The reason we can't drop the article into a computer and have it magically return "true" or "false" is because making computers that even have human intelligence is damn hard as it is, so until we can perfect language parsing and the like and then input millennia of collective mathematical knowledge (correctly, no less), we're not going to have any particular progress on that front.

What does this have to do with "flaws" in contemporary mathematics though? If you're trying to use this as an example of a flaw in anything, it would be the computer industry for not figuring out artificial intelligence, not the mathematics community. So, if this was the author's intention, then I think it's flawed right from the outset, but I don't think this was truly the author's intent. (For one thing, Curry-Howard correspondence is talking about lambda calculus, not what you think when you hear "computer" today.)

Really, the article's entire premise is predicated on extreme misunderstandings of Godel. For instance:

> In fact, Gödel's 1931 Second Incompleteness Theorem insists that it is strictly forbidden to ever claim the truth of any scientific or mathematical statement:

What? Where does it say that? It says (quoting the article here):

> For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.

If a theory includes a statement of its own consistency, then that theory is inconsistent. How does that imply that it is "strictly forbidden to ever claim the truth of any scientific or mathematical statement"? The honest answer is that it doesn't, not in the slightest. When it comes to mathematics, we're not saying the axioms are consistent, we're saying statements within this set of axioms are true. Are the axiomizations true? I don't know, but a lot of work has gone into these theories and so far they look pretty good. But at any rate, the author uses this theorem in a completely incorrect manner. He is right that we don't know if any of the various mathematical foundations are consistent, true, but as I said, so far everything looks pretty good.

Then he throws this bombshell:

> Given Gödel's First Incompleteness we already know that for any claim making use of basic arithmetic, counterexamples must exist.

What? How? Here's Godel's First Incompleteness Theorem, per Wikipedia:

> 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, but not provable in the theory (Kleene 1967, p. 250)

How in the world does that say that "for any claim making use of basic arithmetic, counterexamples must exist." (Counterexamples of what, anyway? Basic arithmetic?) Actually, what the theorem says that you can't have a theory capable of expressing elementary arithmetic that is both consistent and complete. The both is the huge part. So far, for example, we think ZFC is consistent. We know it is not complete; see this: http://en.wikipedia.org/wiki/List_of_statements_undecidable_... . So, the idea that there "must exist counterexamples" (of what, I don't know) as per Godel's First Incompleteness Theorem is simply ridiculous. As a result, we don't know that there are "simply bugs" in the "successor function." As a result, the idea that "all basic claims are more or less false" is completely shot down. Needs better proof.

> The claim above is not Popper-compliant.

Why not? It's falsifiable, so it is. (If it were false, you would need simply to provide an extremely small epsilon and ask for the correspondingly large delta.) It gives the "appearance" of truth because it is indeed true -- relative to the axioms under which it was proven. The author also seems to conflate inductive and deductive disciplines. In mathematics, we create axioms and follow them through to some meaning. In the vast majority of the sciences, the process is very much inductive. So, this seeming focus on making everything mathematical "scientific" is pretty flawed in and of itself. How can you reason about mathematics if you don't have some way to check your results? In the sciences, if I predict gravity will disappear in five minutes for thirty seconds, that is a very testable claim (and a false one, most likely, based on our past experimentation). How can I disprove or prove something in mathematics? Well, we created all these wonderful axioms and they seem consistent (although incomplete, of course), so I can use them to check my results. A little PHP script is not going to cut it. For one thing, the author's script's notion of "division" is predicated on the very idea of division. Where did that come from? For another, the script is not a proof at all. It's just a program to test some values. It doesn't prove anything at all. It's just further evidence towards the proof of something we already know is true.

In sum: the article is bullshit. It misinterprets Godel in terrible ways. For all the fancy French and Latin, it doesn't even provide a programmatic proof of its own claims, the supposed holy grail, after all! The only program it provides is a little test, not anything near a proof. I think if you actually try to provide "Popper-compliant" statements and write programs to prove everything, you end up reverting to all of the normal contemporary mathematical language anyway. I mean, look at this:

> In order to reject this claim, find a number almostZero for which it will not be possible to find a large enough number x for which 1/x is smaller than almostZero.

That sentence is essentially a rewording of an old-fashioned delta-epsilon proof. So, how does that indicate a flaw in contemporary mathematics at all? Why should we stop teaching our kids these "symbols" since they are apparently still so very convenient for reasoning about mathematics? The truth is that there's no reason to. The "executable claims" are only found after all of the "verbiage" is said and done.