Pathological (mathematics) - Pathological Examples

Pathological Examples

Pathological examples often have some undesirable or unusual properties that make it difficult to contain or explain within a theory. Such pathological behaviour often prompts new investigation which leads to new theory and more general results. For example, some important historical examples of this are the following:

• The discovery of irrational numbers by the school of Pythagoras in ancient Greece; for example, the length of the diagonal of a unit square, that is
• The discovery of number fields whose rings of integers do not form a unique factorization domain, for example the field .
• The discovery of fractals and other "rough" geometric objects (see Hausdorff dimension).
• Weierstrass function, a real-valued function on the real line, that is continuous everywhere but differentiable nowhere.
• Test functions in Fourier analysis, which are complex-valued functions on the real line, that are 0 everywhere outside of a given limited interval (hence all derivatives will also be 0 outside of the interval) and inside of the interval, but are still infinitely differentiable everywhere. An example of such a function is the test function,

• The Cantor set is a counterexample to the notion that a measure-zero set must be countable. The Cantor set is both measure-zero (i.e. has "length" 0) and uncountable.
• The Dirichlet function is the function f defined such that f(x) is 1 if x is rational and 0 if x is irrational. This is a counterexample to the idea that every bounded function is (piecewise) integrable.

At the time of their discovery, each of these was considered highly pathological; today, each has been assimilated, which is to say, explained by an extensive general theory. These examples may prompt a reassessment of foundational definitions and concepts. Historically, this has led to cleaner, more precise, and more powerful mathematics.

Such judgments about what is or is not pathological are inherently subjective or at least vary with context and depend on both training and experience—what is pathological to one researcher may very well be standard behaviour to another.

Pathological examples can show the importance of the assumptions in a theorem. For example, in statistics, the Cauchy distribution does not satisfy the central limit theorem, even though its symmetric bell-shape appears similar to many distributions which do; it fails the requirement to have a mean and standard deviation which exist and are finite.

The best-known paradoxes such as the Banach–Tarski paradox and Hausdorff paradox are based on the existence of non-measurable sets. Mathematicians, unless they take the minority position of denying the axiom of choice, are in general resigned to living with such sets.

Other examples include the Peano space-filling curve which maps the unit interval continuously onto ×, and the Cantor set, which is a subset of the interval and has the pathological property that it is uncountable, yet its measure is zero.