tag:blogger.com,1999:blog-20067416.post1533697350996370382..comments2021-05-04T18:49:35.162-04:00Comments on Recursivity: Reply to William Lane CraigUnknownnoreply@blogger.comBlogger47125tag:blogger.com,1999:blog-20067416.post-56012676142327190532019-04-16T15:59:06.714-04:002019-04-16T15:59:06.714-04:00No idea, sorry.No idea, sorry.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-49468866814107557542019-04-14T03:50:30.007-04:002019-04-14T03:50:30.007-04:00Would you happen to know where I could find a vide...Would you happen to know where I could find a video (or audio) of the debate between you and Kirk Durston? The available links are dead or lead to deleted videos.Quinnhttps://www.blogger.com/profile/05115279594632394240noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-69050753837179561452016-02-21T16:36:18.580-05:002016-02-21T16:36:18.580-05:00I completely agree that William Lane Craig has a c...I completely agree that William Lane Craig has a childish understanding of infinity.<br />This is not surprising, as he also has a childish and medieval level understanding of intelligence and consciousness.<br />Step 4 and 5 of his beloved Kalam are completely bogus. They are based on a <b>category error fallacy</b>, resulting from a primitive and nonsensical understanding of <b>intelligence</b> and <b>consciousness</b>.<br />They presupposes the answer by pretending that intelligence is some supernatural <b>thing</b>, in order to prove that you can assign a supernatural, intelligent first cause to the universe.<br />This is a category error fallacy, as intelligence and consciousness are *not* "things" that can exist on their own. They are <b>properties</b> / <b>emergent functions</b> of complex systems, like brains.<br />Intelligence / consciousness can not exist without a system that produces them any more than "wetness" can exist without liquids.<br />Craig/Kalam is trying to pretend that he knows the answer to why the rock in front my house is wet by, claiming an "eternal, immaterial wetness" as the cause, without ever discussing the subject of liquids.<br />It's complete nonsense and displays a childish understanding of reality.<br />Decisions that intelligent systems make, are <b>processes</b> and processes are fundamentally temporal.<br />Functioning itself means a system <b>transitioning from state to state</b>, in an ordered manner, <b>over time</b>, meaning it is a <b>temporal phenomenon</b>.<br />So, you can not have an eternal and timeless intelligent being that makes decisions, like creating universes, because the very property of intelligence and decision making are <b>inherently temporal</b>.<br />Timeless and conscious/intelligent/decision making are contradictory concepts.<br />And the free will part is complete bullshit. Free will is nothing, but the ability of an intelligent system to make decisions, and again, decisions are temporal processes, based on the functioning of the system that produces that intelligence.<br />So, "decisions based on free will" can <b>not</b> be produced by a non-temporal system and they would be simply the result of lower-level non-conscious processes, so they ultimately reduce to the same problem as a non-intelligent eternal cause, which Kalam rejects.<br />Kalam step 4 and 5 essentially try to artificially divorce intelligence from the system of which is an emergent function, in order to try to evade the inevitable problems of temporality , which is like trying to divorce the concept of "wetness" from the substance that causes wetness (liquids).<br />It shows an infantile-level understanding of reality, intelligence, consciousness, systems and processes.Anonymoushttps://www.blogger.com/profile/10142601278420374096noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-7492141171072034732009-02-17T06:06:00.000-05:002009-02-17T06:06:00.000-05:00Newton da Costa and I & coworkers have proved ...Newton da Costa and I & coworkers have proved a theorem that asserts:<BR/><BR/>Consider the set of spacetimes. Then the subset of spacetimes that are exotic and have no `cosmic time' property is topologically and measure-theoretically generic. <BR/><BR/>Spacetime: a 4-manifold, real, smooth, endowed with a smooth 1-foliation (the ``arrow of time'')<BR/><BR/>Exotic: pick up any good recent book on the topology of 4-manifolds and learn from it. <BR/><BR/>Cosmic time property: the property that allows one to say things like, ``the universe began 14 billion years ago.'' Time, on the contrary, can well be a `local' property. Worse: at very very large scales, time may not even exist, and result from a broken symmetry phenomenon at our scale. <BR/><BR/>(I won't even delve on the infinities of set theory...)<BR/><BR/>F. A. DoriaUnknownhttps://www.blogger.com/profile/03537692596767129649noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-90566862882747956472008-07-06T07:49:00.000-04:002008-07-06T07:49:00.000-04:00Kirk: No offense, but you don't know what you're ...Kirk: <BR/><BR/>No offense, but you don't know what you're talking about. You demonstrate this when you say, "I am arguing that, because of the paradoxes, inconsistencies, and contradictions that arise when trying to apply actual infinites to reality, it is irrational to believe they can exist in reality (i.e., a violation of axiom II (i))." <B>But there are no paradoxes (in the mathematical sense) involving the infinities we are talking about</B>. There may be paradoxes involving <B>your</B> intuition about how infinite quantities <I>should</I> behave, but this just indicates your intuition is wrong. People's intuitions have been wrong when confronted with relativity and quantum mechanics, but I don't see you claiming these are impossible. The fact is that <I>mainstream physicists</I> deal with the possibility of infinite quantities existing, as in the example of Malament-Hogarth spacetime. So you can't offhandedly dismiss this as "paradoxical".<BR/><BR/>You're still wrong about the claim that the cardinality of an infinite set is the same as the cardinality of its subsets. I already gave you one example, which you simply dismissed for no reason, so here is another. Consider Z and {0}. {0} is a proper subset of Z, but it does not have the same cardinality as Z.