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#770 Physicists & Philosophers Reply to the Kalam Cosmological Argument (Pt I)

February 13, 2022
Q

Will Dr Craig at any time respond to this https://www.youtube.com/watch?v=pGKe6YzHiME? There is so much in there that is way above my understanding, although a lot of it 'feels' just obfuscation and unprovable.

Paul

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Photo of Dr. Craig.

Dr. craig’s response


A

This video features an impressive array of philosophers and physicists that is at first blush intimidating. But has the kalam cosmological argument been accurately explained to them and are their criticisms on target? In order to see, I’ve asked John Mazzitelli to transcribe the video so that I might more easily respond to it. My brief responses are in blue italics to set them apart from the text. Here we feature Part I of my response. I have inserted subheadings to organize the material.

INTRODUCTION

NARRATOR: Creation stories of Pacific Islanders tell us the world was made from a volcano. Other origin myths say the world came from water. But contemporary science tells us that the world evolved from a big bang. And the outdated view that this marks an absolute beginning has been used by religious philosophers, most notably William Lane Craig, to revive an old argument for God known as the kalam cosmological argument. In this film we'll hear why this argument fails from the very cosmologists quoted by kalam supporters. And we'll also hear from leading philosophers of mathematics who will show why the concept of an infinite universe now being proposed by many physicists is perfectly coherent. And lastly we'll see that if the universe did begin it may not need a cause. The argument is often expressed like this. Whatever begins to exist has a cause. The universe began to exist. Therefore the universe has a cause. And that cause is taken to be God.

PREMISS 2: THE UNIVERSE BEGAN TO EXIST

Let's start at the beginning. Philosophers like Craig have argued that the past must be finite.

Response: I await with bated breath!

A. FIRST PHILOSOPHICAL ARGUMENT

WILLIAM LANE CRAIG: If the universe never began to exist that means that the number of past events in the history of the universe is infinite. But mathematicians recognize that the existence of an actually infinite number of things leads to self-contradictions.

Response: My studied position is that the existence of an actually infinite number of things is metaphysically impossible, even if logically consistent. Some metaphysical impossibilities are also strictly logically impossible, but not all. For a demonstration of the logical contradictions that would ultimately result from the existence of an actually infinite past see Alexander Pruss, Infinity, Causation, and Paradox (Oxford: Oxford University Press, 2018).

NARRATOR: As we shall see, contemporary mathematicians do not think infinity is contradictory although it's true that philosophers of the past were troubled by it. But even they did not banish the concept. For example, Aristotle distinguished between different types of infinity.

1:57 ADRIAN MOORE: The distinction that he was drawing was between an infinity that's present all at once – all at some particular point in time – which is what he meant by an actual infinity contrasted with an infinity that is spread out over time which is what he called a potential infinity. So, for example, if space were infinitely big that would be an example of an actual infinity because the whole of space is there at any given point in time. On the other hand, if you imagine a clock endlessly ticking, the ticks might go on forever. But if they did that would be an example of a potential infinity.

Response: Agreed! I follow Aristotle in thinking that only potential infinites can exist in reality, in contrast to actual infinites, which are merely conceptual.

NARRATOR: But in the 19th century, Georg Cantor revolutionized the mathematics of the infinite.

2:48 DANIEL ISAACSON: What Cantor achieved was to treat infinity as a subject of mathematics itself.

ADRIAN MOORE: It was a whole new branch of mathematics, and it was of breathtaking ingenuity, showed incredible craftsmanship and creativity on Cantor’s part.

Response: Right, Cantor founded infinite set theory.

NARRATOR: What Cantor showed was that an infinite set has a strange feature. We might call this the infinite property. That is, it can be put into a one-to-one correspondence with a subset of itself.

Response: This is rather his definition of the actual infinite.

ADRIAN MOORE: We can pair the even numbers up with all the counting numbers all together. So two gets matched with one, four gets matched with two, six gets matched with three, eight gets matched with four, etc., etc.

3:40 ALEX MALPASS: So for instance you can show that there are just as many even numbers as there are counting numbers when intuition should tell you that there should be half as many.

ADRIAN MOORE: Well, OK, so this just is counterintuitive, and it's counterintuitive because we're used to dealing with finite sets. What we have to say is the case of finite sets just doesn't carry over to the case of infinite sets. But it doesn't follow that there's anything incoherent about what we say in the infinite case. It just follows that we have to start saying different things in the infinite case.

Response: One is not challenging the internal consistency of infinite set theory, given its definitions and axioms. The question rather is whether an actually infinite set of things can be instantiated in reality. Could there be, for example, an actually infinite number of baseballs?

DANIEL ISAACSON: What you get from most mathematicians, I would say almost all mathematicians, is an uncritical acceptance of infinity in the sense of actual infinity. That's what they need for doing the kind of mathematics that they do.

Response: Sure; there are a small number of intuitionists who deny even the mathematical legitimacy of the actual infinite, but that is not the issue before us.

