Given the operation of squaring a number, there are two numbers that are special because the operation doesn't change them: 0 and 1. 0 squared is 0, and 1 squared is 1, but square any other number and the output will differ from the input. These special numbers are called fixpoints (of squaring). In general, a fixpoint is a value (or state of a system) that is left unchanged by a particular operation. This concept, easily definable by the brief equation x=f(x), is at the essence of several other ideas with great practical significance, from Nash equilibria (used in economics and social sciences, to model failures of cooperation) to stability (used in control theory to model systems ranging from aircraft to chemical plants) to PageRank (the foremost web search algorithm). It even appears in work at the foundations of logic, to give definition to truth itself!

A prevalent example in everyday life is the occurrence of fixpoints in any strategic context ("game") with multiple players. Suppose each player can independently revise their strategy so as to best respond to the others' strategies. If we consider this revision process as an operation, then its fixpoints are those combinations of strategies for which the revision operation yields exactly the same combination of strategies: that is, no player is incentivized to change their behavior in any way. This is the concept of a Nash equilibrium, a kind of fixpoint that directly applies to many real-world situations—ranging in importance from roommates leaving dirty dishes in the sink to nuclear arsenals. Moving to a different Nash equilibrium (such as disarmament) requires changing the revision operator (e.g., with an agreement that binds multiple players to change their strategies at the same time).

Fixpoints are also an extraordinarily useful framework to consider ideas that seem to be defined circularly. As a concrete example, the golden ratio phi can be defined as phi=1+1/phi, which actually means phi is the unique fixpoint of the operation that takes the reciprocal then adds one. In general, self-referential definitions can be classified according to the number of fixpoints of the corresponding operation: more than one, and it's imprecise but potentially serviceable; one, and it's a bona fide definition; lacking any, and it's a paradox (like "this sentence is false"). In fact, it is operations that lack fixpoints which underly a whole host of deeply-related paradoxes in mathematics, and led to Gödel's discovery that the early-20th-century logicians' quest to completely formalize mathematics is impossible.

Mathematics is not the only reasoning system we might use; in daily life we use ideologies and belief systems that aren't complete—in the sense that they can't express arbitrarily large natural numbers (nor do we care for them to). We can consider persistent ideologies and belief systems as fixpoints of the revision operation of "changing one's mind"—subject to the types of questions, evidence, and methods of reasoning that the ideology judges as acceptable. If there's a fixpoint here, we're unable to change our minds in some cases, even if presented with overwhelming evidence. As with Nash equilibria, the key to escaping such a belief system is to modify the revision operation: considering just one new question (such as "what questions am I allowed to ask?") can sometimes be enough to abolish ideological fixpoints forever. 

One remarkable feature of the scientific belief system is its non-fixedness: new beliefs are constantly integrated, and old beliefs are not uncommonly discarded. Ideally, science would be complete in the limit of infinite time and experiments without losing its openness, analogous to how programs that produce never-ending lists (such as the digits of pi or the prime numbers) are formally given meaning by infinitely large fixpoints which may only be successively approximated. While an absence of fixpoints in the logical foundations of arithmetic dooms it to "incompleteness," fixpoint theorems have recently been used to show that if we relax our notion of completeness to the almost-equally-satisfying concept of "coherence," there is a revision operation which is guaranteed to have a coherent fixpoint, and can even be approximated by computable algorithms!

Perhaps science, too, aspires to an unreachable, infinite fixpoint in which all knowable facts are known and all provable consequences are proven, such that there would be no more room to change one's mind—and we hope that with each passing year, our current state of knowledge more closely approximates that ultimate fixpoint.