 or larger).  We say that the set of objects is actually infinite if its cardinality is ℵ0 or larger.

Potential infinity is not a cardinality, and does not refer to any particular set of objects.  If we say that the set of all apples in the world is potentially infinite, we are really talking about the set of all possible sets of apples.  And if we say that distance between two stars is potentially infinite, we are really talking about the set of all possible distances between two stars.

We say that a set of objects is potentially infinite if for every finite number M, there exists some possible world where that set of objects has cardinality greater than M.  Similarly, we say that a number X is potentially infinite, if for every finite number M, there exists some possible world where X is greater than M.
Notes on my definition:
  1. Even if the set of all apples in the world is potentially infinite, this does not necessarily imply that there is some possible world in which the set of apples is actually infinite.  It just means that there is no maximum number of apples in the world.
  2. However, if the set of all apples in the world is potentially infinite, this does imply that the set of all possible worlds is actually infinite.  I think this is okay with WLC, because he would not consider possible worlds to be "existing" objects.
  3. Mathematically speaking, this is still poorly-defined because the set of all possible worlds is poorly-defined in mathematics.  However, I think it will suffice for my purposes.
I sincerely hope that my definition is satisfactory to WLC's supporters, but I couldn't know for sure.  It stands to reason that they should be able to tell me whether the definition is satisfactory before I apply the definition.  Therefore, I offer a pause here for objections (though it's a symbolic pause, since realistically I don't expect any supporters to pay attention to a little blog).

----------------------
1. This is relevant to the cosmological argument because WLC contends that a universe without beginning is an actual infinity, and I suppose he contends that a god is not.  But let's not get sidetracked.

2. Equation (1) means that for any positive number ε, there exists some number M such that for all x greater than M, 1/x is between -ε and ε.  Equation (2) means that for any number M, there exists some positive number ε such that if x is between 0 and ε, then 1/x is greater than M.

"A few things wrong about the cosmological argument"
1. Actual and potential infinities
2. Actual infinities in physics
3. What is real?
4. The "absurdity" of Hilbert's Hotel 
5. Interlude: God is infinite 
6. Forming Infinity, one by one 
7. Uncertain beginnings
8. Entropy: The unsolved problem
9. Kalam as an inductive argument
10. Getting from First Cause to God 

62 comments:

Trevor said...

The distinction between actual and potential infinities in math goes back to Aristotle, who didn't believe actual infinities could exist in math or nature. Now pretty much all mathematicians accept the existence of actual infinities in math, but whether there can be actual infinities in nature is still debated. See http://en.wikipedia.org/wiki/Actual_infinity

miller said...

I definitely don't consider Aristotle an authority on modern mathematics. But if Aristotle's definition is inconsistent with mine, I would like to hear it.

Charles said...

"The thing is, in mathematics, there is no distinction between actual and potential infinities."

But in philosophy, there is. This is not really an interesting point; why would mathematicians discuss something that, quite frankly, doesn't concern them? This is an issue of ontology, not mathematics, so I'm not quite clear how this is an argument against Craig, as if to say that Craig's argument trades upon the authority of mathematicians. Any such mathematician who cannot discern between the abstract and the concrete is simply a bad philosopher.

"From a mathematician's perspective, WLC's definition of actual infinity is perfectly well-defined, but his definition of potential infinity is poorly-defined."

And maybe it's not well-defined from a sociologist's perspective, but who cares? This is a metaphysical issue. I don't see anything wrong with his definition.

Charles said...

"The distinction between actual and potential infinities in math goes back to Aristotle, who didn't believe actual infinities could exist in math or nature. Now pretty much all mathematicians accept the existence of actual infinities in math, but whether there can be actual infinities in nature is still debated. See http://en.wikipedia.org/wiki/Actual_infinity"

Exactly. The world of mathematics exists only in the abstract. People often mistakenly presume that there's a one-to-one correspondence between mathematics and the real world, and this is just not the case. Nobody has ever seen the number "7," for example; it's just not something that exists in nature.

In mathematics, you have perfect triangles, numbers, infinite collections, and so forth. In the real world, no shape is perfect, there are no numbers, and every collection has a limit. Any naysayers would open themselves to logical absurdities, such as Hilbert's Hotel or the example of the infinite-paged book.

miller said...

