aleph_0 Posted August 6, 2010 Report Share Posted August 6, 2010 (edited) Okay, this topic has been carrying on for quite some time in no less than two other topics, so I thought it best to organize the conversation and possibly bring it away from other topics in order not to distract from their main points. The controversy is over the question, may we dismiss any hypothesis about the universe which makes claim to an actual infinite quantity, without (additional) empirical investigation. That is to say, is it a necessary falsehood that, for any set S of physical objects, the size of that set is infinite? To make the conversation exact, by "infinity" or "infinite quantity" I just mean an injective map from the natural numbers to some set of physical objects. A map is an assignment of some objects to others. For instance, there is a map which assigns 1 to Washington, 2 to Adams, ..., and 44 to Obama. This map is perfectly clear and presumably uncontroversial. Maps are just functions, which means that for any element in the domain (here, the natural numbers from 1 to 44) there is exactly one associated element in the range (the set of United States Presidents). An example of a relation which is not a function would be the one that assigns 1 to both Washington and Adams. Because 1 gets associated to (or "sent to") two distinct objects in the range, it is not a function. However, the map which sends 1, 2, 3, ... all to Washington is still a map. It's just not an injective map. If we denote the element which gets mapped to by 1 as f(1), and likewise the object which gets mapped to by n as f(n), then an injective map is one that satisfies the following conditional: If f(n) = f(m) then n = m. Obviously the first map we had was injective, for if the map sent n to Hoover and m to Hoover, then n = m, and likewise for all other Presidents. Obviously the map which sent all natural numbers to Washington was not injective, since f(1) = Washington = f(2), but 1 =/= 2. In the case of finite sets, what this means is obvious: If you have an injective map from the domain to the range, then the domain is smaller in quantity (or equal to in quantity) than the range. For instance, 44 is smaller than in quantity (or equal to in quantity) the number of Presidents in United States History. The same holds in the infinite case, in some sense. So the question is, is it necessarily false that there may be some injective map from the set of natural numbers into some set of physical objects, i.e. can we rule out such a hypothesis without your having to check Wikipedia to see if there is any evidence for or against? We will ask the question of sets of physical objects, such that no two elements in the set share any material parts, thus avoiding some annoying gadflies. I've counted five basic arguments for the negative answer on this forum, each of which I find deficient. 1) The concept of "quantity" precludes infinity. I have seen no argument to this effect, merely the bald claim. Even if you have some special notion of what "quantity" means as a term, we may simply look at the definition of "infinite quantity" I gave above. Can there be an injective map from the set of natural numbers into some set of (disjoint) physical objects? What is contained in this question which necessitates an answer, "no"? It is only an answer to this question in which I'm interested, and not any special group jargon about the word "quantity". 2) Argument by induction. The argument goes, "Of all the things you've observed in your life, have any of them been infinite?" Naturally, the answer is no. This is supposed to be convincing for the same reason that one is convinced all ravens are black by the fact that every raven one has observed has been black. I have two objections. First, induction clearly does not apply to quantity (or perhaps more broadly, bijective maps), as one cannot argue "Of all the things you've seen in your life, have you ever seen n of them, for some very large n?" If we let n = 1,000,000, then the answer would be "no", and I would thereby conclude that nothing has quantity 1,000,000 in spite of the fact that I'm convinced of atomic physics and the rough estimate that the table in front of me has a few more than 1,000,000 atoms. Secondly, even if this argument were valid to cast some doubt on an actual infinite quantity, it would not answer the question with which I began this post. Namely, it would not allow me to necessarily rule out any hypothesis on the basis of the fact that it makes claim to some actual infinity. If a given hypothesis were to make such a claim, and were fully capable of accounting for and predicting every single event observed in the past, and from now on, I would come to believe that hypothesis as a verified theory in spite of any such inductive argument. 3) Identity. Some have said that the notion of an actual infinity violates the law of identity, which states that everything which is, is; or put another way, everything has a specific identity. However, I don't see how this law is violated by an actual infinity. On another recent topic about the claim that "existence exists", it was noted that this statement carries no more philosophical weight than to point out that "nothing is not a thing", that one cannot contradict oneself. In the same way, since the law of identity seems to say basically this, I don't see how the law of identity has anything to bear on this topic. It has been said that infinity is not a "specific" quantity, as if it is a quantity, but just not a specific one. But using the notion of infinity stated in the beginning, this is clearly not true. We know exactly what a map is, and exactly what an injective map is, and what the set of natural numbers are, so we should know exactly what an injective map from the natural numbers to some other set is, regardless of which set we are talking about. Thus there is nothing "unspecific" about this conception of infinity. Thus in order to pursue this line of argument, one needs to state how exactly the law of identity precludes the possibility of such an injective map, or how the notion of an injective map as I've described above is somehow not specific. 4) Lack of evidence. One person has pointed out that there is no evidence to suppose that there is an infinite quantity, and I agree to this. It is not my claim that there is an infinite quantity, but only that we cannot rule it out as a possible hypothesis. 5) Axiom. Nobody has stated this, but most seem to think that the finiteness of the universe is an axiom which cannot intelligibly be questioned. Yet I do not have this axiom. I have consciousness, A equals A, and perhaps some others like rules of logic, but nothing about finiteness or infinity. Assuming that finiteness of the universe is not an axiom, then if people on this forum believe it, there must be some argument to that effect. Yet I have not been able to find a sound, non-circular such argument. In your response, if you are pressing one of these five arguments, please identify it in your post to help me identify what your claim is, and exactly what in my response to this argument is incomplete or mistaken. [Edit: Note, this topic is distinct from the other topic labeled "The Finite and the Infinite" or something, because this conversation 1) contains several arguments not therein contained, and is disputing the arguments which are therein contained, and 2) this topic does not make any reference to the point about "eternity" used in that other one. So I believe I'm not guilty of reproducing the same topic in multiple places.] Edited August 6, 2010 by aleph_0 Quote Link to comment Share on other sites More sharing options...

