Professor Everett W. Hall's new book, Modern Science and Human Values, one of the most important to have appeared in the field of Value Theory in the last ten years, shares in rich measure the common characteristic of so many other “prolegomena” to the future discipline of values: it is almost maddeningly frustrating. It sees with crystal clearness the essence of the scientific method and describes it in brilliant detail, from Galileo to Einstein; but it fails to draw a positive moral for value theory. On the contrary, its thesis is thoroughly negative: although value must be known as thoroughly as fact if the world is to survive, the scientific method that has brought us knowledge of fact can never bring us knowledge of value. For fact and value are fundamentally different, fact is known by science in a way that excludes value, and hence, whatever may be the way by which value is to be known, it cannot be science. It is true, science has given the most powerful and incisive formulation to fact while value theory has done nothing of the sort for value. Value theory, therefore, must be brought up to the level of science, otherwise there is danger that we may perish under the impact of science. But how to bring this about, how to construct a value theory as powerful and as representative of value as science is of fact, that is the question which the book can only ask but not answer.

When a body is turned through 180°, reversal occurs in two dimensions. If it is turned about the vertical axis, then left and right sides change places, and also front and back. If it is turned about the horizontal axis which runs from side to side, then top and bottom change places, and also front and back. If, finally, it is turned about the horizontal axis which runs from front to back, then left and right change places as well as top and bottom. All this appears to follow from the properties of three-dimensional space and of rigid bodies. If XOX' is the axis of rotation of a three-dimensional structure, then there are two axes YOY' and ZOZ' at right angles to XOX', and if one of these moves round 180°, then so must the other. It follows that a body is always reversed in two dimensions whenever it is turned about an axis in the third. It follows also that there are always two ways of turning a body in space so as to reverse it in a given dimension: namely by turning it about either of two axes perpendicular to the given dimension.

The problem of the “proper” relation of philosophy and science has been the source of many disputes in our intellectual history. Recently some philosophers and scientists have insisted that technical philosophy is neglecting the results of the social sciences to the detriment of philosophy. The purpose of this paper is to consider the attempts made by contemporary philosophers to utilize material from the behavioral sciences (for a few recent examples, see 2, 15, 19, 20, 23, 24, 29, 38, 41, 50), to review certain of the arguments in favor of the utilization of such material, and to consider the criticisms of that approach.

The question, what a given statement is “about,” often occurs in philosophic discussion. I shall use this question in the hope of illustrating how a relatively simple application of symbolic logic can clarify a problem which might otherwise turn into a maze of complications.

If we were to name the two greatest mathematicians of antiquity, we would probably choose Archimedes and Euclid. The first excelled in research, the second in synthesis or system. The synthesis or system is closely associated with the theory or philosophy of that subject; and Euclid's Elements, which has been characterized as “one of the noblest monuments of antiquity”, is the best concrete instance of the theory of mathematics according to the ancient Greeks. Now Aristotle had a theory of mathematics, although he was not a practicing mathematician. It is the purpose of this article to show that the mathematical method according to Aristotle is superior to and more modern than that according to Euclid. For references to Aristotle's works we shall use pages and lines according to the standard Bekker's edition (Berlin, 1831). By “a number” we shall mean a positive integer or a whole number. We proceed to a brief outline of Euclid's Elements.