Why are mathematical theories based on functions rather than relations?
While the concept of a function is intuitive, it involves a lot of quantifiers that seem arbitrarily chosen. Why must every source be sent to one, and only one target? We don't demand codomains to be in the entire image, so why must all elements in the domain have a corresponding output? Of course, there are practical reasons for this, but said practicality only serves to facilitate computations in existing function-based theories. This is circular reasoning.
The concept of functions is certainly useful, but why not consider it a special case of relations, as we do with injective and surjective functions? For instance, what if we based algebra on "multimorphisms" where if aRb and cRd, then (a+c)R(b+d), R being a relation between two groups. This route would require a rewriting of several fundamental branches of math, and some simple situations may be more complicated to describe, so I doubt we'd see a revision on such a massive scale. But could there be any interesting phenomena to be gleaned?
functions relations
edited Jan 3 at 22:22
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add a comment |
While the concept of a function is intuitive, it involves a lot of quantifiers that seem arbitrarily chosen. Why must every source be sent to one, and only one target? We don't demand codomains to be in the entire image, so why must all elements in the domain have a corresponding output? Of course, there are practical reasons for this, but said practicality only serves to facilitate computations in existing function-based theories. This is circular reasoning.
The concept of functions is certainly useful, but why not consider it a special case of relations, as we do with injective and surjective functions? For instance, what if we based algebra on "multimorphisms" where if aRb and cRd, then (a+c)R(b+d), R being a relation between two groups. This route would require a rewriting of several fundamental branches of math, and some simple situations may be more complicated to describe, so I doubt we'd see a revision on such a massive scale. But could there be any interesting phenomena to be gleaned?
functions relations
edited Jan 3 at 22:22
amWhy
192k28225439
asked Jan 3 at 22:12
dinstruction
554421
1
Relations have their places. Partial functions have their places. Functions are however, to I'd say most people, the easiest thing to define. Common usage also barely differentiates between functions and partial functions, especially in analysis.
– Robert Wolfe
Jan 3 at 22:17
True, functions are special cases of relations; but these in turn are specific subsets of Cartesian products. Each useful concept is named; each is used when it's useful to.
– J.G.
Jan 3 at 22:26
There is a category theoretic approach on this question: we can define the allegory of algebraic structures where the morphisms $Ato B$ are just the homomorphic relations (i.e. subalgebras of $Atimes B$), as you defined, and the whole algebraic theory can be developed there..
– Berci
Jan 3 at 23:04
Functions are more useful.
– William Elliot
2 days ago
add a comment |
While the concept of a function is intuitive, it involves a lot of quantifiers that seem arbitrarily chosen. Why must every source be sent to one, and only one target? We don't demand codomains to be in the entire image, so why must all elements in the domain have a corresponding output? Of course, there are practical reasons for this, but said practicality only serves to facilitate computations in existing function-based theories. This is circular reasoning.
The concept of functions is certainly useful, but why not consider it a special case of relations, as we do with injective and surjective functions? For instance, what if we based algebra on "multimorphisms" where if aRb and cRd, then (a+c)R(b+d), R being a relation between two groups. This route would require a rewriting of several fundamental branches of math, and some simple situations may be more complicated to describe, so I doubt we'd see a revision on such a massive scale. But could there be any interesting phenomena to be gleaned?
functions relations
edited Jan 3 at 22:22
amWhy
192k28225439
asked Jan 3 at 22:12
dinstruction
554421
While the concept of a function is intuitive, it involves a lot of quantifiers that seem arbitrarily chosen. Why must every source be sent to one, and only one target? We don't demand codomains to be in the entire image, so why must all elements in the domain have a corresponding output? Of course, there are practical reasons for this, but said practicality only serves to facilitate computations in existing function-based theories. This is circular reasoning.