<BR/><BR/>You write, "However, when n is an actual infinite, the cardinality of g(n) = the cardinality of s(n) = Aleph-null. There is no difference in their cardinality when n is infinite even though the equation that gives the difference in their cardinality d(n)=8n/10 shows that the difference in their cardinality is infinite when n is infinite. This is a paradox/contradiction/inconsistency." No, there is no paradox - you are simply giving an example of something well-known to every mathematician: a statement that is true for every finite prefix of an infinite set need not hold for the infinite set as a whole. Consider, for example, the well-known theorem that the union of two regular languages is regular. This is also true for every finite union, but not true for infinite unions. This is not a "paradox/contradiction/inconsistency", but merely a well-understood phenomenon of infinite sets. I think you need to take a refresher course in set theory.<BR/><BR/>I claimed that mathematicians don't talk about "potential infinity", and in response you give me a web page by a person who says in the first line, "I am not a leading researcher on infinite sets". Good job! I'll ask again: go look at any book on set theory, and show me where "potential infinity" is defined. <BR/><BR/>You say, "My own response is to consider y=5/x. There is a very big difference between treating x as a potential infinite 'as x approaches infinity' and as an actual (completed/definite) infinite 'when x is infinite'." Kirk, this depends entirely on what your domain is. Are you working in the reals, or the extended reals? <BR/><BR/>No offense, but your understanding of this topic is woefully deficient.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-84096931935649927162008-07-02T18:26:00.000-04:002008-07-02T18:26:00.000-04:00Both Timmy and Bayesian Bouffant have stated a vie...<I>Both Timmy and Bayesian Bouffant have stated a view about space-time that is corrected by IvanM.</I><BR/><BR/>I have no idea what you are writing about. The way to disprove something <I>reductio ad absurdum</I> is to start with a legal proposition within a system. Then you work within the rules of a system until you wind up at a position contrary to your original position. When you say you go back to the beginning of a universe with no beginning, do you really think that is working within the rules of the system? Your time traversal "proof"fails.<BR/><BR/>You then proceed to slide from your time traversal "proof" to comments about the empirical nature of space-time and a presumption that the Big Bang was the beginning of the universe, which is a separate empirical claim. These cannot be part of your time traversal thought experiment. Also, there are already cosmologists considering what may have preceded the Big Bang.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-71738926037814879522008-07-02T18:13:00.000-04:002008-07-02T18:13:00.000-04:00The past history of the universe is finite. Empiri...<I>The past history of the universe is finite. Empirical evidence supports this, the impossibility of having traversed an infinite set of actual years, one year at a time, supports this</I><BR/><BR/>It's impossible at one year at a time? Well then, traverse it two years at a time! Or five. But seriously, it is only "impossible" when you imagine that the traversal has to be done in a finite time, by a creature with a finite lifespan. I.e. you do not have a mathematical proof, but only an argument from incredulity.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-87284072142009913902008-06-02T15:58:00.000-04:002008-06-02T15:58:00.000-04:00Kirk: First, it is not my notion that 'removing a ...Kirk: <I>First, it is not my notion that 'removing a countably infinite number of members from a countably infinite set must result in a null set."</I><BR/><BR/>Sorry, I was trying to guess at the basis for your claim that "d must be zero when n is infinite". Whatever hidden assumption that claim is based on, it's false.<BR/><BR/><I>However, when n is an actual infinite, the cardinality of g(n) = the cardinality of s(n) = Aleph-null. There is no difference in their cardinality when n is infinite even though the equation that gives the difference in their cardinality d(n)=8n/10 shows that the difference in their cardinality is infinite when n is infinite. This is a paradox/contradiction/inconsistency.</I><BR/><BR/>Your scenario involves a particular mapping from S to G -- there are nine elements in G for every element in S. But you can't determine whether two sets have the same cardinality by looking at a single mapping. Other mappings from S to G are bijective, so S and G have the same cardinality. That fact is not contradicted by anything in your scenario, not even by the fact that d=infinity for your 9-to-1 mapping. Your claim that "the difference in their cardinality is infinite when n is infinite" is simply wrong.<BR/><BR/>On a more fundamental note, I find it strange that you think that actual (physical) infinites can be <I>formally</I> disproven. Whatever axioms you base your proof on, you'll have to show that those axioms model reality accurately, and that's an empirical issue. Good luck.<BR/><BR/>On a final note, if you think that your position (i.e. that actual infinites can be mathematically disproven) is a majority position, I'd be interested in references to modern mathematicians who agree with you.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-9725299500367780862008-05-29T19:43:00.000-04:002008-05-29T19:43:00.000-04:00By now, it's probably just you and I reading this ...<I>By now, it's probably just you and I reading this exchange.</I><BR/><BR/>Naw, I like to drop by once in a while to have a nice choke on my beverage. To practice my spit-takes, y'know.<BR/><BR/><I>Hilbert does not offer a formal proof but I think he does offer an informal proof ("…high-level sketches that would allow an expert to reconstruct a formal proof...") ... If one wants to construct a formal proof, then I would suggest...</I><BR/><BR/>Bzzzzt. Wrong. Thanks for playing. I'm sure it's <A HREF="http://en.wikipedia.org/wiki/Crank_(person)" REL="nofollow">fun to pretend</A> you can <A HREF="http://en.wikipedia.org/wiki/Pseudomathematics" REL="nofollow">argue mathematics</A> with the big boys, but this is ridiculous.