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    What About Infinity?
    By Gerhard Adam | December 9th 2011 06:43 AM | 35 comments | Print | E-mail | Track Comments
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    Infinity is a useful concept but it is often used inappropriately by being assigned as a trait to some object or another.  Briefly, nothing can be infinite, since in order for something to "be", it must be defined and measurable.  If it isn't, then the object would exist in a perpetual state of creation and couldn't be said to "be" anything at all ... yet.

    This problem is aggravated by the assumption that since the concept of infinity has utility in mathematics, that it somehow represents something that is translatable and definable.  As an example, in mathematics we understand that pi is an infinite irrational number.  However, in practical terms, this is a meaningless concept.  To claim a number is infinite is to render it incomplete, or "under construction".  It makes it unknowable since its value can never be established.  It is only by making the infinite "finite", that we acquire its utility value.   In short, pi is meaningful only when it isn't infinite.   An infinite number is incomplete and unknown, and only acquires meaning when an "end" is established. 

    Therefore pi is an infinite irrational number, in principle only.  What this suggests is that the process of declaring something infinite, is not in the object itself, but rather in the steps that one can take to determine a particular value or end point.  Essentially infinity is an act of measurement or counting only and describes the condition where there is no procedural restriction on the number of iterations (counting or measuring) that one can engage in.

    Infinity is an imaginary concept that the mind has created (1).  We can't actually imagine it, since that would result in a mentally recursive process.  Even when we say that a set of numbers is infinite, we are only considering this "in principle".  Numbers don't exist except in our minds.  So, if we argue that an infinite set of numbers exists, then what do we mean if we can never actually count them?  We can recognize the absurdity of the situation by considering that an infinite number means that we could have a number larger than all the particles in the universe.  What would it mean to have a number larger than all the objects that one could actually count?  or a number larger than the time available to count it?

    In this context, infinity doesn't specifically define the set of number, but rather defines the process by which we can create the next number in any proposed sequence.  It doesn't describe the numbers; it describes the means by which they can be created.

    This causes a different perspective be applied to situations in which infinity is routinely invoked to make an argument (generally to improve the probability).  As an example, consider how infinity is often used to argue for the inevitability of life being discovered (or existing) on other planets.  After all, in an infinite universe with infinite contents, then the result must always assure that there be an infinite supply of habitable planets and civilizations.  However, if we invoke infinity, then we invoke the requirement that our claimed object (i.e. the universe) is unmeasurable, which further argues that the existence of life on other planets is correspondingly unknowable.  Basically, we're left right where we were all along.  In the absence of actually discovering life on another planet, we can gain no reassurance by the invocation of infinities to make the numbers work better.

    Infinity is an extremely useful concept when it is employed to describe a process were iterative steps, counting, or measurement are involved, but to consider it an actual trait, simply reduces the object to which it is being applied to an imaginary product of the human brain.

    ---------------------------------------------------------
    (1) This is similar to the definition of a line and point in geometry, which are dimensionless, and how it differs from "real" world expressions.

    Comments

    Bonny Bonobo alias Brat
    Infinity is an imaginary concept that the mind has created.  We can't actually imagine it, since that would result in a mentally recursive process.  Even when we say that a set of numbers is infinite, we are only considering this "in principle".  Numbers don't exist except in our minds.  So, if we argue that an infinite set of numbers exists, then what do we mean if we can never actually count them?
    Sorry Gerhard but I have to disagree with you. Infinity has to be the reality of the universe we live in, however I agree that it is a concept that some people cannot conceptualize because for them it means that an infinite set of numbers exists that we can 'never actually count' and for some reason their brains can't handle this. Some people can conceptualize an infinite universe and find it difficult to imagine a finite universe limited by an event horizon and a sign saying 'The universe ends here' when there has to be infinite something or even infinite nothing beyond it, which still has to be imagined.

    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Gerhard Adam
    Infinity CANNOT be real (as a trait), since it would indicate that the object in question is in a perpetual state of creation.  That's the point.  