Mindy Posted August 6, 2010 Report Share Posted August 6, 2010 You need the definition of an everyday term "argued for?" What sort of epistemology underlies that? Look it up in a simple dictionary, in the O.E.D., even. Facts don't need to be argued for. That is the way the word is defined. (See, de-fin-ed, how the word is identified, with an end, not indefinite, lacking an end.) So, you won't respect the conceptual meaning of the term, but you will impose on everybody else your technical, jargonized, operationalized, surjectivized definition? Earth to Aleph. The longer you put it off, the harder it gets. Mindy Quote Link to comment Share on other sites More sharing options...

Nate T. Posted August 7, 2010 Report Share Posted August 7, 2010 Despite the fact that you're a fellow mathematical traveler, Aleph_0, I'm going to have to (in a sense) side with the finite-ists on this one. I think the difference is the fact that you want to pair off natural numbers with physical entities instead of with mental constructs like numbers and such. My argument is closest to your no. 4, but it differs a little from what you had said. We can talk about sets of infinite cardinality in mathematics because we have an unambiguous characterization of all of the sets we often consider in mathematics, such as the set of all perfect squares. Your justification for having set up, once and for all, a one-to-one correspondence between the natural numbers and the perfect squares is that you've exhibited a function f(n) = n^2 which *in principle* matches every natural number to a perfect square. This works great for concepts like numbers, but how would you go about constructing such a general function pairing natural numbers with electrons? You'd have to actually sit down and actually label electrons as you find them-- there's no general formula stipulating that, hey, that electron over there is the 675,598,982nd one! So you're resorting to finding as many electrons as you can and adding them to the list post hoc, making up your one-to-one correspondence as you go along. That is, instead of having *one* rule making the correspondence between your two sets, you have a sequence of correspondences between finite subsets of those sets-- and if you ever finished, you'd have a standard one-to-one correspondence between finite sets, which even Mindy would agree is uncontroversial if you could explicitly exhibit it. Of course you can *imagine* a one-to-one correspondence between natural numbers and electrons, but this would just be you saying, in your head, "Hey, if you give me any number, I can picture the electron it would pair to, why not?" But at this point we've left the realm of reality and gone over to the arbitrary, which is an Objectivism no-no. Quote Link to comment Share on other sites More sharing options...

Grames Posted August 7, 2010 Report Share Posted August 7, 2010 4) Lack of evidence. One person has pointed out that there is no evidence to suppose that there is an infinite quantity, and I agree to this. It is not my claim that there is an infinite quantity, but only that we cannot rule it out as a possible hypothesis. I'll own this one. I will not contradict you, we cannot rule it out as a possible hypothesis. What follows: that which cannot be shown to be true or false cannot be used as a premise in a valid argument to produce a true conclusion. That is what the technical term 'arbitrary' identifies, a useless statement. Quote Link to comment Share on other sites More sharing options...

Alfred Centauri Posted August 7, 2010 Report Share Posted August 7, 2010 (edited) FWIW, there's a series of thought provoking posts concerning the existence, or lack thereof, of infinite sets here. I haven't read them carefully enough to comment but do plan to when (if) I find the time. Edited August 7, 2010 by Alfred Centauri Quote Link to comment Share on other sites More sharing options...