<BR/><BR/>(If I seem too scornful, consider the magnitude of your offenses against sound mathematical reasoning. And since Jeffrey is being the good cop, I figure someone has to be the bad cop.)<BR/><BR/>On a more constructive note, if you really want to learn about this stuff you might check out <A HREF="http://www.mathstat.uoguelph.ca/?action=outline&id=21" REL="nofollow">this course</A>. It's really fascinating material, and the first step in understanding something is to admit what you don't know. Wikipedia and historical essays <I>can</I> be good resources, but often require some nuances of understanding to appreciate properly.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-44198368643119472152008-05-28T08:50:00.000-04:002008-05-28T08:50:00.000-04:00I must admit that you all have gone above my head ...I must admit that you all have gone above my head with the physics and math mix here, lol!<BR/><BR/>Anyway, regarding the comments about g(n) - s(n) = null. Unless I am mistaken, you can add to aleph-null and still have aleph-null. Does it not also work with subtraction? Theoretically can't you subtract an infinity from aleph-null and still have aleph-null? <BR/><BR/>Also, as I've stated before, it is my hypothesis that you can think of every "now" as a zero on a number line. There is infinite time behind us and infinite time ahead. My reasoning is this: if time had a finite starting point, then what was going on before that? I, personally, can't think of an answer to that without committing some kind of logical fallacy, assuming I haven't already done that myself. Assuming that if we could not get to this point if time was infinite is the same fallacy, IMHO, as the old greek argument that motion is impossible from A to B because you would have to go half that distance and before that, half that distance, and before that half that distance, ad infinitum. It is obvious that motion is possible since you simply just move along and your frame of measuring has no bearing on your motion. I propose that is the same way with infinite time. It is obvious that time passes. It passes whether we measure it or not. So, going in reverse, with A being negative-aleph-null, why can't you go back infinitely? I can't see a reason myself.Timmyhttps://www.blogger.com/profile/11430023213232445651noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-31124846014611388172008-05-27T12:22:00.000-04:002008-05-27T12:22:00.000-04:00Jeffrey:By now, it's probably just you and I readi...<B>Jeffrey:</B><BR/>By now, it's probably just you and I reading this exchange. However, I appreciate this opportunity to clarify what I am arguing for and not arguing for.<BR/><BR/><B>Summary of what I have been arguing:</B><BR/>1. The past history of the universe is finite. Empirical evidence supports this, the impossibility of having traversed an infinite set of actual years, one year at a time, supports this, and the impossibility of an actual infinite in reality supports this. Our discussion has centered on the latter, specifically can past history be composed of an actual countable infinite number of years.<BR/>2. I do not take a platonic view of numbers and mathematics. I make a very clear distinction between mathematical equations/models and reality. A mathematical equation fails to be relevant to reality if each variable in the equation does not correspond to a physical property or attribute in our actual physical universe. Thus, neither of your examples are relevant to the real world. Your first example requires the arrow of time to move in reverse. Your second example has a variable n that corresponds to no component in the physical universe.<BR/>3. If something is mathematically impossible then, necessarily, it is impossible in the real world, but that is not a bi-conditional (if and only if) statement. Something may be impossible in our particular universe, but still mathematically possible or possible in another universe. An example of this would be the arrow of time. In mathematics, it can assume the reverse direction, not so in our physical universe, although there may be other universes within which the arrow of time runs in reverse.<BR/>4. I am not (and I want to be crystal clear about this), not, arguing that we cannot or do not use the concept of an actual infinite (e.g., the set of natural numbers) as an idea within mathematics. I am arguing that, because of the paradoxes, inconsistencies, and contradictions that arise when trying to apply actual infinites to reality, it is irrational to believe they can exist in reality (i.e., a violation of axiom II (i)). <BR/>5. I am assuming we are not Platonists, where the set of all natural numbers have actual ontological existence. I see axiom II (i) as the defeating axiom for Platonism.<BR/><BR/><B>A need for caution:</B><BR/>In response to my statement that it is a property of an actual infinite set that the cardinality of an actual infinite is the same as its proper subsets, you wrote, "Of course, this is nonsense. The integers are a subset of the reals, but they don't have the same cardinality." <BR/><BR/>Keep in mind here that we are discussing Aleph-null and, therefore, countably infinite sets. The set of real numbers is not a countable infinite set with cardinality Aleph-null. Care must be taken in distinguishing between a countable and uncountable infinite set. Within the context of the discussion of whether the past history of the universe is composed of an infinite number of years or not, we are talking about a countable infinite. See the discussion on <A HREF="http://en.wikipedia.org/wiki/aleph_number" REL="nofollow">Aleph-null</A>. It states,<BR/><BR/><I>Aleph-null (\aleph_0) is by definition the cardinality of the set of all natural numbers, and (assuming, as usual, the axiom of choice) is the smallest of all infinite cardinalities. A set has cardinality \aleph_0 if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or "one-to-one correspondence", with the natural numbers. Such sets include the set of all prime numbers, the set of all integers, the set of all rational numbers, the set of algebraic numbers, and the set of all finite subsets of any countably infinite set.