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Even Logic Has Limits

  1. Jun 21, 2003 #1
    The laws of nature reflect the intrinsic properties of everything which exists. Logic is the interpretation of those laws. By observing, defining and comparing the nature of that which we seek to understand, we derive knowledge which can be applied to familiar circumstances to predict outcomes. These mental equations, which are simultaneously solved for all known variables, produce conclusions. Valid conclusions usually fit all of the parameters of our empirical observations.

    There is; however, an attribute of nature which does not readily lend itself to rational analysis - Infinity. Infinity is not an existence per se, it is a concept which defies logical interpretation. It is not exempt from the laws of nature and it is not contrary to logic, but it lies beyond the domain of logic because it is not defined - and logic requires definition. It is hard to fathom that although there is a finite distance between every two points in the Universe, there is no furthest point; and the very fact no ‘point of infinity’ exists serves only to validate the concept.

    On the other extreme, we have 'nothing'. In its absolute sense it does not exist - it has no attributes, so it, also, is not defined. There is a relative - or logical - definition of 'nothing'. It is Ø or the empty set.

    Logic is a derivative of reality. When you integrate a derivative (basic calculus) you lose something in the translation (usually a constant).
  2. jcsd
  3. Jun 21, 2003 #2


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    Infinity is quite well defined mathematically, thank you. The only problems that arise are when you naively attempt treat an infinite quantity/collection/whatever as if it was finite.

    There are several mathematical fields that deal with infinite / infinitessimal things routinely, and quite beautifully... though I have a hunch that http://www.ugcs.caltech.edu/~shulman/pub/writings/core1/nonstandard.html would appeal to you the most.
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  4. Jun 21, 2003 #3
    'Xcuse me, but infinity - by its very definition - is UNdefined (else tell me where it is and how much of it I can gather).
    "If, for every X there is an X+1" is a derivative - NOT a definition. What is the highest number??

    Similarly, there is a concept of 'nothing' which is UNdefined. To define 'nothing' in absolute terms is NOT to define. In its relative context 'nothing' is Ø or the 'empty set', but in reality nothing is......
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  5. Jun 22, 2003 #4
    There is one example of infinity in nature which everyone knows and seemingly accepts without question. We generally accept that future time is infinite. Time will continue running on and on and on without end (ie infinitely). Nobody seems to have a problem with this.
  6. Jun 22, 2003 #5
    Yes - this is a concept - not a definition. If the end of time is not limited it is UNdefined (undefined=without limit or boundary)
  7. Jun 22, 2003 #6


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    That sounds like a definition to me! :wink:

    But seriously, if that is your criteria for being undefined, then I retract my objection, since your usage of "undefined" has no relationship to the mathematical usage of "undefined".

    I'm not even sure your definition of "undefined" means what you wish it to mean: For example, I can easily define the sequence of alternating 1's and -1's by saying the n-th term is (-1)n... but this sequence does not have a limit, so it is undefined by your meaning.

    Also, the surface of a sphere is a topological space without boundary, but are you comfortable calling surfaces of spheres undefined?
  8. Jun 23, 2003 #7
    No - such surfaces are defined.

    Every point on (or, for that matter IN) the sphere can be defined by a specific x/y/z position relative to some other given point in space.

    Infinity - being UNdefined, cannot be so described.
  9. Jun 23, 2003 #8
    First off, Messiah, I agree that Logic has limits.

    Secondly, I'd like to point out that one cannot logically find the flaws in logic, because they would be using that which they denounce to denounce itself (Godel's Incompleteness).

    Lastly, I'd like to point out that infinity is, in fact, defined. It may not be conceptualizable (if that's a word) to you, or any other human (perhaps), but it is still defined, otherwise we couldn't speak of "it" (IOW, we would have no concept to refer to as "undefined", unless we had a definite concept...paradox).
  10. Jun 23, 2003 #9
    Infinity is not an existence, per se. It is a derivative - a concept - not an entity. Entities have quality, quantity and position in the Universe. Logic is simply the observation, definition and comparison of those attributes - from which we forumlate 'knowledge' in order to predict 'outcomes' (cause & effect).

