For over a century, drawing from observation, at least at the introductory level, has been integral to many secondary and most post-secondary art school programs in Europe and North America. Its place in such programs is understood to develop an ability to see and interpret on a flat surface the real, three-dimensional world; this skill, in turn, provides support to related mental processes such as memory, visualization, and imagination. Where an artist looks when drawing from observation may not be arbitrary (...) and can be observed, quantified, and analyzed. Our interest in examining the first few minutes of the drawing process takes its lead from the novice’s question, "Where should I begin?" Attempting to understand these first few minutes led to a collaborative study between art educators and cognitive-perceptual psychologists: the former interested in implications for practical pedagogy, the latter in applying expertise in eye movement and scientific methodology in service of a specific real-world question. The stated purpose of the study notwithstanding, contrasting histories and practices in art and science provided contexts for discussion beyond the collection and interpretation of data. This article seeks to report upon and further that discussion. (shrink)

Written by a creative master of mathematical logic, this introductory text combines stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic. Author Raymond Smullyan offers clear, incremental presentations of difficult logic concepts. He highlights each subject with inventive explanations and unique problems. Smullyan's accessible narrative provides memorable examples of concepts related to proofs, propositional logic and first-order logic, incompleteness theorems, and incompleteness proofs. Additional topics include undecidability, combinatoric logic, and recursion theory. Suitable for undergraduate and (...) graduate courses, this book will also amuse and enlighten mathematically minded readers. (shrink)

More on propositional and first-order logic -- More on propositional logic -- More on first-order logic -- Recursion theory and metamathematics -- Some special topics -- Elementary formal systems and recursive enumerability -- Some recursion theory -- Doubling up -- Metamathematical applications -- Elements of combinatory logic -- Beginning combinatory logic -- Combinatorics galore -- Sages, oracles, and doublets -- Complete and partial systems -- Combinators, recursion, and the undecidable -- Where to go from here.

Collects a variety of mathematics and logic puzzles, some based on the theorems of the mathematician Kurt Godel.

A collection of paradoxes, dialogues, problems, and essays discusses aspects of philosophy, including the natures of reality, truth, existence, and death.

The Tao Is Silent Is Raymond Smullyan's beguiling and whimsical guide to the meaning and value of eastern philosophy to westerners. "To me," Writes Smullyan, "Taoism means a state of inner serenity combined with an intense aesthetic awareness. Neither alone is adequate; a purely passive serenity is kind of dull, and an anxiety-ridden awareness is not very appealing." This is more than a book on Chinese philosophy. It is a series of ideas inspired by Taoism that treats a wide (...) variety of subjects about life in general. Smullyan sees the Taoist as "one who is not so much in search of something he hasn't, but who is enjoying what he has." Readers will be charmed and inspired by this witty, sophisticated, yet deeply religious author, whether he is discussing gardening, dogs, the art of napping, or computers who dream that they're human. (shrink)

In this entertaining and challenging collection of logic puzzles, Raymond Smullyan-author of Forever Undecided-continues to delight and astonish us with his gift for making available, in the thoroughly pleasurable form of puzzles, some of the most important mathematical thinking of our time.

Generally speaking, it appears correct to say that in a formulation of first order logic in which a large number of connectives are taken as primitive which allows us to have our cake and eat it too.

Philosophers since antiquity have argued the merits of mathematics as a normative aid in ethical decision-making and of the mathematization of ethics a theoretical discipline. Recently, Anagnostopoulos, Annas, Broadie and Hutchinson have probed such issues said to be of interest to Aristotle. Despite their studies, the sense in which Aristotle either opposed or proposed a mathematical ethics in subject-matter and method remains unclear. This paper attempts to clarify the matter. It shows Aristotle’s matrix of exactness and inexactness for ethical subject-matter (...) and ethical method in the Nicomachean Ethics. Then it probes a resultant puzzle from the matrix, namely, the HL model of the happy life without consideration of mathematical justice and the HJL model of the happy life with such consideration. Finally, it examines Aristotle’s twofold rationale for differentiating these two models in his overall moral feedback loop system: differences in the intellectual virtue of good deliberation; the priority of friendship over justice for the happy life. This suggests Aristotle saw no objection either to using mathematics as an aid to ethical decision-making for a happy life, or to mathematizing at least some parts of an ethical theory of eudaimonism. (shrink)

FRENCH philosophers from Descartes on have been particularly concerned with the philosophical significance of the human body. Perhaps, as Hyppolite declares, Merleau-Ponty most clearly articulates an ontology of the animate body. Such an interest is by no means confined to philosophers in the French tradition. Surprising, as it may seem, the early Hegel also had a strong interest in the theme of embodiment, the significance of which even Merleau-Ponty in his essay ‘Hegel’s Existentialism’ fails to grasp. The French phenomenologist argues (...) that once Hegel embarks on his discussion of Observational Reason the ‘existential’ viewpoint ceases. Thus, the existential problem of death becomes transmuted into a higher form of life through such ‘altitude thinking’. Even Zaner’s The Problem of Embodiment fails to explore the early Hegelian foundations of incarnate-consciousness in the thought of Marcel, Sartre and Merleau-Ponty. It seems appropriate therefore to briefly examine Hegel’s conception of embodiment in The Phenomenology of Mind on the 200th anniversary of his birth. (shrink)

After examining c.s. peirce's concept, taxonomy, and hierarchy of the theoretical sciences of discovery as well as his notion of the economy of research as an objective of science, a study is made of his occasional but distinct use of the term 'system' in conceptual and procedural contexts. the study shows that peirce's views on systems theory resemble current views held by some proponents of the systems concept and the systems approach to problem-solving.

An entertaining series of logic problems and puzzles of increasing difficulty, and all relating important mathematical and logical concepts, includes mind-benders, paradoxes, metapuzzles, number exercises, and a mathematical novel.