Dictionary halting problem makes A=A fallible

It would seem superficially this is correct

A=A, necessarily and infallibly true

but it is false for non-trivial A and where the EQUAL SIGN means equal in essence, not just equal in description. This is a consequence of the dictionary and halting problem that has been well demonstrated by first rate mathematical logicians and computer scientists (like Alonzo Church who was a devout Christian) who actually work with formal languages vs. people clinging to antiquated and non-rigorous theological notions.

Now consider this

Property (4 quarters) = Property (100 pennies)

Depending on how Property operator works, the above is could be true or it could be false.

Similarly we are posed with the same issue here

Property(1+1) = Property(2)

false under modulo 2 arithmetic, true under Peano and Real axioms.

One could construct a moral code whereby…

Property(eating oranges) = KOSHER
Property(eating pork) = NOT KOSHER

or a moral code whereby

Property(eating oranges) = KOSHER
Property(eating pork) = KOSHER

The issue is you can make symbols mean almost whatever you want! So this fact isn’t reassuring for those demanding infallibility.

One work around is to prevent any sort of meaning. So then we allow only identical symbolic strings to be declared as equal. Hence under such a regimen:

“1+1″=”2”

is false since the literal strings are not identical! But such a regimen isn’t very useful. We want to have a system the EQUAL SIGN allows for equality of essence independent of description.

Shakespeare captured the idea there is an essence that transcends description:

A rose is a rose by any other name

But in the world of symbols, the description is the essence. We could create a substitutional machinery that says

1+1 = 2 = 3-1= 5-3 = 1+1-1+5-3-3+2=…..

and thus A=A if we are talking about A’s essence, not its description.

The problem however is if we have a dictionary of symbols, and we can create meaning by substituting all the symbols with other symbols, we have circularity. Nothing becomes clearer by trying to define symbols by other symbols in the same dictionary. In the search for meaning, we end up with a halting problem!.

We try to say the EQUAL SIGN means EQUAL IN ESSENCE but not in description. We could then demand the EQUAL SIGN means only equal in description, but then one has a trivially useless meaning for the EQUAL SIGN.

So if we allow the equal sign to permit substitutions and meaning, and say the EQUAL SIGN refers to equality in essence, not description, then we can say things like this in the axioms of Peano or the Real Numbers:

1+1 = 2
4+3= 7
etc.

But Godel demonstrated, there will be some things of the same essence but different descriptions which one cannot prove as having the same essence. That is, you will not be able to prove A=A even if true for some A in terms of essence because of the halting problem. It is a true but an undecidable proposition, hence it is true but not necessarily true, hence this statement is false

A=A, necessarily and infallibly true

for non-trivial A where the EQUAL SIGN means equal in essence, not equal in symbolic description.

Quod Erat Demonstrandum.

16 Replies to “Dictionary halting problem makes A=A fallible”

  1. OMagain
    Ignored
    says:

    In the search for meaning, we end up with a halting problem!.

    No we don’t. Describe the halting problem in your own words and explain how it applies here.

  2. Neil Rickert
    Ignored
    says:

    The usual view within mathematics, is that “=” means identical.

    So 1+1=2
    is understood to say that “1+1” and “2” are two different names for the exact same entity (the number one which some folk take to be a platonic entity and others take to be a useful fiction).

    However “equal” in ordinary life means something different, as for example “all men are created equal”.

  3. Neil Rickert
    Ignored
    says:

    But Godel demonstrated, there will be some things of the same essence but different descriptions which one cannot prove as having the same essence.

    That’s a creative way of describing Gödel’s result. It’s so creative, that I’m having trouble making sense of it.

  4. keiths keiths
    Ignored
    says:

    Sal,

    Similarly we are posed with the same issue here

    Property(1+1) = Property(2)

    false under modulo 2 arithmetic, true under Peano and Real axioms.

    You’re confusing symbolic representations with the numbers being represented.

    Property(2) is meaningless in modulo-2 arithmetic because 2 doesn’t exist. In modulo-2 arithmetic, 1 + 1 is identical to 0, so any property possessed by 1+1 is also possessed by 0.

    In modulo-3 (or higher) arithmetic, any property possessed by 1+1 is also possessed by 2, since both represent the same underlying number.

  5. stcordova
    Ignored
    says:

    That’s a creative way of describing Gödel’s result. It’s so creative, that I’m having trouble making sense of it.

    Consider:

    1.0 = .999999999999 repeating

    Two different representations of the same thing.

    https://en.wikipedia.org/wiki/0.999

    If we have a non-computable number with two different representations, we won’t be able to prove them to be identical even though they are.

    If they hypothetically have the same representation, no problem, but that’s trivial. Like saying:
    “1+1” = “1+1”

    But they cannot be represented in a formal system, so it’s not provable anyway.