aleph_0 Posted August 7, 2010 Author Report Share Posted August 7, 2010 I think the difference is the fact that you want to pair off natural numbers with physical entities instead of with mental constructs like numbers and such. My argument is closest to your no. 4, but it differs a little from what you had said. We can talk about sets of infinite cardinality in mathematics because we have an unambiguous characterization of all of the sets we often consider in mathematics, such as the set of all perfect squares. Your justification for having set up, once and for all, a one-to-one correspondence between the natural numbers and the perfect squares is that you've exhibited a function f(n) = n^2 which *in principle* matches every natural number to a perfect square. This works great for concepts like numbers, but how would you go about constructing such a general function pairing natural numbers with electrons? You'd have to actually sit down and actually label electrons as you find them-- there's no general formula stipulating that, hey, that electron over there is the 675,598,982nd one! Two points of response: first, I don't believe this is a problem; second, if it were, I think there is an equivalent formulation which is not subject to any such objection. 1) Note that the one-to-one correspondence between the set of natural numbers between 1 and 44, and the set of Presidents in US history is not due to a mental construction. The correspondence between the two is due to the quantity of Presidents, which is independent of human thought. There are many correspondences which exist between numbers and objects which nobody has or will identify. Likewise, a correspondence between the natural numbers and a set of disjoint physical objects could exist though nobody may list every assignment therein. Though numbers do not exist, the facts of mathematical structure are objective, and I'm only appealing to these facts. 2) I may rephrase the question thus: Is it possible for there to be some set S of disjoint physical objects such that for any arbitrary set N of natural numbers, there is some injection from N to S? Quote Link to comment Share on other sites More sharing options...

TuringAI Posted August 7, 2010 Report Share Posted August 7, 2010 I think the main point to address here is that even though no entity or collection thereof can be infinite, not all existents are entities. To claim that entities are the only things that really matter, in this debate or anywhere, is to have a materialistic bias. All numbers exist in the abstract terms of them being potentially reachable. Infinity shares some qualities with numbers, and has identity but not as a number. It is kind of like a 'broken unit' but really it's not broken, just unorthodox. There are a number of unorthodox things, which have tangential usefulness as a concept, which is to say not that they are useless, but just that they don't tie into the mainstream. I think play is necessary for some things, and that includes making concepts. Even if you can't convince other people to accept a concept or make extensive practical use of it doesn't mean the concept is invalid. I think that's where 'infinity' falls into place. Quote Link to comment Share on other sites More sharing options...

Nate T. Posted August 7, 2010 Report Share Posted August 7, 2010 Yes, I probably stated things in a misleading way, now that I look at it. I didn't mean to suggest by the term "mental construct" that the quantity of a collection wasn't an actual property of that collection. After all, concepts are mental entities that your mind constructs (which abstract properties from entities in an objective way, of course), and I was just looking for a term that distinguishes concepts like numbers and such from physical objects. If you have a finite collection of entities, then of course you can assign natural numbers to them in any way you like by explicitly constructing some list-- there need not even be any property or attribute of the finite collection of entities that suggests an ordering of the list, since as you note only the cardinality is important in this context. However, there is a natural ordering to the collection of Presidents (ordinal by term) that makes such a correspondence obvious at once. In the case of a hypothetically infinite set of objects, in order to have a correspondence that works once and for all, you are required to use the knowledge you have of all of the entities you're trying to pair off (such as that they happen to be generated from the natural numbers by the squaring function) otherwise you're never done constructing the correspondence, not even in principle. It's also not enough to say that "well there might be lots of such correspondences that haven't been identified", since the same can be said for those gremlins on Venus, as others have said. Anyway, your reformulated question is easier to answer: you now ask whether the weaker construction I mentioned in my first post of a sequence of nested correspondences actually exists between the first few natural numbers and physical objects. It's precisely this that has to be regarded as arbitrary, since we could never finish constructing such a sequence of correspondences! I suppose I can agree, as Grames has, that one could hypothesize an infinite number of electrons, but I don't see any way of demonstrating the positive claim that the number of electrons is in fact infinite. Quote Link to comment Share on other sites More sharing options...

Plasmatic Posted August 7, 2010 Report Share Posted August 7, 2010 (edited) In fact I have stated 5). As far as I'm concerned it follows from the law of Identity and is unavoidable. Most of the opponents of the axioms call them "arbitrary" tautologies , so I'm not surprised to find that argument leveled at 5). I consider the arguments stated to be exactly like those who say ,"You can't prove that everythig has identity you haven't discovered that by observing the whole of existence". The lack of understanding of the inductive foundation of concepts and axioms and it's role in definitions, is to me, the main cause. The other I think is if some followed this method and accepted it they know a whole host of special science concepts and theories go bye bye.... The refusal to follow the foundational aproach to philosophy/ knowledge. The other point Is that ALL existents that are not entities are attributes of entities! All those epistemological concepts that keep popping up are entity dependant. Concepts like consciousness are attributes of entities. Edit: I consider 3). and 5). the same. "nothing is not" is NOT all it implies. It requires that for everything that is there is something it is not. Multiplicity. Without boundedness no difference exist. One could not differentiate color if green was the only color. Edit: One more thing. "Existence IS Identity" Edited August 7, 2010 by Plasmatic Quote Link to comment Share on other sites More sharing options...

