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Most of those political philosophers who have tried to make sense of claims about degrees of freedom have proposed that the individual options available to the agent be weighted in terms of their values (whether the subjective values of the agent, objective values or the values of the agent’s society). Most prominent among these authors are Charles Taylor, Amartya Sen, Richard Arneson and Richard Norman. This value-based approach to measuring freedom can be shown to conflict with the view that we are interested in measuring freedom only because freedom has non-specific value. It therefore renders degree-of-freedom statements normatively superfluous. Moreover, even if interpreted as a way of making purely rhetorical sense of such statements, the approach has counterintuitive implications.

Some of the authors who adopt the value-based approach to measuring freedom think of freedom as the absence not only of constraints that are external to the agent but also of constraints that are internal to the agent. Most prominent among these authors is Charles Taylor. On Taylor’s view, freedom coincides with self-mastery or self-determination or “positive freedom”. As well as leading to illiberal judgements of degrees of freedom, the tendency to roll together internal and external freedom into a single quantitative attribute can be shown to be logically inseparable from the value-based approach to measuring freedom. Given the arguments of chapter 5, this rules out internal constraints as a kind of constraint that can be relevant in measuring degrees of overall freedom.

Local regression and likelihood methods are nonparametric approaches for fitting regression functions and probability distributions to data. This chapter discusses the basic ideas behind these methods at a level that is of relevance to the analysis of neural data.

This chapter focuses on the relevance of statistics in deterministic science. While most physical phenomena of concern to natural sciences are governed by fundamental, mostly known, physics, their application to such complex systems as the ocean, atmosphere, or ecosystems is monumentally difficult. For most such systems, the full problems—the values of all relevant variables at all space and time locations—are essentially intractable, even with the fastest computers. Hence, there is always more to the focus of inquiry that cannot be modeled; we must somehow fill in the gaps. This is where statistics come in. Until the state of the physical system under investigation is fully quantified (i.e., until the value of every dynamical variable is perfectly known at every point in space and time), there is a certain amount of indeterminacy in every statement made about the state of the system. The remainder of the chapter discusses probability distributions and degrees of freedom.

In Chapter 2 a simple trick—amounting to a change of variables—is introduced that modifies a large class of dynamical systems so that the (autonomous) ω-modified systems thereby obtained are isochronous, namely, they possess an open, fully dimensional sector in their phase space where all their solutions are periodic in all degrees of freedom with the same fixed period. It is then shown how via this trick the isochronous character can be demonstrated of the dynamical systems reviewed in the introductory Chapter 1. When the isochrony sector coincides with the entire phase space the systems are called entirely isochronous. When it does not, a very terse indication is provided of the (nonisochronous, possibly quite complicated) phenomenology displayed by these ω-modified systems outside of their isochrony region, although this interesting subject exceeds the scope of this monograph.

This chapter asks two questions: What are some of the secrets of networks? And what might constitute their poetics, an aesthetic means of revealing their secrets? It leverages the relation between codes and channels, delving into two topics that link them: degrees of freedom and secrets. By degrees of freedom is meant the number of independent dimensions needed to specify the state of a system. This chapter argues that even relatively commensurate systems, which have identical degrees of freedom, can have different secrets—understood as inherent symmetries that organize their sense-making capacities. This chapter also shows how channels as well as codes can have inherent secrets (in addition to their ability to keep and reveal secrets in more stereotypic ways). By extending the notion of poetics, it shows how such systems can be made to reveal their secrets. As will be seen, all this is a way of reinterpreting the Sapir-Whorf hypothesis (i.e., the idea that the language one speaks affects the way one thinks), such that this hypothesis can be usefully applied to media more generally (such as interfaces, algorithms, infrastructure, and networks).

One attractive feature of the Lagrangian method is the ease with which it solves so-called constraint problems. This chapter presents several different ways of solving such problems, with examples of each. In the previous chapter, the generalised coordinates were assumed to be independent variables. However, there are problems of interest in which these coordinates are not independent, but rather are forced into particular relations by constraints. In this chapter, constraints are defined and virtual displacement is discussed, along with virtual work, form of the forces of constraint, general Lagrange equations with constraints, alternate notation for holonomic constraints, reduction of degrees of freedom, recovery of the forces of constraint, generalised energy theorem with constraints, and tractable non-holonomic constraints.

This chapter introduces the calculus of variations in the context of the finite-dimensional configuration space discussed previously. The calculus of variations is concerned with the comparison of line integrals along different paths. The difference between the integral along some chosen path and the integral of the same quantity along other paths is called the variation of that integral. This chapter discusses paths in an N-dimensional space, variations of coordinates, variations of functions, variation of a line integral, finding extremum paths, example of an extremum path calculation, invariance and homogeneity, the Brachistochrone problem, calculus of variations with constraints, example with constraints, reduction of degrees of freedom, example of a reduction, example of a better reduction, and the coordinate parametric method.

This chapter describes the construction of a superconducting artificial atom with two internal degrees of freedom by adding a large inductance to a dc-SQUID phase qubit loop, thus decoupling the junctions’ dynamics.

