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Can a mathematical formula describe its own inventor completely?

  1. Feb 19, 2006 #1
    Can a mathematical formula describe its own inventor completely?
     
  2. jcsd
  3. Feb 19, 2006 #2

    selfAdjoint

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    Well, any mathematical theory that is rich enough to describe arithmetic is necessarily incomplete. Of course that includes infinitary operations like "For All" and "There Exists" applied to infinite sets, and you might not think a human mind does that kind of thing.

    Oh, I take your point about the creator and the theory and the recursiveness thereof, but I don't think that by itself constrains. If the mind is really finitary or constructible, as is suggested by its material foundation on a finite number of cells which might seem to be continuous processors, but in fact themselves depend on finitely many molecules within them, then a finite mathematical theory could describe it.
     
  4. Feb 19, 2006 #3
    So even if the finite number of cells, etc. is true then there is no problem with the formula describing something which contains the origin of itself?
     
  5. Feb 19, 2006 #4

    selfAdjoint

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    Conceptually, I think one could follow Goedel's recursive procedure where he mapped the descriiption of arithmetic (by logical statements) into actual arithmetic statements such that if one logical statement followed from another then you could execute the arithmetic statement for the first and get the second among the results. This is possible and as I said he proved that it leads to an uncompleteness theorem in the case of arithmetic. But I think it would stay consistent if you so described a finite state machine. And if the human mind is such a machine - a big if, but not completely unreasonable to suppose - then you could get the description you ask about.
     
  6. Feb 19, 2006 #5

    arildno

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    I'd like to make a tiny point on "recursiveness"/self-reference that might suggest it isn't a particularly troublesome feature, in accordance, I believe, with SelfAdjoint's first comment.
    Consider the equation:

    x=2x-12 (having the solution x=12)

    Now, we MIGHT regard this equation as defining "x as the number equal to twice itself minus twelve", i.e, containing a self-reference of some sort.

    Furthermore, we can see that in a simple manner, this could lead to a seemingly endless chain of more complicated self-references:
    Since x=2x-12, we evidently must have:
    x=2(2x-12)-12=2(2(2x-12)-12)-12=2(2(2(2x-12)-12)-12)-12 and so on.

    Despite this atrocious behaviour, it is in this case easy to remove the self-referencing definition of x, if that is what one might wish for.

    This was probably off-topic..
     
  7. Feb 20, 2006 #6

    selfAdjoint

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    No, I think it was very relevant to the point I was making. Thank you.
     
    Last edited: Mar 21, 2006
  8. Mar 21, 2006 #7
    Here is a facetious affirmative answer:

    Let M denote the predicate 'being me' (insert your name for "me", and understand this predicate to denoate all of the various features of yourself).

    Let x be a variable that ranges over persons.

    Let n be a term that denotes a proper name that is your name (e.g., Fred, Ethel, etc).

    Here is a "mathematical" formula (using predicate logic) that completely describes its inventor (i.e., me):

    There exists an x such that Mx and x = n.

    In English: There is a person, named "me" (whatever name you picked for "n"), who has all and only the properties that I have (i.e., the properties denoted by the predicate "M").

    Not very informative, but something like this seems reasonable to me.

    But perhaps there cannot be a predicate like "M"? Who knows...
     
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