The Parmenidean Ascent

For the last decade I have been demonstrating that many of the so-called paradoxes generated by the Copenhagen interpretation of quantum mechanics have less puzzling analogs in nonlinear dynamics and chaos theory. This raises questions about the possibility of nonlinearities inherent in the foundations of quantum theory. Since most of us do not think intuitively with nonlinear logic, I take this opportunity to dwell on several peculiarities of nonlinear dynamics and chaos: nonlinear logic itself and the possible connection of infinite nonlinear regressions with free will. Superficially, nonlinear dynamics can be every bit as counterintuitive as quantum theory; yet, its apparent paradoxes are more amenable to logical analysis. As a result, using nonlinear dynamics to resolve quantum paradoxes winds up being more straightforward than many of the alternative interpretations currently being formulated to replace the orthodox interpretation. Chaos theory could be a candidate for bridging the gap between the determinism so dear to Einstein and the statistical interpretation of the Copenhagen School: For deterministic chaos is indeed deterministic; however, intrinsic limitations on precision in measuring initial conditions necessitate analyzing it statistically. Einstein and Bohr both could have been correct in their debates.

338 BOOK REVIEWS necessary propaedeutic for what follows, and some help is needed for the student to accomplish this successfully. The expanded Companion might also touch more on matters of political philosophy and might note the polemic surrounding Popper's attack on the Republic. No Companion to the Republic, however rich in informative explanation and interpretation, can ever exhaust its great subject, and there is much need for a Companion which is as rich as possible. The Catholic University of America Washington, D.C. DOMINIC O'MEARA The Kaliim Cosmological Argument. By WILLIAM LANE CRAIG. New York: Harper & Row, 1979. Contemporary discussions of the cosmological argument generally follow the philosophical model provided by Aristotle and Aquinas, which first invokes an empirical fact about the world (contingent beings exist; there is something in motion) , seeks for a cause or explanation of that fact, notes that an infinite series of causal conditions ordered transitively cannot provide an adequate explanation of that fact, and concludes to the existence of a necessary being or first cause. William Craig helpfully reminds us that this constitutes only one general form the cosmological argument has taken, a form originated by the Arabic practitioners of falsafa. His interest lies in another form developed by practitioners of kaliim, a " whole movement within Arabic thought that might best be called Arabic scholasticism" (4). Craig summarizes the lcaliim cosmological argument, which contrary to the above is concerned with the temporal sequence of events, as follows: 1. Everything that begins to exist has a cause of its existence. 2. The universe began to exist. 3. Therefore the universe has a cause of its existence. (63) The first sixty pages of his book present the kaliim cosmological argument as defended by the 9th and 11th century Arabic philosophers al-Kindi and al-Ghiizali and the 10th century Jewish philosopher Saadia. His concern is to provide a clear presentation of the argument as originally and most forcefully developed, particularly with respect to the defense of premise 2, rather than a detailed critique of it. The remainder of the book is devoted to presenting and defending a contemporary version of the kaliim cosmological argument. With respect to the argument given above, premise 1 is taken by Craig as intuitively obvious. " The first premise is so intuitively obvious, especially when applied to the universe, that probably no one in his right mind really believes it to be false" (141). The villain of the piece is BOOK REVIEWS 889 always Hume, and in reply to Hume's critique Craig argues that " all Hume has really shown is that the [causal] principle ... is not analytic and that its denial, therefore, does not involve a contradiction or a logical absurdity." But it" seems intuitively to be really, if not logically, absurd" (145). Craig's defense of the principle is not so much a defense as an appeal. It would have been profitable to develop a critique of Hume's argument, for example, to show that Hume has confused epistemological with ontological conditions in his argument. The critical premise, however, in the argument is the second, and to this Craig devotes the bulk of his energies. He presents four arguments in its support, two of which are philosophical and two empirical. The first argument is: 4. An actual infinite cannot exist. 5. An infinite temporal regress of events is an actual infinite. 6. Therefore an infinite temporal regress of events cannot exist. (69) Craig spends most of his time defending premise 4, arguing that the actual infinite of mathematics-found especially in Cantor's system and set theory -does not describe the real world. It concerns the mathematical world and was never meant to apply to the real world. Indeed, he argues, were it so applied, absurdities of all sorts result. His argument here is painstaking and (to my mind) unexceptionable. Much less time is devoted to the defense of premise 5. Basically he contends that "the fact that the events do not exist simultaneously is wholly irrelevant to the issue at hand; the fact remains that since past events, as determinate parts of reality, are definite and distinct and can be numbered, they can be conceptually collected into a totality...

