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Hidden assumptions

  1. May 23, 2004 #1

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    As much as I know (and please correct me if I am wrong) no fundamental mathematical systems like ZF or Peano axioms include an examination of our cognition's ability to count.

    In my opinion, if we ignore our abilities to count, then there is a reasonable chance that our fundamental axioms include hidden assumptions.

    For example, let's examine this situation:

    On the table there is a finite unknown quantity of identical beads > 1
    and we have:

    A) To find their sum.

    B) To be able to identify each bead.

    Limitation: we are not allowed to use our memory after we count a bead.

    By trying to find the total quantity of the beads (representing the discreteness concept) without using our memory (representing the continuum concept) we find ourselves stuck in 1, so we need an association between continuum and discreteness if we want to be able to find the bead's sum.

    Let's cancel our limitation, so now we know how many beads we have, for example, value 3.

    Now we try to identify each bead, but they are identical, so we will identify each of them by its place on the table.

    But this is an unstable solution, because if someone takes the beads, put them between his hands, shakes them and put them back on the table, we have lost track of the beads identity.

    Each identical bead can be the bead that was identified by us before it was mixed with the other beads.

    We shall represent this situation by:

    ((a XOR b XOR c),(a XOR b XOR c),(a XOR b XOR c))

    By notating a bead as 'c' we get:

    ((a XOR b),(a XOR b),c)

    and by notating a bead as 'b' we get:

    (a,b,c)

    We satisfy condition B but through this process we define a universe, which exists between continuum and discreteness concepts, and can be systematically explored and be used to make Math.

    What I have found through this simple cognition's basic ability test is that ZF or Peano axioms "jump" straight to the "end of the story" where cardinal and ordinal properties are well-known, and because of this "jump" Infinitely many information forms that have infinitely many information clarity degrees, are simply ignored and not used as "first-order" information forms of Math language.

    In my opinion, any language (including Math) is first of all an information system, which means that fundamental properties like redundancy and uncertainty MUST be taken as first-order properties, but because our cognition's abilities were not examined when ZF or Peano axioms were defined, both redundancy and uncertainty were not included in our logical reasoning or in our fundamental axiomatic systems.

    Also in my opinion, through this simple test we get the insight that any mathematical concept is first of all the result of cognition/object (abstract or non-abstract) interactions.

    My paper http://us.share.geocities.com/complementarytheory/ONN.pdf is a reexamination of some fundamental mthematical concepts that are based on this insight.
     
    Last edited: May 24, 2004
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  3. May 23, 2004 #2

    matt grime

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    if you aren't allowed to use your memory (why not though?) how on earth do you remember that you had to count them, or what the word count means, or why you're sat at a table looking at some beads....
     
  4. May 23, 2004 #3
    X

    ~X

    ~~X

    ~~~X

    ~~~~X

    ~~~~~X

    etc...

    2-valued logic
     
  5. May 23, 2004 #4

    Hurkyl

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    I think this is an unreasonable suggestion.

    The main point of the axiomatic method is to make all assumptions explicit; one is not allowed to use anything that cannot be deduced from the axioms.


    Now, of course, there is the (non-mathematical) question that asks whether a mathematical theory corresponds to some "real life" thing (like counting beads). This brings up a practical point of the axiomatic method; because the axioms are listed explicitly, you can check each of them to see whether or not they correspond to the "real life" thing in question.




    Anyways, now, it almost seems like you're trying to do basic combinatorics
     
  6. May 23, 2004 #5

    Hurkyl

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    Ok, I'll go fetch my stack of stickers.

    The sticker with "1" on it is on top, I'll place it on one of the beads.
    I see a bead without a sticker on it; I'll take the next sticker (which happens to be "2") and place it on the next bead.
    I see a bead without a sticker on it; I'll take the next sticker (which happens to be "3") and place it on the next bead.

    Ah, all the beads have stickers. "3" was the last sticker I used, so there must be three beads!

    And since all of the beads have stickers on them, I can distinguish them too.
     
