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Thread: Problems and Questions in Atheism

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    tWebber Adrift's Avatar
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    Quote Originally Posted by Boxing Pythagoras View Post
    For example, when Craig says, "[Actual] infinity is symbolized by the Hebrew letter aleph (ℵ)," he is wrong. The symbol ℵ in set theory refers to the cardinality of infinite sets. It is not a symbol for the concept of infinity, actual or otherwise.
    Doesn't the fuller quote give context to exactly that,

    By contrast with that, the actual infinite is an infinite which is, as it were, complete. The number of items in the collection is not growing toward infinity; it is infinite! It is complete and static and involves an actually infinite number of things.

    This type of infinity is symbolized by the Hebrew letter aleph (ℵ) and is used in set theory. In set theory, mathematicians talk about sets like the set of natural numbers which have an actually infinite number of members in the set. The collection is not growing toward infinity as a limit. It is infinity. There are an actually infinite number of natural numbers in this set.



    Quote Originally Posted by Boxing Pythagoras View Post
    For another example, look to how Craig's description of Hilbert's Hotel from that same talk consistently treats infinity as if it is a number, despite the fact that Craig recognizes earlier in the lecture that infinity is not a number.
    What are you referring to exactly? I don't see anywhere in which Craig uses Hilbert's Hotel differently from Hilbert. And they both come to the same conclusion. As Hilbert states, "The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking – a remarkable harmony between being and thinking."


    Quote Originally Posted by Boxing Pythagoras View Post
    Or how, after discussing Hilbert's Hotel, he says, "Someone might say that you can’t do inverse operations with mathematical quantities. Not on paper perhaps, but there is no way you can stop people from checking out of a real hotel." Now, obviously, he meant to say "infinite quantities," here, not "mathematical quantities;" but even so, he's incorrect. Subtraction and division, for example, are not defined in Transfinite Arithmetic, as Craig notes earlier in the talk; but Transfinite Arithmetic doesn't add, multiply, or exponentiate cardinal numbers. Since cardinal numbers are the focus of Craig's Hilbert Hotel discussion, Transfinite Arithmetic is not really relevant. There certainly ARE number systems which include infinite numbers, are logically consistent, and define subtraction and division on infinite numbers-- for example, the Hyperreals or the Surreals. I gave examples of how the Hyperreals quite easily resolve the absurdities which Craig alleges result from Hilbert's Grand hotel.
    Craig is not the one who alleges that absurdities result from Hilbert's Hotel. That is, he did not come up with this idea that Hilbert's Hotel results in absurdities. You're acting like he came up with all of this all by himself, and is making a fool of himself for having discovered these absurdities, but as I've already pointed out, he's simply mimicking others who've come to similar conclusions. In the end, I can't help but feel that your issue with Craig is less to do with his misuse of math, and more that you disagree with those mathematicians that he relies on. There are mathematicians, good mathematicians, that do not believe that actual infinities can exist in the real world. I imagine even they would not accept that the use of hyperreals could resolve absurdities with infinities in real life.

    I'm no mathematician though, and I know enough to know when I'm out of my depth. So I suppose there's something really obvious here that I'm missing. Have you emailed Dr. Craig your solution to Hilbert's Hotel using hyperreals? He answers emails all the time on both his website and podcast. I couldn't find anyplace where Craig discusses hyperreals to solve Hilbert's Hotel Paradox, but I did find this interesting discussion between the skeptic mathematician Jeffrey Shallit, and a poster named Wade. Wade makes a number of very interesting points that you might want to look over.

    Quote Originally Posted by Boxing Pythagoras View Post
    I am certainly disputing that Hilbert's Hotel is a paradox. There's nothing self-contradictory about a hotel with an infinite number of rooms.
    I realize you are disputing that Hilbert's Hotel is a paradox. Do you dispute that other mathematicians refer to it as a paradox? It appears as though even The Concise Oxford Dictionary of Mathematics (2009) considers it a paradox.