Charles,

Many of the disputes over WLC's argument have to do with the actual/potential distinction. Isn't it best, then, to have a more precise definition? WLC seemed to think so himself, or he wouldn't have referred to calculus or cardinalities (though the less charitable interpretation is that he only referred to these as a way of trading on the authority of mathematicians).

Math is my way of making the definition more precise. Besides, don't you agree that math is fun?

Of course, I should not presume that WLC speaks for you. For example, you called Hilbert's Hotel a "logical absurdity", while WLC specifically says that it is a "real or factual impossibility", rather than a logical impossibility.

drransom said...

Charles: I agree that a philosophical construct is not required to have a mathematical interpretation, but Craig's definition is quite clearly trading on mathematics. He defines potential infinities in terms of limits, and claims it can be identified with a concept in calculus:

Crudely put, a potential infinite is a collection which is increasing toward infinity as a limit, but never gets there. Such a collection is really indefinite, not infinite. The sign of this sort of infinity, which is used in calculus, is ∞.

So if the mathematical definition isn't coherent, then Craig's whole argument is shot unless he has an alternative, non-mathematical definition up his sleeve. And then he needs to come up with an argument applicable to that definition.

(Incidentally, I think the symbol ∞ can have meaning on its own in analysis, as the supremum of the set of extended real numbers, but that's not what Craig is talking about.)

Charles said...

I see nothing wrong with his definition at all. The original posters only point is that the actual/potential distinction does not exist in mathematics, to which I am replying that this is irrelevant and thus inconsequential. He only mentions calculus to give the symbol.

drransom said...

Charles--you're right that there's a syntactic ambiguity and that he could merely be saying that the symbol is used in calculus, not that "this sort of infinity" is used in calculus. But he precedes the mention of calculus by using a definition in terms of "limits," a calculus concept. Then he goes on to contrast "possibile infinity" with "actual infinity," which he says is employed in set theory.

You're right about the syntactic ambiguity, but I don't think your interpretation is very plausible.

miller said...

"The original posters only point is that the actual/potential distinction does not exist in mathematics"
The main point was to offer a precise definition for purposes of future posts. Do you have a concrete disagreement with the definition itself, or do you only disagree with the act of making a precise definition?

Charles said...

My point is that Craig's argument is not unclear, and therefore there's no need for him to be anymore precise. A potential infinity is a finite quantity that is increasing without bound; an actual infinity is a collection of things that has reached infinity as a limit--a truly infinite quantity, so to speak.

I guess precision and plenitude--in this case, anyway--are just in the eye of the beholder. Craig's notions, to you at least, are nebulous. But they make perfect sense to me, and I am persuaded by his argument.

As far as "limit" being restricted to calculus, I can find that same word being used in books regarding embroidery.

If, indeed, the distinction between potential and actual infinities exists in set theory, then what miller said is incorrect. I'm not making the claim one way or the other.

drransom said...

So a potential infinity is a "finite quantity that is increasing without bound," that sounds like a never-ending series of events in time. (Which would mean that the status of something as potentially infinite can never be verified, since there's no way of establishing that a series of events will never terminate.) Do I understand you correctly? Or can a potential infinity be something other than a series of events in time?

Charles said...

No, potential infinitude applies to any collection of objects the quantity of which can steadily increase towards infinity but will never get there. So, for example, there can possibly be more stars in the universe than there already are; however, the collection will never stop at infinity.

Perhaps an easier way to discern potential and actual infinities is to say that the former is such that the parts are less than the whole, whereas the latter is such that the parts are the whole.

Craig's point, I think, is that actual infinities exist abstractly, but there's no way they can exist concretely. In reality, no part can be equal to the whole.

miller said...

I think precision is definitely context-dependent. Informal chats have a lesser standard than blogs, which have a lesser standard than philosophy papers, which have a lesser standard than math papers. It is arguable that WLC is meeting the standards of his chosen medium.

All the same, I think the cosmological argument itself calls for a more precise definition. Otherwise we'll just be making worthless assertions at each other. "I think WLC's definitions make perfect sense to me." "I think they are vague and open us up to equivocation." Progress: zero.

(I must also reiterate that math = fun.)

Charles said...

I think part of the confusion stems from "increasing." I meant to write "can increase without bound." Sorry about that.

drransom said...