ADRIAN MOORE: So one of the first things that mathematicians come to appreciate is that infinite collections do have different properties from finite collections. Things that we absolutely take for granted in the finite case simply don't carry over to the infinite case. And if it's the finite case that is providing us with our basic intuitions then that means that some of the results about infinite sets are going to be counterintuitive. But that's just what you would expect. Why should the finite and the infinite be the same? As long as we're prepared for these differences, again the very worst that it will be is counterintuitive. It won't be contradictory. It won't be inconsistent.

Response: Again, the claim is not that infinite set theory is self-contradictory or inconsistent. The question is whether the counterintuitive consequences of the real existence of an actually infinite number of things justifies skepticism about the possibility of real actual infinites.

5:04: ALEX MALPASS: One way to bring out the difference between the finite and the infinite is to think about somebody counting. So imagine George is counting to ten and he's got as far as five – how many numbers has he got left to count? He's got five numbers left to count – six, seven, eight, nine, ten. But now imagine George is going to count all of the numbers. Every single counting number. He's got up as far as five. How many numbers has he got left to count? He's got infinitely many numbers still left to count. What about when he gets to ten? How many numbers has he got left to count? He's still got infinitely many numbers left to count. So his task isn't going down over time but when he's counting a finite set his task is going down over time.

Response: So turn the example around. Suppose George has been counting from eternity past. Is it really possible for someone to count all of the negative numbers one at a time ending at 0?

NARRATOR: One classic example that Craig has given to show infinity cannot exist is that of Hilbert's Infinite Hotel.

RF VIDEO: The mathematician David Hilbert illustrates the problem by imagining a hotel with an infinite number of rooms all of which are occupied. There's not a single vacancy. Every room in the infinite hotel is full. Now suppose a new guest shows up and asks for a room. The manager says, “Sure. No problem.” Then moves the guest who was in room number one to room number two, and the guest who was in room number two to room number three, and so on to infinity. As a result of this shuffling, room number one becomes vacant and the new guest happily checks in even though all the rooms were already full and nobody has checked out.

NARRATOR: It seems Craig is claiming infinity is problematic because the Hilbert Hotel is both full and able to admit new guests. But the problem may simply be the way “full” is being defined.

ALEX MALPASS: If what we mean by “full” is every room is occupied then that's true but it doesn't prohibit us being able to accommodate new guests because we can shuffle them all up in an infinite hotel. If what we mean by full is “can't accommodate new guests” then it's just false that it's full.

Response: Right; so do you think that it is really possible that an actual hotel could be full in the first sense and yet accommodate infinitely more guests just by moving people around? There is no logical contradiction involved in such a scenario, but is it really possible metaphysically?

NARRATOR: In Craig's video, a contradiction is supposedly found by considering what happens when the guests leave the infinite hotel.

RF VIDEO: Suppose all the guests in the odd numbered rooms check out. In that case, an infinite number of people have left the hotel. And yet there are no fewer people in the hotel. But suppose instead all the guests in rooms numbered four and above check out. In that case only three people are left. And yet exactly the same number of people left the hotel this time as when all the odd-numbered guests checked out. Thus we have a contradiction. We subtract identical quantities from identical quantities and get different answers.

DANIEL ISAACSON: The trouble . . . that video is so below threshold. The complaint in that clip – in particular the one that says that if there are only three people left then they're still the same number of people that have checked out as were there originally – that is a feature of infinity that has to be taken on board very importantly which is that any finite initial segment of an infinite set if it's linearly ordered is such that what's left is infinite. So no matter how far you go in fact you're still leaving infinitely many elements ahead of you.

Response: This is a solution? It simply restates in a confused way the counterintuitive consequences of the real existence of an actually infinite number of things. You can take away infinity from infinity and get any result, from 0 to infinity.

ALEX MALPASS: There's a formal theory of arithmetic which is characterized by the Peano axioms and that's a language where you can express any sentence of arithmetic – like 7 plus 3 equals 10 or something. But when it comes to this Cantorian transfinite arithmetic there's just no notion of subtraction. It's not well-defined to say infinity minus 3. That's not a sentence of either the Peano arithmetic or the Cantorian arithmetic. So while someone might say it informally, there's no formal language where that really makes any sense.

Response: Right; that’s my point. Inverse operations like subtraction and division are prohibited mathematically for transfinite (infinite) numbers; but you can’t stop people from checking out of a hotel. Bar the door, and they’ll jump out the windows!

NARRATOR: One of the strange things that Cantor showed was there are infinities of different size. The first infinity is known as aleph naught. The next largest is aleph one and so on.

