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There was a situation in my class where one person has asked "what's the difference between those two curves," (of a diagram, displayed on the whiteboard). One person has said "one is blue and the other is green," then the person asking the question has responded "it's not a difference, those are just two definitions." And maybe not my obsession about the definition of a definition I would just have ignored this occurrence.

I understand the concept of as a definition as a "class," a blueprint, to which instances (objects) of this class belong to. I also understand that definitions have no validity within themselves, they're just statements.

My question is, what did the person asking the question mean by saying "it's not a difference, those are just two definitions"? Where does the difficulty in understanding this concept lie? What picture do you have in your mind when you're trying to imagine the concept of a definition?

PS. I am aware of the diversity and structures of definitions. What I am interested in, is the essence of this concept, not its details.

closed as unclear what you're asking by Joseph Weissman Jan 19 '16 at 0:17

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  • Definitions are abstractions: they are things we use to qualify, describe, or interpret objects (that is, "what is" if anything that is can be taken as being an object). Now, there is no unicity: you can be blue and green at the same time (basically what cyan is), round and rectangular (a cylinder), or any other thing one can think about. – sure Nov 10 '15 at 16:13
  • Why did the questioner respond "it's not a difference, those are just two definitions," when the person saying "one is blue and the other is green," is clearly not defining anything? How do you call such statement (the one of the questioner)? – Joe Nov 10 '15 at 16:19
  • I think there may be more context involved here. I'm comfortable with the meaning of definition (as Mauro uses it), but I cannot quite determine what the response means. A follow up statement from them might clarify. In their head, there seems to be two "unbound" terms which are being defined somehow by the colored curves, but there's not quite enough context to nail down what they were thinking. – Cort Ammon Nov 10 '15 at 16:30
  • @CortAmmon Isn't it that the questioner interpreted the colors of the curves as being a part of the definition of these curves? Therefore he interpreted that the person putting color on these curves was defining those this way? But even then, those two varying colors are still differences between the two curves? – Joe Nov 10 '15 at 16:33
  • That might be clear given your experience, but it is less clear with the recounting of it. For example, I might interpret the curves as the abstract curves of data, rather than the inked lines on the whiteboard, literally using a different definition of "curve" which does not include the color of ink used to portray them. I could also have been trying to make the point that both abstract curves are actually portraying some definition within the data (which is not shown in this post), and he's trying to point out a situation where the actual curve isn't the important part at all. – Cort Ammon Nov 10 '15 at 16:43
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In logic and mathematics, with a definition we "introduce" a new term [the definiendum] as an abbreviation for a statement involving previous introduce terms [the definiens].

The basic rule is that the new term must not occur into the definiens.

If so, to state :

"a curve is blue"

is not to state a definition.

The possible source of the equivocation is the multiple usage of "is" into natural language.

From a logical point of view, we can identify three different "contexts" :

  • "Plato is a philosopher"; this context is relative to an object (or individual) belonging to a set or class. In modern math (set theory), this is expressed as : Plato ∈ philosphers.

  • "a man is a male"; this context is relative to a "concept" to be part of a "more general" one. In modern math it is expressed with set inclusion : men ⊆ males.

  • "2+2 is 4"; this context is relative to identity, i.e. the relation between two names denoting the same "thing". In modern math it is expressed as : 2+2 = 4.

A definition of a new term use the identity "is", while in the statement "a curve is blue" we are using the belongs to "is", because we are asserting that the curve drawed on the whiteboard belongs to the class of blue things.

Neither it is a case of inclusion, because it makes no sense to assert that the concept "curve" is part of the more general concept "blue things".


You can see Definitions.

  • The person "one is blue and the other is green," wasn't defining anything, they were pointing out a difference. Obviously, why did the other person say "it's not a difference, those are just two definitions"? – Joe Nov 10 '15 at 16:16
  • @Joe - so what is the purported "definition" ? You are asking if the second person answering " those are just two definitions" intended to mean : "one curve is blue" is a def (and the same for "one is green") ? – Mauro ALLEGRANZA Nov 10 '15 at 16:19
  • I guess the person asking was trying to say that "one curve is blue" and "one curve is green" are no differences but definitions. Am I wrong? – Joe Nov 10 '15 at 16:26
  • @Joe - Ok; I've updated accordingly my answer, but its "meaning" does not change... of course, this does not mean that I'm right :-) – Mauro ALLEGRANZA Nov 10 '15 at 16:35
  • Isn't the statement "it's not a difference, those are just two definitions" of the person asking nonsensical? – Joe Nov 10 '15 at 16:40
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A definition is a baptism: One chooses a name for a certain thing, a matter of fact or a situation. One is free which name to choose. Afterwards one can use the definition as an abbreviation for the whole matter of fact.

The questioner did not want as answer to hear a name, the blue curve versus the green curve. Probably he wanted to hear a characteristic distinguishing property, e.g. one curve is closed, the other is open.

  • "he wanted to hear a characteristic distinguishing property," one curve had a green color the other had a blue color, color is a property which allows us to distinguish between the curves, it's a difference of a property they both share (color). Why would you say that "did not want as answer to hear a name"? – Joe Nov 10 '15 at 16:11
  • Because the colour which one uses to draw a mathematical curve is not important for the mathematical properties of the curve. – Jo Wehler Nov 10 '15 at 16:19
  • It wasn't a math class, it was a biology class. Even if the person wasn't to hear that, that what the questioner said doesn't make any sense at all. Green and blue ARE the differences between those curves, it seems that the questioner wasn't able to construct a question to address whatever he wanted to ask. – Joe Nov 10 '15 at 16:22
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Definition: A:=B, I define that every time B is the case, it is the case that A.

unicorn:=a white, horned horse

bachelor:=unmarried man

...

I would say that the questioner used another use of "definition" in his statement about colors: Instead of using it in a narrow sense, it is used in a very broad sense, because as I take it, he says "You are just defining a predicate to these curves in addition to the very being of the curves."

Color is not inherent to the curves, therefore it has to be assigned to them. Assignments are in some sense definitions, because as they become part of the definition of the occurence of a thing. But not of the thing itself, because it is some kind of weird to say that the curve would not be the same curve if it was not green.

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