    To illustrate the point, go ahead an calculate the circumference of a circle using pi as an infinite irrational number.  Let me know your answer.

    Since it clearly can't be done, then while we may talk about pi as being infinite, it isn't practical as anything other than a concept.  We can certainly say that there are no bounds to the number of steps we can take to calculate a value, but it is meaningless to attempt to use it as an actual infinite number. 

    Similarly with the concept of the universe.  To argue that the universe is "infinite" is to argue that it is unknowable, since it is impossible to measure or to say anything meaningful about it, because it isn't the same object from one second to the next.  
    ...some people cannot conceptualize because for them it means that an infinite set of numbers exists that we can 'never actually count' and for some reason their brains can't handle this.
    There's no "some people" about it.  No one's brain is capable of understanding an infinite number, since to do so would require an infinite amount of time to even express it.  Therefore people may accept the concept of an infinite number, but they can't realize an infinite number.    As I said, there's no problem in arguing that there is no limit to the processes that can be used to generate any particular number, so they are "infinite" in that sense.  But the idea of a physical expression of an infinite number is just silly.  It has no meaning.  If it can't be expressed, can it truly be said to exist?      
    Mundus vult decipi
    Bonny Bonobo alias Brat
    Infinity CANNOT be real (as a trait), since it would indicate that the object in question is in a perpetual state of creation.  That's the point.   
    I agree that infinity indicates that the universe is in a perpetual state of creation but I don't see why that can't be real? Time is an important dimension in describing an infinite universe in a perpetual state of creation because time along with space also has to be infinite. Time can never end and the universe can also never have a finite end because what lies beyond these endings, infinite no time and infinite nothing? Infinite no time and infinite nothing are still infinite, therefore an infinite universe has to be both real and in a perpetual state of creation or evolution. 
    To illustrate the point, go ahead an calculate the circumference of a circle using pi as an infinite irrational number. Let me know your answer. Since it clearly can't be done, then while we may talk about pi as being infinite, it isn't practical as anything other than a concept....There's no "some people" about it. No one's brain is capable of understanding an infinite number, since to do so would require an infinite amount of time to even express it. Therefore people may accept the concept of an infinite number, but they can't realize an infinite number. 
    Even though Pi is an infinite, irrational and transcendental number, it is still used extensively and usefully in scientific and engineering calculations and constructions, so it is still real and comprehensible. I guess that what I'm saying here is that theoretically, pretty accurate approximations could also be used to measure an infinite and perpetually creating universe at any one point in time, even if these approximate measurements become synonymous to an undefinable, infinite irrational number. They are therefore still useful for creating approximate measurements as useful concepts in our minds, to even hypothetically one day fairly accurately describe a perpetually creating universe's size, again only at one specific point in time. Or don't you agree with that either? 

