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.

  • Definitions are not absolute as you think they are. Definitions serve MULTIPLE PURPOSES--not just one. There are even distinct types of definitions --not just one type. Some definitions apply to reality while others don't. Some definitions simply swap & substitute with the word. In this way we can use shorter sentences by using a word instead of the literal attributes that word refers to in reality. So I could just use the term Rhetoric instead of writing ". . . The faculty of observing all of the available means of persuasion." One word as opposed to many right? Just 1 example for you. – Logikal May 21 '20 at 19:10
  • There are different types of definitions, the ones that are just abbreviations are called stipulative, mathematical definitions within an axiomatic theory are of this sort. It is implicit "definitions" of the theory itself (through axioms) that do more than just abbreviate. But whether mathematicians thereby "create" new objects or "discover" them in a pre-existing "landscape" (or, perhaps, a mixture of both) is a perennial question that depends on one's metaphysics, and the answer makes no difference to the substance of mathematics. – Conifold May 21 '20 at 19:12
  • Some definitions are conceptual & not sense verifiable. I can combine ideas that are true & not reality: I have an idea of a man; I also have an idea of a horse distinctly. If I combine them I can define a Cenataur which is not true in reality. 2 true ideas brought forth an idea not verifiable in science for instance.some terms have no legit definition such as human being. A dictionary will report a person. What is a person? You get told a human being. Circular reasoning in the dictionary. So the dictionary is NOT AN AUTHORITY. You as a human being should know words are used in a context. – Logikal May 21 '20 at 19:18
  • Context express definitions not the dictionary entries. The dictionary is a guide not something you should be pointing to & say "the dictionary says . . . " This means no literal reading of every sentence you read. Some sentences are ironic, analogous satire, rhetorical etc.Context defines what idea is being brought forward to others . You can read the Stanford entry or other philosophical sources about definitions: plato.stanford.edu/entries/definitions. You will find that philosophy materials go into rules of definition as well. We can't just do what we like with words! Well we OUGHT NOT. – Logikal May 21 '20 at 19:24
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    You may have a look at " Theory Of Definition " in Suppes, Introduction to Logic <web.mit.edu/gleitz/www/…> – user37859 May 22 '20 at 14:20

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

Further reading:

  • 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
  • "This distinction presumes some sort of an ontological realm." I think this makes sense. When we define e.g. a function is somewhat like giving a name for what already exists right? Like when we give names to a particular set of people? – ado sar Jul 27 '20 at 12:26
  • I would say the introduction of objects with implicit definitions is an exception to that. The axioms of ZFC together arguably bring sets to life and define what they are. No realm presupposed in this case. The logicists took this hard but they finally had to give in. – Mr. White Jul 27 '20 at 16:00
  • So is the definition who "creates" the mathematical objects? But I still don't understand something. When someone asks "Does an even function exists?" I could answer e.g. x^2 but then could he say me "first you have to define that function in order to exist"? In other words if we didn't define that funciton would that function didn't exist? – ado sar Jul 27 '20 at 16:08
  • It is a bit subtler than that. Mathematical objects exist because they are part of our successful scientific theory of the world. The axioms of ZFC, ZFC being part of this scientific theory, claim the existence of sets and say what they basically are (implicit definition). Even functions exist because they can be defined based on these axioms (explicit definition). It is not necessary for somebody actually to define even functions for them to exist. Consider the analogy: there exist many physical objects which have never been called out. – Mr. White Jul 27 '20 at 16:21
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    Whatever is explicitly definable based on the axioms exists. An example: I explicitly define mybelovedset := {x in R; x is prime or the checksum of x is prime}. This set is at best relevant to me and that's why have given it a name (explicitly definded it). Yet, it was there all along as nobody cared.---The ontological weight gets carried by the axioms and they can carry it because mathematics is such a succesful part of our theory of the world. – Mr. White Jul 27 '20 at 16:55

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