</I><BR/><BR/>As one examines this definition of Aleph-null, and reflects on the requirement that the cardinality of the set of all natural numbers is the same as the cardinality of the set of all prime numbers, one can see why contradictions can arise, as I illustrated using Hilbert's Hotel. This is where Secondclass's objection needs to be addressed.<BR/><BR/><B>Secondclass wrote:</B> " No, the word "indeterminate" is used because it's indeterminate. You resolved the indeterminacy by algebraically subtracting s(n) from g(n), showing that g(n)-s(n) goes to infinity, which is correct. This contradicts your notion that removing a countably infinite number of members from a countably infinite set must result in a null set. But that notion is incorrect, so there is no contradiction."<BR/><BR/>First, it is not my notion that 'removing a countably infinite number of members from a countably infinite set must result in a null set." I showed that the difference in cardinality between g(n) and s(n) increases as n approaches infinity according to d(n)=8n/10. However, when n is an actual infinite, the cardinality of g(n) = the cardinality of s(n) = Aleph-null. There is no difference in their cardinality when n is infinite <I>even though the equation that gives the difference in their cardinality d(n)=8n/10 shows that the difference in their cardinality is infinite when n is infinite.</I> This is a paradox/contradiction/inconsistency.<BR/><BR/><B>Jeffrey wrote:</B> "The distinction between "actual" and "potential" infinity does not exist in mathematics. There is a well-established theory of infinite quantities, and "potential infinite" is not one of them. Infinite quantities do exist in mathematics, and are manipulated all the time; therefore, to claim that Hilbert had a mathematical proof that there is no infinity in nature is simply silly"<BR/><BR/><A HREF="http://www.math.vanderbilt.edu/~schectex/courses/thereals/potential.html" REL="nofollow">Eric Schechter writes</A> in the section titled 'History and Controversy', "Nearly all research-level mathematicians today (I would guess 99.99% of them) take for granted both 'potential' and 'completed' infinity, and most probably do not even know the distinction indicated by those two terms." My own response is to consider y=5/x. There is a very big difference between treating x as a potential infinite 'as x approaches infinity' and as an actual (completed/definite) infinite 'when x is infinite'. We may use the idea of the actual infinite in mathematics all the time, but as I pointed out in (2) and (3), we must be very careful to make a distinction between a useful idea or concept in mathematics and the physical universe. It really does not matter what modern physics has discovered recently (and it certainly has not discoved an example of an actual infinite). Discoveries and advances in modern physics do not trump mathematical axioms. I.e., if axiom II (i) is accepted as true in mathematics (which it must if it is to practice formal proofs), then it must be accepted in the real space-time continuum in which we find ourselves (see (3) of my initial summary). Physicists certainly use the idea of infinity, but they get into trouble when they attempt to apply them to reality, whether it be the infinitesimally small (e.g., a singularity) or the infinitely large. I did a quick survey of the web and I would have to say that you are very much in the minority in arguing that actual infinites can exist in reality. I think you need to provide an example in the physical universe (and not a mathematical model …. I already dealt with the Malament-Hogarth spacetime and showed how it breaks down at the speed of light). A mathematical model is not to be confused with reality. A chronic problem often encountered in physics with mathematical models is their incompleteness and/or approximation. Some variables transpose beautifully and others either have no counterpart in reality, or they violate axiom II (i), the principle of non-contradiction (as in the singularity).<BR/><BR/><B>Jeffrey wrote:</B> " I completely dispute your claim that "But Hilbert has shown that, given his theory of proof, one of the axioms of which is the law of contradiction, an actual infinite is impossible in reality." Please support this by providing a citation to the exact page number where this is "shown".<BR/><BR/>That depends upon whether we are talking about an <A HREF="http://en.wikipedia.org/wiki/Proof_theory" REL="nofollow">informal proof or formal proof</A>. Hilbert does not offer a formal proof but I think he does offer an informal proof ("…high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, given enough time and patience.", Wiki, ref above). I will outline it here, including page numbers as requested:<BR/><BR/><B>Principle axiom:</B> II (i), p. 198 (see also last paragraph on p. 199 for verification that this is the principle axiom)<BR/><B>Problem:</B> Treating the actual infinite as anything more than a useful idea or concept for mathematics leads to violations of axiom II (i), ('… the literature of mathematics is glutted with inanities and absurdities which have had their source in the infinite.' p. 184 …. there are also several other references to the paradoxes, contradictions, and inconsistencies that arise out of treating the infinite as real, or something more than just a useful mathematical concept or idea.)<BR/><B>Conclusion:</B> The infinite "neither exists in nature nor provides a legitimate basis for rational though …", p. 201 (and rational thought includes mathematical discussions of the infinite).<BR/><BR/>I would think that Hilbert's informal proof is sufficient, given that axiom II (i) coupled with the paradoxical 'problem of the infinite' he mentions, makes it evident that the actual infinite cannot exist in reality. I would also not use the term 'exists in mathematics' when discussing the actual infinite, unless one very carefully defines what one means by 'exist', for mathematical 'existence' is certainly different from ontological, or 'real' existence.<BR/><BR/>If one wants to construct a formal proof, then I would suggest using the last paragraph on p. 199 as the starting point, with the objective of showing that 1≠1 if one assumes an actual infinite to be real. <BR/><BR/><B>Jeffrey wrote re. Hilbert's Hotel thought experiment:</B> "But this is supposed to be an actual physical experiment, not just a thought experiment. How can we, in physical space, "notify the entire hotel" in a finite time? We can't, because the speed of light is finite. So your thought experiment fails immediately, right there."