This chapter contains some paradoxes that arise without any syntax theory. It sheds some light on how Russell’s paradox and the liar paradox are connected.

This chapter introduces matrix factorizations – somewhat like the reverse of matrix multiplication. It starts with the eigendecomposition of symmetric matrices, then generalizes to normal and asymmetric matrices. It introduces the basics of the singular value decomposition (SVD) of general matrices. It discusses a simple application of the SVD that uses the largest singular value of a matrix (the spectral norm), posed as an optimization problem, and then describes optimization problems related to eigenvalues and the smallest singular value. (The “real” SVD applications appear in subsequent chapters.) It discusses the special situations when one can relate the eigendecomposition and an SVD of a matrix, leading to the special class of positive (semi)definite matrices. Along the way there are quite a few small eigendecomposition and SVD examples.

This chapter starts out by introducing the energy eigenvalue equation – the time-independent Schrödinger equation. Firstly, the notion of energy quantization is introduced by semi-analytical means. A particle is confined within a rectangular well. It is seen, eventually, that the corresponding solutions to the time-independent Schrödinger equation can only exist for a few, specific energies. This phenomenon is also seen for other physical quantities, such as angular momentum and charge. Next, the situation in which the potential experienced by a quantum particle is periodic is given particular attention. It may be studied numerically by minor adjustments of the framework already developed. The case of a periodic potential is important as, to a large extent, it forms the basis for understanding solid state physics. Since the Hamiltonian is Hermitian, its eigenvectors form an orthonormal set in which any state or wave function may be expanded. This is exploited in the last part of the chapter, which is dedicated to determining ground states – the energy eigenstate of minimal energy. This is done in two ways – by using what is called the variational principle and by so-called propagation in imaginary time. These methods are implemented for several examples and compared to full solutions.

This appendix collects a series of analytical methods that are needed in various parts of the book. All the tools are oriented toward the diagonalization of Hamiltonian, including some cases that allow a complete analytical diagonalization (e.g., coupled resonators) as well as perturbation theory methods for cases where the fully interacting model is too complex for an exact solution.

This chapter considers two possible explanations for the paradoxes.One is Lawvere’s diagonal theorem from category theory. Theother is the inclosure schema, proposed by Priest as the structureof many paradoxes and a step toward a uniform solution to theparadoxes. Inclosure suggests that paradoxes arise at the limits ofthought because the limits can be surpassed, and also not. Theconsequences of accepting Priest’s proposal are explored, andit is found that, from a thoroughly dialetheic perspective, (i) somelimit phenomena cannot be contradictory, on pain of absurdity, and(ii) true contradictions are better thought of as local, not“limit,” phenomena. Dialetheism leads back from theedge of thought, to the inconsistent in the everyday.

The algebraic eigenproblem is the mathematical answer to the physical questions:What are the principal stresses in a solid or fluid and on what planes do they act?What are the natural frequencies of a system?Is the system stable to small disturbances?What is the best basis with respect to which to solve a system of linear algebraic equations with a real symmetric coefficient matrix?What is the best basis with respect to which to solve a system of linear ordinary differential equations?What is the best basis with respect to which to represent an experimental or numerical data set?