Charles, I think I sort of have a grasp of what you mean by "actual infinity," but now we're stuck at the conceptual problem of what it means to exist "abstractly" versus "concretely." There are paradigmatic cases (the set of integers is "abstract," while my computer is "concrete") but once you get past the paradigmatic cases I think the distinction you end up with is hopelessly fuzzy. Let's consider some examples:

* The collection of points between my head and my computer monitor.
* A Cayman Islands shell company that does no business and exists solely to hold title to another entity to acquire tax advantages.
* The common law.
* My shadow.

Are these things abstract or concrete? I don't have the slightest idea how you would go about deciding.

Charles said...

Something is abstract just in case it is ontologically dependent upon some mind or other; something is concrete if it's not.

What you've raised is an interesting epistemological issue, and we could debate all day on the plausibility of idealism or reductive materialism (I'm not sure which one applies to you).

But would you be willing to grant that *if* there are concrete objects, then they cannot actually be infinite?

drransom said...

I tried posting a reply yesterday, but for some reason it didn't go through, so hopefully it will this time:

I'm definitely not going to concede that there can be no "concrete" actual infinities according to your definition of "concrete." In fact, I contend that such infinities are not merely possible, but exist in the real world. For example, the interior of the sun is not ontologically dependent on a mind, but it contains an infinite number of points. Similarly, the volocity of a falling rock (expressed as a percentage of the speed of light) is not ontologically dependent on a mind, but takes on an infinite number of values.

The obvious way to refute my examples is to argue that the interior of the sun does not, in fact, contain an infinite number of points, or that a falling rock does not take on an infinite number of velocities. And that might be true: perhaps space and time are quantized in such a way that the interior of the sun only contains an finite number of points and a falling rock does not take on an infinite number of velocities (or its velocity is not even well-defined at the micro level). But contemporary physics, as I understand it, typically assumes that space and time are not quantized. Physicists might change their mind, much as their changed their minds about Euclidean space, but that's a physical question, not a philosophy question.

Philosophical arguments that scientists are wrong have a very bad history: consider the philosophers who argued that Einstein misunderstood the nature of time, or that Newton was wrong because action at a distance is impossible. They look pretty absurd now.

miller said...

Later posts in the series will consider examples of actual infinities in physics, and what a "concrete" object is. So I'll save my thoughts on that for later.

Charles said...

I would take it that any physicist who assumes this is either a bad philosopher or means something different by "point."

I really don't care what some physicists *assume*. What I care about is their reasons for their assumptions; are they good or bad?

Interestingly, scientists (such as Vic Stenger) seem to have the most difficult time debating Dr. Craig.

drransom said...

Charles--I don't understand your claim here. Are you claiming that physical theories based on continuous space and time are bad physics? And if they're good physics but bad philosophy, then why? Because "concrete" actual infinities cannot exist? That's hopelessly circular. And what makes you think physicists are employing a non-geometric meaning of "point"?

Of course, our host, an Actual Physicist, can correct me if I'm wrong.

Charles said...

Also, this is not meant to be disparaging towards both of you guys specifically, but it's something worth pointing out nonetheless.

I find it quite striking that whenever an actual scientist--say, Robert Jastrow or Michael Behe--expresses theistic underpinnings within a scientific discovery, s/he is deemed a bad scientist and shunned by the scientific community, whilst atheists--whose worldview is typically championed as being defensible only under the auspices of science--give every possible nit-picky argument to show that their interpretations are wrong.

Yet as soon as a theist expresses disapproval of ideas such as virtual particles emerging out of actual nothingness, the existence of multiple universes, and/or the reification of mathematical points, the theist is unreasonable for doubting the epistemic authority of scientists.

Why does this not work both ways?

Charles said...

I'm saying that any physicist whose theory requires the reification of mathematical objects such as points, number and what have you, is a bad philosopher whose philosophical incompetence has given way to slovenly scientific methodology. I wouldn't say he or she is a bad physicist overall (I'm not a physics expert--I can only tell you what good philosophy is), but that s/he made a mistake.

miller said...

drransom,

As I understand it, Charles has a few ways out. He can claim that the physics is mistaken (ie space and time are discrete). Or he can argue that positions and instants are abstract objects, not real ones. Or he can claim that they are only potentially infinite. Try to guess which way Charles is taking.

drransom said...

Charles, I don't think the social treatment of theistic scientists has any particular relevance to this debate.

As for "reifying" points in space, you haven't said anything about what that means or why it's bad philosophy. (What is the difference between points in space and "reified" points in space?)

The existence or nonexistence of numbers as Platonic objects has nothing to do with physics, but the existence of continuous space with an infinite number of points vs. quantized space where a finite volume consists only of a finite number of points is actually quite important.