    For every entity in the Universe, you may calculate coordinates of location, you may determine a volume, you may describe its physical properties (even inertness is a property). You cannot point to a location, distance or volume in the cosmos and say "this is infinity" - it is UNdefined.
  11. Jun 23, 2003 #10
    Oh really? Then the Universe is not an entity either, is it?

    Erm...nuh-uh. Logic is the use of a reasoning system (to put it basically). Science is the observation, definition, and comparison of those attributes.

    Don't you see that you are defining properties of that which you say cannot be defined. Even if all we could do was say what "infinity" wasn't (though we actually can say what it is, and I'll attempt that in a moment), we would still be defining it. Much like true "irrationality". It can only be defined by what it isn't, but it is still defined (see last few pages of "I think therefore I am", and the thread, "what is irrational?").

    Now, the definition of infinity: That which has no end.

    Pretty simple, isn't it? I don't see anything wrong with that definition, do you?
  12. Jun 23, 2003 #11
    If 'an entity' denotes singular, then of course not. It is ALL entities.
    Science and mathematics are both ways encoding logic to facilitate the transfer if ideas and information.
    Yes. The term 'that' indicates a thing - a physical reality. A circle has no end, yet it is not infinite - it is definable. The best definition of infinite is 'not finite' - which also means not defined (from the same word root)

    Function: adjective
    Etymology: Middle English 'finit', from Latin 'finitus', past participle of 'finire'(to finish)
    Date: 15th century
    1 a : having definite or definable limits <finite number of possibilities> b : having a limited nature or existence <finite beings>
    2 : completely determinable in theory or in fact by counting, measurement, or thought <the finite velocity of light>
  13. Jun 23, 2003 #12

    That is mind boggling isn't it? If we were to accept that there is no infinty. That everything has an ultimate end. Such as space, we would then have to accept that time also, has an ending. That there will come eventually a finality where time ends and stands still. When all things cease to exist.

    I have to agree here, that though we give a definition to infinity, it is by definition the connotation of things we cannot put a definitive answer on. We say that something is infinite because we cannot find a finite solution. If we were to travel to the edge of the universe and discover the end, then the universe becomes finite. Until we do find the end of something, it is infinite. It's just the answer to all things we cannot answer.

    Then of course the question arises: Mathmatically aside, How can we disprove infinity? It becomes Paradoxical, because if we find the answer, it is no longer infinite, but as long as we cannot find a solution or and end, it still remains infinite. It is by definition of itself, the unproven variable. And if something is truly infinite, it can never be disproved. And thus, we except it.
    Last edited: Jun 23, 2003
  14. Jun 23, 2003 #13
    Cosmologists and countless others have a problem with this. Some believe, for example, that time began with the big bang and will end with the big crunch. Infinity is commonly accepted by the public at large, but then, they accept a lot of demonstrably irrational ideas. Notably, one of the most common sources of belief in infinity is among the religious. Whether or not infinity is a rational aspect of nature is a question of debate, especially among physicists who consider infinities cropping up in their equations to be a sign they did something wrong.

    There is, however, one pointed limit to logic as a description of nature and that is the power of reason. My computer, for example, can perform logical operations way beyond my human capacity but it has no capacity to reason. No capacity to give anything meaning and context.

    Logic itself is derived from reasoning and is ultimately based on reductio ad absurdum, reduction to the absurd. Without the context and meaning of the absurd logic has no structure much less meaning. To assert that logic and infinity describe everything then, is to imply that infinity itself is absurd. That I can agree with.
  15. Jun 24, 2003 #14
    Will that be something like the king-size Crunchie - mouth watering honeycomb surrounded by luscious chocolate?
  16. Jun 24, 2003 #15


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    First off, your definition has nothing to do with having a boundary.

    Secondly, it's very easy to give coordinates to spaces that include coordinates for point(s) at infinity. For a simple example, consider the extended real numbers; that is the ordinary real numbers plus two extra points, +&infin; and -&infin;.

    I can re-coordinatize the real numbers as follows: for every real number r, I assign it the coordinate y = arctan(r).

    This recoordinatization condenses all of the real numbers into the interval (-&pi;/2, &pi;/2)... and it maps +&infin; to the coordinate &pi;/2 and -&infin; to the coordinate -&pi;/2.

