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Kurt Gödel

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English: Portrait of Kurt Gödel, one of the mo...

Portrait of Kurt Gödel, one of the most significant logicians of the 20th century via Wikipedia

Kurt Gödel (1906-1978) was a mathematician and philosopher born in Brünn, Austria-Hungary (currently Brno, Czech Republic).

Gödel’s “Undecidability Theorem” demonstrated that at some point a mathematical system based on axioms and operational rules becomes self-referential. This means that the mathematical system cannot prove nor disprove the original axioms from which subsequent statements are derived.

Furthermore, Gödel’s “Incompleteness Theorem” revealed that a mathematical system, if taken far enough, begins to generate contradictory statements.

These theorems suggest that mathematics is always an extension of original axioms. Rules and procedures within a given system merely extend the original axioms.

This presents problems for mathematics as a whole and, particularly, for those who wish to integrate its various branches.

Gödel suggests that if each mathematical branch stems from different core assumptions, all branches cannot be artificially pasted onto a single body of assumptions.

A mathematician at IBM, Gregory Chaitin, says that Gödel’s work, along with the uncertain, unpredictable elements in quantum physics, demonstrates the limits of mathematics, in particular, and of scientific knowledge, in general.

But what does this mean to us on the everyday level? For one thing, it means we should take many so-called “scientific” truth claims with a fair degree of caution. Although many scientists (and those who benefit from the worldview) seem to enjoy spouting off scientific research as if it were the gospel truth, those of us who can think for ourselves are not so quick to jump on the bandwagon.

English: Square root of x formula. Symbol of m...

Image via Wikipedia

Sadly, however, many people just aren’t able to see the relative nature of scientific research.¹ And they remain wide-eyed and virtually brainwashed by science, the new religion for the 21st century. This might in part be due to the visible success of the scientific method in diverse areas, such as health, construction, travel, electronics and communications technologies, to name a few.

But it’s still a fundamental (and common) mistake to confuse technological achievements with truth.

¹ Even some with graduate degrees uncritically parrot the latest scientific ideas, which doesn’t say much about their intellectual development or the institutions that granted their degrees.

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3 thoughts on “Kurt Gödel

  1. Well, I don’t know how to phrase it nicely but this article does not at all reflect Gödel’s work. Let’s do it point by point.

    — “Godel’s “Undecidability Theorem””:
    Gödel produced 2 theorems about “incompleteness” and that’s usually how people refer to it.

    — demonstrated that at some point a mathematical system based on axioms and operational rules becomes self-referential …
    Rather, he used self-referent expressions to demonstrate his theorems. Formal systems (the “mathematical system based on axioms and operational rules”) don’t “become” self-referential

    –“cannot prove nor disprove the original axioms upon which subsequent statements are derived”
    By definition axioms are never to be proved. Those are the original starting points from which subsequent derivations (the proofs of the theorems) are made.

    — “revealed that a mathematical system, if taken far enough, begins to generate contradictory statements.”
    No. The first Incompleteness theorem just states that within any formal system that includes the number theory and that is consistent (meaning one cannot demonstrate one thing and its negation) one can find true propositions/statements (“true” meaning that we can demonstrate they are true) that cannot be demonstrated/derived in *this* formal system; hence the name “incompleteness”: not all “truths” for a given system can be demonstrated *in* this system.

    “These theorems suggest that mathematics is always an extension of original axioms.”
    Formal demonstrations are always derivations from axiomatic systems. It’s not what is suggested by Gödel at all, it’s just the way mathematicians are building formal systems.

    “This presents problems for mathematics as a whole and particularly for those who wish to integrate its various branches”
    I don’t understand this argument

    Hope this is useful.

    jm

  2. Thanks… I’m going to have to look over my sources for this again and also ask mathematics people to continue with this thread.

    I appreciate the time you’ve taken to comment on this.

  3. Foundations of The Quantum Logic is a blog-site for raising awareness of a newly discovered foundation to Quantum Physics. The new research finds that conventional Quantum Theory does not wholly implement the mathematics, and that there is no missing physics: no hidden variables, no many worlds. When Quantum Mechanics is implemented, not as an applied mathematics, but as a first-order logic, we find that Quantum Mechanics has an intuitive basis that makes good logical sense of quantum experiments.

    This resolves paradoxes present in Quantum Philosophy since the 1920’s. After so long, such a discovery may seem unlikely and too good to be true. Nevertheless the results are meticulously researched and well developed in a theory that does provide the full story of Quantum Physics: the various parts of the theory supporting each other with integrity.

    If you are interested in the origins of indeterminacy in Quantum Mechanics, read my newly finished paper titled:

    “The Mathematical Undecidability within Quantum Physics: The Origin of Indeterminacy and Mechanism of Decision at Measurement”

    To get a copy click on the following link:

    http://steviefaulkner.files.wordpress.com/2010/04/undecidability-in-qm_current.pdf

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