Frequently mathematical errors can be attributed to applying a theorem or using a formula in a situation which does not meet all the initial requirements. An obvious example would be the use of the Pythagorean theorem to compute the length of the longest side of a scalene triangle. Such trivial mistakes are rarely made. However, when considering mathematical theory together with the physical universe, this is essentially what some mathematicians have done. Instead of determining theoretical relationships to describe the universe, they idealize the universe to conform to a mathematical model. In doing this some errors slip by under the guise of extrapolation and simplification. To avoid these sources of error and to obtain significant results relevant to the physical universe, it has become necessary for numerical analysis to transcend the bounds of mathematical rigor.