Charles said...

Reification is fallacious by definition, so your question is like asking why it's bad philosophy to violate the law of noncontradiction.

But I think what you're really asking is why we should assume that points in space are non-physical objects. For me to answer that, you'll have to tell me what you mean by "point in space."

drransom said...

Charles--you defined "concrete" as "not ontologically dependent on a mind," which the points in the space in the interior of the sun are not. Likewise with the velocities taken on by a falling rock. Now you're switching to asking about whether points are "physical," whatever that means. (Is the gravitational field "physical"?)

I assumed that reification included treating actually concrete things as concrete (e.g., reifying my computer), but if it only involves treating abstractions as "concrete," you're assuming your conclusion.

Charles said...

A reification is a fallacy whereby an abstract entity is treated as concrete.

http://en.wikipedia.org/wiki/Reification_(fallacy)

Let me rephrase what I said, for I would like to move forward with the discussion:

I think what you're really asking is why we should assume that points in space are abstract objects. For me to answer that, you'll have to tell me what you mean by "point in space."

drransom said...

I don't think I can offer a set of necessary and sufficient conditions (one rarely can for anything useful) and I don't think you'll be satisfied by a general description, but here goes:

A point is a zero-dimensional bit of space.

Charles said...

Then your contention is that there's an infinite number of dimensionless (therefore, unextended) spaces existing in reality. But you a difficulty on your hands:

Provided their concrete existence, there are no discernible differences between any two points, which means that they are not different at all. There is nothing true of any zero-dimensional bit of space that isn't also true of any other zero-dimensional bit of space; therefore, by Leibniz's law, they are all one and the same. So it follows, there are not an infinite number of zero-dimensional bits of space.

This was actually recognized by Kant early on, as he states in the Critique, "For, first of all, we can imagine one space only, and if we speak of many spaces, we mean parts only of one and the same space. Nor can these parts be considered as antecedent to the one and all-embracing space, and, as it were, its component parts out of which an aggregate is formed, but they can be thought of as existing within it only (Transcendent Aesthetic)."

On the other hand, it is easy enough to discern between different mathematic points so long as they are conceived only as abstractions. In your mind, you can discern them according to divisions, additions, subtractions and so forth; for example, point A could be the half-point between two walls, point B could be that half-point's half-point, and so on. To surmount this objection, you would have to show that numbers--such as "7"--exist in reality, which is an even more daunting task than what you're currently attempting.

You also cannot discern points according to their locations, because, firstly, an unextended thing cannot be located anywhere, and secondly, two unextended things are not such that one cannot be where the other one is. For instance, if two mathematical points approaching one another, they are not going to trespass upon each other's location, for only extended objects can occupy only one location at a time.

miller said...

I think of a point as an elementary object, and space as a collection of points with some relationships to each other. For example, you can consider the distance between two points, which is zero if and only if the two points are identical. Each point also has a collection of properties such as its electromagnetic field and metric.

Your philosophy is cool too though, Charles. Nice touch with all points being one, and yet discernible in the abstract. Not gonna bother arguing with that one.

drransom said...

I don't think an argument about the existence or nonexistence of points in space is going to get us much of anywhere, but let me get down to the center of the disagreement. Your objection is premised on the idea that a point's location, and the physical properties of that location (e.g., the electromagnetic field), are insufficient to distinguish it from other points. Suffice it to say that I don't agree.

(I also don't see what's wrong with saying numbers exist in "reality," but I'm not going to place any philosophical weight on that particular opinion.)

Charles said...

miller, I don't what you hope to accomplish with snide sarcastic comments. I take it you invite people who disagree with you to comment on your blogs. Making snide comments only closes down the lines of communication. It's not very productive.

If you have a disagreement, please state what it is and why you disagree. Yes, points, so defined as dimensionless spaces, are discernible only as abstractions. Why is that a problem?

Charles said...

drransom, how can a dimensionless space have any location?

drransom said...

Charles, I'm not sure what would even count as an answer to that question. It seems like asking how light can be both a wave and a particle: it just is.

Similarly, a point has a location by virtue of having one. I don't think I can offer any further explanation unless you understand why the concept is problematic.

drransom said...

Unless you explain why it would be problematic, I mean

Charles said...

drransom, I don't know enough about light to be able to comment on that example. But I'm totally sympathetic to scientific discoveries being counterintuitive. I've read a little bit about quantum mechanics, special and general relativity, and what have you.

