Lexicon of Arguments

 

Thiel I 18/19
Mathematics/Ontology/Mathematical Entities/Thiel: Def "Logicism" attributes mathematics (any object) to logic. The object of mathematics is then the object of logic. What is then the object of logic: logicism must say: actually no material object at all, but "all objects" in the sense that its statements apply to all objects in the world. We create the objects ourselves. "In itself" such an object never exists!
>Mathematical entities, >Platonism.
DubislavVs: Every convention must be made about something. So one must ask the conventionalist about which structures his axioms have to be regarded as consistent.
>Beginning, >Axioms.
I 312
In modern mathematics one speaks not only of "the" addition, but of "an addition" and introduces linking signs. For example, one writes addition as "$" if it is associative and commutative, if it is not the case, one might prefer to write the operation as multiplication "§" or something else.
>Axioms/Hilbert.
I 312/313
Ontology/object/mathematics/Thiel: the validity of such laws does not turn the subject area into a number area, just as the validity of any set-theoretical laws transforms the (ranges of) numbers into (ranges of) sets.
>Numbers, >Sets.
The recording of the possible types of operations does not provide any fundamental discipline.
I 314
It may be that the universality of mathematics is based on the ever new applicability of the very general operations and not on the fact that mathematics deals with particularly general objects.
>Generalization, >Generality.
Although we always carry out the same set-theoretical operations in the different fields of mathematics, this does not mean that there are "sets" as autonomous objects. At most, a fundamental discipline should be envisaged that fulfils this task as a fundamental canon for "dealing with everything and everyone".