    Now, if we're infinityphobes, this construction maps +&infin; and -&infin; down to ordinary finite numbers and we can manipulate them as such... though I imagine it's much easier to manipulate them according to the definition of the extended real numbers.
  17. Jun 24, 2003 #16
    That would be GREAT; however &infin; is NOT a point. If it was a point, it would be defined. &infin; is a symbol, not a value.
  18. Jun 24, 2003 #17


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    I've defined it as a point, thus it is a point. :smile:

    But I think the point of your objection is that you think that your concept of infinity does not correspond to what I've defined.

    So I ask you, what is wrong with it? As I've defined it, +&infin; is to the right of every point on the (ordinary) number line, and -&infin; is to the left of every point on the number line. Any sequence of real numbers that increases without bound converges to +&infin;, and any sequence of real numbers that decreases without bound converges to -&infin;.

    In the context of spatial position (in which you were using the term "infinity"), what does my definition of &infin; lack?
  19. Jun 24, 2003 #18
    I think the point that is being made is that infinity, though definable in abstract mathmatical terms, has no real world definition.
  20. Jun 24, 2003 #19


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    I would accept mathophobia if Messiah wasn't directly claiming math couldn't define infinity. :smile: I've used infinity (and infinitessimals) countless times, and they behave exactly like I think I should behave (including disappointing me because they don't behave exactly like finite things).
  21. Jun 25, 2003 #20

    Both Messiah and Hurkyl are both right and wrong. It is an argument about language. Here are online Merriam-Webster dictionary entries for 'infinite' and 'infinity'.

    Main Entry: in·fi·nite
    Pronunciation: 'in-f&-n&t
    Function: adjective
    Etymology: Middle English infinit, from Middle French or Latin; Middle French, from Latin infinitus, from in- + finitus finite
    Date: 14th century
    1 : extending indefinitely : ENDLESS <infinite space>
    2 : immeasurably or inconceivably great or extensive : INEXHAUSTIBLE <infinite patience>
    3 : subject to no limitation or external determination
    4 a : extending beyond, lying beyond, or being greater than any preassigned finite value however large <infinite number of positive numbers> b : extending to infinity <infinite plane surface> c : characterized by an infinite number of elements or terms <an infinite set> <an infinite series>
    - in·fi·nite·ly adverb
    - in·fi·nite·ness noun

    Main Entry: in·fin·i·ty
    Pronunciation: in-'fi-n&-tE
    Function: noun
    Inflected Form(s): plural -ties
    Date: 14th century
    1 a : the quality of being infinite b : unlimited extent of time, space, or quantity : BOUNDLESSNESS
    2 : an indefinitely great number or amount <an infinity of stars>
    3 a : the limit of the value of a function or variable when it tends to become numerically larger than any preassigned finite number b : a part of a geometric magnitude that lies beyond any part whose distance from a given reference position is finite <do parallel lines ever meet if they extend to infinity> c : a transfinite number (as aleph-null)
    4 : a distance so great that the rays of light from a point source at that distance may be regarded as parallel

    Some of these preclude assigning the words to a definite object and some allow it (infinite or transfinite number).

    A dictionary tells us how words are actually used, not how they (for some reason) ought to be used.

    George Cantor contemplated this and chose to go with the word 'transfinite'. He contemplated an absolute infinity that would transcend all actual sets, a perfect continuum. It doesn't really exist. A set theoretic approach handles this nicely. The ultimate infinity for ordinal numbers is ordinal but not a proper set. Therefore, one cannot add 1 to it{1}.

    same source:
    Main Entry: trans·fi·nite
    Pronunciation: (")tran(t)s-'fI-"nIt
    Function: adjective
    Etymology: German transfinit, from trans- (from L) + finit finite, from Latin finitus
    Date: 1902
    1 : going beyond or surpassing any finite number, group, or magnitude
    2 : being or relating to cardinal and ordinal numbers of sets with an infinite number of elements

    {1} more accurately: 1 cannot be added to it and produce something ordinally beyond it, unlike actual ordinal numbers.
    Last edited: Jun 25, 2003
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