• Fuzzball Baggins
    12
    The measure problem is an issue that happens with infinite sets, where the ratio of different categories in the infinite set is measured differently depending on which way you order the set. Take for example the task of measuring the ratio of odd to even integers. If you order all numbers with all the odd ones first, and then set a cut-off point (say the first ten numbers) and count the ratio in that sub-set, you will get 100% odd. If you order the set in a different way, you may get a different ratio.

    I've read one physicist claiming that this means that existence, time, space, everything must be finite, because infinite sets are logically contradictory, as you can apparently change their ratios by changing the order in which you look at them.

    However it has occurred to me that the measure problem would apply equally well to a large finite set, say a set of a billion integers which it would take a very long time to actually count in order to determine the correct ratio of odd-to-even. So I suspect that the measure problem doesn't imply that there is something inherently illogical about infinite sets at all, I think that infinite sets have concrete ratios between their categories just like a finite set would, and the measure problem simply represents the fact that if you order the set in a way that doesn't reflect its real ratios you will get an incorrect answer. Similar to what would happen if you sent our a survey to a sample of people who don't accurately represent the population.

    What do you guys think? Anything wrong with my reasoning? Anything I've missed?
  • SophistiCat
    579
    I've read one physicist claiming that this means that existence, time, space, everything must be finite, because infinite sets are logically contradictory, as you can apparently change their ratios by changing the order in which you look at them.Fuzzball Baggins

    I wonder who would say such a thing. Where did you read this?

    What do you guys think? Anything wrong with my reasoning? Anything I've missed?Fuzzball Baggins

    Yeah, you missed, or rather forgot, your own argument showing that some measures just aren't well-defined. This doesn't imply anything logically contradictory, of course, only that not every measure that you care to describe is well-defined.

    The measure problem in cosmology is not that you can't come up with some well-defined measure - there is no lack of candidates. The problem is in coming up with a physical justification for a specific measure - and that's a scientific problem.
  • Fuzzball Baggins
    12
    Hi SophistiCat, the physicist who said it was Max Tegmark. The first half of his book Our Mathematical Universe is very good, but in the second half he starts extrapolating a bit too much from the measure problem, and seems to think that any set that has the measure problem can't actually exist.

    Your definition of the measure problem makes much more sense!
  • Metaphysician Undercover
    4.7k
    I've read one physicist claiming that this means that existence, time, space, everything must be finite, because infinite sets are logically contradictory, as you can apparently change their ratios by changing the order in which you look at them.Fuzzball Baggins

    "Infinite set" is self-contradictory. "Infinite" implies unbounded, and set implies "bounded". To say that there is an infinite set is like saying that there is an infinite object, the two concept "infinite" and "object" contradict each other, such that this is impossible.
  • Fuzzball Baggins
    12
    Why do sets have to be bounded? What about the set of all integers - is that not a proper set?
  • Metaphysician Undercover
    4.7k

    We went through this recently on a different thread. Let's say that "set" is defined as a "well-defined collection", as Wikipedia suggests. A "collection" in the sense of a noun implies having been collected, so an infinite collection is impossible because the act of collecting cannot be complete, and such a collection cannot exist. "Collection" in the sense of a verb, meaning the act of collecting, cannot be construed as an object, a "set", because this would be a category mistake. So an "infinite set", as an infinite collection in the sense of an object, is impossible by contradiction, and it is impossible as a "well-defined" activity because it is an incomplete activity.
  • ssu
    774
    However it has occurred to me that the measure problem would apply equally well to a large finite set, say a set of a billion integers which it would take a very long time to actually count in order to determine the correct ratio of odd-to-even.Fuzzball Baggins
    Perhaps with numbers and mathematics one should stick to the logic of math itself and not bother about physical time and physical doing, of what kind of numbers our present day computers or computers of the future can handle. Even a atural number that is one hundred thousand digits long can be problematic for us to handle and our Computers to handle, yet the logic of the number is totally similar to a natural number that is two digits long, basically one between 0 and 99. Otherwise you will start looking for the quite illogical "first too big number that cannot be handled by a computer".
  • frank
    1.8k
    However it has occurred to me that the measure problem would apply equally well to a large finite set, say a set of a billion integers which it would take a very long time to actually count in order to determine the correct ratio of odd-to-even.Fuzzball Baggins

    It takes a lot longer if you're counting backwards.
  • Terrapin Station
    5.2k
    in the second half he starts extrapolating a bit too much from the measure problem, and seems to think that any set that has the measure problem can't actually exist.Fuzzball Baggins

    Is this a good time to point out thar sets are something we make up, so they only exist insofar as someone is imagining them?
  • Fuzzball Baggins
    12
    Yeah, it probably is. Mathematics represents reality so well that I think sometimes people get carried away.

