Ernst Mach: Quotes about mathematics

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

“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

“The student of mathematics often finds it hard to throw off the uncomfortable feeling that his science, in the person of his pencil, surpasses him in intelligence,—an impression which the great Euler confessed he often could not get rid of. This feeling finds a sort of justification when we reflect that the majority of the ideas we deal with were conceived by others, often centuries ago. In a great measure it is really the intelligence of other people that confronts us in science.—Mach, Ernst.”

Ernst Mach

Source: 20th century, Popular Scientific Lectures, (Chicago, 1910), p. 196: Mathematics seems possessed of intelligence

People , Feeling , Intelligence , Mathematics

“Mathematical and physiological researches have shown that the space of experience is simply an actual case of many conceivable cases, about whose peculiar properties experience alone can instruct us.”

Ernst Mach

Source: 20th century, Popular Scientific Lectures, (Chicago, 1910), p. 205; On the space of experience.

Mathematics