But the notion of a locatable non-extended object is prima facie contradictory. What is it to be located other than having some extension in space? How do you discern one point from another?

miller said...

Charles, there is nothing I can hope to accomplish with you. You are full of opinions and short on justifications. I can't imagine any third party being persuaded by your position just because you've stated it (and unclearly at that).

drransom said...

But the notion of a locatable non-extended object is prima facie contradictory. What is it to be located other than having some extension in space? How do you discern one point from another?

I don't buy that it's prima facie contradictory, and I don't agree with your general style of argument. You've proposed a set of necessary and sufficient conditions and seem to be implying that your definition should stand unless I can come up with an alternative, superior definition. I don't accept this procedure. Developing necessary and sufficient conditions for a natural language concept is pretty much impossible.

As for how you distinguish one point from another, you distinguish them by virtue of their position in space relative to other points, and by the physical characteristics of their locations. Say one point is in the interior of the Sun, and another is in the middle of my living room. Among other things, the gravitational and electromagnetic fields at the two points are different.

(And I presume by "extension" you mean something other than the opposite of intension? If you mean the opposite of intension, then of course the extension of 'the point at the center of the sun' is the point at the center of the sun, while the point at the center of the sun has no extension, just as my computer has no extension.)

miller said...

To be fair, I obviously find some entertainment in this, so at least I'm accomplishing that.

"What is it to be located other than having some extension in space?"
I think of "extension" as meaning that it is located at multiple points. Define your terms.

"How do you discern one point from another?"
If all points are "one and the same", how do you discern one point from itself?

Charles said...

Miller, the irony here is that I can easily imagine agnostics swayed by your pedantic usage of algebraic symbols, phrases such as “from a mathematician’s perspective,” esoteric terminology and/or articles pertaining to arcane subject matter. To be clear, I’m not imagining your website to be some sort of catalyst towards personal intellectual epiphanies, for that would require that such onlookers actually understand the sophisticated literature in discussion; rather, such folks simply need a representative whose purported credentials and recondite literary style are apt to intimidate theists untrained in apologetics.

I gave several arguments, and your only tactic seems to be mockery or ambiguating terms whose meanings are clear to ordinary people the reading levels of whom are at least ninth grade.

I think it’s about time you drop out of the discussion and say that I have the final word, while I continue this with drransom.

Charles said...

“Extension,” as defined in Webster, refers to a property whereby something occupies space.

I don’t quite follow what you’re objecting to in terms of my debate style; you’ve defined “point” in a certain way and I’m asking you follow up questions, based on my understanding of words contained in the definition you’ve put forth. What’s wrong with that?

Otherwise, would you rather I let you go on the offensive and have you ask me questions about my worldview? Because I’m totally open to that.

And again, you are assuming that a point actually has a position in space; but that clearly cannot be if a point is dimensionless.

drransom said...

Charles, I think it's entirely clear that a point can be located a position in space--say, the center of the sun. (I'm not sure I'd say it "occupies" a position, but I don't think much hangs on that.) I think we've hit an impasse.

Sorry if I read too much into your argumentative style though!

miller said...

LOL, okay, Charles. I don't really see myself as trying to convince theists. I see myself as finding ridiculous excuses to talk about math and physics. So... you don't like math, eh?

It's my custom to give adversaries the last word in blog comments. So shoot.

Charles said...

This would just be assuming the real existence of a mathematical point(the sun's center) to support the real existence of mathematical points. But yes, I agree that we've hit an impasse.

So is your rebuttal to the cosmological argument that there is an infinite regression of causes and effects?

Charles said...

I love math and physics, though I'm not generally a fan of mathematicians and physicists acting as the go-to people for theological/metaphysical issues.

Anyway miller, thanks for the discussion. I do enjoy your blogs, even though I might disagree with you.

drransom said...

There could be an infinite regress, though if there is I think it would consist of time corresponding to the interval (0, ∞) rather than (-∞, ∞). So on my hypothesis there's an infinite regress without there being a first instant. I don't claim this is true, though I believe it is, since it's the option that sounds most plausible to me.

Or there might not be an infinite regress: I really don't know. Maybe time corresponds to [0, -∞). (But note that here there's a Zeno-style infinite regress, since between t = 0 and any t > 0 there are still an uncountably infinite number of times). Or there could be a first instant and quantized time, so there's no form of infinite regress at all.

