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A Little Quiz -truth, existence, proof

  1. Apr 22, 2009 #1
    Defend your answers.


    1) Does the number 1 exist?

    2) Is it true that 1 + 1 = 2?

    3) Is it possible that 1 + 1 = 2 is false?

    4) Can you doubt that 1 + 1 = 2?

    5) Can you doubt that 1 exists?

    6) Can you prove that a statement is true without knowing the meaning of the statement? [For example, Prove that "XqXX$amNNmM yyty3wjj: kXzQQp"]

    7) If a statement cannot be proven true, does that mean that the statement is false?

    8) Can you prove the following statement? "There is no proof of this statement."

    9) Is it possible that number theory is inconsistent (contains contradictions)?

    10) Is the mind capable of making a mistake?

    11) Is it possible to doubt a statement is true even though the mind claims that statement is true?
     
  2. jcsd
  3. Apr 22, 2009 #2

    Hurkyl

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    I'll answer this because it's a fun one.

    Formal proofs are a purely syntactic affair; they boil down to nothing more than manipulating strings of symbols according to some prescribed rules. So, in fact, you can prove a statement without knowing its "meaning" -- as long as you know what rules of inference you're allowed to use.

    I omitted "is true" on purpose, because that's an issue of semantics, not syntax. And, of course, you can't say anything about "truth" if you don't know anything about semantics.
     
  4. Apr 23, 2009 #3
    Very nice :approve:
     
  5. Apr 23, 2009 #4
    For 7, 8, and 9 see Godel's work.
     
  6. Apr 23, 2009 #5
    Here is my attempt to answer the questions. Naturally, per my answer in 10, I could be mistaken.

    1. Yes. It is a term we apply to configurations of matter that contains exactly one of these items.
    2. Yes, by definition.
    3. No, by definition.
    4. Yes, but it is an unjustified doubt in that case.
    5. Yes, but it is an unjustified doubt in that case.
    6. No. Meaningless statements cannot exist in reality. You need to know the meaning of a statement to know which rules of inference you are allowed to use. If I assert that Jaxyplonk dangfur frt, then you dont really know what I am talking about. Is it a encryption of a statement in formal logic? Religious statement from another culture? If you where told that it is a encrypted statement in formal logic, you gain some, albeit very limited understanding of what it is.
    7. Generally speaking, no, but if this statement implies that there should be evidence for it, but it is not, then the statement is false per modus tollens.
    8. No, since such a proof would lead to its own demise.
    10. Yes.
    11. If your mind claims that a statement is true, you do not doubt it by definition, providing you do not have some sort of personality disorder.
     
  7. Apr 23, 2009 #6
    This one reminds me of: "Have you stopped beating your wife?"

    I can't see how you avoid being snookered whether you answer "Yes" or "No".

    Circular loopiness of annoyance looms.
     
  8. Apr 23, 2009 #7
    I'll bite.
    1) Define 1 and define what it means for a number to exist. I take 1 to be the multiplicative identity which exists in non trivial rings, fields, groups and so on. It does not exist in the field of characteristic 1 for example.

    2) Given the usual definitions for 1,+,2 and =, yes. Follows from the construction of natural numbers.

    3),4),5) Look above.

    6) see Hurkyl

    7) No, e.g. continuum hypothesis from ZFC

    8) Depends on the system, if you can construct it and prove it, it is inconsistent (the system).

    9) Yes

    10) Yes

    11) What other thing do you have to determine the truth of statements other than the mind? If the mind claims a statement is true, then you cannot doubt it as you have no other means of assessing its validity.
     
  9. Apr 23, 2009 #8
    my answer got deleted :(..

    Anyways, my answer to #1 was yes 1 DOES exist. It necessarily exist because you are using it in your questions... i count 9 times you use 1 on its own (that is excluding 11)
     
  10. Apr 24, 2009 #9
    1) as an abstract concept
    2) in certain circumstances
    3) 1 human plus 1 human can sometimes equal 3 humans
    4) you can doubt anything, doesn't mean much
    5) sure, doubting is easy
    6) Hurkyl
    7) depends on the statement
    8) silly word game
    9) any theory can be inconsistent
    10) depends on the mind
    11) see #10
     
  11. Apr 27, 2009 #10
    4) just reminds me of a particularly nice quote from Descartes:
    "Is doubting doubt doubtable"

    As for the rest... most depend on your interpretation and your own set of assumptions and personal axioms about reality. As we can't even prove the existence of other minds using reasoning alone, and can't discount that an external entity is manipulating us into believing that incorrect maths is actually correct then we can't discount any possibility. But we can function as though maths is correct and our reasoning about it is adequate even if that's not the case :)
     
  12. Apr 29, 2009 #11
    If other minds do not exist, who are you talking to? Having a rational debate with a figment of your imagination is by definition irrational.
     
  13. Apr 29, 2009 #12

    Hurkyl

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    No, something not governed by or according to reason is, by definition, irrational.
     
  14. Apr 29, 2009 #13
    Perhaps I should have said something along the lines of trying to have a rational discussion with the intent on convincing a figment of your imagination is pragmatically unwise because you are just arguing with yourself. The general idea with a rational discussion is trying to convince the person you are arguing with. If a person cannot be convinced of something, one is seemingly wasting ones time.
     
  15. Apr 29, 2009 #14
    Since there is no way to prove other minds exist, all one can really do is operate rationally from 'the premise' that they either do, or do not.

    Which premise one chooses will depend on what one wants to accomplish. The fact that a theory of other minds is useful in dealing with many aspects of reality, doesn't make its truth or untruth more rational.

    And if we're going to talk about rationalism in terms of philosophy, one has to acknowledge that the word means 'deriving truth through reasoning', with the explicit commitment to the idea that this is the 'way to find truth', something that implicity excludes all but the self, as really relevant.

    Empiricism, the opposite of rationalism, is the philosophy more closely associated with modern ideas of scientific truth.
     
  16. Apr 29, 2009 #15
    If we don't exist, who are you talking to?
     
  17. Apr 29, 2009 #16
    I never said you don't exist. You're confusing ontology with epistemology.
     
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