## Theology and Indeterminate Infinity

Apologists often claim that actual infinites are logically impossible. One of the arguments which they utilize to support this claim deals with subtracting quantities from infinite quantities. One example of this comes from Blake Giunta’s Belief Map:

Infinity minus an infinity yields logically impossible scenarios. Notably, one can take away identical quantities from identical quantities and arrive at contradictory remainders.

On the face of it, this claim appeals to our intuitive understanding of subtraction. If I were to claim that there exists some Integer, , such that and , then we stumble upon the contradiction that . Subtracting identical quantities from identical quantities should yield identical results.

Belief Map further cites the work of William Lane Craig and James Sinclair for a particular example of how infinities can yield such contradictions.

For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and infinity as the remainder. For this reason, subtraction and division of infinite quantities are simply prohibited in transfinite arithmetic – a mere stipulation which has no force in the nonmathematical realm.

Unfortunately, this example is not quite so logical as its authors would like it to be. The reason for this is that they don’t seem to have a very good grasp of the mathematics which they are trying to discuss. The final statement of this paragraph makes a blatantly false claim: “For this reason, subtraction and division of infinite quantities are simply prohibited in transfinite arithmetic.” It is not the case that subtraction and division are “simply prohibited in transfinite arithmetic.” Rather, subtraction and division are not defined in these cases. This is a very important distinction.

Imagine, for a moment, that I have asked you to perform the calculation “seven vike four.” You would rightly ask me what ‘vike’ means? How does one ‘vike’ two numbers? Until there is a definition for viking numbers, it is not possible for anyone to vike those numbers. It is not that we are prohibited from viking numbers. It is simply that the phrase is undefined and, therefore, meaningless.

Similarly, let’s say that I asked you to perform the calculation “seven divided by zero.” Now, despite the fact that division is well-defined for almost all numbers, you may recall from your grammar school math classes that division is not defined when dividing by zero. In this case, once again, we are not prohibited from dividing by zero. Dividing by zero is simply not a meaningful concept.

Similarly, in the field of mathematics known as “transfinite arithmetic,” subtraction and division are not defined operations. Despite the similar name, “transfinite arithmetic” is not the same arithmetic which children learn in grammar school. It is a collection of rules for manipulating sets. It doesn’t matter that these operations are well-defined for other areas of mathematics– they are not defined for transfinite arithmetic. Again, these operations are not prohibited, as Dr. Craig claims; they are simply meaningless.

But doesn’t Dr. Craig’s example cite clear ways of subtracting from infinite sets? How can it be undefined if there exist ways to do it?

We are not saying that one cannot choose and eliminate any arbitrary number of elements from a given infinite set. Rather, what we are saying is that there is no general definition which can account for all cases of subtraction between two different sets in transfinite arithmetic. The manner in which one eliminates the elements from an infinite set can change the result, and therefore there is no single, correct way to do it. So, contrary to Dr. Craig’s claim, it is not the case that we are prohibited from removing elements of an infinite set, in mathematics; we simply do not have a general operation for subtracting any given set from any other given set, in transfinite arithmetic. Special cases can always be considered by themselves.

When different special cases of performing an operation can be done in order to produce different results, we say that the more general statement is *indeterminate* in that case. For example, as we mentioned earlier, division by zero is not defined in the arithmetic which we all learned in grammar school. However, when mathematicians see the statement they immediately recognize it as an indeterminate form. Depending on the special case, the mathematical phrase can be equal to any number.

Take, for instance, the phrases and . In both cases, these limits resolve to . However, a simple application of L’Hopital’s rule tells us that while . I can just as easily show cases where is equal to 5 or to 17 or to π or to any other number. This is not inconsistent or contradictory, because is not a number; it is a mathematical phrase.

Similarly, we can imagine two numbers and which are both infinite. The phrase is indeterminate. Depending on the particular case, could result in any number. This is not a contradiction. This is not an inconsistency. This is simply an indeterminancy.

The other examples which apologists usually bring out in support of their opposition to actual infinites tend to fall to similar problems. They are misunderstandings or misstatements of the underlying mathematics. The simple fact of the matter is that the idea that actual infinities are logically and metaphysically possible has not even been controversial in the Philosophy of Mathematics for a century. As such, when an apologist makes pronouncements upon the nature of mathematics, we should be more than a little skeptical.