Mindy Posted August 7, 2010 Report Share Posted August 7, 2010 (edited) Yes, I probably stated things in a misleading way, now that I look at it. I didn't mean to suggest by the term "mental construct" that the quantity of a collection wasn't an actual property of that collection. After all, concepts are mental entities that your mind constructs (which abstract properties from entities in an objective way, of course), and I was just looking for a term that distinguishes concepts like numbers and such from physical objects. If you have a finite collection of entities, then of course you can assign natural numbers to them in any way you like by explicitly constructing some list-- there need not even be any property or attribute of the finite collection of entities that suggests an ordering of the list, since as you note only the cardinality is important in this context. However, there is a natural ordering to the collection of Presidents (ordinal by term) that makes such a correspondence obvious at once. I believe you were right in the first place, Nate. The collections are themselves abstractions. So, when you say, I didn't mean to suggest by the term "mental construct" that the quantity of a collection wasn't an actual property of that collection, you are referring to a logical property of a mental construct. A collection is objective only if it is defined in reference to some particular. "All the cows in this field," is objective, (objective and finite,) but "All the numbers that could be conceived of," isn't. So, for example, the largest cow in the field can be a part of that collection, of the collection of all the cows on the farm, a part of the collection of all the cows in the county, etc. It may be thought about in an unlimited number of ways, but it is just one existent. So, you succeeded at distinguishing concepts from physical things. Mindy p.s. That was an excellent argument you made. Edited August 7, 2010 by Mindy Quote Link to comment Share on other sites More sharing options...

aleph_0 Posted August 7, 2010 Author Report Share Posted August 7, 2010 (edited) I think the main point to address here is that even though no entity or collection thereof can be infinite, not all existents are entities. I don't think this is going to help out, really, since I'm just talking about sets of disjoint objects. These could be electrons or asteroids for all I care. If you have a finite collection of entities, then of course you can assign natural numbers to them in any way you like by explicitly constructing some list-- there need not even be any property or attribute of the finite collection of entities that suggests an ordering of the list, since as you note only the cardinality is important in this context. However, there is a natural ordering to the collection of Presidents (ordinal by term) that makes such a correspondence obvious at once. Note, however, that I do not need to appeal to this ordering, nor does my question appeal to any ordering. In the case of a hypothetically infinite set of objects, in order to have a correspondence that works once and for all, you are required to use the knowledge you have of all of the entities you're trying to pair off (such as that they happen to be generated from the natural numbers by the squaring function) otherwise you're never done constructing the correspondence, not even in principle. Again I emphasize that correspondences exist whether people are aware of them or not. Thus, a pairing that some human being could not, in finite time, represent or think of, is still a pairing. The pairing does not need to be given as a computational procedure, for any arbitrary input--it can be an arbitrary association of some integers with some objects. That no person will ever know all of the correspondences is unimportant for the same reason that it is unimportant (to the question of quantity) that no person will ever think of the correspondence between the number of atoms in my computer and the set of that many natural numbers. It's also not enough to say that "well there might be lots of such correspondences that haven't been identified", since the same can be said for those gremlins on Venus, as others have said. If this is the ultimate response, then I think you concede my point, because a scientific theory which posits gremlins cannot be rejected out of hand--it can only be rejected on the basis of lack of evidence, but if evidence were ever presented one would then accept the hypothesis. I have said elsewhere how I think such evidence might conceivably be given, but that's not even important. The point that I want to establish is that such a notion is not self-contradictory or unintelligible. I just want to establish that one cannot dismiss as meaningless, a theory of the universe which incorporates some claim of an infinite quantity. Anyway, your reformulated question is easier to answer: you now ask whether the weaker construction I mentioned in my first post of a sequence of nested correspondences actually exists between the first few natural numbers and physical objects. It's precisely this that has to be regarded as arbitrary, since we could never finish constructing such a sequence of correspondences! I suppose I can agree, as Grames has, that one could hypothesize an infinite number of electrons, but I don't see any way of demonstrating the positive claim that the number of electrons is in fact infinite. That we could not finish such a construction does not entail that such a correspondence wouldn't exist. Again, the correspondences are not purely mental, and they're not contingent upon us in any way. That we could not, for instance, ever actually count out e^(e^(e^79)) of anything does not thereby entail that the number is not a quantity. In fact I have stated 5). As far as I'm concerned it follows from the law of Identity and is unavoidable. If it is an axiom, then it shouldn't follow from anything. If it follows, then there must be an argument. Most of the opponents of the axioms call them "arbitrary" tautologies , so I'm not surprised to find that argument leveled at 5). I'm not criticizing the axiom, I'm disputing that the axiom tells us anything about this particular subject. In the same way, if somebody told me that bunnies are asexual because of the law of identity, I would challenge that the two have nothing to do with each other, and before moving on in argument I would need some demonstration of the connection. Without boundedness no difference exist. One could not differentiate color if green was the only color. If there is no boundary in the sense that no two things can be distinguished, then yes, this is correct. However, nothing about the supposition of an infinite quantity implies the impossibility of distinguishing two things. These are two different senses of "boundary". One is a boundary between two distinct things, and another is an end-point. The "boundary" between red and green, if we are to speak in such odd terms, is their different properties. Take away this "boundary" and there is no way to distinguish the two. The boundary between me and my seat is a line in space. Take this away, in some sense--i.e. if there were no line between me and my seat--then it'd be a fluid melding of my butt and the chair, and one couldn't distinguish between them. The boundary on a set of numbers is just the number which is greater than or equal to all numbers therein. What is it bounded from? Nothingness? That isn't a thing, so taking away the boundary doesn't cause a melding of two things, where the distinction between them can no longer be made. The set can be distinguished from other sets, and from the lack of any set. Edited August 7, 2010 by aleph_0 Quote Link to comment Share on other sites More sharing options...

