13 quotes
The Pythagoreans associated good and evil with the limited and unlimited, respectively.
Mathematical Thought from Ancient to Modern Times (1972)
The Greeks failed to comprehend the infinitely large, the infinitely small, and infinite processes. They "shrank before the silence of the infinite spaces."
Mathematical Thought from Ancient to Modern Times (1972)
Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits.
Mathematical Thought from Ancient to Modern Times (1972)
Aristotle says the infinite is imperfect, unfinished, and therefore unthinkable; it is formless and confused. Only as objects are delimited and distinct do they have a nature.
Mathematical Thought from Ancient to Modern Times (1972)
Laplace created a number of new mathematical methods that were subsequently expanded into branches of mathematics, but he never cared for mathematics except as it helped him to study nature.
Mathematical Thought from Ancient to Modern Times (1972)
For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension.
Mathematical Thought from Ancient to Modern Times (1972)
In arithmetic the Arabs took one step backward. Though they were familiar with negative numbers and the rules for operating with them through the work of the Hindus, they rejected negative numbers.
Mathematical Thought from Ancient to Modern Times (1972)
Aristotle had considered the question of whether space is infinite and gave six nonmathematical arguments to prove that it is finite; he foresaw that this question would be troublesome.
Mathematical Thought from Ancient to Modern Times (1972)
The concept of the infinitely small is involved in the relation of points to a line or the relation of the discrete to the continuous, and Zeno's paradoxes may have caused the Greeks to shy away from this subject.
Mathematical Thought from Ancient to Modern Times (1972)
The usual courses present segments of mathematics that seem to have little relationship to each other. The history may give perspective on the entire subject and relate the subject matter of the courses not only to each other but also to the main body of mathematical thought.
Mathematical Thought from Ancient to Modern Times (1972)
Because they [the ancient Greeks] feared infinite processes they missed the limit process. In approximating a circle by a polygon they were content to make the difference smaller than any given quantity, but something positive was always left over. Thus the process remained clear to the intuition; the limit process, on the other hand, would have involved the infinitely small.
Mathematical Thought from Ancient to Modern Times (1972)
He had doubted the correctness of the law of refraction of light but when he found in 1661 that he could deduce it from his Principle, he not only resolved his doubts about the law but felt all the more certain that his Principle was correct.
Mathematical Thought from Ancient to Modern Times (1972)
When an equation...clearly leads to two negative or imaginary roots, [Diophantus] retraces his steps and shows by how by altering the equation, he can get a new one that has rational roots. ...Diophantus is a pure algebraist; and since algebra in his time did not recognize irrational, negative, and complex numbers, he rejected equations with such solutions.
Mathematical Thought from Ancient to Modern Times (1972)