    I suppose this is really more a discussion of the definition of the word set rather than whether the universe could be infinite, so I'll agree with you that with the definition that humans have given the word set, the term 'infinite set' is illogical :P
  • Metaphysician Undercover
    4.7k
    I suppose this is really more a discussion of the definition of the word set rather than whether the universe could be infinite, so I'll agree with you that with the definition that humans have given the word set, the term 'infinite set' is illogical :PFuzzball Baggins

    Therefore, we have "the measure problem". Doesn't it make sense to rid our mathematics of such illogical axioms? When we realize that such things are illogical, we can apply the same principle in other areas. Consider "the universe" for example. As such, it is an identified and named object. It cannot be infinite according to a very similar contradiction as mentioned above. If it were infinite it could not be individuated and identified as an object, it's existence would be indefinite and therefore not an object which has definite existence. So to speak of 'the universe" is to speak of an object, and an object cannot be infinite. Therefore by the same reason that it is illogical to talk about an infinite set, it is also illogical to talk about an infinite universe.
  • SophistiCat
    579
    Be ware that MU's "definitions" are his own. If for some reason you want to know what MU thinks about mathematics (or anything else for that matter), then by all means read what MU has to say about it. If you want to know something about the subject as such, look elsewhere.
  • Fuzzball Baggins
    12
    I'm not sure I agree that an infinite thing is ill-defined, or that the multiverse is an object, or that objects have to be bounded. It might be difficult to imagine an infinite thing but I don't think it breaks the laws of physics. Not that I'm saying you are wrong, just that I personally am unconvinced.
  • Metaphysician Undercover
    4.7k

    There is surely an indefinite aspect of infinite, which is not so commonly developed in talk of "infinite". One definition of indefinite is limitless, and, something which continues indefinitely is infinite.

    An object must be bounded, because it is an individual, a unity, a whole. Without these conditions it is indefinite. It's not an issue of what can be imagined, or the laws of physics, but it's an issue with the laws of logic, specifically the law of identity. When we identify an object, we point it out, then proceed to describe it by assigning properties or attributes. "Indefinite" refers to what we cannot grasp, what is beyond our apprehension. So, when we assign to an object, the property or attribute of "indefinite", we are saying that there is something about that object which is impossible to apprehend.

    This is an act of judgement which is made, the object is judged as incomprehensible. It does not mean that the object really is incomprehensible, it has just been judged as incomprehensible. This is a self-defeating judgement. It impairs the will to understand the object, by identifying it as not understandable. Further, if any aspect of the object appears to be incomprehensible, illogical, or logically inconsistent with another aspect of the object, we can accept these logical inconsistencies of the object, by concluding that they are due to the indefiniteness of the object.

    Therefore it is completely unreasonable to identify an object as indefinite, or to in any way assign "indefiniteness" to an object. We must assume that the appearance of indefiniteness is due to our inability to understand, and not part of the object itself. This will inspire us to continue to try and understand the object, to develop our minds rather than just assuming it is impossible to understand. And, from the other perspective, if the object really is indefinite, and therefore impossible to understand, we would never be able to know this with certainty, because this would require knowing the object which cannot be known. So it is completely unreasonable to assume that any object is indefinite, or infinite, no matter how you look at it.
  • Fuzzball Baggins
    12
    Okay, I agree with you that an object should be bounded. But I don't consider the multiverse to be an object. In fact, I think that the multiverse is a mathematical construct and that all the objects that we perceive around us are just emergent behaviours of the mathematical laws governing our reality.
  • Metaphysician Undercover
    4.7k

    Actually I don't really know what a multiverse is. In one way, "multi" implies a multiplicity of objects, but also in another way "the" implies a single object. It's probably self-contradictory like infinite set.
  • Fuzzball Baggins
    12
    I think you're taking the English language a little too far, using its structure to decide what can and cannot exist in reality. Think about your reasoning: I chose to say 'the' multiverse, which leads you to think that the multiverse is an object, and therefore bounded, and therefore an infinite multiverse is a self-contradictory concept. So you're deciding that an infinite multiverse doesn't exist simply because I used the word 'the'?
  • Metaphysician Undercover
    4.7k

    No it's not quit like that. What I said is that "the" implies a single object, while "multi" implies a multitude. I could have just as easily said that the name "multiverse" identifies a single object, so it's not really "the" which is the problem, "the" was just an indicator. It indicated that "multiverse" is a name which identifies an object.

