Ernst Mach: Operating

Ernst Mach was Austrian physicist and university educator. Explore interesting quotes on operating.
Ernst Mach: 24   quotes 3   likes

“The mental operation by which one achieves new concepts and which one denotes generally by the inadequate name of induction is not a simple but rather a very complicated process. Above all, it is not a logical process although such processes can be inserted as intermediary and auxiliary links. The principle effort that leads to the discovery of new knowledge is due to abstraction and imagination.”

Ernst Mach

3rd edition, p. 318ff, As quoted by Phillip Frank, Philosophy of Science: The Link Between Science and Philosophy (1957)
20th century, "Erkenntnis und Irrtum: Skizzen zur Psychologie der Forschung" (1905)

Imagination , Knowledge , Effort

“There is no problem in all mathematics that cannot be solved by direct counting. But with the present implements of mathematics many operations can be performed in a few minutes which without mathematical methods would take a lifetime.”

Ernst Mach

Source: 19th century, Popular Scientific Lectures [McCormack] (Chicago, 1898), p. 197; On mathematics and counting.

Mathematics , Problem

“Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar,—we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one.”

Ernst Mach

Source: 20th century, "Populär-wissenschafliche Vorlesungen" (1908), pp. 224-225: On thought-economy in m., 203.

Mathematics , Nature , Science , Time