First-order

In mathematics and other formal sciences, first-order or first order most often means either:

"linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of higher degree", or
"without self-reference", as in first-order logic and other logic uses, where it is contrasted with "allowing some self-reference" (higher-order logic)

In detail, it may refer to:

Mathematics

First-order approximation
First-order arithmetic
First-order condition
First-order hold, a mathematical model of the practical reconstruction of sampled signals
First-order inclusion probability
First Order Inductive Learner, a rule-based learning algorithm
First-order reduction, a very weak type of reduction between two computational problems
First-order resolution
First-order stochastic dominance
First order stream

Differential equations

Exact first-order ordinary differential equation
First-order differential equation
First-order differential operator
First-order linear differential equation
First-order non-singular perturbation theory
First-order partial differential equation, a partial differential equation that involves only first derivatives of the unknown function of n variables
Order of accuracy

Logic

First-order language
First-order logic, a formal logical system used in mathematics, philosophy, linguistics, and computer science
First-order predicate, a predicate that takes only individual(s) constants or variables as argument(s)
First-order predicate calculus
First-order theorem provers
First-order theory
Monadic first-order logic

Chemistry

First-order fluid, another name for a power-law fluid with exponential dependence of viscosity on temperature
First-order reaction a first-order chemical reaction
First-order transition

Computer science

First-order abstract syntax
First-order function
First-order query

Other uses

First-order desire
First-order election, in political science, the relative importance of certain elections
First order Fresnel lens

gollark: On many issues.

gollark: The common consensus is kind of bad.

gollark: (you have to pick the opinions to put on it of course)

gollark: https://osmarks.net/stuff/political_opinion_calendar.html

gollark: So instead of being bad™ and picking your political opinions based on either your social environment or "actual underlying beliefs", you can just read your opinion for the day off the calendar.