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The Foundations of a Non-Naive Mathematics

  1. Jun 20, 2004 #1
    Hi,

    Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (include its links).

    I'll be glad to get your detailed remarks and insights.

    Thank you,

    Lama

    -----------------------------------------------------------------------------------

    Edit (11/8/2004):

    Here is a list of my axioms:

    Tautology:
    x implies x (An example: suppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.
    (tautology is also known as the opposite of a contradiction).

    (EDIT: instead of the above definition, I change Tautology to: The identity of a thing to itself.

    It means that in my framework we do not need 'if, then' to define a Tautology)


    Set:
    A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.

    Multiset:
    A set-like object in which order is ignored, but multiplicity is explicitly significant.

    Singleton set:
    A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.

    Urelement:(no internal parts)
    An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.

    It means that in my framework we do not need 'if, then' to define a Tautology)

    A definition for a point:
    A singleton set p that can be defined only by tautology* ('='), where p has no internal parts.

    A definition for an interval (segment):
    A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts.

    (*more detailed explanation of the first two definitions:

    ---------------------
    Remark:
    In Standard Math we had to write:

    Point proposition:
    If a content of a set is a singleton and a urelement and has no directions, then it is a point.

    Segment propositon:
    If a content of a set is a singleton and a urelement and also has directions, then it is a segment.

    But since in this framework a Tautology is the identity of a thing to itself,
    we do not need an 'if, then' proposition for tautology.
    ---------------------

    Let us examine {.} and {._.} definitions by using the symmetry concept:

    1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.

    It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.

    2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'

    In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .

    In short, in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise (mathworld.wolfram.com/Tautology.html)

    As we can see, in my framework '<','>' symbols have a deeper meaning then 'order'.

    Actually, in order to talk about 'order' we first need a 'direction')


    The axiom of independency:
    p and s cannot be defined by each other.

    The axiom of complementarity:
    p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).

    The axiom of minimal structure:
    Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.

    The axiom of duality(*):
    Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.

    The axiom of completeness:
    A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels (lowest bound and highest bound are the ends of some given element, or a collection of more than one element, where beyond them it cannot be found*.)

    --------------------------------
    *Let us clarify the 'finite' concept in my framework:

    In my system I have 4 building-blocks, which are:

    {}, {.}, {._.}, {__}

    The cardinal of {} is 0.

    The cardinal of {.} is one of many.

    The cardinal of {._.} is one of many.

    The cardinal of {__} is The one.

    The bounds of lowest and highest existence (the ends) of these building-blocks
    are determined by their cardinality, for example:

    (in this example I omitted {.}_AND_{._.} and used only their building-blocks)

    The lowest and highest bounds of {.} are cardinals 1 to 1.

    The lowest and highest bounds of {._.} are cardinals 1 to 1.

    The lowest and highest bounds of {} are cardinals 0 to 0.

    The lowest and highest bounds of {__} are cardinals The 1 to The 1.

    The lowest and highest bounds of {{.},{._.},{.}} are cardinals 1 to 3.

    The cardinals beyond {.} are 0, n>1 and the 1.

    The cardinals beyond {._.} are 0, n>1 and the 1.

    The cardinals beyond {} are n>0 and The 1.

    The cardinals beyond {___} are any cardinal which is not The 1.

    The cardinals beyond some n are 0 and any j where j>n.
    --------------------------------



    The Axiom of the unreachable weak limit:
    No input can be found by {} which stands for Emptiness.

    The Axiom of the unreachable strong limit:
    No input can be found by {__} which stands for Fullness.

    The Axiom of potentiality:
    p {.} is a potential Emptiness {}, where s {._.} is a potential Fullness {__}.

    The Axiom of phase transition:
    a) There is no Urelement between {} and {.}.
    b) There is no Urelement between {.} and {._.}.
    c) There is no Urelement between {._.} and {__}.

    Urelement (http://mathworld.wolfram.com/Urelement.html).


    The axiom of abstract/representation relations:
    There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.


    (*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.


    (By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html )



    The Axiom of the paradigm-shift:

    Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system.
     
    Last edited: Aug 11, 2004
  2. jcsd
  3. Jun 21, 2004 #2
    Ok, let us examine the duality concept.

    In my system:

    1) Each element of the Real-Line is both some unique element, and a scale factor of the entire Real-Line.

    Strictly speaking, each element has both local and global properties of the Real-Line system.

    There is an important graphic model at page 5 of No-Naïve-Math.pdf
    that can help you to understand the duality idea.