Nate T. Posted August 7, 2010 Report Share Posted August 7, 2010 (edited) (i) Again I emphasize that correspondences exist whether people are aware of them or not. Thus, a pairing that some human being could not, in finite time, represent or think of, is still a pairing. The pairing does not need to be given as a computational procedure, for any arbitrary input--it can be an arbitrary association of some integers with some objects. That no person will ever know all of the correspondences is unimportant for the same reason that it is unimportant (to the question of quantity) that no person will ever think of the correspondence between the number of atoms in my computer and the set of that many natural numbers. ... (ii) If this is the ultimate response, then I think you concede my point, because a scientific theory which posits gremlins cannot be rejected out of hand--it can only be rejected on the basis of lack of evidence, but if evidence were ever presented one would then accept the hypothesis. I have said elsewhere how I think such evidence might conceivably be given, but that's not even important. The point that I want to establish is that such a notion is not self-contradictory or unintelligible. I just want to establish that one cannot dismiss as meaningless, a theory of the universe which incorporates some claim of an infinite quantity. ... (iii) That we could not finish such a construction does not entail that such a correspondence wouldn't exist. Again, the correspondences are not purely mental, and they're not contingent upon us in any way. That we could not, for instance, ever actually count out e^(e^(e^79)) of anything does not thereby entail that the number is not a quantity. (i) If you want to use the concept "bijective map" as it is used in mathematics to establish quantities of collections which are not sets in mathematics, you must justify this usage in the broader context of physical objects. For finite sets (even very large ones) this is done by enumeration, and is uncontroversial. Since we do not have any referents of literally infinite collections of physical objects (for reasons already mentioned) we cannot apply this concept to physical objects. In this sense, you do need to construct such a correspondence, or you literally are talking about nothing. That you would go ahead and stipulate such a pairing using these concepts outside the mathematical context is really what is arbitrary. Put another way, you're smuggling in the premise that collections of physical objects satisfy the axioms of set theory. Unless you want to do mathematics only with finite sets (which you certainly don't), you need to show that the Axiom of Infinity Holds, which is precisely the point in question. So far, though, this is an arbitrary assertion on your part. So you'd either need to exhibit a literally infinite collection of entities for us, or start from another foundation of mathematics besides ZFC. The onus, either way, remains on you. (ii) A theory with absolutely no evidence is to be dismissed out of hand, and precisely for that reason-- that's the theory/practice unity. But (and especially in light of my challenge in (i)) I'd be interested to see how you might provide such evidence. (iii) It doesn't for mathematical objects since we have induction, but it does for physical objects. This is the meaning of the qualifier "in principle." I can whimsically say that e^(e^47) pairs to e^(2e^47), though Edited August 7, 2010 by Nate T. Quote Link to comment Share on other sites More sharing options...

dream_weaver Posted August 8, 2010 Report Share Posted August 8, 2010 3) Identity. Some have said that the notion of an actual infinity violates the law of identity, which states that everything which is, is; or put another way, everything has a specific identity. However, I don't see how this law is violated by an actual infinity. On another recent topic about the claim that "existence exists", it was noted that this statement carries no more philosophical weight than to point out that "nothing is not a thing", that one cannot contradict oneself. In the same way, since the law of identity seems to say basically this, I don't see how the law of identity has anything to bear on this topic. It has been said that infinity is not a "specific" quantity, as if it is a quantity, but just not a specific one. But using the notion of infinity stated in the beginning, this is clearly not true. We know exactly what a map is, and exactly what an injective map is, and what the set of natural numbers are, so we should know exactly what an injective map from the natural numbers to some other set is, regardless of which set we are talking about. Thus there is nothing "unspecific" about this conception of infinity. Thus in order to pursue this line of argument, one needs to state how exactly the law of identity precludes the possibility of such an injective map, or how the notion of an injective map as I've described above is somehow not specific. Just how do you apply one to one correspondence to two groups which have no specific multiplicity? Are you equivocating on the concept of set by treating an open ended classification as if it were a concrete collection? Quote Link to comment Share on other sites More sharing options...