  2. #262
    tWebber Roy's Avatar
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    Quote Originally Posted by 37818 View Post

    Hilbert's paradox actually explains paradox quite well I think.
    Thanks - have been sampling the others too. They're all good.
    JohnMartin: "My assertions are fact. They are so fact that even when you deny them, you assert them by implication as shown above.
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  3. #263
    tWebber Boxing Pythagoras's Avatar
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    Quote Originally Posted by Adrift View Post
    Doesn't the fuller quote give context to exactly that,

    By contrast with that, the actual infinite is an infinite which is, as it were, complete. The number of items in the collection is not growing toward infinity; it is infinite! It is complete and static and involves an actually infinite number of things.

    This type of infinity is symbolized by the Hebrew letter aleph (ℵ) and is used in set theory. In set theory, mathematicians talk about sets like the set of natural numbers which have an actually infinite number of members in the set. The collection is not growing toward infinity as a limit. It is infinity. There are an actually infinite number of natural numbers in this set.
    The fuller quote doesn't correct the problem to which I pointed. The symbol ℵ does not represent a "type of infinity," as Craig claims. Rather, it represents the cardinality of infinite sets. The cardinality of an infinite set isn't a "type of infinity." The cardinality of an infinite set is a description of an algorithm which one might use in listing the elements of that set.

    What are you referring to exactly? I don't see anywhere in which Craig uses Hilbert's Hotel differently from Hilbert. And they both come to the same conclusion. As Hilbert states, "The overall result is then: The infinite is nowhere realized. Neither is it present in nature nor is it admissible as a foundation of our rational thinking – a remarkable harmony between being and thinking."
    I'm referring particularly to when Craig says this:

    Suppose all the people in the odd-numbered rooms check out – 1, 3, 5, 7, and so forth. How many guests are left? Well, all the even-numbered guests. An infinite number of guests are still left in the hotel even though an equal number has already checked out and left the hotel. But now let’s suppose instead that all of the guests in the rooms 3, 4, 5, 6, 7, out to infinity checked out. How many guests are left now? If there is a room #0, just three are left. Yet, the same number of guests checked out this time as when all of the odd-numbered guests left. You subtract identical quantities from identical quantities and you get non-identical results, which is absurd.


    I have highlighted the particularly offending portion. It is not the case that the same number of guests checked out in each of those cases. Each case describes an infinite number of guests checking out, to be sure, but not the same infinite number. This is not a case of subtracting identical quantities from identical quantities to receive non-identical results, as Craig alleges.

    Craig is not the one who alleges that absurdities result from Hilbert's Hotel. That is, he did not come up with this idea that Hilbert's Hotel results in absurdities. You're acting like he came up with all of this all by himself, and is making a fool of himself for having discovered these absurdities, but as I've already pointed out, he's simply mimicking others who've come to similar conclusions.
    Whether Craig came up with them himself or is simply repeating something he's heard others claim is irrelevant. The fact of the matter is that he presents a misunderstanding of the mathematics in order to support his claim that actual infinites do not exist.

    In the end, I can't help but feel that your issue with Craig is less to do with his misuse of math, and more that you disagree with those mathematicians that he relies on.
    Nope, it's due to his misuse of math. If Craig were making legitimate mathematical arguments, I might disagree with him, but I wouldn't accuse him of being ignorant of mathematics. Our disagreement, in that case, would likely be a philosophical one, and I would cite those reasons for my disagreement. I have done exactly that, in fact, when I've discussed Dr. Norman Wildberger's view on the subject. Dr. Wildberger is a mathematics professor at the University of New South Wales. He understands the mathematics under discussion, but objects to some of the axioms underlying that mathematics.

    However, Craig doesn't understand the mathematics. He makes claims about the mathematics which are incorrect in order to support his claims about actual infinites.

    There are mathematicians, good mathematicians, that do not believe that actual infinities can exist in the real world. I imagine even they would not accept that the use of hyperreals could resolve absurdities with infinities in real life.
    Those mathematicians would argue that the axioms underlying the Hyperreals do not apply to the real world. They would not argue that Hilbert's Hotel on the Hyperreal number system results in mathematical absurdities.