Quantum logic originated with the Birkhoff-von Neumann paper of 1936.2 The authors argued that the transition from classical to quantum mechanics involves the transition from a propositional calculus forming a Boolean algebra to a propositional calculus which constitutes “a sort of projective geometry”.* For von Neumann, the significance of this thesis for the foundational problems of the theory lay in the possibility of regarding the projective structure as the logic of a quantum set theory -- a non-Boolean theory of sets for properties of quantum mechanical systems in which dimensionality plays the role of cardinality and transition probabilities between events are induced by automorphisms of the logic.** Thus, probabilities are logical: they are not defined, as in the classical case, by measures over events represented as sets of maximal truth value assignments, parametrized by the points of a phase space. Indeed, the quantum sets have no “points” in the general case; there are no “hidden variables” underlying the statistics associated with a quantum logic or quantum set theory. This is a radically different account of the “irreducibility” of the quantum statistics from that provided by the Copenhagen interpretation.

The four-color theorem in standard mathematics and the four-letter theorem in Hilbert mathematics are juxtaposed. The former is a topological theorem thus not allowing for its elementary metric proof as the latter. This is interpreted as an example of “Modernity’s bottle” forcing mathematics not to use any tools entering reality for complying with the taboo for the gap between mathematical models and reality. The same relation in Hilbert arithmetic is interpreted as passing through the “threshold” between two and three dimensions after the Gleason and Kochen - Specker theorems. The representation of mathematical duality and physical complementarity is discussed. The philosophical concept of “the totality” translated in ontomathematics implies for mathematical models to be “concave” in order to obey the absoluteness of the totality. A principle of the equivalence of “concave” (philosophical or “metaphysical”) and “convex” (proper mathematical) explanations being inherent from the viewpoint of ontomathematics is investigated by the example of the correspondence of the Bohmian (non-local and metaphysic: thus “concave”) versus “Copenhagen” (local and agnostic: thus, “convex”) interpretations of quantum mechanics. The eventual representation of quantum contextuality/ holism as fractal structures/ functional (Hausdorff) dimensionality is conjectured. The third dimension of reality (in the sense of the cited theorems) is researched in its two versions: “concave” (metaphysic, in particular, Bohmian) and “convex” (traditional and mathematical, particularly, “Copenhagen”). An illustration by the “four letters” of DNA (RNA) is considered. An interpretation of Peano arithmetic as the “half” of Boolean algebra unifying propositional logic and set theory is linked.

Philosophy of mathematics is arguably one of the oldest branches of philosophy, and one that bears significant connections with core philosophical areas, particularly metaphysics, epistemology, and (more recently) the philosophy of science. This entry focuses on contemporary developments, which have yielded novel approaches (such as new forms of Platonism and nominalism, structuralism, neo-Fregeanism, empiricism, and naturalism) as well as several new issues (such as the significance of the application of mathematics, the role of visualization in mathematical reasoning, particular attention to mathematical practice and to the nature of mathematical explanation). Excellent work has also been done on particular philosophical issues that arise in the context of specific branches of mathematics, such as algebra, analysis, and geometry, as well as particular mathematical theories, such as set theory and category theory. Due to limitations of space, this work goes beyond the scope of the present entry.

In this chapter we will provide some implementation of some basic functions of rough set theory. Implementation of RST functions can be found in other libraries as well. The major aspect here is that the source code is also provided with each and every line explained. The explanation in this way will help research community to not only easily use the code but also they can modify as per their own research requirements. We have used Microsoft Excel VBA to implement the function. The reason behind is that VBA provides easy implementation and almost any of the dataset can easily be loaded in to the Excel. We will not only provide implementation of some of the basic RST concepts but also complete implementation and explanation of the source code of some of the most common algorithms like PSO, GA, QuickReduct, etc.

Quantum mechanics is usually defined in terms of some loosely connected axioms and rules. Such a foundation is far from the beauty of, e.g., the `principles' underlying classical mechanics. Motivated, in addition, by notorious interpretation problems, there have been numerous attempts to modify or `complete' quantum mechanics. A first attempt was based on so-called hidden variables; its proponents essentially tried to expel the non-classical nature of quantum mechanics. More recent proposals intend to complete quantum mechanics not within mechanics proper but on a `higher (synthetic) level'; by means of a combination with gravitation theory (R Penrose), with quantum information theory (C M Caves, C A Fuchs) or with psychology and brain science (H P Stapp). I think it is fair to say that in each case the combination is with a subject that, per se, suffers from a very limited understanding that is even more severe than that of quantum mechanics. This was acceptable, though, if it could convincingly be argued that scientific progress desperately needs to join forces.