    2) A point is a Real-Line building-block that can be defined only by using =

    A segment is a Real-Line building-block that can be defined by using < , > or =

    No segment {._.} can be a point {.} exactly as no < or > can be =

    It means that no segment can be constructed (defined) by finite or infinitely many points.

    Because the Real-Line has at least {._.} and {.} building blocks, we get an absolute/relative system that has also properties of a fractal, because of a simple reason:

    A point is a 0-dimension element that is not affected by the “over-all” scale factor.

    An interval is a 1-dimension element that is affected by the “over-all” scale factor.



    The result is an interaction between two opposite properties of the Real-Line:

    1) The relative property can be defined as infinitely many unique intervals along the Real-Line and also in infinitely many different scales of it.

    2) The absolute property can be defined as infinitely many points along the Real-Line.


    Not one of these properties can satisfy the definition of the Real-Line.

    Strictly speaking, the Real-Line is at least an absolute/relative system.

    Because of this duality of each element in the Real-Line, the Real-Line has an invariant cardinality over infinitely many different scales of itself.

    This self-similarity over infinitely many different scales is the most basic property of a fractal.
     
    Last edited: Jun 21, 2004
  4. Jun 21, 2004 #3

    matt grime

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    One presumes you have demonstrated that your new definition of the real line is equivalent to the set of all cauchy sequences of rational numbers "modulo convergence", or the set of dedekind cuts. That is, there is a point where you show how one construction can be used to derive the other. To save me time, what page number of the pdf will that be on?
     
    Last edited: Jun 21, 2004
  5. Jun 21, 2004 #4
    In my theory (please read page 11 of No-Naive-math.pdf)
    {.} is a potential {} and {._.} is a potential {__}.

    {} AND {__} are the unreachable limits of the Language of Mathematics.

    The Language of Mathematics become meaningful only if it uses the products of the interactions between {.} and {._.}.

    It means that any meaningful thing in the Language of Mathematics it at least {.} and {._.}.

    Strictly speaking, The Language of Mathematics is at least an absolute/relative system.
     
    Last edited: Jun 21, 2004
  6. Jun 21, 2004 #5

    matt grime

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    That doesn't answer the question; you could try answering the question, say, I realize that would be setting a precedent for you of course.
     
  7. Jun 21, 2004 #6

    NateTG

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    It's Organic.

    So, continue posting at your own risk.
     
  8. Jun 21, 2004 #7
    This is the precise answer to your question Matt.

    But there is another question according to your response, which is: "Do you understand my answer?"
     
    Last edited: Jun 21, 2004
  9. Jun 21, 2004 #8

    matt grime

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    I know it is he, as is www, shemesh and some others, and I don't intend to get into a 'debate'. Unless my screen is broken there still doesn't appear to be a page reference given, from which one presumes I don't understand the answer. Cranks, eh?
     
  10. Jun 21, 2004 #9
    Matt please show us that you understand my answer.
     
  11. Jun 21, 2004 #10

    matt grime

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    An answer to the request would look something like: if S is the set of reals as defined by you then <argument> implies it is the set of reals accoridng to the proper definitions, conversely, if you take the proper definitions then <argument> which implies my construction. What you wrote is utter utter rubbish.
     
  12. Jun 21, 2004 #11
    So, you demostrated that you do not understand my answer.
     
    Last edited: Jun 21, 2004
  13. Jun 21, 2004 #12

    matt grime

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    your 'answer' did not contain any reference to the definition (any of them) of the real numbers; it was thus not an answer to the question that explicitly asked you to prove your view was equivalent to any of these. but we long since stopped expecting you to understand such things.

    and the question only asked you for a reference to a page in the articles.
     
  14. Jun 22, 2004 #13
    My definition of the Real-Line is better then Dedekind's Cut or Cauchy sequences of rational numbers, because of a simple reason:

    Your absolute-only system cannot deal with real complexity because redundancy and uncertainty are not its "first-order" properties, my absolute/relative system can.

    In short, standard Math system of the Real-Line is trivial because Dedekind's Cut or Cauchy sequences of rational numbers are trivial (and by using the word "trivial" I do not use the interpretation of current community of mathematicians that use this word for “self-evidence” or “extremely simple” thing).

    If you can prove that < or > are =, then and only then my system is a superfluous system.
     
    Last edited: Jun 22, 2004
  15. Jun 22, 2004 #14

    matt grime

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    You promise it's superfluous? ok, in the reals x=y iff (x>y)and(x<y) is false.

    Does this mean that your set, what ever it is, isn't the real numbers? If you can't produce an equivalence to the proper real numbers.