    Many things around us in every day life are real but still immeasurable and perpetually changing, like the number of grains of sand in our world for example. Why should that be any more or less real than the number of grains of sand in an infinite universe? They are both approximatable, like irrational numbers but still infinite and incalculable as is an infinite, perpetually creating universe.
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    Gerhard Adam
    ...it is still used extensively and usefully in scientific and engineering calculations and constructions, so it is still real and comprehensible.
    No, which is precisely the point.  It is only useful when it is no longer an infinite number and we've arbitrarily approximated it's finite value.  It is useless as an infinite number, therefore it is infinite in principle only.
    They are both approximatable, like irrational numbers but still infinite and incalculable as is an infinite, perpetually creating universe.
    If they are incalculable or unmeasurable, then they aren't science.  They are an act of faith because they represent a trait that you claim exists that can never be demonstrated.
    Time can never end and the universe can also never have a finite end because what lies beyond these endings, infinite no time and infinite nothing.
    I'm not sure these questions have any meaning, but I'm equally sure that you have no basis for making such claims beyond your own belief system.  What is time to a photon?  What does it mean to travel backwards in time?  If physics doesn't demand a particular direction to time for interactions, then how is it you feel fine in defining that it has an "existence"?
    Mundus vult decipi
    Bonny Bonobo alias Brat
     It is only useful when it is no longer an infinite number and we've arbitrarily approximated it's finite value. 
    Yes, I agree, I was also saying that.
    I'm equally sure that you have no basis for making such claims beyond your own belief system.What is time to a photon?  What does it mean to travel backwards in time?  If physics doesn't demand a particular direction to time for interactions, then how is it you feel fine in defining that it has an "existence"?
    Oh well, I believe that one day I will have a basis for making such claims, if I keep studying physics and applying my logic, until then I'll shut up.
    My latest forum article 'Australian Researchers Discover Potential Blue Green Algae Cause & Treatment of Motor Neuron Disease (MND)&(ALS)' Parkinsons's and Alzheimer's can be found at http://www.science20.com/forums/medicine
    blue-green
    "Mathematics ... would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude."
    Nietzsche in Human, All Too Human section 11 (1878)
    Gerhard Adam
    I disagree, since from the beginning it was understood that mathematics was an idealized interpretation of the world.  It is a system of logic, not a means to represent the world around us.  Isn't this precisely why we refer to such mathematical expressions as "models"? 
    Mundus vult decipi
    blue-green
    No Gerhard, only very recently has mathematics been understood as being model inventing or various strains of logical thinking. Mathematical training used to be very much coupled with the real world or heavenly ones behind the scenes or in the stars.
     
    You have to remember, that the division of labor of mathematics into specialized fields is a very recent and modern invention. Mathematics, science, medicine, religion and good form used to be rolled into a single ball of wax. 

    Nietzsche statement is an encapsulation of the history of mathematics and remarkably prescient since it was written before Einstein, Lorentz et al. overturned the not-so physical basis of Euclidean Geometry.

    Today, we all be skeptics and cynics ... so every representation is just another bad model. Art and Science used to not be that way. Not even a little.




    As an example, consider how infinity is often used to argue for the inevitability of life being discovered (or existing) on other planets. After all, in an infinite universe with infinite contents, then the result must always assure that there be an infinite supply of habitable planets and civilizations.
     
     Well, there's a lot of controversy as to whether infinity can be used as a number. However, in many cases it means "as big as you need. So a well-contructed argument would be "After all, in a big enough universe with big enough contents, then the result must always assure that there be a big enough supply of habitable planets and civilizations." It's still implicitly in-finite because there is no limit to how big "big enough" can be.

    I don't think the use of infinity invalidates the argument. However the "big enough" form does reveal where the argument itself falls down. I would rebut it by saying:
    You say: If the expected density of inhabited planets is D and the volume of the univere is V then the expectation value is DV inhabited planets. You then let V go to infinity (see? no need for actual infinity!) and assume that DV increases without limit. But you are assuming that D is constant. There may be enough variable conditions in the universe for the integral Tobe o finite.




     
    Gerhard Adam
    I'm not sure I see how infinity helps here.  After all, my point is that infinity simply represents an object in the perpetual state of creation.  So, in effect, the argument is simply ... wait around long enough and something will be created that accounts for the phenomenon we're looking for.

    Mundus vult decipi
    I'm kind of agreeing with you although I don't agree that the "argument within the argument" is valid. You have to make certain assumptions about the probability, namely that the integral over time increases without limit. That forces us to consider which scenarios do this and which don't. Bandying the word "infinity" around doesn't give us licence to ignore the probabilty function.
    Larry Arnold
    What the real problem is, is that we with our cognitive and structural limitations cannot conceive of a way to describe what is out there that does not involve the rather neat "fiction" of mathematical relationships.

    I wonder what the fact that I can conceive of the possibility of there being alternate and incomprehensible (in a Rumsfeldian sense) alternatives, says about human cognition? Probably that it is profoundly faulty in some way.

    When things work, it is convenient, when they don't we go hunting for explanations, even to the extent of setting up huge experiments, however we don't have a lot of choice in the matter, but for all we know it might look very different if we had a cognitive process capable of modelling in this currently mythical unknown way.