    Assume a system obeys FieldAxioms:
    http://mathworld.wolfram.com/FieldAxioms.html

    SquareRoot(A) = SquareRoot(A)

    True if the field are the Reals, meaningless if the field are the Rationals.

    If a non-computable number has no representation in a system, it’s sort of moot to be saying it can be compared since the comparison is undecidable.
    https://en.wiktionary.org/wiki/noncomputable

    Like saying
    Smallest Fraction = Smallest Fraction

    When StephenB says “same ontological sense”, that’s non-rigorous philosophical mush, certainly doesn’t have much utility in symbolic representation.

  6. Neil Rickert
    Ignored
    says:

    stcordova: Consider:

    1.0 = .999999999999 repeating

    Two different representations of the same thing.

    But how does that relate to Gödel? His proof used only integers, and depended on the unique factorization into products of primes.

  7. stcordova
    Ignored
    says:

    Under the axioms of the reals, even for non-computable numbers:

    A + (-A) = 0

    But that is an axiom, not a proposition, so it is defined as true by assumption, that is different than true by necessity.

  8. stcordova
    Ignored
    says:

    Keiths,

    You are correct, my mistake. Thanks for pointing it out. I should have said:

    Property(1+1) = Property(0)

    under modulo 2.

  9. stcordova
    Ignored
    says:

    First off, I’m not disputing A=A in reals as an axiom, I’m disputing it as a proposition. It is true by assumption, not by necessity.

    But how does that relate to Gödel? His proof used only integers, and depended on the unique factorization into products of primes.

    Would you prefer Tarski’s undefinability of TRUTH theorem?
    http://www.math.hawaii.edu/~dale/godel/godel.html

    The Liar Paradox. “Truth” for English sentences is not definable in English.

    Tarski’s Undefinability of Truth Theorem. TRUTH, the set of numbers which encode true sentences of number theory, is not definable in number theory.

    I’m just trying to say, A=A can be accepted as true as an axiom, not as a proposition.

  10. shallit
    Ignored
    says:

    No offense, Sal, but you’re so confused that it would take hours trying to sort it out. For one thing, you’re mixing up two different senses of the word “undecidable”. Godelian undecidability is not the same as “undecidable” in computation theory (which is nowadays more usually called “uncomputable”).

    See, for example, https://en.wikipedia.org/wiki/Undecidable_problem and read the paragraph starting “There are two distinct senses of the word “undecidable” in contemporary use. “

  11. stcordova
    Ignored
    says:

    Jeff!

    Hey thanks for the criticism! No offense taken. Thanks for reading.

    Sal

  12. Mung Mung
    Ignored
    says:

    shallit: No offense, Sal, but you’re so confused that it would take hours trying to sort it out.

    I admit I was lost reading that OP. So I too thank you.

    Aren’t Dembski and Marks the greatest though!? Cant’ wait for the second edition of NFL.

  13. stcordova
    Ignored
    says:

    Jeff,

    I know you are a recognized expert on computational number theory. I’m indebted to you for pointing out my misunderstanding. Thank you for the wiki link:

    if a decision problem is undecidable (in the recursion theoretical sense) then there is no consistent, effective formal system which proves for every question A in the problem either “the answer to A is yes” or “the answer to A is no”.

  14. Elizabeth Elizabeth
    Ignored
    says:

    Just having one of my random-with-respect-to-function post purges. Some posts moved to Guano.

    Please try to stick to the rules guys

    ta.

  15. stcordova
    Ignored
    says:

    But Godel demonstrated, there will be some things of the same essence but different descriptions which one cannot prove as having the same essence

    In light of Jeff Shallit and Niel’s criticisms, this sentence must be withdrawn and reworded since as Jeff pointed out there two notions of Undecidable and one sense of the word applies to computation number theory.

    Proposed rewording along the following lines:

    1+1 = 2
    What we really mean is something like
    A = Property(1+1) = Property (2) = A
    which reduces to
    A =A

    that is because the meaning of the EQUAL SIGN goes deeper than mere identical description, but some substitution of symbols, i.e. we can substitute the string “1+1” with the string “2” under our rules.

    What is not under dispute in the real numbers is that A=A is pretty much an axiom. I claim it cannot be offered up as a proposition. A=A is true by assumption, no necessity that can hypothetically be proven or disproven.

    If we have non-computable numbers, their representation is undecidable. The reals have far more non-computable numbers than computable numbers. If we have say the following numbers, B, C, D,E:

    A= B+C = D+E = A
    that hypothetically could reduce to
    A=A
    if we could actually compute the value, but we actually can’t because A is non-computable, then we have a case where A=A is true but cannot be proven as such.

    There will be some A=A that we can only assert by assumption, not by necessity.

  16. Neil Rickert
    Ignored
    says:

    stcordova: Proposed rewording along the following lines:

    The new version is a big improvement.

    I have limited keyboard access over the next few days, so I won’t give detailed comments.

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