I generally try to avoid flamebait, but I saw this article linked off of Twitter. I should have stopped reading after the first section where it's clear that the author is a troll. Evolution and science denialism aside, the misrepresentation of mathematics in the article is inexcusable.

After attacking Darwin and scientific thought in general, an appeal to emotion, he proceeds into a second hand quote from a philosopher on the subject of "fallacies". It's kind of ironic that the inclusion of this quote would serve as an appeal to authority.

Next, he goes into intelligent design saying:

we could find incontrovertible evidence that reality, matter, life, has been designed, but that interpretation of the evidence would be discarded because naturalism dictates the exclusion of anything which might lead outside of a naturalistic explanation.

This is absolutely false. Scientific theories are necessarily falsifiable. If the evidence implied a "design", that's what the scientific theory would be. The fact is that the evidence points to the contrary. Biology shows a picture of "unintelligent design", consistent with a process of genetic mutations occurring over time. The naturalistic explanation is the one that the evidence supports.

Then he claims that Gödel's Incompleteness Thereom proves this.

He managed to get Gödel's basic background information right, but incorrectly describes the Incompleteness Theorem.

From the article:

Any system that is consistent is incomplete.The consistency of axioms (axioms=assumptions that cannot be proven) cannot be proved from within the system.

The real Incompleteness Theorems:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,^{[1]}but not provable in the theory (Kleene 1967, p. 250).For any formal effectively generated theoryTincluding basic arithmetical truths and also certain truths about formal provability,Tincludes a statement of its own consistency if and only ifTis inconsistent.

Notice how the part about "basic arithmetic" is conveniently left out of the definition? That's because the author doesn't want you to know that there can exist axiomatic systems which are both complete and consistent. First-order predicate logic was proven to be both complete and consistent by none other than Gödel himself. Furthermore, saying that the Incompleteness Theorem "utterly [destroyed] atheist Bertrand Russell’s logical system for arithmetic" doesn't give Russell the credit he deserves. Gödel's technique was based on the same idea as Russell's Paradox to begin with. Despite its incompleteness, the development of Russell's work into Zermelo-Fraenkel set theory was an important building block in the foundation of later mathematics. By referring to him as "atheist Bertrand Russell", it's clear that the author is more concerned about religion than the actual mathematics.

Next we have a very weak analogy. He describes three items on a table and says:

Now draw a circle around those items. We will call everything in that circle a system. Gödel’s first theorem tells us that nothing in the circle can explain itself without referring to something outside the circle.

It's true that Gödel's theorem succeeded in "stepping out of basic arithmetic", but here's where that omitted condition of a "formal system capable of basic arithmetic" comes into play. Are a half-full cup of coffee, a fishing pole and a jacket capable of arithmetic? If the answer is no, then Gödel's theorem doesn't apply. Capable of self reference? Maybe if the coffee mug says "I'm a half full cup of coffee" on it.

The analogy of a computer is a much better example. Computer programs are capable of basic arithmetic. What Gödel's theorem implies for computers is that there exist certain programs which are computationally irreducible. The only way to determine the output of such a program is to run it. If we think of Nature like a computer program, the only way to be certain of the future "output" is to let Nature run its course. This result does not prevent science from making conjectures about the structure of Nature, but requires that science adopt a Black-box testing procedure which entails experimentation and observation. There are certainly unanswerable questions in science, such as the precise position and momentum of elementary particles, but evolution isn't one of them. The evidence for evolution is incontrovertible.

The final second shift the analogy to the universe and the claim is that what's outside the universe is unknowable. Just because we can't see what's outside the universe, which would be white-box testing, doesn't mean we can make and test hypotheses about it as a "black-box". The Many-worlds interpretation of quantum theory is one such example which predicts that our universe is but one of many possible universes. Similarly, M-theory predicts the existence of hidden dimensions beyond space and time. Just because some questions are unanswerable, doesn't mean all questions are.

The article ends by claiming that evolution and naturalism are "fallaciously circular", but here's the real circular fallacy:

- Author misinterprets Gödel's theorem to imply that all axiomatic systems are incomplete or inconsistent.
- Author mistakenly assumes that science is an axiomatic system.
- Based on this misinterpretation, author concludes that science must be incomplete or inconsistent.
- Since author concludes that complete scientific knowledge is incomplete or inconsistent, author ceases to look for empirical evidence of scientific claims.
- Since author ceases to look for evidence, author does not find any evidence.
- Since author does not find any evidence, author concludes that scientific knowledge is incomplete.
- As a consequence, the author's incomplete knowledge becomes a self-fulfilling prophesy.

This whole article is a Proof by Intimidation. The "average Joe" doesn't know enough about contemporary math and science to go through and verify each detail. The use of mathematics vocabulary in the article is deliberately being used to distract the reader from the real issue -- the overwhelming evidence for evolution. The references to Gödel's Incompleteness Theorem are nothing more than a red herring, and the author even misstates the theorem to boot.