    Now where did I put my index of books that don't index themselves?

    There is of course an infinite number of things that can be considered infinite is there not? (nowhere does it say in the book of rules that any of these infinite things have to actually exist. I can name a new number that contains properties that no existing number contains, (for that is it's definition) but having named it, I haven't done any more than create another creature in the mediaeval bestiary)
    Gerhard Adam
    Which is precisely why I suggested that infinity describes a process we can invoke that has no bounds and which can be used to create the object of our interest.  However, to assign infinity as a trait to an object places us (and the object) in the untenable position of being in a perpetual state of creation.  There is no point where we can say ... 'that's the object I mean', since it is never the same object from moment to moment.
    Mundus vult decipi
    vongehr
    "nothing can be infinite, since in order for something to "be", it must be defined and measurable."
    That is all you needed to write. ;-)
    If you consider that infinity pops up in the form of singularities (e.g. infinite density in black hole solutions of classical general relativity), you can add strong support for your thesis here by pointing out that singularities are unobservable (would need infinite resolution) and thus cannot belong to proper operational science as more than a mathematical trick. In fact, refusing singularities always turns out to be the correct next step (e.g. quantum mechanics and for example string theory have shown that indeed general relativity is wrong about the singularities in black holes.)
    Gerhard Adam
    You're just trying to get my posts to be shorter and shorter :)
    Mundus vult decipi
    singularities are unobservable (would need infinite resolution) and thus cannot belong to proper operational science as more than a mathematical trick
    "Operational", of course, is Sascha Vongehr's private definition of science :) On the first other hand, I am sure he will not object when a ToE turns out to be tautology, neither suggesting nor requiring experimental verification, indeed incapable of same. On the second other hand, the proposition "the universe is infinite" does not seem to depend on a singularity in the sense used here.
    blue-green
    1/2 + 1/4 + 1/8 + 1/16 .... = 1
    This is one of the easiest infinite series to sum. Geometrically, one can visualize it by starting with the answer, one object, and then cutting it in half, and then cutting one of the two halves in half, and then cutting one of the remaining 1/8 th bits in half .... ad nauseum.

    This little trick allows one to see how any single object can be the sum of many, even infinitely many. One could have of course used less precise cutting tools and realized the number 1 as the sum of all of the bits of any process in which you kept cutting away at the remainder. Really, that's the whole point of calculus: the sum of an infinite number of terms can be finite ... to wit, anything finite can be expressed an infinite sum .... and it can be done so in an infinite number of ways.

    Physically we know that we cannot do this subdividing ad nauseum. Democritus (the Laughing Philosopher) postulated the existence of atoms on the hunch that subdivision would come to an end. 
    Today we know that if have a sufficiently powerful hammer, we can keep smashing into bits much smaller than the level of atoms .... although the blow itself introduces energy and bits. 
    With quantum Electro-Dynamics QED, the simplest process of an electron going from one state to another (completely intact) is the sum of ever tinier subprocesses which can be summed up to create a sequence that converges on a real number, that can be tested and confirmed experimentally. An electron has no difficulty with all of this becoming even though an infinite number of subtle processes are involved.

    Just saying all of this .... for a little more balance in the reporting.

    And mind you, infinitely large things can easily be reduced to something small and finite by applying a simple mathematical transformation. The compactification of an infinite plane into a sphere is an example to keep in mind. So whether something is infinite or finite is more a choice of how you wish to describe it. Fundamentally, it's neither.












    Gerhard Adam
    Geometrically, one can visualize it by starting with the answer, one object, and then cutting it in half, and then cutting one of the two halves in half, and then cutting one of the remaining 1/8 th bits in half .... ad nauseum.
    I'm not sure what your point is, since I was quite clear that infinity serves quite well as a concept.  It is when it is used as a trait that it fails.  You're simply arguing about the concept, which I never disagreed with.
    Mundus vult decipi
    blue-green
    From my post, you can see how even Finiteness is questionable as a trait, even though it is a useful concept.
    It felt like a revelation to me when I first saw it.
    We say that "pi" is an infinite number. How can something be infinite in one direction only? What is the difference between a line infinite in one direction and line infinite in both directions? If one could illustrate this in reality, there should be no difference. How can something infinite have a beginning?