    Note that your set of real numbers doesn't actaully have any numbers in it. It is a set of "global scale factors", which you've not defined, with some other properties and operations and stuff. The set could equally well have bananas in it.
     
  16. Jun 22, 2004 #15
    What I say is very simple:

    A point is a Real-Line building-block that can be defined only by using =

    A segment is a Real-Line building-block that can be defined by using < , > or =

    No segment {._.} can be a point {.} exactly as no < or > can be =


    No > or < can be constructed by finite or infinitely many =


    Conclusion: Real-Line building-blocks are at least {._.} and {.}


    In Standard Math the Real-Line building-block is only {.} and this is the reason why it is an absolute-only system.

    You do not understand the duality idea (where each Real-Line number is both some well-defined element and a scale factor of the entire Real-Line), because you look at it only from an absolute-only (or fixed in your language) point of view.

    Warning: There is no return to an absolute-only point of view, after you understand the Real-Line from an absolute/relative point of view.

    Please show us an explanation (not a thechnical use of some function) by Standard Math, that can clearly show us why a proper subset of the Real-Line can have the cardinality of the entire Real-Line?
     
    Last edited: Jun 22, 2004
  17. Jun 22, 2004 #16

    matt grime

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    Two sets have the same cardinality if there is a bijection between them. That is the explanation because that is the definition. That you don't think cardinality ought to be invariant under bijections is your misunderstanding of how mathematics works.

    As you reject proper maths you must define what the reals are without using proper maths, otherwise your argument is vacuous. So until you define from your first principles only how to construct the set of real numbers you are not on solid ground.

    You are also wrong to say that no interval may be constructed by using infinitely many equalities, but then I don't suppose you know about o minimal structures and tarski's construction (any subset of a real finite dimensional vector space defined by a finite number of inequalities may be given by a finite set of equalities) , of course we can get a simpler refutation of your position by using an uncountable number of equalities.

    Besides you contradict yourself in that post by saying a point is a segment because segments are defined using < > OR = (ie it includes all points), and then saying NO segment is a point.

    You've still not defined what 'scale factor' means.
     
  18. Jun 22, 2004 #17
    No, I say that a point and a segment are different things exactly as '>' or '<' cannot be '=' .

    A point is a Real-Line building-block that can be defined only by using '='

    A segment is a Real-Line building-block that can be defined by using '<' , '>' or '=' (the use of '=' here is the tautology of a segment to itself and there is nothing here which is related to points).

    It means that no '>' or '<' can be constructed by finite or infinitely many '=' .

    In short, no segment (interval) can be represented by points and vise versa, and we need at least segments {._.} and points {.} to define the Real-Line (again, no one of them can be defined in the terms of the other).


    A bijection is the result of your measurement, so how can you use it to explain why a proper subset of some set can have the same cardinality of the entire set?
     
  19. Jun 22, 2004 #18

    matt grime

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    *cough* seeing as a cardinal number is an equivalence class of sets modulo the relation 'there is a bijection between them' then I think we can explain why two sets have the same cardinality by using a bijection.

    The other bits of your post demonstrate that your initial explanation was, and still is, in need of rewriting because it is inconsistent when read literally, that you intended to mean something else is immaterial (your illexplained use of 'used to define')


    One notes you do not refute the comments about o-minimal structures.
     
  20. Jun 22, 2004 #19
    Now all you have to is to explain to us why there can be a bijection between a set to its proper subset.
    Look, we are in 'theory development' where people some times need to use their own abilities to understand another points of views. So, please put aside your rigorous well-defined standards and move out of the limits of your spot light from time to time.

    Believe me, it will be a good exercise for your brain mussels.
    Ordered-minimal structures because the definable subset of R are exactly those that must be there because of the presence of '<'.

    I'll be glad if you show me where I can find in Standard Math my fractal point of view of the Real-Line, when I use an o-minimal (<,>) and a point (=) as the minimal (must have) building-blocks of the Real-Line.
    A multiplication operation of each well-defined R number with the entire Real-Line.
     
  21. Jun 22, 2004 #20

    matt grime

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    What one earth do you mean explain how there can be a bijection? I can write one down and demonstrate it's a bijection, eg N to N\{1} given by x -> x+1, so look, by example there can be.

    o minimal does not as far as i am aware mean ordered minimal, though there are ordered o minimal structures

    a scale factor is a mulitplication operation that multiplies an element of R the R? that makes no sense., you've not described how to do this.

    and there is a difference between changing definitions, or rather deciding that which you first examine is not what you want and then defining something else (this happens a lot in developing theories when done properly), and not defining something properly which means all subsequent deductions are invalid.
     
    Last edited: Jun 22, 2004
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