<BR/><BR/>No, it's not supposed to be a physical experiment. It is a thought experiment to see if an actual infinite can exist in reality. If the thought experiment produces a violation of axiom II (i), then it will fail in reality as well (see my (3) at the outset of this post). Of course, there isn't enough mass in the universe to build Hilbert's Hotel, humans will never produce an infinite number of offspring to fill Hilbert's Hotel, the speed of light limits things (as you point out), etc. (By the way, we could use a PA system that depends on quantum simultaneity to get around the limitation imposed by the speed of light.)<BR/><BR/><B>Conclusion:</B> <BR/>Jeffrey, I am not sure whether you are just playing the devil's advocate here (and I do not use that term pejoratively, as I apply that term to myself sometimes in discussions) or you truly belief that:<BR/><BR/>a) actual infinites can exist in reality (i.e., map to some real component of the physical universe),<BR/>b) an infinite regression of causes can occur where the arrow of time runs only forward.<BR/><BR/>I see very little support for your position in my brief survey of the web, although my sampling may well be insufficient and truth is not decided by numbers. I think for (a), you will need to supply a physical example where each variable in your mathematical model maps to a real component of the universe. For (b), I don't think you actually hold this position, but are playing the devil's advocate; it's indefensible in reality.<BR/><BR/>Again, I appreciate your hospitality in permitting me to clarify and defend my position here. It is an interesting topic, though it may be unrealistic to think we are going to resolve it here.Kirk Durstonhttps://www.blogger.com/profile/00678032887521146961noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-49881803799919181542008-05-22T11:13:00.000-04:002008-05-22T11:13:00.000-04:00Kirk: The word 'indeterminate' is used precisely ...Kirk: <I>The word 'indeterminate' is used precisely because violations of Hilbert's axiom II (i) arise...</I><BR/><BR/>No, the word "indeterminate" is used because it's indeterminate. You resolved the indeterminacy by algebraically subtracting s(n) from g(n), showing that g(n)-s(n) goes to infinity, which is correct. This contradicts your notion that removing a countably infinite number of members from a countably infinite set must result in a null set. But that notion is incorrect, so there is no contradiction.<BR/><BR/>Craig makes the same kind of mistake, imputing properties of finite numbers to transfinite numbers (along with other fallacies such as question-begging and appeals to intuition/incredulity).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-69224320707796865642008-05-22T00:34:00.000-04:002008-05-22T00:34:00.000-04:00"Infinite quantities do exist in mathematics, and ..."Infinite quantities do exist in mathematics, and are manipulated all the time"<BR/><BR/>Caveat: the existence of an infinite set is an axiom of ZFC, but is not part of the Peano axioms.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-41068364587162436242008-05-21T07:04:00.000-04:002008-05-21T07:04:00.000-04:00Kirk:The distinction between "actual" and "potenti...Kirk:<BR/><BR/>The distinction between "actual" and "potential" infinity does not exist in mathematics. There is a well-established theory of infinite quantities, and "potential infinite" is not one of them. Infinite quantities <I>do</I> exist in mathematics, and are manipulated all the time; therefore, to claim that Hilbert had a mathematical proof that there is no infinity in nature is simply silly.<BR/><BR/>Hilbert was writing 80 years ago, before many discoveries of modern physics. Modern physicists <I>do</I> consider infinite quantities, such as the example of Malament-Hogarth spacetime I cited.<BR/><BR/>In my example of a past time existing at -n for all positive integers n, this is not a "potential" infinity, but an actual infinity, in the sense that mathematicians use: I can make a one-one correspondence between the positive integers (an infinite set) and time coordinates. <BR/><BR/>I completely dispute your claim that "But Hilbert has shown that, given his theory of proof, one of the axioms of which is the law of contradiction, an actual infinite is impossible in reality." Please support this by providing a citation to the exact page number where this is "shown".<BR/><BR/>You say, "it is a property of an actual infinite that the cardinality of an actual infinite set (denoted by the famous aleph-null) is the same as its proper subsets." Of course, this is nonsense. The integers are a subset of the reals, but they don't have the same cardinality.<BR/><BR/>You say, " Finally, the concierge realizes that he will never notify everybody if he does it one room at a time, so he gets on the PA system and notifies the entire hotel. " But this is supposed to be an actual physical experiment, not just a thought experiment. How can we, in physical space, "notify the entire hotel" in a finite time? We can't, because the speed of light is finite. So your thought experiment fails immediately, right there.<BR/><BR/>I also dispute your claim "If n is infinite, so is d, but if the cardinality of g and s must be Aleph-null, then g and s are equal and so d must be zero when n is infinite, but since d is also an actual infinite, its cardinality must be aleph-null as well; a violation of the law of contradiction." There is no contradiction here, only your misunderstanding about how to manipulate infinities.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-86533789211953996552008-05-20T12:15:00.000-04:002008-05-20T12:15:00.000-04:00Sorry for taking so long to respond. I find it dif...Sorry for taking so long to respond. I find it difficult to squeeze in the time to participate in internet discussions, but this is an interesting topic, and the readership here is not necessary in favor of what I'm arguing for, which I like. I'll briefly address several points raised here, and especially the question of whether an infinite regression of a causal chain is mathematically possible.