    I'm no mathematician though, and I know enough to know when I'm out of my depth. So I suppose there's something really obvious here that I'm missing. Have you emailed Dr. Craig your solution to Hilbert's Hotel using hyperreals?
    I have not. Even if Dr. Craig were to answer such a question, it would likely be a one-off reply, on his part. Given the complexity of the matter, and given Dr. Craig's numerous misunderstandings of the mathematics with which he is somewhat aware, I have no confidence that such a one-sided correspondence would be at all useful, on my part.

    I couldn't find anyplace where Craig discusses hyperreals to solve Hilbert's Hotel Paradox, but I did find this interesting discussion between the skeptic mathematician Jeffrey Shallit, and a poster named Wade. Wade makes a number of very interesting points that you might want to look over.
    Wade doesn't seem to have any understanding of the Hyperreals, either. When the user Gareth McCaughan mentions them, Wade misunderstands and thinks the gentleman is referring to Transfinite arithmetic.

    Beyond that, the discussion definitely illustrates some of the problems I have with Craig's discussion of Hilbert's Hotel. Wade mistakes the fact that subtraction is not defined on Transfinite arithmetic for implying that it is impossible to perform subtraction on infinite numbers. Then, after noting that subtraction is not defined, he tries to draw conclusions about that subtraction, which is preposterous. That'd be like bringing up an Onomblattive in a conversation, noting that you have no definition for what an "Onomblattive" actually is, then proceeding to claim that the Onomblattive therefore proves your point.

    I realize you are disputing that Hilbert's Hotel is a paradox. Do you dispute that other mathematicians refer to it as a paradox? It appears as though even The Concise Oxford Dictionary of Mathematics (2009) considers it a paradox.
    I certainly don't! People often use the phrase "paradox" colloquially to refer to something which is counter-intuitive or which can easily be misunderstood as self-contradictory. For example, the Twin Paradox of Special Relativity isn't actually a paradox, but because a simple misunderstanding can make it appear to be paradoxical, the name has stuck despite the problem having been resolved. Similarly, even some people who don't actually consider Hilbert's Hotel to be self-contradictory have referred to it as a "paradox," in this colloquial sense.

    If there are mathematicians who do still consider the thought experiment to be an actual paradox, I will be happy to disagree with them.
    "[Mathematics] is the revealer of every hidden truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
    --Thomas Bradwardine, De Continuo (c. 1325)

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    tWebber
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    Quote Originally Posted by Boxing Pythagoras View Post
    Hilbert's hotel isn't actually a paradox. Nor is it an argument to show that actual infinites cannot exist.

    Dr. Craig's problem, when discussing infinity, is that he constantly makes the mistake of trying to use infinity as a number. Infinity is not a number, and as such, you cannot perform mathematical operations on infinity. There are numbers which are infinite, however, and one can perform mathematical operations on these numbers. Unfortunately for Dr. Craig, these numbers can resolve the alleged problems which he cites for infinity fairly easily.

    For example, insofar as Hilbert's Hotel is concerned, let's say that the Hotel in question has N rooms, where N is an infinite Hyperreal such that N=(1,2,3,4,5,...). This defines a situation in which the Hotel has the same number of rooms as there are Natural numbers, and let's say that all of these rooms are occupied by a single patron. That means there are N patrons staying in the Hotel. Now, let's say 3 people check out of the Hotel. Sure enough, the number of patrons remaining in the Hotel is still infinite, but it is not the same number of patrons as it had originally. Now, there are N-3 patrons, where N-3=(-2,-1,0,1,2,...). It is not true that N=N-3.

    Now let's say that instead of 3 patrons, all of the patrons in the even numbered rooms check out. Sure enough, this means that an infinite number of patrons are checking out of the Hotel-- but again, it is not the same infinite number as we had originally. In this case, we see that N/2=(0.5,1.0,1.5,2.0,2.5,...) patrons have left, and again, it is not true that N=N/2.

