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I have always been fascinated by how well conclusions drawn from mathematical models could fit reality, so I wondered if there are any counter examples.

In "Gödel, Escher, Bach" I could already find some pathological examples where axiomatic models seem to fit something like addition, but its implications do not fit anymore–however, I would like to know if there are any (historical) real world examples of something like this.

Wall of text for clarification: I would be very grateful if answers could be split up into the two categories "assumptions were not correct" (e.g. pre-relativistic physical models when gravity was thought to be exerted instantly over any arbitrary distance or economic models that did not take Newcomblike-problems into account) and "the mathematical models fit observed reality, however their implications do not anymore" (or the ones that cannot be made correct by adjusting assumptions). Latter might seem impossible as we are talking about implications, but equivalence or implications are only defined within our "arbitrary" axiomatic system. So while you may argue that our models are defined in order to reflect reality (e.g. our concept of natural numbers), they can only do so marginally as even the existence of their majority (e.g. irrational (Pythagoreans), negative and imaginary (pretty much every mathematician at the time) numbers) has been disputed to a great extent. So I came to the conclusion that it is a nice "coincidence" that these models, relying on so many little helpers, really seem to be equivalent to observed reality.

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I'd say that given the right premises, then the model works. If it does not work, it's not the logic's fault but the premises'. At least in the "real world". –  geodude Mar 28 at 16:40
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Every scientific theory (which at least as of the last 400-500 years have been mathematical models) that has ever been supplanted (proved wrong) made predictions that did not describe reality. Newton's theory of gravity perhaps isn't a great example, but still by not predicting things like Mercury's precession, it essentially predicted the wrong thing. Perhaps a more convincing "blunder" is classical statistical mechanics which predicted the so-called ultraviolet catastrophe (which everyone knew was nonsensical--hence "catastrophe"): en.wikipedia.org/wiki/Ultraviolet_catastrophe –  Jared Mar 28 at 16:51

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Euclidean geometry.

I am asked to enter at least 30 characters, hence this additional sentence.

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As geodude said, if the premises are correct then the conclusions are correct. However, some models used in mathematics do not reflect reality.

A Turing machine is one example. Even if infinite memory is theoretically possible, accessing any part of memory in constant time is not. This is why most mathematical analysis of runtime aren't exactly correct (off by a factor of a polylogarithm), but you probably won't notice unless you do hardware design.

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I have a hard time being able to relate to this - if premises are correct, then the conclusions are correct within the formal system we are working in, but not necessarily for "reality". (I edited the original question accordingly) (I hope geodude is not sore, but I am answering to this point here, as it is hard to keep track of a discussion going on in the original question's comment section) –  Peter Mar 28 at 17:35
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I am having a hard time understanding exactly the point of the question in the first place. First off, this answer doesn't make sense to me. "If the premises are correct then the conclusions are correct". No, if the premises are correct and your argument is valid, then the conclusions are correct, i.e. your argument is both sound (the premises are correct) and valid (the logic is correct). The problem with reality is that there is no starting point from which you can claim any premise to be absolutely correct and thus no matter how valid your argument, it can never be logically sound. –  Jared Mar 28 at 18:12
    
If you could elaborate on why this question is pointless, this would be an answer as well I guess - to be honest I'm not feeling overly proficient here :) The problem you are addressing in the last paragraph pretty much sums up my amazement that theories can describe reality so well and even lead to progression and as there are so many examples where it "just works" nonetheless, I was wondering if there are counterexamples as well. –  Peter Mar 28 at 18:25
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The distinction of whether or not you consider rules of inference to be assumptions or not is arbitrary, arguing over them is just arguing vocabulary. To your question Peter, there is not a universal consensus on logic. Different logics have different rules of inferences. If you want, feel free to make up a logic whose rules of inferences don't reflect reality; no one is stopping you. If you are interested in established logics that have been found in error, look up paradoxes, there have been a few, I believe Churchhill's logic had one. –  DanielV Mar 29 at 1:21
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@Peter oops, I meant Alanzo Church, not Churchhill –  DanielV Mar 31 at 2:21

Doesn't Gödel's second incompleteness theorem mean that even arithmetic cannot describe reality perfectly ? I am making the assumption that in reality everything is either true or false. I know this is sloppy wording and I feel like putting most of my words in quotes. What I mean is that my gut says that for example, in reality the Continuum hypothesis is either true or false. There's either a set with a higher cardinality than natural numbers, but lower than reals, or not.

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If we make the assumption that all well-formed formulas are part of reality, then yes. But that is a rather bold claim; I don't feel comfortable saying that any infinite set is "real" in (e.g.) the Platonic sense. –  Eric Stucky Apr 8 at 8:51
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Fair enough. So are there Gödel sentences that have to do with something that can be mapped to something "tangible" ? –  Balazs Rau Apr 8 at 16:44

The paradox known as the Banach-Tarski theorem would have it that you can take a unit ball, subdivide it into 5 parts, move those parts around by rigid motions, and reassemble them into... a pair of unit balls. This tends to go counter to physical assumptions such as conservation of mass, etc. The official explanation usually given is that the "parts" in question are of a particularly nettlesome kind known as "nonmeasurable". This does not take away from the fact that the paradox tends to go counter to what natural scientists would expect to happen. The "mathematical existence", as it were, of such sets is a consequence of a nonconstructive set-theoretic axiom known as the axiom of choice.

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