<BR/><BR/><B>On Bayesian Bouffant's objection to the use of first names</B><BR/>I see blog discussions like this as akin to going for coffee after class to hash out some interesting issue, where the use of first names contributes to an informal and collegial atmosphere. In no way do I mean disrespect to anyone in this forum and especially to Jeffrey. Dr. Shallit is a professor at one of Canada's top-ranked universities and automatically commands respect for that accomplishment. Of course, I'm happy to call anyone whatever they wish. I'll continue to use first names unless otherwise requested. One additional thing; I especially respect anyone who uses their real name in these discussions, even it it's only their first name.<BR/><BR/><B>Quote mining to misrepresent vs. quoting to support a summary</B><BR/>IvanM, I think I did the latter. Hilbert sets out to write his paper to address the problem of the infinite. On page two he writes, "It is, therefore, <I>the problem of the infinite</I> in the sense just indicated which we need to resolve once and for all." In his summary at the end of the paper, he believes he has resolved the problem and writes, <BR/><I>Our principle result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought – a remarkable harmony between being and thought.</I><BR/>What happened in between? First he lays out the problem, violations of the law of contradiction, also referred to as paradoxes and inconsistencies. Second, he lays out a series of axioms that need to be applied to this problem. Third, he emphasizes the need for consistency as the "one condition" and an "absolutely necessary one" which is ensured by axiom II (i), the law of contradiction. He does present a theory of proof, as IvanM mentions, but IvanM overlooks the primary reason and context within which Hilbert presents that theory, the problem of the infinite. Speaking of the sketchy use of quotes, I would also question how IvanM uses the quote at the end of his response. See my longer quote above for the context of that quote.<BR/><BR/><B>On the nature of time in this universe</B><BR/>Both Timmy and Bayesian Bouffant have stated a view about space-time that is corrected by IvanM. I want to add something and use Bayesian Bouffant's N+1 disproof of the notion that time had a beginning (where N=the age of the universe). Einstein's General Theory of Relativity requires that space-time is inseparable. When astronomers and cosmologists talk about the beginning of the universe, they are talking about the beginning of space-time itself. In other words, there is no time = N+1. There is no 'before'. The controversy is the nature of the beginning.<BR/><BR/><B>On the beginning of time</B><BR/>During my years doing an undergraduate degree in physics at the U of Manitoba, I learned that most physicists do not think in terms of an A-theory of time or a B-theory of time, indeed, most would have never heard of those terms. To physicists, time is a physical property of the universe, inseperable from space, and seems to flow only in one direction on the macroscopic level (see the arrow of time http://en.wikipedia.org/wiki/Arrow_of_time ). Some hold (I call this the Newtonian view of time, which may or may not be correct) that physical time is not quantized and that the universe began with a singularity at time t=0, where t denotes a point. The singularity, itself, is a violation of axiom II (i), which is why many physicists have a problem with it. Others believe that at the microscopic scale, time may be a quanta within which there is no arrow of time. In this view t=0 is not a point but a quanta. In other views, such as the pea-instanton theory, time might begin as an extended surface on a small sphere, to use the pea analogy. In all cases, time has a beginning, 'before' which there is no space-time. Only a Newtonian view of time insists that t=0 must represent a point. t=0 can represent the first quanta, the extended surface of space-time on the pea instanton, or the beginning of the arrow of time whatever the nature of that beginning.<BR/><BR/><B>On the use of 'indeterminate' when it comes to infinite set theory</B><BR/>Secondclass wrote, "Infinity minus infinity is an indeterminate form, so there is no contradiction." Well, that might be true so long as we did not worry about why the word 'indeterminate' is used in these cases. The word 'indeterminate' is used precisely because violations of Hilbert's axiom II (i) arise, as illustrated in my previous post about Hilbert's Hotel if we take the application of actual infinites to reality seriously. I want to remind the reader that we are talking about the cardinality of aleph-null when it comes to actual infinites.<BR/><BR/><B>Problem with Jeffrey's examples:</B><BR/>In his opening post in this discussion, Jeffrey suggests that 'we could define times -1, -2, -3, etc. without having to define an actual point called "-infinity".' In his second example, Jeffrey suggests that we could have an event at time 1/(n+1) causing an event at time 1/(n). There is a problem with this in that neither example can apply to the causal chains of the history of the universe. If the history of the universe had no arrow of time, or the arrow of time could run in reverse, then Jeffrey would be right. In both his examples, time relies on a potential infinite. In the first example, one works backward through time and, of course, never reaches infinity. This is a classic example of Hilbert's concept of a potential infinite (page 189). In Jeffrey's second example, as n approaches infinity, t approaches 0. But since n never reaches infinity, the curve never reaches zero on the t-axis. <BR/><BR/>I was talking about the history of space-time, and whether it had a beginning. In the actual world, the arrow of time does not run backward, which entails that the sequence of causal events does not run backward either. Causal chains unfold only in the forward direction (which does allow for simultaneity in macroscopic quantum events). Thus, Jeffrey's first example fails to show that the past history of space-time could be an actual infinite. The second example is also a potential infinite, but this time it is the variable n. Since in actual space-time, the arrow of time flows only in the positive (forward, or future) direction, this forces the variable n to descend from infinity. For a mathematical model to apply to actual space-time, every critical component of the model must have some correspondence with something in the real world. What does n represent in the actual world? If n represents an actual infinite in reality, then it violates Hilbert's axiom II (i), and n cannot be a potential infinite given the arrow of time for this universe. Thus Jeffrey's second example, although valid for potential infinites, fails to show that the past history of space-time could be infinite.