    Dr. Craig doesn't have a very good understanding of the mathematics which he attempts to describe.
    See, I'm not a math guy and can't go into this kind of detail. My argument against Craig is philosophical.

    The series of numbers is an accidentally ordered series, that is, a series which does not depend on the continued existence of any of the prior elements. The existence of 3 is not dependent on the continued existence of 2. A classical example is fathers and sons: a father has to bring a son into existence, but after the son's conception, the father does not have to continue to exist. Craig has to demonstrate that there is a logical incomprehensibility of having an infinite series, which he can't.

    What Craig would have to show is that the past-finitude of the universe is an essentially ordered series, one where there cannot logically fail to have been some sort of first cause. For him to do that, he has to assume A-theory of time, which, while not necessarily in conflict with modern physics, isn't as self-evident as Craig seems to believe.

  5. #265
    tWebber 37818's Avatar
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    What is in evidence that our known universe has an infinite past?
    . . . the Gospel of Christ, for it is [the] power of God to salvation to every [one] believing, . . . -- Romans 1:16.

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  6. #266
    tWebber Boxing Pythagoras's Avatar
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    Quote Originally Posted by 37818 View Post
    What is in evidence that our known universe has an infinite past?
    Other than mathematical models? Nothing.

    Incidentally, that's exactly the same amount of evidence as we have for our known universe having a finite past.
    "[Mathematics] is the revealer of every hidden truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
    --Thomas Bradwardine, De Continuo (c. 1325)

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    tWebber 37818's Avatar
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    Quote Originally Posted by Boxing Pythagoras View Post
    Other than mathematical models? Nothing.

    Incidentally, that's exactly the same amount of evidence as we have for our known universe having a finite past.
    Actually that is not true. The prevailing view is that our known universe began 13.8 billion years ago is that of a finite past based on the evidence.
    . . . the Gospel of Christ, for it is [the] power of God to salvation to every [one] believing, . . . -- Romans 1:16.

    . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3, 4.

    Whosoever believeth that Jesus is the Christ is born of God: . . . -- 1 John 5:1.

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    tWebber Boxing Pythagoras's Avatar
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    Quote Originally Posted by 37818 View Post
    Actually that is not true. The prevailing view is that our known universe began 13.8 billion years ago is that of a finite past based on the evidence.
    That prevailing view is based upon mathematical models. The oldest physical data available to us are from hundreds of thousands of years after the Big Bang. While this is incredibly close to the event, on cosmological scales, the simple fact of the matter is that we have no physical data earlier than this.

    All cosmological claims about times prior to the CMB are simply based upon mathematical models. There exist models in which the universe is past finite, and others in which the universe is past infinite.
    "[Mathematics] is the revealer of every hidden truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
    --Thomas Bradwardine, De Continuo (c. 1325)

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    tWebber 37818's Avatar
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    Quote Originally Posted by Boxing Pythagoras View Post
    That prevailing view is based upon mathematical models. The oldest physical data available to us are from hundreds of thousands of years after the Big Bang. While this is incredibly close to the event, on cosmological scales, the simple fact of the matter is that we have no physical data earlier than this.

    All cosmological claims about times prior to the CMB are simply based upon mathematical models. There exist models in which the universe is past finite, and others in which the universe is past infinite.
    Show the math for a red shift and an infinite past. It does not exist.
    . . . the Gospel of Christ, for it is [the] power of God to salvation to every [one] believing, . . . -- Romans 1:16.

    . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3, 4.

    Whosoever believeth that Jesus is the Christ is born of God: . . . -- 1 John 5:1.

  10. #270
    tWebber Boxing Pythagoras's Avatar
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    Quote Originally Posted by 37818 View Post
    Show the math for a red shift and an infinite past. It does not exist.
    Are you claiming that mathematical models of a universe with a past-infinite time dimension cannot account for red shift in the CMB? Or perhaps red shift in the motion of galaxies?
    "[Mathematics] is the revealer of every hidden truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
    --Thomas Bradwardine, De Continuo (c. 1325)

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