  7. May 23, 2004 #6
    To my point of view you have adequately described a "hidden Variable" system in thought processes. Think about what thinking is. First memory comes and goes a stimulated. Comupter models fail from memory problems of how does the self-starting computer make the kinds of decisions and accomplish the kinds of thought processes, such as intuitive leaps, as the mind. That thought is restricted to the brain is an unverified assumption. Does the brain have the storage capacity necessary to retain all the thoughts back to childhood when properly stimulated and still be able to think and rationalize? I say no. When you learn to drive, you are aware of the mental and physical coordination when learning, now try to tell yourself how you drive. You haven't enough time left in you life to do this. I mean read your memory, not just make a logically sounding speech on how you drive.

    What is not hidden by the thinking mind? I have been trying to capture the secret of how a mind can stack some very complicated words and pharases uttered by a person ad lib, as in a heated discussion, but not know exactly what words will exit the lips.. Are yo aware of the next words or do you have just a hint?

    Hidden variables, nonlocal force centers, you name it. Try a book by Rupert Sheldrake "The Presence of the Past| Morphics Resonance |the Habits of Nature." Excellent book on every thing you wanted to know about this and that . . .

    Mathemtical models seem helpful, but I assume the modles are relativiely useless when getting to the nitty gritty of thought and consciousness..

    What about the body, including the brain, as a mere antenna absorbing and proccessing information fields? There is data that suggest this.
    :smile:
     
  8. May 24, 2004 #7

    matt grime

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    You (WWW) might also like to consider the thoughts of othere such as Russell about this and the nature of numbers and what they are when we count. But that's philosophy, not maths.
     
  9. May 24, 2004 #8

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    I wrote:

    Limitation: we are not allowed to use our memory after we count a bead.

    It means that in this case we always start to count from the begining, and the result is that without our memory we cannot count beyond 1.
     
    Last edited: May 24, 2004
  10. May 24, 2004 #9
    I hope you all do not mind the intrusion here?

    What lies beneath is a interesting question. At planck length things can become very different? What is memory in this instance?

    So we have to find this "point" where the math arrives(expands), where it can be reduced to a point and then "spreads out" in terms of probabilistic determinations. But to do so, would produce infinities. So what would determine such pathways in the marble drop? What mathematical discription would reveal the math involved in Pascal's triangle?

    George Lakoff and the Cognitive Science of Mathematics What is the Steering Behavior that would calculate all these probabilities and then, move it through that point.

    Were the infinities existant?

    How woud such things retain the memory as "information" in solid discrete objects?
     
    Last edited: May 24, 2004
  11. May 24, 2004 #10

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    Hi geistkiesel,

    First, thank you for your interesting post.

    I think that a part of the answer of our cognitive abiliteis to deal with comlexity somehow connected to an inherited abilites to associate between opposite (abstract or real) things.

    This (in my opinpon) important property give us the ability to explore in non-trivial ways a lot of Simultaneous input sources which constructed as a very complex probabilistic wave function, which within milliseconds we have to take some useful information out of them.

    It cannot be done if we cannot deal with parallel multi-opposite situations and find our non self destructive unique real time way to express ourselves.

    Shortly speaking, the memorey of the evolution process is in our genes and it is the companion of every living thing for better or worse.
     
    Last edited: May 24, 2004
  12. May 24, 2004 #11

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    I have a very simple claim which is: If a mathematician does not include its own basic abilities when he use fundamental concepts, then there is a reasonable chance that he can make some "automatic" (unconscious) shortcuts that ignore valuable properties of these fundamental concepts.

    I gave an example of how Pano ignored properties like redundancy and uncertainty and his own ability to count when he defined his natural number system.

    Because Mathematics is first of all a form of language, both redundancy_AND_uncertainty and our cognition's abilities to count must be included right from the beginning of any fundamental definition process.
    To make it clearer, we do not start mission B, before we finish mission A.

    But this was not my goal, my goal was to give you an insight about cognition/object interactions.

    If you try to be "smart" by using shortcuts, then you miss the universe of infinitely many information forms that I suggesting, which can be used by us to make Math.

    By this attitude you repeat on Peano's shortcut.
     
    Last edited: May 25, 2004
  13. May 24, 2004 #12

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    Hi sol2,

    Your post is very intersting.

    I'll read and then reply to you.
     
