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tl;dr Incompleteness does not say what you thought it says.
Incompleteness theorems state that "For any *formal system* whose theory is recursively enumerable (including Peano Arithmetic), if it *is* consistent, then first, it is incomplete, and second, it can't *prove* its own consistency."
1. Here we are talking about arithmetic as a formal system, that is one that can be performed by a machine autonomously, not the arithmetic you do in school.
2. This does not mean that arithmetic is inconsistent, but as a matter of fact, it assumes the consistency of arithmetic. You might've mistaken its inability to prove its own consistency with its inconsistency. These are not the same. -
3. Incompleteness means there is a *sentence* that is not a theorem of this specific formal system, nor is its negation. But this doesn't mean that it is not a theorem in another system; in fact we can always devise a system that proves the consistency of another (somehow simpler) system, but (if it satisfies the mentioned conditions) it also eventually fails to prove something else, like its own consistency, now and forever. It got nothing to do with what "current science" can't prove that "future science" can. It also doesn't mean that we could decide for an arbitrary given sentence, like P=NP, if it is one of those unprovable sentences or not.
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4. This kind of ideal "science" you're talking about should be based on a *formal system*, which is, dude, like, none. To understand the difference, see it as something that can be done solely by a machine. This just could be remotely applied to automated theroem provers, which only constitute to (a rather small part of) the pure mathematics, not online holey wars.
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5. On the other hand the kind of science that *proves* "Earth is not flat", "Earth orbits Sun", "Santa isn't real" or even "P=NP", works by *observation* and *deduction*, both can be different from what a machine does. For example one could first observe some evidence (like measuring the sum of angles of a giant triangle, or just walking a straight line and returning to the starting point) and after series of (possibly intuitionistic) deductions, conclude that Earth is not flat. Or one could just be thrown out to the outer space and observe it directly, without (or with simpler) further deductions. Both are valid scientific proofs. So the notion of a proof could also be different from that of a formal system. (Although, I can tell by experience that online debates over these topics are also far from being scientific, even in this sense.)
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6. Furthermore, a formal system is intended to be a machinery for manipulating strings, without considering what they could actually *mean*. In fact the whole point of formalism is that you can keep out those ridiculous beliefs by separating the syntax of a language from its semantics, preventing ambiguous and inconsistent interpretations. This comes at the price of narrowing down what we are actually able to talk about in a formal language to a very small and specific group of notions. And I tell you, the formal language doesn't care at all.
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7. So even if we are sure that some formal system works just as we humans deduce and think logically, we are yet to be sure that it is in fact talking about the real world (if there's such thing), so we can take its theorems as actual facts. This depends on how we define it (like by different axioms, say one that states that "Santa is real"), and how we interpret its sentences (by different models, so we can interpret "Santa is real" to mean something different from its usual meaning in English). For example the peano arithmetic here can be true in many different interpretations, i.e. it can describe different mathematical structures, many differ drastically from what we intuitively think about natural numbers. So to devise a formal system which is able to prove something about the physical world, we first need a concrete basis for our interpretation of (what is true in) the physical world.
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8. Even if there was such a formal system, it would be consistent by the virtue of it being satisfiable in some model (the real world). So this science can't just prove something and disprove it later, and that there are sentences that are true and false does not even makes sense.
9. Not all those ridiculous beliefs need to seem "unscientific". Many of them, quite contrary, have a legit look and are treated as the scientific revelation, when many of them are wrong in the very basics. So they are even harder to detect and prevent. IMO this is just another religion for the majority to both feel intellectual and also safe from what they still can't accept. I guess we have a goog example here. -
@beleg your points heavily rely on your understanding of formal systems and their use. It is probably better for you to rrad more on that than me to try to explain bullet by bullet.
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@ravijojila well, incompleteness is about formal systems, you can't generalize it to arbitrary notions. That's why I asked for reference to the rant you said, to understand your understanding of formal systems.
And sorry, I didn't quite get your last sentence.
On science and religion. Inspied by a comment in another rant, credits to @Commodore and @cjbatz
According to Godel's incompletness theorems, aritmetics is incomplete and inconsistent. Therefore, any science based on aritmetics (dude, like, every) is also.
Therfore, as a mathematician, I must accept that there are things that cannot be proven by current science, and that there are statements that are true and false at the same time in current science.
So, science can't prove religious beliefs? It cant prove P vs NP either. It might someday. Science couldn't prove earth wasn't flat for a looong time. Or Pythagoras theorem.
But more importantly, if science can prove something, doesn't mean it can't prove the exact oposite.
This way of thinking allows for any and all ridiculous beliefs, under the shield of "it might be proven one day" or "doent't mean opposite isn't true also" but kerp in mind that there are complete and consistent sciences and proofs in them. Check if something's been proven to exist or not exist without doubt.
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