Soft question: Examples where implications derived from mathematical models failed to describe reality

Question

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.

Answer 1

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.

Answer 2

Euclidean geometry.

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

Answer 3

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.

Answer 4

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.