IOW, I'm both going to assert that there can be actual infinities in reality and deny that a finite regress requires a first cause with properties that make it identifiable as a god.

Charles said...

I think the notion of a beginningless series of causal events makes the argument worse. In this case, you are saying that there's a series of causal events already maximized at infinity; that is, the infinitienth event was not arrived at via successive addition from a first event.

This being the case, every causal potentiality would be actualized already and, as such, nothing could change. However, things do change; we experience this every single day.

The only way to surmount this, from what I can tell, is to say that changes in states of affairs are not caused, and therefore change-events are not part of the infinite causal series. But then it's hard to see what currency we should place on causal events, for what is a causal event if it does not entail an agent bringing about a state of affairs not previously instantiated? We would have to say that there is no causation, and that everything just is.

drransom said...

Charles--I don't think I understood any of the terminology you used, to the point where I can't say if I agree or not. Key terms where I'm not sure what you mean: "causal potentiality," "successive addition," "maximized at infinity," "actualized."

I would add that if I'm right about space and time, then every event, except one or more possible first events, would be the "infiniteith" event. Even that language is misleading, since talking about an nth event implies that the set of events is countable, but I believe it has cardinality at least aleph-one.

Charles said...

A causal potentiality is a hypothetical causal event that hasn't happened, but could happen. More concisely, causal event A is a potentiality if and only if A occurs in some possible world W but not in the actual world G, and G has access to W.

In mathematics, successive addition is any instance of numbers getting added together in a sequence; e.g., in the sequence "1 + 2 + 3 = 6," the number "6" was arrived at via successive addition. In the case of the KCA, Craig borrows the term and uses it with regard to a series of interrelated causal events--a "causal chain," so to speak. For example, one event causes some other event, which causes another event; hence, the third event in the example was arrived at via successive addition from the first event.

"Maximized at infinity" means the same thing as "actual infinite"; it's a collection of items infinite in number. I use the term "maximized" insofar as the cardinality of an infinite set--whether concrete or abstract--can never be larger than it already is.

Something is "actualized" if it becomes actual from a previous state of potentiality. For example, when I was four years old, I was potentially thirty years old; however,that potentiality was actualized later on.

drransom said...

Charles--now that you've defined your terms, I think the key issue in your argument is here:

This being the case, every causal potentiality would be actualized already

which is a non sequitur. The only way it holds is if there's a merely finite number of causal potentialities, and this suppressed premised has not been established.

I think there are other problems with the argument, but that's the key one.

(Of course, none of this stuff is directly relevant to what I see as the principal flaw in the cosmological argument. Perhaps there is at least one first cause, but the existence of one or more first causes does not entail that any of them has the properties of something we would identify as a "god." I think most other arguments for the existence of a god suffer from the same problem.)

drransom said...

(I'll add that the linked article has an argument that the first cause must be a personal creator, but the argument is no more than the argument from incredulity:

For how else could a temporal effect arise from an eternal cause? If the cause were simply a mechanically operating set of necessary and sufficient conditions existing from eternity, then why would not the effect also exist from eternity? For example, if the cause of water's being frozen is the temperature's being below zero degrees, then if the temperature were below zero degrees from eternity, then any water present would be frozen from eternity. The only way to have an eternal cause but a temporal effect would seem to be if the cause is a personal agent who freely chooses to create an effect in time.

And even the supposed personalty of the first cause doesn't establish that it has any other properties people typically want to ascribe to gods, such as large amounts of power to intervene in the physical universe, concern for human affairs, some level of moral goodness, worthiness of worship, or even continued existence. Nobody is particularly interested in establishing the existence of a sadistic god, an indifferent god, a dead god, a weak god, or any of the other multitude of possibilities. (The only thing we can be quite sure of is that the first cause in question is either evil, indifferent to human affairs, or incapable of creating a better universe than the one that already exists.)

drransom said...

Correction:

The only thing we can be quite sure of is that the first cause in question is either evil, indifferent to human affairs, or incapable of creating a better universe than the one that already exists.

Another possibility is that the first cause is nonevil, not indifferent to human affairs, and capable of creating a better universe than the one that exists, but created this universe rather than that one by mistake. Should have mentioned that one too!

Charles said...

Again, you've assumed, for the sake of argument, that there is an infinite regress of causes and effects. In other words, you are granting that there's an actual infinitude of causal events.
Now if you survey the cardinal number of causal events, you'd have to conclude that, as an aggregate, it cannot be larger.

