definition

definition

The

definition

of a mathematical term is a meta-mathematical

construct

or statement that specifies as precisely as possible the meaning of that term. For example, a definition of “

sphenic number

” is “a

composite

integer

with three

distinct prime factors

.” A mathematical

concept

is

well-defined

if its content can be formulated independently of the form or the alternative representative which is used for defining it. Furthermore, one should distinguish between mathematical definitions and mathematical descriptions of a real system; mathematical definitions express completely and meaningfully a mathematical concept in terms of other, related mathematical concepts that have been already defined, and also

primary

or primitive concepts that can no longer be defined in terms of other mathematical concepts and logical operands. For example, the primitive concept of ‘

collection

of

elements

or members’ or ensemble, has no explicit definition even though it is employed to mathematically define the concept of set. A definition of a fundamental concept, such as set,

category

, topos,

topology

,

homology

etc., also contains several axioms, or basic

assumptions

/conditions imposed on the auxilliary concepts employed by such a fundamental concept definition. For example, the category of sheaves on a site is called a (Grothendieck) topos; however, a topos can also be defined directly by specifying only a few (

Grothendieck topos

) axioms.

Therefore, ultimately, a mathematical definition depends on the choice of the

mathematical foundation

selected, e.g., set-theoretical, category-theoretical, or topos-theoretical, as well as the type of logic adopted, e.g.,

Boolean

, intuitionistic or

many-valued logic

. Thus, the general definition of a mathematical definition is not simply a mathematical concept, but it is instead a

meta-mathematical construct

, or the

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⁢

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⁢

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⁢

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⁢

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⁢

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⁢

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⁢

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⁢

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of a construct. In the case of topos-theoretical

foundations

the Brouwer-intuitionistic logic is explicitly assumed in the construction/definition of the topos. As an example, the

category of sets

–subject to certain axioms, including the

axiom of choice

– may be considered a

canonical

example of a Boolean topos, but it is not the only one possible, as different axioms may be selected to avoid several known antimonies in

set theory

.

Thus, a mathematical concept is well-defined only when its mathematical foundation

framework

is also specified either explicitly or by its context. For reasons related to apparent ‘simplicity’, many a mathematician prefers only Boolean logic and a set-theoretical foundation for definitions, in spite of severe limitations, known inherent

paradoxes

and

incompleteness

.

Alternative definitions of the same concept often offer additional insights into the meaning(s) of the concept being defined, as well as added flexibility in solving problems and discovering proofs.

A definition of a

constant

is an equation with a symbol (or some other notation) on the left and either an exact value or a

formula

for a value on the right. For example, the definition of the

golden ratio

is

ϕ

=

1

+

5

2

.

A definition of a

function

is an equation, usually with function notation on the left (a symbol, with an

argument

or a list of arguments in parentheses) and on the right a formula using the arguments to calculate the value of the function for those arguments. Optionally, the equation could be accompanied by statements of what acceptable arguments are. For example, Euler’s

totient function

is defined as

ϕ

⁢

(

n

)

=

n

⁢

∏

p

|

n

(

1

-

1

p

)

for

n

∈

ℤ

. To give one more example: the

factorial function

is defined as

n

!

=

∏

i

=

1

n

i

for

0

<

n

∈

ℤ

. Euler’s

gamma function

Γ

⁢

(

x

)

is sometimes said to extend the definition of

factorials

to any

x

∈

ℂ

.

Title

definition

Canonical name

Definition

Date of creation

2013-03-22 17:32:19

Last modified on

2013-03-22 17:32:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

18

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 00A05

Synonym

mathematical description

Synonym

meta-construct

Related topic

SetTheory

Related topic

CategoryTheory

Related topic

Topos

Related topic

AxiomOfChoice

Defines

mathematical definition

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