    So yes, infinite is a mathematical concept as far as I can understand this. Trying to apply it to the "real world" just does work.

    Gerhard Adam
    How can something be infinite in one direction only?
    Within the framework of concepts, it's not impossible.  Clearly the set of all even numbers is a smaller "infinite" set than the set of all real numbers. 

    However, in the end, it all begs the question, because it isn't about whether we can conceive of something that is infinite, or how we try to imagine it.  It's only about whether it has a "real world" meaning, which it doesn't.
    Mundus vult decipi
    Clearly the set of all even numbers is a smaller "infinite" set than the set of all real numbers.
    Really?  It's not obvious at all to me.
    It's only about whether it has a "real world" meaning, which it doesn't.
    It has all the meaning it needs no matter how often you assert that it does not.
     

     
    Mathematically, the set of all even numbers is defined as being smaller than the set of all real numbers. It is part of the way math works. Infinite is mostly a mathematical concept. It's also useful in some philosophical discussions. However, human beings have no way to picture "real" infinity, assuming there is such a thing, because everything humans do is finite.
    I agree that it seems that a subset of an infinite series should be infinite also. Mathematically, that's not true. If you are talking more to philosophy, you might be able to win the argument. It's all about definitions. Like black holes, quantum mechanics, etc. These are things humans really cannot grasp so we define the phenomena, give it rules and experience it within these rules.
    What real-world meaning does infinite have? There is nothing infinite in our lives.

    Mathematically, the set of all even numbers is defined as being smaller than the set of all real numbers.
    Of course it isn't. It may be smaller but it certainly is not defined that way.
    I agree that it seems that a subset of an infinite series should be infinite also.
    I don't know whom you imagine you are agreeing with but I have not said such a thing (neither would I).
    These are things humans really cannot grasp so we define the phenomena, give it rules and experience it within these rules.
    Oh puh-lease!
    There is nothing infinite in our lives.
    That's settled then. The universe must be finite because... because there is nothing infinite in our lives! Anonymous has spoken.
     

    I stand corrected. There is at least one thing that is infinite......

    ... and it isn't the universe.

    Thank you Einstein.
    How can something infinite have a beginning? 
    Very easily.

    Before 6000 years ago there was no universe.
    For all time since creation there has been and will continue to be a universe.
    So when did God begin?

    Beer.

    Peyote?

    You are welcome (from Einstein).

    Yes, that is exactly what I mean--and I like your example. Infinity is a mathematical concept.

    Hello Adam, i´m very sorry to bother you with a question that is not related to your article itself, and i´m especially sorry to bother you with my bad english skills (i´m just a german teenager searching for an answer) but i would really appreciate it if you could help me with a question that popped in to my head yesterday:

    If you would generate random numbers between 1 and infinity, would the result always be an infinite number?

    I think that it has to be that way, but i can´t really formulate it, especially not in english.
    Thank you in Advance.

    Gerhard Adam
    If you would generate random numbers between 1 and infinity, would the result always be an infinite number?
    Yes.  Mathematically the issue of infinity is a bit different since there can be many different kinds and some can be larger than others.  For example the set of all real numbers is infinite and yet it includes the set of all even numbers which is also infinite.  This involves set theory, so for a basic explanation perhaps this link will help.
    http://en.wikipedia.org/wiki/Infinity

    http://en.wikipedia.org/wiki/Cardinality

    BTW, your English is very good.
    Mundus vult decipi
    thank you very much for your answer, because this question was crucial to my "understanding" of infinity- don´t get me wrong I know that we humans can only grasp the concept of infinity to a certain degree, but exploring the nature of infinity is always mind-blowing for me :D