Reblogged this on James' Ramblings and commented:

Reblogging for future reference:

Yea, I have lots of problems with this again. Sorry to be a gadfly but I can’t let this go by unchallenged. And your use of ‘the appellation ‘apologist’ is just annoying. It’s just such a rhetorically pejorative word. Sure, Blake might be one, but Craig isn’t JUST an apologist; he’s a renowned, accomplished philosopher. Heck! I could just refer to you as an ‘apologist’ (after all, you’re defending your position against people who you think are misunderstanding infinity, right?) In fact, I’ll do that and speak about you in the third-person and see how you like it.

First, the APOLOGIST BP tries to represent arguments against the infinite in logical terms. Nope. Craig is pretty clear that these are arguments demonstrating the ‘metaphysical’ impossibility of an actual infinite, the idea that if there were a one-to-one correspondence between the set of all natural numbers and CONCRETE objects in the world, you get the Hilbert-type paradoxes involving subtraction/division. Craig is NOT a finitist (he thinks the infinite has mathematical legitimacy).

Second, the APOLOGIST BP gives us more lectures on how we don’t understand mathematics. Let’s see what we dummies get wrong. The apologist BP says that subtraction/division aren’t prohibited in transfinite arithmetic because subtraction/division aren’t defined and so it’s meaningless. Uh, yea. It’s meaningless and THEREFORE it’s prohibited. What’s the problem? Does mathematics ever permit meaningless operations? Well, if it doesn’t, then they’re, uh, prohibited . . . This just seems like a pedantic point. Silly apologist Craig seems to be on the same page (on THIS point) with Oppy (1995), Huemer (2016 – he actually makes the same exact point as me on page 87), Takeuti & Zaring (1982), and on and on.

Third, the APOLOGIST BP brings up the silly point that Craig does believe the operations are permitted because the Hilbert-type scenarios have people leaving the hotel. I really must confess that this point is bewildering to me. Uh, yes. The scenarios involve, say, subtraction. Uh, yes. Using the operations (as we used them as wee little lads in elementary school, where BP thinks Craig skipped school to go graffiti the lemniscate on underneath overpasses) in the real world leads to absurdities (such absurdities are metaphysically impossible, so there’s no feasible world in which there could be an actually infinite number of concrete objects/events in the REAL world). It’s not that there’s presently no definition of subtraction – it’s that there will ‘never’ be such a definition BECAUSE its indeterminate. And yet – per impossible – if we assumed a Hilbert-type hotel, someone could leave (make the operation infinitely determinate)! Anyone could leave. An infinite number (aleph-null) could leave the hotel (and there’s still the same ‘number’ of people in the hotel: that’s absurd). Someone could do something for which there’s no meaningful mathematical function to describe it. And if it’s not meaningful, something tells me it’s going to be prohibited. 0/0 doesn’t matter. Is it indeterminate or meaningless? Or both? You tell me an operation (subtraction) is meaningless and you make a comparison to something that isn’t even an operation! 0/0 isn’t an operation. Is it meaningless because its indeterminate? Is it indeterminate because it’s meaningless? Are meaninglessness and indeterminacy synonymous?

Fourth, the APOLOGIST BP thinks Craig doesn’t believe in the mathematical legitimacy of the actual infinite. The point is that he would AGREE with you about the indeterminacy/meaninglessness point. Craig is talking about the REAL world. If all the people left the hotel from room #3 and up, the SAME cardinality of people would leave had the people in all the even-numbered rooms had left – and there’d be an INFINITE gap between the number of people left in the hotel in the first case and the number of people left in the hotel in the second case. This is metaphysically impossible in the real world. Do whatever you want in Math-land – this thing ain’t’ happenin’ in the real world (or any realizable world).

Fifth, the APOLOGIST BP thinks it has never been controversial in the philosophy of mathematics whether an actual infinite was metaphysically possible. Sorry. I was laughing so hard, milk came out my nose. Do you know what metaphysical possibility means? Please distinguish for me narrow and broad logical possibility. I could literally cite hundreds of different philosophers that deny that an actual infinite is metaphysically possible (especially if the mathematician is a Platonist or a modal realist or thinks time is discreet or wants to say there’s a one-to-one correspondence between the natural numbers and CONCRETE objects in the world).