    So it appears, at first glance, that there may be an issue with self-contradiction, because it is suggested that a multitude of objects is a single object. However, we do commonly speak of a multitude of objects as a single object, that's what happens in arithmetic; 2, 3, 4, etc., are each representative of a single object, a number, but each number defines a multitude as well. What happens with "infinite" is that the multitude is undefined, and even specified as undefinable. But the object, the particular number, 10, 15, 25, or whatever, only has existence because it defines a multitude. Its very existence, as a number, is completely dependent on its capacity to define a multitude. If any such number which is signified by a numeral, "6", "7", "8" etc.,, did not define a multitude, it would not exist as an object. "Infinite" signifies an undefined multitude. So by the very fact of what it signifies, the possibility of it being an object is denied. What "infinite" signifies is "it is impossible that I am an object like a number", because a number necessarily defines a multitude while "infinite" necessarily does not.. .
  • tim wood
    1.3k
    A "collection" in the sense of a noun implies having been collected, so an infinite collection is impossibleMetaphysician Undercover

    How so impossible? You and I can each think about all the positive numbers. That well-defines a set, collected in thought. If you mean collectible in hand or basement or warehouse somehow, then you needn't resort to concepts of infinity to have uncollectible sets.
  • Metaphysician Undercover
    4.7k

    I haven't collected them all yet in my mind. So how could they be collected in my thought? Furthermore, "all the positive numbers" does not qualify as "well-defined" in a mathematical sense, because how many positive numbers that there are is indefinite.
  • tim wood
    1.3k
    I haven't collected them all yet in my mind.Metaphysician Undercover
    I have them collected in my mind. But I do not understand your claim. A large number of grains of sand of sand is certainly collectible. But in terms of your apparently empirical criteria, they're uncountable - I guess you can't have a large pile of sand, yes?

    Furthermore, "all the positive numbers" does not qualify as "well-defined"Metaphysician Undercover
    Says who besides you? Your opportunity to educate.
  • Metaphysician Undercover
    4.7k
    I have them collected in my mind.tim wood

    You have all the positive numbers collected in your mind!? Can you list them then?

    A large number of grains of sand of sand is certainly collectible.tim wood

    My claim is quite simple. A large number of grains of sand is collectible. An infinite number is not.

    I guess you can't have a large pile of sand, yes?tim wood

    Do you understand that there is a significant difference between "a large pile of sand", which obviously has a finite number of grains of sand, (as any pile of sand does), and "an infinite number of grains of sand"? The latter is not a pile of sand.

    Your opportunity to educate.tim wood

    If you will listen, I will oblige.
  • tim wood
    1.3k
    You have all the positive numbers collected in your mind!? Can you list them then?Metaphysician Undercover
    Which ones would you like me to list?

    Do you understand that there is a significant difference between "a large pile of sand", which obviously has a finite number of grains of sand, (as any pile of sand does), and "an infinite number of grains of sand"? The latter is not a pile of sand.

    Your opportunity to educate.
    — tim wood

    If you will listen, I will oblige.
    Metaphysician Undercover
    I know the difference - I'm by no means sure you do. And your "if you listen" in response to my question to you as to who besides you says so - your answer in response to that question - is simply an example of what I've experienced as your toxic style of discussion. You made a claim,
    My claim is quite simple. A large number of grains of sand is collectible. An infinite number is not.Metaphysician Undercover

    I ask you who besides you says so. And you do not answer. Answer or Hitchen's razor awaits you.
  • Fuzzball Baggins
    12
    Your claims seem a little arbitrary. Especially your claim that the multiverse being seen on the one hand as a multitude and on the other as a single object makes it self-contradictory. A bunch of bananas is both a single object and a multitude of bananas.

    I don't think it's a good idea to rely on human intuition when it comes to physics. The human brain has evolved to cope with our everyday experiences, not with the laws of physics. Think about relativity and quantum mechanics - very unintuitive!

    Do you have any logical reasoning (not involving human intuition, but based on the laws of physics or mathematics) for why an infinite thing could not exist in reality?

    I don't think the concept of a set having to be 'collected' quite applies to what can and cannot exist in reality. I may not be able to create an infinite collection, or even imagine all the members of an infinite set, but reality doesn't have to 'collect' anything - infinite things can exist simultaneously without having to be created one by one.
  • Metaphysician Undercover
    4.7k
    Which ones would you like me to list?tim wood

    All of them of course. I want you to prove to me what you claimed. "I have them collected in my mind."