TuringAI Posted August 8, 2010 Report Share Posted August 8, 2010 Just how do you apply one to one correspondence to two groups which have no specific multiplicity? Are you equivocating on the concept of set by treating an open ended classification as if it were a concrete collection? We do this by taking a short cut rather than counting every single instance. Since counting is by definition a finite process we can't use that method. Rather, we exhaust the set or sets under consideration by applying mathematical tools that apply to both finite an infinite sets. Of course I don't think 'mathematical induction' (which isn't the same as logical induction) applies because it requires a sequence and all sequences are finite otherwise it implies a contradiction. Rather we have to use something else. Quote Link to comment Share on other sites More sharing options...

dream_weaver Posted August 8, 2010 Report Share Posted August 8, 2010 An infinite sequence would imply a contradiction? Hmmm. Quote Link to comment Share on other sites More sharing options...

TuringAI Posted August 8, 2010 Report Share Posted August 8, 2010 An infinite sequence would imply a contradiction? Hmmm. Let me rephrase that. A sequence that reaches infinity would be a contradiction. Numeric sequences that apply to finite numbers that don't terminate aren't contradictions, but mathematical inductions only applies to the finite numbers since that's one of the premises, whether it's realized or not, of the logical chain. What I mean specifically is that logical chains that begin at some number and apply to all subsequent numbers do not apply to infinity because there's a fundamental discontinuity of the type of 'number' under consideration. Quote Link to comment Share on other sites More sharing options...

dream_weaver Posted August 8, 2010 Report Share Posted August 8, 2010 An infinite sequence would imply, while a sequence that reaches infinity would be a contradiction? Fundamental disontinuities? Hmmm. Quote Link to comment Share on other sites More sharing options...

aleph_0 Posted August 8, 2010 Author Report Share Posted August 8, 2010 (i) If you want to use the concept "bijective map" as it is used in mathematics to establish quantities of collections which are not sets in mathematics, you must justify this usage in the broader context of physical objects. For finite sets (even very large ones) this is done by enumeration, and is uncontroversial. Since we do not have any referents of literally infinite collections of physical objects (for reasons already mentioned) we cannot apply this concept to physical objects. In this sense, you do need to construct such a correspondence, or you literally are talking about nothing. That you would go ahead and stipulate such a pairing using these concepts outside the mathematical context is really what is arbitrary. Put another way, you're smuggling in the premise that collections of physical objects satisfy the axioms of set theory. Unless you want to do mathematics only with finite sets (which you certainly don't), you need to show that the Axiom of Infinity Holds, which is precisely the point in question. So far, though, this is an arbitrary assertion on your part. So you'd either need to exhibit a literally infinite collection of entities for us, or start from another foundation of mathematics besides ZFC. The onus, either way, remains on you. Even using mere finite sets, you are still assuming that ZFC axioms apply to sets of physical objects, so this much you and everybody else here have already conceded. Moreover, my point is not to prove that there are infinite sets of disjoint physical objects, but to ask what is contradictory about supposing it. Thus, I do not need to prove the axiom of infinity for physical objects, but to ask what is contradictory about it's hypothetically being true. So the burden of proof is actually not on me, since I am merely denying the claim that people on this forum make, that there can be no infinite quantity. I refuse to accept this because the case has not been made to satisfaction, and so I am demanding a more thorough argument. (ii) A theory with absolutely no evidence is to be dismissed out of hand, and precisely for that reason-- that's the theory/practice unity. But (and especially in light of my challenge in (i)) I'd be interested to see how you might provide such evidence. A theory with absolutely no evidence is to be ignored, not dismissed as demonstrably untrue. My question is how can people state that it is impossible (not just not-yet-proven) for there to be infinite quantities. (iii) It doesn't for mathematical objects since we have induction, but it does for physical objects. This is the meaning of the qualifier "in principle." I can whimsically say that e^(e^47) pairs to e^(2e^47), though I don't take the point. Are you asserting that there cannot be e^(e^(e^79)) of anything? Just how do you apply one to one correspondence to two groups which have no specific multiplicity? I don't know what could possibly be a "lack of specific multiplicity," but each set is well-defined. We know what the natural numbers are and we know what a set of disjoint physical objects is. You assign one element of one set to one element of another--that's all you do for a mapping. One instance of this is the mapping from the set of natural numbers to the even numbers: f(n) = 2n. It is actually a bijective mapping, and each set is infinite. The fact that I gave the mapping by using a computational procedure is insignificant, since I have only used this to name the map rather than create it. The map (and all correspondences, for that matter) exist whether they are named or not. All I'm asking is whether it is self-contradictory to suppose that a map from the natural numbers to some set of disjoint physical objects could be injective. Are you equivocating on the concept of set by treating an open ended classification as if it were a concrete collection? What could this mean? Do you deny that there are sets of physical objects? Do you deny that the set of chairs in the room I am occupying is a set? Quote Link to comment Share on other sites More sharing options...

