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I've heard numerous articles and videos state that when it comes to black holes and singularities our "math breaks down".

Is there any other areas in physics where math similarly "breaks down", where the equation returns zero or unexpected values?

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    $\begingroup$ This question (v2) seems like a list question. $\endgroup$ – Qmechanic Mar 16 '16 at 10:10
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    $\begingroup$ You can construct such examples in any theory. Self-energy of charges is a trivial question that has no useful answer in electrodynamics. Scattering on 1/r potentials (gravitational or electrostatic) leads to undefined results for small distances as well as for scattering cross section at large distances. Neither the ideal gas nor coupled harmonic oscillators satisfy a naive notion of the ergodic hypothesis in statistical mechanics... the joke is really on everyone who expects that even near-trivial physical systems behave "nicely". $\endgroup$ – CuriousOne Mar 16 '16 at 12:02
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It simply means that the mathematical models used for things like black holes give stupid results when some numbers become extreme. It is an indication that the model is not good enough to handle those situations. Typically, the equations return "infinity" - like one divided by zero, or zero when there should be a non-zero value.That's why you end up with statements like the interior of a black hole is an infinitely dense point of zero dimension.

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    $\begingroup$ exactly. Math only "breaks down" in the sense of Goedel's incompleteness theorem. $\endgroup$ – Carl Witthoft Mar 16 '16 at 15:26
  • $\begingroup$ Has physics come up against Godel in any meaningful sense (beyond limits to computation)? $\endgroup$ – user56903 Mar 16 '16 at 15:39
  • $\begingroup$ Not sure that question makes sense, since physics is what it is, and we use whatever math we've learned/discovered to try to model our universe. $\endgroup$ – Carl Witthoft Mar 16 '16 at 16:35
  • $\begingroup$ Well, one possible consequence might be the impossibility of ever proving that any TOE we develop is the best even if it passes all tests forever $\endgroup$ – user56903 Mar 16 '16 at 16:50
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Just because an equation returns infinity doesn't mean that it's not "good enough" to describe the situation. One had to differentiate between removable singularities (singularities that depend on the choice of coordinates) and and non-removable singularities.

For example with the schwarzschild metric as r-> 2m there is a singularity but it can be removed by introducing Kruskal coordinates. However, this doesn't remove the singularity as r->0.

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