    I know the difference - I'm by no means sure you do. And your "if you listen" in response to my question to you as to who besides you says so - your answer in response to that question - is simply an example of what I've experienced as your toxic style of discussion. You made a claim,tim wood

    If what I say is true, then what difference does it make, whom, besides me, says so? Can you not read, and judge what I say, for yourself, without requesting an appeal to authority?

    I ask you who besides you says so. And you do not answer. Answer or Hitchen's razor awaits you.tim wood

    I'll take the razor, and here's my proof. I'll reproduce from above, as it appears like you haven't read the thread. Tell me which part you dislike

    So it appears, at first glance, that there may be an issue with self-contradiction, because it is suggested that a multitude of objects is a single object. However, we do commonly speak of a multitude of objects as a single object, that's what happens in arithmetic; 2, 3, 4, etc., are each representative of a single object, a number, but each number defines a multitude as well. What happens with "infinite" is that the multitude is undefined, and even specified as undefinable. But the object, the particular number, 10, 15, 25, or whatever, only has existence because it defines a multitude. Its very existence, as a number, is completely dependent on its capacity to define a multitude. If any such number which is signified by a numeral, "6", "7", "8" etc.,, did not define a multitude, it would not exist as an object. "Infinite" signifies an undefined multitude. So by the very fact of what it signifies, the possibility of it being an object is denied. What "infinite" signifies is "it is impossible that I am an object like a number", because a number necessarily defines a multitude while "infinite" necessarily does not..Metaphysician Undercover

    Your claims seem a little arbitrary. Especially your claim that the multiverse being seen on the one hand as a multitude and on the other as a single object makes it self-contradictory. A bunch of bananas is both a single object and a multitude of bananas.Fuzzball Baggins

    Yes, I went through this, I reposted it above in case you didn't read it through.

    Do you have any logical reasoning (not involving human intuition, but based on the laws of physics or mathematics) for why an infinite thing could not exist in reality?Fuzzball Baggins

    I went through this already. It is unreasonable to assume that any thing is infinite because such an assumption impedes our capacity to know that thing, and it is also impossible to know that a thing is infinite. So it's not the case that it is impossible that an infinite thing exists, in reality, but it is impossible to know that any given thing is infinite, and detrimental to the understanding of that thing, to assume that any given thing is infinite. Therefore it is unreasonable to assume that there is an infinite thing in reality.

    I don't think the concept of a set having to be 'collected' quite applies to what can and cannot exist in reality.Fuzzball Baggins

    Of course it applies. We create descriptive terms, and the laws of logic to reflect reality. When something is contradictory, like a square circle, we say that it is impossible because it cannot exist in reality. So, if we produce a concept of descriptive terms which contradict (a contradictory description), we say that this thing cannot exist in reality. That is the case with "infinite collection". As a noun "collection" implies having been collected, as a verb "collection" implies the act of collecting. The noun contradicts "infinite" and the verb when qualified by "infinite" signifies an indefinite act.

    I may not be able to create an infinite collection, or even imagine all the members of an infinite set, but reality doesn't have to 'collect' anything - infinite things can exist simultaneously without having to be created one by one.Fuzzball Baggins

    You can say whatever you want about the "infinite thing", that's the problem with assuming an infinite thing. Because the thing is indefinite, it cannot be properly identified, and laws of logic cannot be applied. That is why assuming an "infinite thing" is detrimental.
  • tim wood
    1.3k
    All of them of course. I want you to prove to me what you claimed. "I have them collected in my mind."Metaphysician Undercover

    all positive numbers of the form 2n, n being any integer.

    I ask you who besides you says so. And you do not answer. Answer or Hitchen's razor awaits you.
    — tim wood

    I'll take the razor, and here's my proof. I'll reproduce from above, as it appears like you haven't read the thread. Tell me which part you dislike
    Metaphysician Undercover

    The razor, then. "What is averred without evidence can be dismissed without evidence." Btw, a request for a respectable source is perfectly reasonable.
  • Metaphysician Undercover
    4.7k
    all positive numbers of the form 2n, n being any integer.tim wood

    I asked for the list, not a description of it.
    The razor, then. "What is averred without evidence can be dismissed without evidence." Btw, a request for a respectable source is perfectly reasonable.tim wood

    Did you read my proof? I do believe that a logical proof qualifies as "evidence".
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal
youtube
tweet
Add a Comment