The concept of functions is certainly useful, but why not consider it a special case of relations, as we do with injective and surjective functions? For instance, what if we based algebra on "multimorphisms" where if aRb and cRd, then (a+c)R(b+d), R being a relation between two groups. This route would require a rewriting of several fundamental branches of math, and some simple situations may be more complicated to describe, so I doubt we'd see a revision on such a massive scale. But could there be any interesting phenomena to be gleaned?
functions relations
functions relations
edited Jan 3 at 22:22
amWhy
192k28225439
asked Jan 3 at 22:12
dinstruction
554421
edited Jan 3 at 22:22
amWhy
192k28225439
asked Jan 3 at 22:12
dinstruction
554421
edited Jan 3 at 22:22
amWhy
192k28225439
edited Jan 3 at 22:22
amWhy
192k28225439
edited Jan 3 at 22:22
amWhy
192k28225439
192k28225439
asked Jan 3 at 22:12
dinstruction
554421
asked Jan 3 at 22:12
dinstruction
554421
asked Jan 3 at 22:12
dinstruction
554421
554421
1
Relations have their places. Partial functions have their places. Functions are however, to I'd say most people, the easiest thing to define. Common usage also barely differentiates between functions and partial functions, especially in analysis.
– Robert Wolfe
Jan 3 at 22:17
True, functions are special cases of relations; but these in turn are specific subsets of Cartesian products. Each useful concept is named; each is used when it's useful to.
– J.G.
Jan 3 at 22:26
There is a category theoretic approach on this question: we can define the allegory of algebraic structures where the morphisms $Ato B$ are just the homomorphic relations (i.e. subalgebras of $Atimes B$), as you defined, and the whole algebraic theory can be developed there..
– Berci
Jan 3 at 23:04
Functions are more useful.
– William Elliot
2 days ago
add a comment |
1
Relations have their places. Partial functions have their places. Functions are however, to I'd say most people, the easiest thing to define. Common usage also barely differentiates between functions and partial functions, especially in analysis.
– Robert Wolfe
Jan 3 at 22:17
True, functions are special cases of relations; but these in turn are specific subsets of Cartesian products. Each useful concept is named; each is used when it's useful to.
– J.G.
Jan 3 at 22:26
There is a category theoretic approach on this question: we can define the allegory of algebraic structures where the morphisms $Ato B$ are just the homomorphic relations (i.e. subalgebras of $Atimes B$), as you defined, and the whole algebraic theory can be developed there..
– Berci
Jan 3 at 23:04
Functions are more useful.
– William Elliot
2 days ago
1
1
Relations have their places. Partial functions have their places. Functions are however, to I'd say most people, the easiest thing to define. Common usage also barely differentiates between functions and partial functions, especially in analysis.
– Robert Wolfe
Jan 3 at 22:17
Relations have their places. Partial functions have their places. Functions are however, to I'd say most people, the easiest thing to define. Common usage also barely differentiates between functions and partial functions, especially in analysis.
– Robert Wolfe
Jan 3 at 22:17
True, functions are special cases of relations; but these in turn are specific subsets of Cartesian products. Each useful concept is named; each is used when it's useful to.
– J.G.
Jan 3 at 22:26
True, functions are special cases of relations; but these in turn are specific subsets of Cartesian products. Each useful concept is named; each is used when it's useful to.
– J.G.
Jan 3 at 22:26
There is a category theoretic approach on this question: we can define the allegory of algebraic structures where the morphisms $Ato B$ are just the homomorphic relations (i.e. subalgebras of $Atimes B$), as you defined, and the whole algebraic theory can be developed there..
– Berci
Jan 3 at 23:04
There is a category theoretic approach on this question: we can define the allegory of algebraic structures where the morphisms $Ato B$ are just the homomorphic relations (i.e. subalgebras of $Atimes B$), as you defined, and the whole algebraic theory can be developed there..
– Berci
Jan 3 at 23:04
Functions are more useful.
– William Elliot
2 days ago
Functions are more useful.
– William Elliot
2 days ago
add a comment |
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1
Relations have their places. Partial functions have their places. Functions are however, to I'd say most people, the easiest thing to define. Common usage also barely differentiates between functions and partial functions, especially in analysis.
– Robert Wolfe
Jan 3 at 22:17
True, functions are special cases of relations; but these in turn are specific subsets of Cartesian products. Each useful concept is named; each is used when it's useful to.
– J.G.
Jan 3 at 22:26
There is a category theoretic approach on this question: we can define the allegory of algebraic structures where the morphisms $Ato B$ are just the homomorphic relations (i.e. subalgebras of $Atimes B$), as you defined, and the whole algebraic theory can be developed there..
– Berci
Jan 3 at 23:04
Functions are more useful.
– William Elliot
2 days ago