<BR/><BR/><B>On the mathematical impossibility of an infinite regression of causes in actual history</B><BR/>I will invoke the mathematical concept of Hilbert's <I>actual infinite</I> and Hilbert's axiom II (i) in his mathematical theory of proof.<BR/>1. Because the arrow of time in actual space-time moves only in the forward direction for macroscopic events, an infinite regression of causes requires that some real component of past history is an actual infinite.<BR/>2. An actual infinite cannot occur in reality (due to violations of axiom II (i))<BR/>therefore, an infinite regression of causes in actual space-time are mathematically impossible from the mathematical concept of an actual infinite and violations of axiom II (i) if an actual infinite is applied to reality.Kirk Durstonhttps://www.blogger.com/profile/00678032887521146961noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-1349820587591533822008-05-16T09:28:00.000-04:002008-05-16T09:28:00.000-04:00Bertrand Russell, from The Scientific Outlook (193...Bertrand Russell, from <I>The Scientific Outlook</I> (1931) as included in <I>Bertrand Russell on God and Religion</I> (ed. Al Seckel, (1986):<BR/><BR/>The purely intellectual argument on this point may be put in a nutshell: is the Creator amenable to the laws of physics or is He not? If He is not, He cannot be inferred from physical phenomena, since no physical causal law can lead to Him; if He is, we shall have to apply the second law of thermodynamics to Him and suppose that He also had to be created at some remote period. But in that case He has lost His <I>raison d'etre</I>. It is curious that not only the physicists, but even the theologians, seem to find something new in the arguments from modern physics. Physicists, perhaps, can scarcely be expected to know the history of theology, but the theologians ought to be aware that the modern arguments have all had their counterparts at earlier times. Eddington's argument about free will and the brain is, as we saw, closely parallel to Descartes's. Jeans's argument is a compound of Plato and Berkeley, and has no more warrant in physics than it had at the time of either of these philosophers. The argument that the world must have had a beginning in time is set forth with great clearness by Kant, who, however, supplements it by an equally powerful argument to prove that the world had no beginning in time. Our age is rendered conceited by the multitude of new discoveries and interventions, but in the realm of philosophy it is much less in advance of the past that it imagines itself to be.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-71195262385865010552008-05-14T19:21:00.000-04:002008-05-14T19:21:00.000-04:00Craig (and apparently Durston) are presuming what ...Craig (and apparently Durston) are presuming what J. M. E. McTaggart called A-series time, where "now" is an objective moment that traverses time, passing through past events into the future. Most physicists advocate B-series time, where all events, past, present, and future, are part of space-time. (Perhaps it's not fair to say "presuming," as Craig does argue for it.)<BR/><BR/>Richard Sorabji's 1983 book, _Time, Creation, and the Continuum_ has a critique of Craig's kalam cosmological argument which focuses on his arguments about the alleged impossibility of an infinite past.<BR/><BR/>Graham Oppy has also had several exchanges with Craig regarding the argument, and in <A HREF="http://www.infidels.org/library/modern/graham_oppy/gifford.html" REL="nofollow">"Time, Successive Addition, and Kalam Cosmological Arguments,"</A> he specifically takes issue with the idea that past events must be a temporal series produced by successive addition.Lippardhttps://www.blogger.com/profile/16826768452963498005noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-30279411958913071152008-05-14T12:12:00.000-04:002008-05-14T12:12:00.000-04:00Going in a slightly different direction, Quentin S...Going in a slightly different direction, Quentin Smith works a bit on this topic and has reprints <A HREF="http://www.infidels.org/library/modern/quentin_smith/bigbang.html" REL="nofollow">here</A>.<BR/><BR/>The arguments move quickly into the arcana and hermeneutics that professional philosophers love so well, but it's nice wrapping one's head around Putnam or Kripke from time to time.Erdos56https://www.blogger.com/profile/04426474525236405685noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-45253019003996695252008-05-14T09:13:00.000-04:002008-05-14T09:13:00.000-04:00IvanM:Thanks for that link! That was interesting. ...IvanM:<BR/>Thanks for that link! That was interesting. But, one reason I am not a theoretical physicist is that there are things that I can't process for myself. The idea that the universe has a shape of any kind and a finite area and volume is something that I can't process. My problem lies here: if the universe is finite, and somehow I get close to the edge, what is there to stop me from going further and what is beyond that boundary? I hope my question makes sense :-P <BR/><BR/>Maybe there is some explanation for this where I have to think differently, like when I found out about the Riemann Spiral(I think that's what it is called). For those that don't know what I mean, I am talking about this "spiral" that Riemann came up with that spirals through all 3 dimensions and needs a 4th dimension to spiral into to exist. When I first read that it I couldn't think anymore the rest of the day, like it drained my brain-battery. Maybe I would have to think like that. I guess that's what I'm getting at. <BR/><BR/>It's questions like these: infinite time and space; that make me wish some super-evolved smart and wise aliens(I think statistically they do exist) would come along and give us a hand with these things ;-)<BR/><BR/>Sorry, Jeffrey, for getting off topic. I do tend to do that.Timmyhttps://www.blogger.com/profile/11430023213232445651noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-16882287191608653232008-05-13T15:09:00.000-04:002008-05-13T15:09:00.000-04:00Anonymous:You say, "You've made ad hominem remark...Anonymous:<BR/><BR/>You say, <I>"You've made ad hominem remarks that Craig and Kirk were acting childish with respect to their understanding of mathematics."