  14. May 24, 2004 #13

    matt grime

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    and after we've counted one bead, how are we to remember anything at all? what our name is, what our langauge consists in, what we are doing....
     
  15. May 24, 2004 #14

    matt grime

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    but you are unable to justify why peano needs to incorporate uncertainty or redundancy into his arithemetic, which is by definition something which apparently does not require these objects. Objects one migh say that you have not defined without referenece to the natural number system that is insufficient. That is a circular argument, by the way.

    Nor do you understand that something is exactly what it does. if you are saying they, the naturals as we all know them, are insufficient to count things then perhaps you could offer some evidence to back this up, and more importantly you might explain why it is that we shoudl give a toss? seeing as it is just an axiomatic and self consistent theory where is it wrong? there ought to be some evidence to back up your opinion. so where is it? if you are once more to claim (not clime) it is wrong please justify why, offer some example where it is insufficient. just one example, that isn't unreasonable. and bear in mind the structural nature of mathematics.
     
  16. May 24, 2004 #15

    matt grime

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    justify that statement. what is a language, what is a form of such and why is mathematics one of these
     
  17. May 25, 2004 #16

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    As for your question about language, in my opinion language is first of all interactions among systems that cause internal or external, actual or potential, direct or indirect influences on these systems.

    Some systems can be aware to these interactions (and then the word "information" is used) and some are not.

    In our case, the language of Mathematics has a tremendous influence on our systems, and therefore any form of information must be examined right from its first-level.

    When Peano defined his axiomatic system he used his natural ability to go straight to the forms of information where cardinality, ordinality and the unique identity of each object are well-known.

    But in my opinion, when you define the fundamental level of some language, you have to examine your own system's abilities during this process, otherwise there is a chance that you will ignore fundamental properties that can have a significant influence on the "character" of your basic products, and in this case the products are the Natural numbers, and also addition and multiplication operations.

    In my system, where redundancy_AND_uncertainty are first order properties, Natural numbers have internal complexity, which is based on cognition/object interactions.

    And because of this internal complexity more information forms can be defined and ordered by their symmetrical degrees and information clarity degrees within any given finite quantity.

    Also addition and multiplication have two words of operations: internal and external.

    By Peano system the Natural numbers world is only the external world, where each operation changing the quantity of n.

    Let us look on - and + operations on n from the eyes of the symmetry concept:

    The external result of ((((1),1),1),1) - 1 is (((1),1),1)

    The internal result of ((((1),1),1),1) - 1 is (((1,1),1),1)

    The external result of (((1),1),1) + 1 is ((((1),1),1),1)

    The internal result of (((1,1),1),1) + 1 is ((((1),1),1),1)

    So as you see - and + do not changing the quantity but the symmetry degree of each structure.

    Here is a simple example of a theorem and proof which are out of the scope of Peano system:

    Theorem: 1*5 not= 1+1+1+1+1

    Proof: 1*5 = {1,1,1,1,1} not= {{{{1},1},1},1},1} = 1+1+1+1+1

    To understand this example please look at: http://us.share.geocities.com/complementarytheory/ONN.pdf

    You take my analogy as it is instead of use your creative mind and check the "what if...?" Question that I suggesting here.

    So (for you) let us say that after we count a bead, all we remember is that we have a mission which is: to count beads.

    Also, if you don't mind, I think it will be good for you to read De-Bono's book on parallel thinking: http://members.optusnet.com.au/~charles57/Creative/Books/B20471.htm
     
    Last edited: May 25, 2004
  18. May 25, 2004 #17
    hmm my post got deleted for excess { } thingies i think :O

    crap
     
  19. May 25, 2004 #18
    hmmm when i post on this thread it gets deleted.

    how very odd
     
  20. May 25, 2004 #19
    k i'll try once more.

    is what you're saying:

    5*1 makes the assumption that we want the sum of the items listed, such that 1+1+1+1+1 which creates the union {{{{{1},1},1},1},1)

    but without that assumption 5*1 could mean any number of things as:

    1,1,1,1,1

    1+1, 1, 1, 1

    1+1, 1+1, 1

    1+1, 1+1+1

    1+1+1, 1, 1

    1+1+1+1, 1

    or

    1+1+1+1+1
     
  21. May 26, 2004 #20

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