I am having a hard time to understand what a definition does. Is it an abbreviation we use instead of using too many words? But then why mathematicians define mathematical objects? Does it mean they "create" a new abstract object or the object already exists and they give it a name? Are the objects (abstract or concrete) that come first or the definitions? I apologise if the question is silly.
The concept of definition is broad. So broad in fact, that you can bring to life a whole ontological realm with definitions.
This is the case with set theory and all the mathematical objects you can define upon set theory. The abstract object of set is implicitly defined by the theory's axioms. Further objects are defined with explicit and contextual definitions. For more, see below.
I can offer a brief outline of distinctions traditionally made in philosophy when it comes to definitions. Not all of them are strictly tenable but this gives you an impression of what might occur in literature.
Philosophers have distinguished between definitions according to
- their metaphysical content
- their normative claim
- the class of the defined object
- the class of the defined symbol
- the linguistic means used to define
When it comes to metaphysical content:
real definitions are definitions which capture the essence of the defined (genus proximum, differentia specifica). Example: human := intelligent animal
nominal definitions are definitions which sort out without capturing this essence. Example: human := earth based animal that builds airplanes
This distinction obviously presumes some sort of a metaphysical theory.
When it comes to normative claim:
a definition with normative claim intents to sort out newly from the time of its uttering on. Example: super-prime := prime with no prime in the distance of 5 numbers
a definition with no normative claim repeats what had been defined with normative claim before. Example: prime := number with exactly two factors
When it comes to the class of the defined object:
Definitions for objects might for instance differ from definitions of concepts.
This distinction presumes some sort of an ontological realm.
When it comes to the class of the defined symbol:
Definitions for names might for instance differ from definitions of logical constants.
This distinction presumes some sort of a realm of symbols.
When it comes to the linguistic means used to define:
an explicit definition defines by asserting the exchangeability of words by words (salva veritate). Example: vixen := female dog
a contextual definition defines (here: log_b(a)) by the exchangeability of sentences by sentences. Example: log_b(a)=x <=> b^x=a
an implicit definition defines by a set of axioms. Example: What "set" means is defined by the axioms of Zermelo-Fraenkel set theory
- The source mentioned above by Ray LittleRock is good.
- For a deep scientific approach: Suppes, Patrick; Luce, R. Duncan; Krantz, David; Tversky, Amos (2007) (1972). Foundations of Measurement, Vols. 1–3. Dover