A definition is a speech which says what is a thing or what name means. From where division enters the real definitions and the nominal definitions .
A definition poses a déséquivalence between a term ( Meaning ) and direction ( Signifié ). It authorizes to replace the second by the first and revêt thus a practical utility . It is also the result of an operation, and thus introduced the time (the definite direction is finished , last , en-soi ), as well as an actor (often implicit).
The definition falls under the order of the denotation, but a term also connotes directions, and this without explicitly calling upon time or an actor. It does it thanks to an external structure of space of meaning, but there exists also an internal structure which is expressed through the etymology.
It is seen, the Concept of definition does not assert itself: it falls under a structured totality. It implies and it indicates choices: which uses (among others) serves the new definition? which actors is it useful? It is a tool useful, but not indifferent.
The definition establishes a border between the defined word, and the words used to clarify it. It establishes thus an ordered structure, a tree structure by levels between classes of words. It is seen well that this structure is however local , that this order is not preserved if one unrolls the structure gradually. This order, quite useful, is only one illusion.
To define the definition: problems
a definition is the determination of the limits of the extension of a Concept (definition of Lalande, in its critical dictionary). More deeply, the definition exposes in a speech articulated (compound minimalement of two words) the comprehension of a concept. To say that an animal is a living endowed with significant knowledge , for example, it is to articulate between them two concepts ( alive and endowed with significant knowledge ) which enter the constitution and which make it possible to seize the nature of a third ( animal ).
There is obviously a circle to define the concept of definition: the attempt even supposes the solved problem, and seems to deny the interest of the step (why define definition so by là-même one supposes the known definition?). It is what philosophy anglo-saxone calls a blind point of the reason.
Thus the definition suggested above of the word definition employs itself of other words, which one supposes that they have themselves a definition. But the problem is initially that of the direction: how can one apprehend the direction of the words? The answer varies considerably from one author to another. For example, for Plato, the direction is immutable, and it is used as base to our knowledge; for Quine, on the other hand, the direction is unspecified, and always depends on a unit on theories and concepts.
The problem thus relates to the theory of knowledge and the reference.
In the quoted example, the author subordinates the definition to the concept , his extension and his determination . The paradox created, will one ask, is not it a dialectical movement of the thought? Undoubtedly, if this thought is action , and asserted as such, but not if it is result. A truth immanente which would contain paradoxes is only one negation of the reason, a base for the réenchantement of the world which max Weber denounces. But a denunciation is undoubtedly based itself on a truth immanente, unless claiming with a transcendent prospect. One can then come to a form from relativism (cf Scepticisme or post-modernism), failing to find a rationality minimal which ensures us that words that we use have feel and thus a definition.
The question is to know for which direction of the word “definition” a speech is judicious.
Designs of the definition
The inventor of the definition would be, according to Aristote, Socrate. Socrat seeks indeed with the result that a thing is such as it is: for example, in the major Hippias , why this beautiful thing is beautiful? It would be there thus common to the beautiful things, a gasoline, whose formulation is the definition.However, the starting point of Socrate is existential: it is a question of becoming aware of what we say and of what we make when we follow of the designs morals or scientists. The definition makes it possible to put to the test our alleged knowledge, especially when Socrate shows with its interlocutors that they cannot produce a coherent definition of what they think: they thus do not think anything definite, anything which does not have a precise and well defined extension. In the best of the cases, they are ignoramuses, in worst impostors.
The problems involved in the definition (in particular the problem of the paradox given higher) were motivations in research for all the philosophers. Indeed, the analysis of the concepts and what one wants to say, the research of the extension of the concepts which we use, is one of the major aspects of philosophy, Plato and Aristote with Locke, Hume and all Anglo-Saxon philosophy in particular.
Logic
In Logical, a definition is a statement which introduces a symbol called term indicating the same object as another symbol, or associated with a continuation called assembly , symbols whose significance is already known.
Certain symbols like those of the existence, the membership, negation etc which cannot be defined, are coarsely introduced by calling upon words of the natural language and the intuitive idea that the man can have some. These primitive terms belong to the “basic intuitive field”. (Concept of the nondefinite words used by Alfred Korzybski)
In Mathematical, a definition is a statement written in natural language or in formal Language (of logic), which introduces a new word or symbol associated with an abstract object describes by an assembly of other words or symbols whose direction was already précisé.
The idea that we have object thus defined, is called a mathematical concept.
These words or symbols are “abbreviations”, intended to represent such assemblies of letters and symbols. These abbreviations make it possible a mathematician to use the mathematical object thus built without having with the spirit its complete and detailed definition. In practice, the abbreviations are alphabetical letters, ordinary signs or words, for example:
- π represents a number
- E represents the exponential one of 1
- “not” and “right” is geometrical objects
- the signs + and × is “laws”
It would be possible to write all mathematics only in formal language, but that would make their use difficult and according to Roger Godement, a number as simple as 1 would require an assembly of approximately ten thousand symbols.
Let us give some examples of definitions now:
- Either has a positive integer. Let us pose B=A.
- Is D and two nonparallel lines. That is to say I the point of intersection of D and Of.
We define item I and we are supposed to know what are a line, the parallelism and a point of intersection.
- a natural integer is known as first if it is different from 1 and if it admits like dividers only 1 and itself.
" " Some remarques" " :
A definition is not a Théorème, it gives simply a denomination to mathematical objects but does not describe rules of use of these objects or properties checked by these objects (others only those which define it).
When we define an object, we use in general “if” who means “so by definition”, “when” or “when”, as in the following definition:
An integer relative N is even if ∃k ∈ ℤ, N = 2k.
Some use awkwardly “if and only so” in the place of “if”, but that does not have a direction, since they write in this case an equivalence between a term which is not a proposal which, moreover, is not defined yet and a proposal.
If the definition of a given object supposes that a proposal P is checked, then the assertion “by definition” or “under the terms of the definition” the proposal P is checked means that we use the intrinsic proposal P with the object. Let us consider the following definition:
Définition: A square is a quadrilateral whose sides are of the same length and whose angles are droits.
It is obvious that all the sides of a square are equal length because this property belongs to the definition. We can say in this case “by definition”, a square is all its sides equal length.
If the same mathematical object (or “mathematical being”) receives several definitions and that all the properties of the one of them are equivalent to those of the others, then these definitions are known as equivalent.
A definition does not have a direction that within the framework of a given mathematical theory and for example it is impossible to consider a derivable function definite on the whole of the natural entireties with values in ℝ.
In a mathematical talk, it happens that an “intuitive” definition is given before the mathematical definition; its role is to highlight the motivations of such a definition.
For example, of the definitions of dictionary: explanation of a mot.
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