Quantities are mathematically represented as real multiples of a unit in the context of a real vector space generated by a unit . The multiplication of the unit x by a real number r is also called scalar multiplication and is written as rx . The choice of unit is indicative of the type of size, for example, for everyday sizes such as lengths with the unit meter (m), masses with the unit gram (g), monetary values with the unit euro (€) or intervals with the unit octave . The largest area of application is physics with a large number of physical quantities .
In ancient times, sizes were already implicitly defined by Eudoxus of Knidos . His theory of size has been handed down in Euclid's Elements . In it he generalized the Pythagorean theory of numbers in such a way that irrational proportions are also included. His axioms and calculation rules, which he applied in proofs, ensure that ancient sizes are embedded in a modern size range. The so-called Archimedean axiom already belongs to the Eudoxian axioms of magnitude . Even before Euclid, various quantities were used in ancient sciences, such as length, area and volume in geometry , time in Aristotle's physics , and duration and interval size in Aristoxenus' music theory . Via Euclid's elements, the concept of size then gained canonical validity until the end of the 19th century. Still Peano stood in the Euclidean tradition and spoke of sizes (quantitates) instead of positive real numbers.
In the mathematics of the 20th century, however, the concept of size was replaced by the concept of the real number , which is an abstraction of the concept of size because it neglects the respective unit. In modern physics, quantities with units still play an important role. There the term was tailored to physical quantities; On the one hand, this is a generalization that also includes more complex quantities with direction ( vectors ), but on the other hand, it is a restriction that does not take into account quantities from other areas.
- Nicolas Bourbaki : Elements of the History of Mathematics, Göttingen 1971, Chapter 12
- Euclid, Elements, Book V
- Peano, Arithmetices principia nova methodo exposita, 1889, §10