SO, when a mathematician-APOLOGIST like BP makes pronouncements about the nature of philosophy, we should be more than a little skeptical.

By all means! I always welcome any commentary on my blog. Please, always feel free to challenge anything which you find to be problematic.

I have absolutely no intention of using the word as a pejorative and I see no reason why one should think it is pejorative. Both Mr. Giunta and Dr. Craig self describe as apologists, quite proudly; and I quite frequently acknowledge Dr. Craig’s accomplishments in the Philosophy of Religion and in Theology. I honestly don’t understand why you might think I am using the term pejoratively.

On that note, if you would like to refer to me as a “mathematician-apologist,” I would absolutely welcome it. I quite like the ring of that.

The claim to which I was responding was the one on Belief Map to which I link at the beginning of the article. That claim is explicitly in regards to the logical impossibility of actual infinites, while arguments about the metaphysical impossibility are made elsewhere on Belief Map. Similarly, the example drawn from Craig & Sinclair is not a metaphysical example but rather a mathematical and logical example.

No, that’s nonsensical. One cannot prohibit something which is meaningless. After all, what would it even mean to prohibit something meaningless? In order for an operation to be prohibited in any meaningful way, that operation itself must be meaningful. As I mentioned, subtraction and division are not defined on transfinite arithmetic. It is not the case that we are prohibited from performing these operations. Rather, there are no operations called “subtraction” and “division” in transfinite arithmetic which can even be prohibited.

As I mentioned in the article, there are specific ways by which elements can be removed from an infinite set. This is not the same thing as a generic subtraction operation. So, while the latter is not defined on transfinite arithmetic, the former is absolutely possible. It is a fallacy of equivocation to refer to this sort of specific element removal as subtraction.

A set with infinite cardinality could leave the hotel while a set with infinite cardinality remains in the hotel. What is absurd about this? No logical contradictions arise. The mathematics is perfectly valid. The fact that it is counter-intuitive does not imply that it is absurd.

Why not? Do you have anything besides a bald assertion of your intended conclusion to support this?

Quite the opposite. I’ve acknowledged on numerous occasions that the idea of actual infinites was highly controversial in the past. What I have claimed in this article– and what I will continue to maintain– is that the logical possibility of actual infinites has not been very controversial for the past century. Yes, there do exist Constructivists who still deny the logical possibility of actual infinites, but they make up only a tiny element of philosophers of mathematics. The majority of philosophers of math do not reject the logical possibility of actual infinites.

The metaphysical possibility of actual infinites is a different question. It is not the one which I was addressing in this article. I will say that I see no good reason to accept that actual infinites are metaphysically impossible, either, but I will certainly acknowledge that this question is more open to questioning.

I agree! Skepticism is a healthy position to maintain. I don’t ask that anyone should ever accept my claims simply because I have made them. I am certainly not infallible, and I have certainly made mistakes in the past. If I have made any, here, I absolutely welcome their correction.

Ok, I’ll narrow this down to a few issues.

(i) The apologist-appellation is used all the time as a pejorative label. But since you’re using it non-pejoratively, I’ll drop the point.

(ii) On the logical impossibility of actual infinities, are you talking about narrow or broad logical impossibility (and I hope you know that both of these are different from mathematical possibility)?

(iii) Meaningful prohibitions of meaningless operations – I confess your points make no sense to me. First, take the logical positivists (that meaningful synthetic statements are those that can be confirmed/verified or implied by those that have been confirmed/verified). According to the positivists, to say that God is good (call this statement P) is to utter a meaningless statement. If they’re right, am I prohibited from saying it? In a sense, no. I can still ‘say it’. I can utter words into the aether. But there’s another sense in which I am prohibited from saying it: it’s at the cost of saying anything meaningful. So, P is meaningless and yet it’s prohibited (per the criterion of meaningfulness laid down by the positivists). So, that’s what it means to prohibit something meaningless.

