Christian Thiel on Mathematics - Dictionary of Arguments
Thiel I 10
Mathematics/Object/Thiel: The object does not coincide with the question "What is mathematics? The latter is about the way of thinking.
Old definitions of "mathematics" 19th century: she was "ultimately" science of numbers.
Mathematics/Bolzano: it is the science of the quantities.
20th century Paul Lorenzen (1962) "essentially nothing other than the theory of the infinite itself." Also Weyl, 1926
Third view: Empiricists: they have difficulties with infinity. In the strict sense in the 20th century it is no longer represented this way. Validity is conceded, but the content is denied.
Thiel I 23 et seq.
Mathematics/Tradition/Thiel: Aristotle, Kant and Plato adopt an object, an area of mathematics. More important to them seems to be the question of how the human relates to it.
Distinguishing between inventing and discovering. Plato: Euthydemos: Geometers, mathematicians and astronomers are like hunters, they explore what is already there.
AristotleVsPlato: he had joined Kratylos and Heraclitus to the extent that even after him there could be no science of the sensual, since everything was in flux. Thus, a definition of objects is not even possible.
Plato: there are always many of the same kind of mathematical objects, while the idea is always only one.
Thiel: one will be allowed to think of the four-time occurrence of the isosceles triangle in a square.
AristotleVsPlato: denies the existence of mathematical objects independent of body things. They exist on or in objects and can be isolated by abstraction. Mathematical objects are not themselves concrete, real objects. But they also have no "separated being". Each number is always only the number of something._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Philosophie und Mathematik Darmstadt 1995