This chapter examines implicit assumptions that philosophers make regarding fundamental ontology: atomism and the fundamentality of the micro scale (and so the fundamentality of physics). Along with these assumptions comes the commitment that degrees of freedom belong only to atoms. This chapter puts forward an alternative conception of fundamentality: systemism. According to Systems theory, bonds between entities are the most fundamental characteristic of objects. The principles of Systems make clear the importance of situated entities; a System results from the application of a sheath. The properties of a System quite frequently have very little to do with the properties of the constituents when they are not integrated in the System. There are significantly fewer degrees of freedom in a System than one would expect if reductionism is true.

A number of interesting mechanical systems have one or more essentially stable equilibrium configurations. When disturbed slightly, they vibrate about equilibrium in characteristic patterns called normal modes. This chapter presents the Lagrangian theory of these small vibrations for the simple case of systems with a finite number of degrees of freedom. The theory has wide application. For example, the normal mode oscillations of crystalline solids underlie both the overtone structure of a church bell and the definition of phonons in solid state physics. A similar formalism leads to photons as the quanta of modes of the electromagnetic field. This chapter defines equilibrium points in the configuration space of a mechanical system and discusses how to find them, along with small coordinates, normal modes, generalised eigenvalue problem, stability, initial conditions, energy of small vibrations, single mode excitations, and zero-frequency modes.

In this chapter we introduce the concepts of symmetry, dimensions and scaling, which are fundamental to a principled construction and interpretation of minimal models in this book. We also introduce the formal models for production and preferences, which we call the “pre-institutional society”. Society defines the context within which the institutions of the polity and the economy exist. The pre-institutional society is meant to represent the shared layers of constraint within which the different economic institutions must function. Economic life is potentially complex in all its dimensions. In comparison to reality, the space of tractable models is small. The subject of this chapter is to develop criteria for categorizing models and criteria by which to judge the validity of using them as sufficient representations of economic phenomena.

This chapter applies the teleological account of agency developed in the first part of the book to free will. The chapter argues that free actions should be identified with behaviors for which we are responsible, and that these in turn should be identified with intentional actions or goal-directed behaviors—that is, the teleologically explicable behaviors. It is further argued that those who make distinctions between these categories are motivated to do so by a prior acceptance of the causal theory of action. The chapter makes a preliminary argument for the claim that, on the non-causal, teleological account of action explanation, determinism is irrelevant to free will, thus yielding a non-causal compatibilist view of freedom. Since teleological explicability comes in degrees, the view also implies that freedom of the will itself comes in degrees, and the chapter argues that this is a good result.

This chapter presents extended forms of Hamilton’s principle and the phase space Hamilton’s principle based on the extended Lagrangian and Hamiltonian methods developed earlier. It also discusses Noether’s theorem, a method for using symmetries of the extended Lagrangian to identify quantities that are conserved during the motion of the system. Noether’s theorem is a powerful technique for discovering conserved quantities in complex Lagrangian systems. The basics of the method are analysed in the simple context of Lagrangian systems with a finite number of degrees of freedom.

There are some tough problems in comprehending the control of head movements. The head-neck system is multijointed and the posture and the movement of the head can be controlled by distinct pairs of muscles that may subserve the same functions or help to perform a particular task. There seems to be considerable redundancy. The behavioral degrees of freedom are few, yet simple movements such as rotating the head may result from the contraction of many muscles acting in a coordinated manner manifesting the necessity for some constraints. Another problem is that different tasks may need to be performed and the organization of the sensory inputs and the motor outputs must be appropriate for a particular task, such as controlling gaze or posture or both at the same time.

Like many biologic systems, one of the differentiating features of the head and neck is its mechanical intricacy. The head-neck system includes approximately thirty muscles; each spans multiple joints, and each joint has multiple degrees of freedom. The sensory system includes several radically different types of sensory organs. At first, this intricacy may seem tough, yet it must be confronted squarely if a deep comprehension of sensory-motor coordination is to be made. One aspect of sensory-motor coordination that is epitomized by the head-neck system is the general problem of coordinate transformations. The root of the problem is that several parts of the process of doing an action in response to sensory stimuli are each largely described in their own terms.

Materials are made up of regions of space that are homogeneous in structure and properties, called phases. The number of different phases in a material depends on its temperature, pressure and composition, as given when the material is at equilibrium by the Gibbs phase rule. This was discovered by the American scientist J. Willard Gibbs during his ground-breaking investigations in the late 19th century into the thermodynamics of heterogeneous materials. This chapter explains the differences between solutions, mixtures and compounds; the use of phase diagrams to determine the structure of a material; and the way in which phase transformations can be used to change the structure of a material. Gibbs grew up in an academic family at Yale University in New Haven at the time of the American Civil War. He was the first person to receive an engineering doctorate in the United States, and he later became a fundamental theoretician of thermodynamics, statistical mechanics and vector fields.

This chapter focuses on the electron spin degree of freedom in semiconductor spintronics. Exploitation of the spin degree of freedom for the conduction carriers provides a key strategy for finding new functionalities. In particular, the electrostatic control of the spin degree of freedom is an advantageous technology over metal-based spintronics. However, carriers in semiconductors are not spin-polarised, and generation of spin-polarised carriers is crucial for semiconductor spintronics. The main point of the chapter is spin–orbit interaction (SOI), which gives rise to an effective magnetic field. The essence of SOI is that the moving electrons in an electric field feel an effective magnetic field even without any external magnetic field. In particular, the Rashba spin–orbit interaction is important since the strength is controlled by the gate voltage on top of the semiconductor's two-dimensional electron gas.