By Jean Dieudonne, H.G. Dales, J.C. Dales
Introduction.- arithmetic and Mathematicians.- the character of Mathematical Problems.- items and techniques in Classical Mathematics.- Appendix.- a few difficulties of Classical Mathematics.- Appendix.- New items and New Methods.- Appendix.- difficulties and Pseudo-Problems approximately "Foundations".-Appendix.- Index.
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Additional resources for Mathematics — The Music of Reason
This method has remained the same for mathematicians of all time. But reflections on the nature of the "hypotheses" of which Plato speaks (it will be noted that he never uses the word "truth") and on the entities to which they are applied have never ceased since then to occupy the minds of mathematicians and of philosophers. This is what is known as the problem of the "foundations of mathematics". Many of Plato's dialogues have as their theme the attempt to elucidate "abstract" words used in current speech without any clear conception of what they mean, such as beauty, courage, love, piety, justice, virtue, etc.
It is assumed, however, that the procedure is being applied to an "exact" number X which is already known. But when Z > 0 is a known number, can the same be said of X = /Z? The construction of the "proportional mean" given by Euclid (Book VI, 13) (see Figure 17) answers this question by the intersection of a semi-circle and a line. From the fifth century, however, the Greeks faced a similar problem with respect to what we call "cube roots". We shall see in §8 how they "constructed" them, and why their geometric conceptions prevented them from making similar "constructions" for "nth roots" in the case where n ~ 4.
Another observation taken of the moon enabled them to find out that the distance ES is much greater than EM 3 • fu fact, during the first quarter, the plane separating the illuminated part of the moon from the part in shadow passes through E, and the straight line EM is therefore perpendicular to ES (Figure 3); hence EM ES = cosa. 003. As sightings on the sun are difficult, the Greeks thought that a was near to 87°, and therefore obtained too small a value for ES. Thus we make only a very small error if we consider that at all points of the earth's surface all the rays coming from the centre of the sun's disc are parallel.
Mathematics — The Music of Reason by Jean Dieudonne, H.G. Dales, J.C. Dales