Yet if you wish to great that this situation still includes causal potentialities--that is, causal events that could happen, but haven't happened yet--then you are granting that the aggregate of causal events, which is supposed to be infinite in magnitude, could have an even greater magnitude, which is logically incoherent.

And somehow, you want to say that a necessary precondition of the realization of every causal potentiality is that there only be a finite number of them. I don't quite get how you've reached this conclusion, since an implication of an infinite set A whose members include only objects of type B is that no type-B-object can be included in A's complement; for example, given the infinite set of natural numbers N, no natural number can be included in the complement of N.

I'll try addressing the other parts of your post(s) later.

Larry, The Barefoot Bum said...

Yet if you wish to great that this situation still includes causal potentialities--that is, causal events that could happen, but haven't happened yet--then you are granting that the aggregate of causal events, which is supposed to be infinite in magnitude, could have an even greater magnitude, which is logically incoherent.

Your conclusion does not follow from your premises. I do not think you understand transfinite arithmetic.

drransom said...

I think the easiest way to address this issue is point-by-point:

Now if you survey the cardinal number of causal events, you'd have to conclude that, as an aggregate, it cannot be larger.

I don't think I understand what you mean by "cardinal number" in this context. Do you mean the mathematical concept of cardinality (where, e.g., aleph-null is a "cardinal number")? If so, then an infinite set can trivially be larger, since there's no largest infinite cardinality. Or do you mean some other conception of cardinality?

then you are granting that the aggregate of causal events, which is supposed to be infinite in magnitude, could have an even greater magnitude, which is logically incoherent.

I don't think I understand what you mean by "magnitude" either.

If you are saying that the concept of adding elements to an infinite set is logically incoherent, then you've just dismissed practically all of mathematics. I suppose you will fall back on the concrete/abstract distinction, but why would adding elements to an "abstract" infinite set any less incoherent than adding elements to a "concrete" infinite set?

IOW, is the concept of taking the union of {0} and the positive integers to get the natural numbers incoherent?

And somehow, you want to say that a necessary precondition of the realization of every causal potentiality is that there only be a finite number of them.

I can see that my wording was ambiguous here. I meant that your inference doesn't follow unless there's a merely finite number of causal potentialities. If there's an infinite number of causal potentialities, they might stop being realized, or they might not.

an implication of an infinite set A whose members include only objects of type B is that no type-B-object can be included in A's complement; for example, given the infinite set of natural numbers N, no natural number can be included in the complement of N.

In general, "A's members include only objects of type B" means

if x is in A, then x is also in B.

It does not follow from this statement that all members of B are also in A: that's just affirming the consequent. Did you mean to refer to a set A whose members include all objects of type B? If so, the inference is trivial regardless of whether or not A is infinite.

If you need an example of an infinite set A consisting only of members of B, but there are members of B in the complement of A, let A be the even natural numbers and let B be the natural numbers. Then A consists only of natural numbers, but there are natural numbers in the complement of A (namely, the odd natural numbers).

Charles said...

A "cardinal number" in mathematics refers to any natural numbers used to describe the magnitude of a set.

No, an infinite set cannot be larger than it is--unlike, say, the set of all human beings, which is not an infinite set and therefore can be larger than it is.

"Magnitude" refers to size. I'm saying that it is logically incoherent for you to increase the magnitude of a set whose members include only elements of a specific type just in case you presuppose that it's an infinite set including all members of that type.

We're past the concrete/abstract distinction, at this stage. What I'm saying is true in mathematics as well as in reality: there's no natural number that is not a member of the set of all natural numbers.

Mind you, I'm just assuming your premise that there is an infinite regress of causal events.

No, the union of {0} and {1, 2, ...} is not incoherent. However, "A & ~A" is incoherent if A = {0, 1, 2, ...} and ~A = {3}.

I've never heard an interpretation of the statement "A's members include only objects of type B" according to which x belongs to B if x belongs to A. I was referring to a set A whose members include all objects of type B (I really don't find this to be ambiguous at all). But it's not trivial; for example, the set of all human beings can be larger, whereas the set of all natural numbers cannot be larger--or can you posit a single set, or even a union of many sets, whose cardinality is greater?