Nate T. Posted August 8, 2010 Report Share Posted August 8, 2010 (edited) ... my point is not to prove that there are infinite sets of disjoint physical objects, but to ask what is contradictory about supposing it. Thus, I do not need to prove the axiom of infinity for physical objects, but to ask what is contradictory about it's hypothetically being true. So the burden of proof is actually not on me, since I am merely denying the claim that people on this forum make, that there can be no infinite quantity. I refuse to accept this because the case has not been made to satisfaction, and so I am demanding a more thorough argument. ... A theory with absolutely no evidence is to be ignored, not dismissed as demonstrably untrue. My question is how can people state that it is impossible (not just not-yet-proven) for there to be infinite quantities. I think you can accept *as a hypothetical* that an infinite number of electrons exist, in that you can form the words: "What if there were a mapping from the natural numbers to the electrons?", or even imagine in your head a bunch of electrons being labeled with various numbers without end. It's just that, being an arbitrary assertion, it won't tell you anything about anything, being based on a notion of completed infinity ripped from its context as a concept of method, seeing as you manifestly cannot produce an example of an infinite collection of objects. If not being able to positively disprove the existence of the arbitrary (infinite numbers of electrons or God) was your point, that's true, I guess. Anyway, I didn't say theories with no evidence should be dismissed as demonstrably untrue (in that assertions about various constructs posited, etc., must not exist), I just said they should be rejected, or ignored, if you like. It doesn't stand in a positive correspondence with reality nor does it contradict any known facts, it's simply as though nothing has been said. Same as any assertions about infinitely many electrons-- which has the further disadvantage of being impossible in principle to check, not just "not-yet-proven". Edited August 8, 2010 by Nate T. Quote Link to comment Share on other sites More sharing options...

aleph_0 Posted August 8, 2010 Author Report Share Posted August 8, 2010 (edited) I mean more than this. For instance, sure, I can form the words, "There is a round square at the bottom of the ocean," but this is an outright contradiction and so we can reject any such hypothesis without further empirical investigation--and here I mean actual rejecting, not just ignoring. Can we do the same with the hypothesis of an injective mapping from natural numbers to disjoint physical objects? If yes, then we can take this notion to define the phrase "infinite quantity", and people on this forum have no justification for the claim that all quantities are finite, and cannot dismiss physical theories which claim it; if no, then people on the forums have been right. And again, just because I cannot produce an explicit mapping doesn't mean that one doesn't exist, so this argument has no force in the question I'm posing. Now if I were trying to positively prove that there actually is an infinite quantity, you might respond this way, but that is not my intent. Now beyond this, here is a scenario in which one could prove (insofar as the theory of atoms has been proved) an infinite quantity: A hypothesis about physics has, as an essential component, the assumption that there are infinite quantities. It explains all of past recorded events and facts, and it perfectly predicts all future events, with accuracy as close as it is possible for us to measure. Rather than reject the hypothesis because it makes claim to an infinite quantity, I would accept the hypothesis as proven theory. Alternately, suppose that we found that the Cosmic Background Radiation were constant everywhere, and emanating from a single direction. Since our best hypothesis is that this radiation is a product of the Big Bang, and so indicates the location of the Big Bang, then it would make sense to think that for any distance traveled toward the Big Bang, one will not reach its origin (the constant frequency of the radiation indicating that, for any given point, that point is just as distant from the origin of the Big Bang as any other point to which we might travel, in principle) and continue to encounter this radiation. Thus one could count an infinity of this radiation, whatever it is. I'm rather certain that the frequency of the Cosmic Background Radiation is not constant, but the point being that such facts could indicate an infinite quantity. In any case, if you concede that your only objection is the lack of evidence, then you agree with me. It is not a self-contradictory hypothesis to suppose that there is an infinite quantity. Edited August 8, 2010 by aleph_0 Quote Link to comment Share on other sites More sharing options...

Tensorman Posted August 8, 2010 Report Share Posted August 8, 2010 I don't see why there could not be an infinite number of physical objects. The argument that this is an arbitrary assertion doesn't hold, as its negation "every set of physical objects is finite" is equally arbitrary. It cannot be proved and the question whether there exists infinite or only finite numbers of objects will probably always remain unprovable, as there exist numbers that are so large that we in practice will never be able to determine the difference between such a number and infinity. So this remains an undecidable question, there could be an infinite number of physical objects, but we'll never be able to prove it, nor its negation. Quote Link to comment Share on other sites More sharing options...

aleph_0 Posted August 8, 2010 Author Report Share Posted August 8, 2010 (edited) I would expect no less a post from a hybrid between tensor and man. Though I'm not quite so certain that such a thing is immune to proof. We have some pretty awesome techniques for discovering things that seem impossible to discover. Whatever. An accompanying voice of reason is welcome. Edited August 8, 2010 by aleph_0 Quote Link to comment Share on other sites More sharing options...