Second, I have no idea why I can’t affirm meaningful prohibitions of meaningless operations by virtue of the fact that the operation is meaningless. That seems perfectly coherent to me. This same kind of thing is going on in the positivist example. Instead of operations, the positivists are talking about certain statements. I mean, you said yourself that subtraction doesn’t exist in transfinite arithmetic. That would seem to mean that the operation of subtraction (when it’s postulated as existing in such an arithmetic) would be meaningless ‘there’. I can mention P all day while at Church (and it may have this surface-level, ordinary language meaning), but the positivist is going to come along and say that mentioning P is prohibited. Why? Because it’s meaningless. Your matter-of-fact dismissal of this is just bewildering to me.

Third, you make this element-removal/generic subtraction distinction. Who cares? Craig is talking about the latter and how it’s prohibited. If that’s too strong for you, just do a general perusal of his Reasonable-Faith articles and you’ll find locutions like ‘conventionally prohibited’. Right! The operation of generic subtraction isn’t allowed in transfinite arithmetic (as you say, it doesn’t exist there!). It isn’t defined there (if it’s not defined, it’s meaningless, right?). —– Please unpack the idea of element-removal without using generic subtraction.

(iv) I don’t know what you mean by ‘absurd’. Logically? Mathematically? Metaphysically? The idea that a set with infinite cardinality would be in the hotel after a set with infinite cardinality exited the hotel wouldn’t be mathematically absurd! It wouldn’t be logically absurd either (what’s logically absurd is having an actually infinite number of sets of infinite cardinality which can exit the hotel, and getting an actually infinite number of different quantities as a result. And so far, no one has been able to explain how this idea’s mathematical legitimacy implies its broad logical or metaphysical possibility.

I see no reason to think that actual infinities are logically impossible either in the narrow or the broad sense. The original article on Belief Map to which I was responding does not state in which sense it opposes the logical possibility of actual infinities, but that seems to be a moot point given my previous statement.

No, you are not prohibited from saying it. It does not follow from “to utter P is to utter a something meaningless” that “therefore, one cannot utter P.” Nothing prohibits us from uttering something meaningless, here. Now, if we add the premise, “one cannot utter something meaningless,” then we could validly come to the conclusion which you claim, but this additional premise is quite clearly false. We could put in place an artificial prohibition, but if we do our artificial prohibition is actually prohibiting something meaningful (eg, the utterance of a particular class of things). We are not prohibiting a meaningless thing.

If it is meaningless, what is it that we are meant to be prohibiting?

Once again, “X is not allowed” is not the same statement as “X does not exist.” It’s certainly not even the case that subtraction is conventionally prohibited in transfinite arithmetic– that would similarly imply that subtraction is a meaningful concept which is nonetheless avoided due to conventions held by mathematicians. In order to put in place conventions preventing mathematicians from performing subtraction in transfinite arithmetic, there must exist a thing called “performing subtraction” in transfinite arithmetic. That is not the case.

You were the one who initially utilized the term, and it seemed by your context that you meant “metaphysically absurd.” However, I’ll once again say that I do not believe that there is anything absurd about actual infinities, whether one means that logically, mathematically, or metaphysically.

Are you here referring to Craig’s example of all the even-numbered-room guests exiting the hotel versus all the guests in rooms numbered greater than 3? If so, what is logically absurd about this scenario? There are no contradictions or inconsistencies implied.

I don’t believe that I’ve ever claimed that mathematical legitimacy implies metaphysical impossibility. I’m simply saying that I see no reason to think that actual infinities are metaphysically impossible, as has been claimed by Mr. Giunta and Dr. Craig and numerous others. The appeal to mathematical legitimacy simply shows that there do not seem to be any inconsistencies in the concept of actual infinity, itself. If one wants to show that actual infinities are metaphysically impossible, then the burden is upon that person to show what contradictions arise from that concept given a broader set of implications.

1. Can you define for me broad logical possibility? I don’t think you know what it means. Thanks.

2. I don’t know what your ‘previous statement’ is referring to. I don’t care what Belief Map is saying. What matters is how the argument is presented by its progenitors.

