From this point on, main issues in system theory are tackled. The very first, considered in this chapter, is the all-important question of system identification. This is perhaps the most basic question in system theory and related linear algebra, with a large pedigree starting from Kronecker's characterization of rational functions to its elegant solution for time-variant systems presented here. Identification, often also called realization, is the problem of deriving the internal system’s equations (called state-space equations) from input–output data. In this chapter, we only consider the causal, or block-lower triangular case, although the theory applies just as well to an anti-causal system, for which one lets the time run backward, applying the same theory in a dual form.

The bridging concept between the abstract and geometric is the theory of realizations. This chapter concentrates on symmetric sets, namely, finite sets on which a group of permutations acts transitively. After a discussion of their basic properties, the concept of their realizations is introduced, with operations on them (such as blending) showing that the family of their congruences classes has the structure of a convex cone. A key idea is that of the inner product and cosine vectors of realizations, which define them up to congruence. The theory up to this point is then illustrated by some examples. It is next shown that, corresponding to the tensor product of representations, there is a product of realizations. Another fundamental notion is that of orthogonality relations for cosine vectors. The different realizations derived from an irreducible representation of the abstract group may form a subcone of the realization cone that is more than 1-dimensional. These are looked at more closely, leading to a definition of cosine matrices for the general realization domain. There follows a discussion of cuts and their relationship with duality. Cosine vectors may have entries in some subfield of the real numbers, with implications for the corresponding realizations. The chapter ends with a brief account of how representations of groups are related to realizations.

Classical regular polytopes are those whose symmetry groups are generated by hyperplane reflexions; these were the subject of Coxeter’s monograph ‘Regular Polytopes’, which first appeared in 1948. This chapter classifies these polytopes, and describes relationships among them. A new condition is given here to be a classical regular polytope. First treated are the three sequences of polytopes – simplex, staurotope and cube – that occur in all dimensions. The 24-cell is looked at next, introducing the useful role played by quaternions in several places. Next are described the familiar icosahedron and dodecahedron. Then the regular 600-cell is constructed and its realization domain found. The dual 120-cell is then described, followed by finding the regular star-polytopes and their abstract automorphism groups. Then the discrete regular apeirotopes or honeycombs are classified. The regular compounds of polytopes are then described, including some that are relatively new. Throughout, the Petrie polygons of these regular polytopes are found by elementary means, that is, not by solving trigonometric equations. The realization domain of the 120-cell is left until last, because describing it draws on much of the earlier material in the chapter.

The chapter uses the principle according to which difference-making or counterfactual dependence is sufficient for causation to show that there are physical effects of mental causes. If non-reductive physicalism is true, applying the principle is straightforward. The principle also yields higher-level causes that are not mental but might be considered problematic. These causes are best diagnosed as causes that have little explanatory relevance. If dualism is true, applying the principle about causation in order to show the existence of mental causation is less straightforward, but still possible. In order to avail themselves of the principle, dualists need to assume that the laws that connect the mental and physical realms have a special status. Rival approaches according to which mental causation or human agency require the transference of a physical quantity or of a power are in conflict with empirical results. The account of mental causation by counterfactual dependence, by contrast, squares with these results.

The chapter deals with the exclusion problem. How can mental events have physical effects if these effects already have physical causes? Even if this is possible in principle, would this not yield a situation like in a firing squad, where the victim’s death is overdetermined by the firings of the squad members? And would it not be implausible that the situation is like this whenever there is mental causation? If a difference-making approach to causation is adopted, these questions can be answered in a satisfactory way. Although mental and physical causes might nominally overdetermine their physical effects, cases of mental causation are sufficiently dissimilar to typical cases of overdetermination not to be problematic. Unlike in cases of mental causation (on the account offered here), in typical cases of overdetermination, the individual causes do not make a difference to the effect. The exclusion problem is harder to solve if it is formulated in terms of sufficient causes, but no commitment to sufficient causation follows from the account of mental causation in terms of difference-making.