ADRIAN MOORE: So there's this infinite number which is called aleph 7 and there's also an infinite number which is called aleph 5. We can add those. If we add aleph 7 to aleph 5 you might think that what we're going to get is aleph 12. But it doesn't work like that. The infinite case is different from the finite case. In fact what happens if you add two infinite numbers together is that the bigger one swallows up the littler one. The smaller one is insignificant compared with the bigger one. And aleph 7 plus aleph 5 is just aleph 7. Aleph 7 just takes over and it's as if the aleph 5 wasn't even there. So aleph 7 plus aleph 5 is aleph 7. And similarly aleph 7 plus aleph 3 is aleph 7. If you add these smaller infinite numbers to aleph 7 you just get aleph 7. Now suppose we tried to perform a subtraction. Suppose we start with aleph 7 and ask what is it that you need to add to aleph 7 to get aleph 7? Now, unfortunately there's no clear answer to this because we saw that if we added aleph 3 that would just give us aleph 7. If we added aleph 5 that would give us aleph 7. There's no longer any such thing as the number that we need to add to aleph 7 to get aleph 7. And this is an example of how subtraction just isn't well defined in the infinite case.

Response: The higher alephs only reinforce the suspicion that we are dealing here with a purely conceptual realm, not something that can exist in reality. Indeed, the real existence of a plurality of things numbered by some of these higher alephs may well be impossible because there would not be enough space-time points to accommodate them all! The kalam cosmological argument, however, concerns itself only with the least transfinite number, aleph naught.

11:08 ARIF AHMED: You could say the same about 0. That, you know, 0 divided by 0 isn't defined. Anything divided by 0 isn't defined in normal arithmetic. But that doesn't mean division is meaningless or that zero is contradictory.

Response: That’s right mathematically, but questions remain about the real existence of zero number of things. If I say that there are zero elephants on the quad, does that mean that there are elephants on the quad and their number is zero? Do I not rather mean that there are just no elephants on the quad?

NARRATOR: As we saw earlier, when presenting to a lay audience Craig has claimed that the infinite is contradictory.

WILLIAM LANE CRAIG: But mathematicians recognize that the existence of an actually infinite number of things leads to self-contradictions.

Response: Yes, that is a short-cut to communicate these extraordinarily difficult concepts to laymen. Some, but not all, metaphysically impossible situations involve logical contradictions.

NARRATOR: But when talking to philosophers, a different claim is made.

WILLIAM LANE CRAIG: Now Alex is certainly right that when we appeal to these absurdities we are not talking about logical contradictions or incoherences. Jose Benardete in his book on infinity says that there's no logical contradiction involved in these monstrosities but you have only to look at them in their concrete reality to see that this is metaphysically impossible.

Response: Right; illustrations like Hilbert’s Hotel and Benardete's book may not be logically contradictory but they are metaphysically impossible. Others like Pruss’s Grim Reaper paradox do involve logical contradictions.

ALEX MALPASS: In contrast to other notions of possibility, metaphysical possibility is far less clear what the definition of that is supposed to be. And many of us are dubious that just simply pointing that something seems absurd is enough to forbid it from existing in reality. Philosophers have to be a lot bolder than that. And a lot more open-minded.

Response: Fair enough; the boundaries of the metaphysically impossible are unclear, since they may not involve logical inconsistencies. For example, is it metaphysically possible that gold might have had a different atomic number than 79, or that your desk could have been made of ice? Intuitions may differ, but most philosophers would say that these logically consistent scenarios are metaphysically impossible. Similarly, given the counterintuitive consequences of the real existence of an actually infinite number of things, skepticism about their metaphysical possibility is surely justified.

DANIEL ISAACSON: A hotel case is so basic that that's no place for anybody to dig in their heels against infinity. Even if you only accept potential infinity, the Hilbert Hotel and those results about it are completely incontrovertible.

Response: A strange response! He must mean merely that Hilbert’s Hotel is a good illustration of the existence of an actually infinite number of things. Of course, we should expect that David Hilbert, perhaps the greatest mathematician of the 20th century, knew how to illustrate accurately the actual infinite. His hotel is not merely potentially infinite.

ALEX MALPASS: The examples that are supposed to be problematic always involve admitting new guests, shuffling guests from one room to the other. And if we imagine a hotel where the doors were sealed and nobody could move from one room to the other it's hard to think of a similar example that could bring out anything that looked absurd about it. It would just be a hotel with infinitely many rooms in. And if that's right, it does lead you to wonder whether the problem is the infinity involved or whether it's the manipulation of those infinite elements that's the problem. And that's helpful because, I mean, it's plausible to suppose that what's done is done and you can't change that. The past is fixed and unchangeable. So that if the past is supposed to be like an infinite hotel then it's more like one where the guests can't shuffle around than one where they're free to move to different rooms. I mean, it's just impossible that yesterday didn't happen given that it has happened.

Response: This reply is bizarre. We can use any concrete reality to illustrate the existence of an actually infinite number of things, e.g., baseballs, coins, stars, etc. There is nothing about the absurdity involved in Hilbert’s Hotel that hinges upon the illustration’s involving a hotel with rooms and doors!

 

That’s it concerning the philosophical argument against the infinitude of the past based on the impossibility of the existence of an actually infinite number of things! Next week we’ll look at criticisms of the second philosophical argument against the infinitude of the past based on the impossibility of forming actually infinite number of things by successive addition.

- William Lane Craig