Marx's Labor Theory of Value: A Defense

he present book and its companion volume The Tenseless Theory of Time: a T Critical Examination are an attempt to adjudicate what one recent discussant has called "the most fundamental question in the philosophy of time," namely, "whether a static or a dynamic conception of the world is correct.

This book treats the mechanisms of growth and cycles in capitalist economies in a unified manner, incorporating a highly original macro-dynamic theory based on Marxian micro-foundations and historical perspectives. That theory was developed about 50 years ago by Nobuo Okishio (1927–2003) and included the ideas of Keynes and Harrod. In mainstream economics, it used to be standard to analyse long-term economic growth and business cycles in different frameworks. That approach has been changing recently, but it still tends to be common to discuss them separately.

In general terms, the shape of an object, data set, or image can be de­ fined as the total of all information that is invariant under translations, rotations, and isotropic rescalings. Thus two objects can be said to have the same shape if they are similar in the sense of Euclidean geometry. For example, all equilateral triangles have the same shape, and so do all cubes. In applications, bodies rarely have exactly the same shape within measure­ ment error.

Matric algebra is a mathematical abstraction underlying many seemingly diverse theories. Thus bilinear and quadratic forms, linear associative algebra (hypercomplex systems), linear homogeneous transformations and linear vector functions are various manifestations of matric algebra. Other branches of mathematics as number theory, differential and integral equations, continued fractions, projective geometry etc. make use of certain portions of this subject. Indeed, many of the fundamental properties of matrices were first discovered in the notation of a particular application, and not until much later recognized in their generality.

Theory of Computation is a unique textbook that serves the dual purposes of covering core material in the foundations of computing, as well as providing an introduction to some more advanced contemporary topics. This innovative text focuses primarily, although by no means exclusively, on computational complexity theory: the classification of computational problems in terms of their inherent complexity. It incorporates rigorous treatment of computational models, such as deterministic, nondeterministic, and alternating Turing machines; circuits; probabilistic machines; interactive proof systems; automata on infinite objects; and logical formalisms. Although the complexity universe stops at polynomial space in most treatments, this work also examines higher complexity levels all the way up through primitive and partial recursive functions and the arithmetic and analytic hierarchies.

This series of volumes, a list of which is given on pages vn and VIU, takes up mathematics at the beginning, and gives complete proofs. In principle, it requires no particular knowledge of mathematics on the reader's part, but only a certain familiarity with mathematical reasoning and a certain capacity for abstract thought. Nevertheless, it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course.