Your example doesn't apply to what I said. I did not say that all infinite sets have the same members; I said, firstly, that no infinite set can be smaller than any other set, and secondly, that any set containing all members of a specific type cannot be such that any of those elements are in its complement. In your example, B is the set of natural numbers, which means that there's not a single natural number in the complement of B.

Collin said...

If spacetime is continuous, then it would seem to imply that there is no first event. However, what it actually means is that the concept of "event" itself is ill-defined. As Charlie pointed out, there is no way to insert "and X happened" into a potentially continuous action.

If spacetime is discrete, each event could be traced back to a first cause, but that cause would not be unique. And in the context of the CA, that would suggest as many gods as would fit inside the "head of a pin" that began the universe -- obviously not what WLC had in mind!

So I'd say arguing about the CA is a dead-end for both sides of the debate.

BTW, Charlie, two comments on your last paragraph:

1. You say an infinity can't be smaller than anything. This means you reject the CDA. Why?

2. You refer to "the complement of a set" as a complete phrase. (My math primer in elementary school made the same mistake.) Actually, a complement is a difference between two sets.

drransom said...

Collin--why does continuous spacetime imply the nonexistence of a (possibly nonunique) first event? (And what is the CDA? Wiki is not helpful.)

Charles--you write that "any set containing all members of a specific type cannot be such that any of those elements are in its complement," but this is a tautology, and is true whether or not the set in question is infinite. I'm not sure why you think it establishes anything.

You also state that no set can have higher cardinality than the natural numbers, but this is just straightforwardly false. The natural numbers have cardinality aleph-null, but the real numbers have cardinality aleph-one, the power set of the real numbers has a higher cardinality than that (aleph-two, I think, but I'm not sure), and so forth. This is not actually relevant to this argument, but is true.

Charles said...

Drransom-- It establishes that if there's an infinite regress of causal events, then there are no causal potentialities. Suppose that "S" represents the infinite set whose members consist solely of causal events in the actual world G, that "S1" represents the set of causal events in some possible world W, and that W is unequal and mutually accessible to G; it follows that S1 must necessarily be a subset of S. Otherwise, either there are causal events in ~S or the cardinality of S is finite.

My main point was that no infinite set can be larger than it is, which is true as far as I can tell; for instance, you cannot increase the magnitude of |N| via the addition of a natural number. On the other hand, the cardinality of the set of human beings increases as soon as a baby is born.

You are right about aleph-null and aleph-one, however. I stand corrected.

Collin-- I'm not sure what "CDA" is. Sorry. The book I have on NBG set theory does use the phrase "the complement of A," or "~A", where "~A" equals the class of elements not belonging to A; I think what you have in mind is the complement B relative to A--symbolized in NBG as "A~B"--which is the class of elements belonging to A but not to B, or the union between A and B's complement.

Michelle said...

(Charles)"And again, you are assuming that a point actually has a position in space; but that clearly cannot be if a point is dimensionless."

I'm sorry if I'm moving this discussion hopelessly backwards, but I had to stop reading at this point and comment. A "point" is of course an abstract thing that only takes on physical meaning when we ascribe certain properties to it. Thus, the definition of "point" is entirely context-dependent. "Points" in the context of the interior of the sun or the distance between stars have been ascribed the property of position. The points ARE positions in space. Positions in space ARE points.

Charles, I am intrigued by your definition of concrete versus abstract. You say that anything concrete is that which does not ontologically depend on the mind. However, I contend that you haven't defined "mind," and that you are implicitly relying on a hierarchy of philosophical thought firmly implanted in Western culture. I, on the other hand, would contend that there is no thing which is not ontologically dependent on mind, which is an idea more firmly implanted in certain Eastern philosophies.

Also, hi miller! I met you at Caltech a few summers ago and I've just reconnected with your blog. Very interesting.

miller said...

Oh, hi Michelle! I was reading your comment thinking what a nice contribution it was to the thread, when I realized who it was. Good to hear from you!

Charles said...

Actually, as far as I know, idealism reached its apex in the West, where it was most famously propounded by philosophers such as Berkeley, Kant, and Hegel. I don't know if this is relevant to the issue of infinities, though. However you wish to parse the language--whether you wish to say that infinities are abstract as opposed to concrete, immaterial as opposed to material, ideal as opposed to real, and what have you--there is an important distinction to be made between infinities in mathematics and those in nature. I think even a staunch idealist would be willing to concede Craig's point that a hotel with infinitely many rooms is problematic.

Are positions in space abstract or concrete? If concrete, how many could there be?