Freedom, generally, is having an ability to act or change without constraint. A thing is "free" if it can change its state easily and is not constrained in its present state. In philosophy and religion, it is associated with having free will and being without undue or unjust constraints, or enslavement, and is an idea closely related to the concept of liberty. A person has the freedom to do things that will not, in theory or in practice, be prevented by other forces. Outside of the human realm, freedom generally does not have this political or psychological dimension. A rusty lock might be oiled so that the key has freedom to turn, undergrowth may be hacked away to give a newly planted sapling freedom to grow, or a mathematician may study an equation having many degrees of freedom. In mechanical engineering, "freedom" describes the number of independent motions that are allowed to a body or system, which is generally referred to as degrees of freedom."
Advocates of free will regard freedom of thought as innate to the human mind, while opponents regard the mind as thinking only the thoughts that a purely deterministic brain happens to be engaged in at the time.
In political discourse, political freedom is often associated with liberty and autonomy in the sense of "giving oneself one's own laws", and with having rights and the civil liberties with which to exercise them without undue interference by the state. Frequently discussed kinds of political freedom include freedom of assembly, freedom of association, freedom of choice, and freedom of speech.
In some circumstances, particularly when discussion is limited to political freedoms, the terms "freedom" and "liberty" tend to be used interchangeably. Elsewhere, however, subtle distinctions between freedom and liberty have been noted.John Stuart Mill, differentiated liberty from freedom in that freedom is primarily, if not exclusively, the ability to do as one wills and what one has the power to do; whereas liberty concerns the absence of arbitrary restraints and takes into account the rights of all involved. As such, the exercise of liberty is subject to capability and limited by the rights of others.
Wendy Hui Kyong Chun explains the differences in terms of their relation to institutions:
Liberty is linked to human subjectivity; freedom is not. The Declaration of Independence, for example, describes men as having liberty and the nation as being free. Free will--the quality of being free from the control of fate or necessity--may first have been attributed to human will, but Newtonian physics attributes freedom--degrees of freedom, free bodies--to objects.
Freedom differs from liberty as control differs from discipline. Liberty, like discipline, is linked to institutions and political parties, whether liberal or libertarian; freedom is not. Although freedom can work for or against institutions, it is not bound to them--it travels through unofficial networks. To have liberty is to be liberated from something; to be free is to be self-determining, autonomous. Freedom can or cannot exist within a state of liberty: one can be liberated yet unfree, or free yet enslaved (Orlando Patterson has argued in Freedom: Freedom in the Making of Western Culture that freedom arose from the yearnings of slaves).
Another distinction that some political theorists have deemed important is that people may aspire to have freedom from limiting forces (such as freedom from fear, freedom from want, and freedom from discrimination), but descriptions of freedom and liberty generally do not invoke having liberty from anything. To the contrary, the concept of negative liberty refers to the liberty one person may have to restrict the rights of others.
In purely physical terms, freedom is used much more broadly to describe the limits to which physical movement or other physical processes are possible. This relates to the philosophical concept to the extent that people may be considered to have as much freedom as they are physically able to exercise. The number of independent variables or parameters for a system is described as its number of degrees of freedom. For example the movement of a vehicle along a road has two degrees of freedom; to go fast or slow, or to change direction by turning left or right. The movement of a ship sailing on the waves has four degrees of freedom, since it can also pitch nose-to-tail and roll side-to-side. An aeroplane can also climb and sideslip, giving it six degrees of freedom.
Degrees of freedom in mechanics describes the number of independent motions that are allowed to a body, or, in case of a mechanism made of several bodies, the number of possible independent relative motions between the pieces of the mechanism. In the study of complex motor control, there may be so many degrees of freedom that a given action can be achieved in different ways by combining movements with different degrees of freedom. This issue is sometimes called the degrees of freedom problem.
Some equations have many such variables. This notion is formalized as the dimension of a manifold or an algebraic variety. When degrees of freedom is used instead of dimension, this usually means that the manifold or variety that models the system is only implicitly defined. Such degrees of freedom appear in many mathematical and related disciplines, including degrees of freedom as used in physics and chemistry to explain dependence on parameters, or the dimensions of a phase space; and degrees of freedom in statistics, the number of values in the final calculation of a statistic that are free to vary.