</I><BR/><BR/>You are incorrect. I said (and it is easy to verify) that "In general, Craig seems to have an extremely naive, almost childish view of infinity", and I stand by that. Having a "childish view" is not even remotely like "acting childish". And you don't even know what "ad hominem" means, either, since my comment was not "ad hominem".<BR/><BR/>Sorry, Mr. Anonymous, you're out of here.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-71669847435702325622008-05-13T15:01:00.000-04:002008-05-13T15:01:00.000-04:00Jeff,If you're going to comment on the blog, pleas...Jeff,<BR/><BR/><B>If you're going to comment on the blog, please exhibit some politeness.</B><BR/><BR/>May I ask that you do the same, too? You've made ad hominem remarks that Craig and Kirk were acting childish with respect to their understanding of mathematics. That is not polite, sir. You're obviously exhibiting some hypocrisy.<BR/><BR/><B>"You claimed he said Hilbert said an infinite regress of causes was mathematically impossible."<BR/><BR/>No, I said Durston said that. And I said Durston was misled by Craig, which is correct.</B><BR/><BR/>So, let me get this straight, you NOW admit that Craig never said it, but that Durston said it. But then Durston was misled by Dr. Craig who, uh, never said that an infinite regress of causes was mathematically impossible, anyway! So, how in the world can Dr. Craig mislead Durston if Dr. Craig never conceded it? LOL! You really don't know what you're talking about, do you? No, Jeffrey, that is not correct.<BR/><BR/><B>Even assuming I quoted Craig incorrectly, why do you think this would be a "lie" as opposed to a simple mistake?</B><BR/><BR/>So, you made a mistake, then? Well, you just admitted that Dr. Craig never said an infinite regress of causes was mathematically impossible via Hilbert. Can you at least apologize for making that mistake then? And can you apologize to Durston for saying he was misled by Dr. Craig? Since you admitted Dr. Craig never made said claim about Hilbert.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-72055543968546057742008-05-13T13:53:00.000-04:002008-05-13T13:53:00.000-04:00Anonymous:If you're going to comment on the blog, ...Anonymous:<BR/><BR/>If you're going to comment on the blog, please exhibit some politeness.<BR/><BR/>" You claimed he said Hilbert said an infinite regress of causes was mathematically impossible." <BR/><BR/>No, I said Durston said that. And I said Durston was misled by Craig, which is correct. <BR/><BR/>Even assuming I quoted Craig incorrectly, why do you think this would be a "lie" as opposed to a simple mistake?Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-20067416.post-88793503816317947802008-05-12T22:38:00.000-04:002008-05-12T22:38:00.000-04:00Timmy: Space can be finite without having a bounda...Timmy: Space can be finite without having a boundary wall, in exactly the same way as the two-dimensional surface of a sphere has finite area but no boundary. There are many possibilities for the <A HREF="http://en.wikipedia.org/wiki/Shape_of_the_Universe#Global_geometry" REL="nofollow">topology of the universe</A>, whether finite or infinite in volume.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-79261849141194344782008-05-12T15:25:00.000-04:002008-05-12T15:25:00.000-04:00Kirk,Thank you for responding. I thought your deba...Kirk,<BR/><BR/>Thank you for responding. I thought your debate with Shallit was masterfully done. Not only did you beat him in a verbal debate, but you finished him off quite well with your literary skills here in his blog!<BR/><BR/>Jeffrey,<BR/><BR/>I hope you would at least be humble enough to admit you deliberately misquoted Dr. Craig. You claimed he said Hilbert said an infinite regress of causes was mathematically impossible. Dr. Craig never said that. Instead of admitting your bold-faced <B>lie</B> you make the laughable excuse that he misunderstood you. As we can see he's not the one with that problem.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-20067416.post-86674433558564387082008-05-12T12:34:00.000-04:002008-05-12T12:34:00.000-04:00Kirk, a few responses:Now imagine that an observer...Kirk, a few responses:<BR/><BR/><I>Now imagine that an observer is coming toward us from the infinite past in her own time machine that moves forward through time. When will the two time machines pass each other? Never.</I><BR/><BR/>Why never? You seem to be assuming your conclusion, namely that the other observer cannot have already traversed infinite time.<BR/><BR/><I>For example, we use imaginary numbers in everyday three phase electrical calculations, but imaginary numbers and imaginary infinites cannot, logically, exist in reality. No one, for example, would argue that I can have the square root of -1 apples in a basket.</I><BR/><BR/>It's true that number of apples in a basket must be a real number, but it's also true that the impedance of an inductor must be an imaginary number. The idea that imaginary numbers can't exist in reality, with the implication that real numbers <I>can</I>, doesn't make sense. Neither real nor imaginary numbers exist as physical objects, but both are useful for modeling physical phenomena.<BR/><BR/><I>If n is infinite, so is d, but if the cardinality of g and s must be Aleph-null, then g and s are equal and so d must be zero when n is infinite, but since d is also an actual infinite, its cardinality must be aleph-null as well; a violation of the law of contradiction.</I><BR/><BR/>There is no reason that d must be zero. Infinity minus infinity is an indeterminate form, so there is no contradiction.Anonymousnoreply@blogger.com