3. Your reasons against the possibility of prohibiting something meaningless are meaningless to me (lol). I AGREE that I’m not prohibited from SAYING something meaningless. Who disputed that? What’s prohibited is that whatever it is I am saying isn’t going to be magically meaningful just because I said it, right? I have no idea what import you think this artificial prohibition has on our discussion. I’m not talking about artificially prohibiting the utterance of something meaningful (like some Orwellian nightmare); I’m talking about its semantic prohibition because its various utterances (which you can SAY, but not MEAN) semantically cut its own throat. So, I don’t follow you’re reasoning here at all. You keep understanding prohibition in terms some barbed-wire fence that encloses some jewel that Indiana Jones is trying to get to. That’s not what I’m talking about. Something can be semantically prohibited because it’s meaningless. It’s prohibited for me to be multiply located (barring weird time travel scenarios). If I claim that I want to be multiply located, I’d expect you to say that it’s prohibited because it’s not logically possible in the broad sense. If I object that it’s senseless to prohibit me from doing something that’s already impossible, you might respond that I’m being unreasonable because the sense in which I’m prohibited isn’t in some subjunctive sense according to which my wish will be fulfilled if the laws of nature were tweaked with or if matter was arranged differently. That’s not the domain for which the prohibition is designed or concerned. Sorry. This just seems obvious to me.

4. Who argued that because X isn’t allowed, X doesn’t exist? I sure haven’t. It’s a far cry from arguing for the conclusion that a certain operation is meaningless (and therefore prohibited) to using this conclusion in a premise in another philosophical argument leading to the conclusion that actual infinities are broadly logically impossible.

5. I have no idea what you’re talking about regarding your objection to the idea that subtraction could be conventionally prohibited in certain domains. Craig wouldn’t deny that subtraction is meaningless in ALL domains. What mathematician would claim that? My claim that it’s meaningless is obviously pertaining to one domain: transfinite arithmetic. And yes. There is such a ‘thing’ as “performing subtraction” in transfinite arithmetic, but it’s on the same level as above: when the metaphysician talks about possible worlds not knowing that the logical positivist has rendered all talk of possible worlds meaningless because of their verificationist criterion of meaning. The metaphysician can still SAY things about possible worlds, and in that sense, it’s not prohibited (we don’t live in an Orwellian dystopia where metaphysicians suffer punitive damages for uttering meaningless propositions). That’s what I’m talking about regarding subtraction in transfinite arithmetic. I can go ahead and DO it (1. All the natural numbers minus all the numbers greater than 2 is 2. 2. All the natural numbers minus all the even numbers is infinity: all the odd numbers). Cool! I DID it. But that doesn’t mean I’ve done anything meaningful, and for that reason, even though I could DO it (I could “PERFORM” subtraction), I’m DOING something that’s PROHIBITED. I don’t know how much clearer I can make this.

6. And just so we’re on the same page, I need you to define logical, mathematical, and metaphysical possibility. I need to know you know the meanings of these terms. Until you do this I can’t even begin to understand your question about whether it’s LOGICALLY absurd to believe what follows from Hilbert’s Hotel, because I don’t know what you’re talking about regarding ‘logically’. Of course, I’m going to claim it’s not logically absurd IN A SPECIAL SENSE.

7. Who has argued that mathematical legitimacy IMPLIES metaphysical impossibility? What’s typically argued is that mathematical legitimacy implies metaphysical POSSIBILITY. These are two different claims. As things are, I see no reason to think that actual infinities are NOT metaphysically impossible, and you’ve given me no reason to think otherwise. And who cares what mathematical legitimacy shows or doesn’t show. It’s a moot point. Craig already believes in its mathematical legitimacy. So, don’t you think Craig would know whether or not what he’s going to write after he makes that point is going to be empty gas? Of course, he does. He’s talking about a completely different modal category. He’s not saying that Hilbert-Hotel type scenarios lead to logical contradictions (NARROW logical impossibility); he’s saying the scenarios lead to absurdities, and thinking such is served by the intuition pumps provided, pumped by the scenarios. The burden is on the critic to demonstrate that there’s some relevant modal bridge that wouldn’t inhibit the seamless transition from strict logical (narrow) possibility to broad logical possibility with no bumps or hiccups. Plantinga’s ideas in The Nature of Necessity have established the bumps and hiccups substantially. The burden is on you to show that the bumps are chimeras.