Nate T. Posted August 8, 2010 Report Share Posted August 8, 2010 (edited) I mean more than this. For instance, sure, I can form the words, "There is a round square at the bottom of the ocean," but this is an outright contradiction and so we can reject any such hypothesis without further empirical investigation--and here I mean actual rejecting, not just ignoring. Can we do the same with the hypothesis of an injective mapping from natural numbers to disjoint physical objects? If yes, then we can take this notion to define the phrase "infinite quantity", and people on this forum have no justification for the claim that all quantities are finite, and cannot dismiss physical theories which claim it; if no, then people on the forums have been right. And again, just because I cannot produce an explicit mapping doesn't mean that one doesn't exist, so this argument has no force in the question I'm posing. Now if I were trying to positively prove that there actually is an infinite quantity, you might respond this way, but that is not my intent. ... In any case, if you concede that your only objection is the lack of evidence, then you agree with me. It is not a self-contradictory hypothesis to suppose that there is an infinite quantity. Okay, I think we both agree that to make a positive claim of an infinite number of electrons would be an arbitrary claim. It is your opinion that, if we can imagine any kind of phenomena, stipulate a definition describing it, and find no logical contradiction in its terms of definition, we can therefore define a new concept based on such imaginings? This would seem to be an appeal to the analytic/synthetic dichotomy-- and it is is not how concepts work. First you need referents, then you form concepts, then you form definitions to capture the essentials. The lack of a self-contradiction in your stipulated definition doesn't make the concepts involved any less vacuous. Now there *are* referents of "completed infinity", but all of them are concepts of method taken from advanced mathematics, not physical objects as in physics, which is why I say the notion of "one-to-one mapping" between infinite collections is being taken out of context. Thus if it is not your intent to exhibit an infinite collection of physical objects, you are using a floating abstraction and thus aren't really saying anything. Of course you're still free to imagine it, if you like, but it isn't a serious construction. Regarding infinities that may arise in the application of scientific theories such as the Big bang, these are artifacts of the modeling equations, and I think you'd be hard pressed to find scientists arguing for the existence of actual infinities based on the mathematical structure of their governing equations (although I bet there are some, such as general relativity theorists and black holes). Edited August 8, 2010 by Nate T. Quote Link to comment Share on other sites More sharing options...

aleph_0 Posted August 8, 2010 Author Report Share Posted August 8, 2010 Can I not form the concept of a dragon? I have no referent, but surely I understand the concept. Of course, I have abstracted this concept from other concepts, like lizard-like features, but that can't be the point at issue since my correspondence between infinite sets of numbers and sets of physical objects is just an abstraction from its use in pure mathematics. As for your last paragraph, it seems to confuse two distinct scenarios I provided. I wasn't appealing to their use of mathematical equations which employ infinity in order to argue that there is an infinite quantity--again, I am not arguing that there is an infinite quantity. In the first case, I was supposing a scientific theory which essentially claims that there are infinite quantities. In the second, I was providing a slightly different scenario that built on some things that we actually do know about the Big Bang. Quote Link to comment Share on other sites More sharing options...

Nate T. Posted August 8, 2010 Report Share Posted August 8, 2010 Can I not form the concept of a dragon? I have no referent, but surely I understand the concept. Of course, I have abstracted this concept from other concepts, like lizard-like features, but that can't be the point at issue since my correspondence between infinite sets of numbers and sets of physical objects is just an abstraction from its use in pure mathematics. As for your last paragraph, it seems to confuse two distinct scenarios I provided. I wasn't appealing to their use of mathematical equations which employ infinity in order to argue that there is an infinite quantity--again, I am not arguing that there is an infinite quantity. In the first case, I was supposing a scientific theory which essentially claims that there are infinite quantities. In the second, I was providing a slightly different scenario that built on some things that we actually do know about the Big Bang. Good dictionaries define dragons as *fictional* animals like big lizards that breath fire, etc. These refer to the imagination, which exists as a mental entity, not literal dragons existing somewhere. If you want to abstract the notion of one-to-one correspondence between mathematical sets and physical objects, you need one example of a completed infinity of physical objects to abstract from. Otherwise, like the dragon, your referent is imaginary, which as I've mentioned I have no problem with. Similarly, merely supposing a scientific theory or modifying an existing one to fit your argument says nothing as to the fact of the matter. If you do find an actual example of such a thing I'd be interested to see it, though. Quote Link to comment Share on other sites More sharing options...

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