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Shortest distance between two points is a line proof
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Two questions (please bear with my curiosity as I am still somewhat a beginner) - is this considered to be a theorem or just an axiom?
In general, how does one come to know if a statement is an axiom or if there is a proof to it? Just over time?
soft-question proof-explanation axioms
asked Jul 19 at 3:12
wannabemathmajor
556
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up vote
1
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favorite
Two questions (please bear with my curiosity as I am still somewhat a beginner) - is this considered to be a theorem or just an axiom?
In general, how does one come to know if a statement is an axiom or if there is a proof to it? Just over time?
soft-question proof-explanation axioms
asked Jul 19 at 3:12
wannabemathmajor
556
Actually, I do not understand the statement. What does "Shortest distance between two points is a line" mean? Do you mean something like "shortest path between two points is a line" ?
– Suzet
Jul 19 at 3:20
The notion that we measure distance by the euclidean norm is an arbitrary convention and we can get different results with different (but intermally consistent) methods of "distance". A line is defined by sets of points that follow solve an equation. You can use calculus to prove this is the shortest path between two points. So it's a theorem, but what a "distance" and what a "line" is definitions and they are so chosen to force "line = shortest distance" to be a foregone conclusion.
– fleablood
Jul 19 at 3:23
You asked how one knows if a statement is an axiom or if there is a proof for it. These two properties are not exclusive. A collection of axioms is simply the foundation of a theory. It lays the groundwork about how primitive notions can be manipulated and which facts are to be accepted as understood when discussing that particular theory, made up of theorems which rely on those axioms. These axioms can absolutely be proven, in another theory, but they don't have be. Basically what I'm trying to say is that whether something is an axiom is purely relative to what theory you're working with.
– nasdaq
Jul 19 at 3:24
You may notice that some books refer to things like "the axioms of the real numbers" or "axioms of group theory" for what you might consider to be merely properties of these things or their definitions. And indeed the existence of mathematical objects which have these properties CAN be proven. But calling these properties "axioms" is not a misnomer because an axiom simply means the foundation for the theory in which models of these objects can be created. Defining each different theory with axioms allows us to create individual rigorous theories consistent up to the point of their axioms.
– nasdaq
Jul 19 at 3:35
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Two questions (please bear with my curiosity as I am still somewhat a beginner) - is this considered to be a theorem or just an axiom?
In general, how does one come to know if a statement is an axiom or if there is a proof to it? Just over time?
soft-question proof-explanation axioms
asked Jul 19 at 3:12
wannabemathmajor
556
Two questions (please bear with my curiosity as I am still somewhat a beginner) - is this considered to be a theorem or just an axiom?
In general, how does one come to know if a statement is an axiom or if there is a proof to it? Just over time?
soft-question proof-explanation axioms
asked Jul 19 at 3:12
wannabemathmajor
556
edited Jul 19 at 3:28
asked Jul 19 at 3:12
wannabemathmajor
556
asked Jul 19 at 3:12
wannabemathmajor
556
asked Jul 19 at 3:12
wannabemathmajor
556
556
Actually, I do not understand the statement. What does "Shortest distance between two points is a line" mean? Do you mean something like "shortest path between two points is a line" ?
– Suzet
Jul 19 at 3:20
The notion that we measure distance by the euclidean norm is an arbitrary convention and we can get different results with different (but intermally consistent) methods of "distance". A line is defined by sets of points that follow solve an equation. You can use calculus to prove this is the shortest path between two points. So it's a theorem, but what a "distance" and what a "line" is definitions and they are so chosen to force "line = shortest distance" to be a foregone conclusion.
– fleablood
Jul 19 at 3:23
You asked how one knows if a statement is an axiom or if there is a proof for it. These two properties are not exclusive. A collection of axioms is simply the foundation of a theory. It lays the groundwork about how primitive notions can be manipulated and which facts are to be accepted as understood when discussing that particular theory, made up of theorems which rely on those axioms. These axioms can absolutely be proven, in another theory, but they don't have be. Basically what I'm trying to say is that whether something is an axiom is purely relative to what theory you're working with.
– nasdaq
Jul 19 at 3:24
You may notice that some books refer to things like "the axioms of the real numbers" or "axioms of group theory" for what you might consider to be merely properties of these things or their definitions. And indeed the existence of mathematical objects which have these properties CAN be proven. But calling these properties "axioms" is not a misnomer because an axiom simply means the foundation for the theory in which models of these objects can be created. Defining each different theory with axioms allows us to create individual rigorous theories consistent up to the point of their axioms.
– nasdaq
Jul 19 at 3:35
add a comment |Â
Actually, I do not understand the statement. What does "Shortest distance between two points is a line" mean? Do you mean something like "shortest path between two points is a line" ?
– Suzet
Jul 19 at 3:20
The notion that we measure distance by the euclidean norm is an arbitrary convention and we can get different results with different (but intermally consistent) methods of "distance". A line is defined by sets of points that follow solve an equation. You can use calculus to prove this is the shortest path between two points. So it's a theorem, but what a "distance" and what a "line" is definitions and they are so chosen to force "line = shortest distance" to be a foregone conclusion.
– fleablood
Jul 19 at 3:23
You asked how one knows if a statement is an axiom or if there is a proof for it. These two properties are not exclusive. A collection of axioms is simply the foundation of a theory. It lays the groundwork about how primitive notions can be manipulated and which facts are to be accepted as understood when discussing that particular theory, made up of theorems which rely on those axioms. These axioms can absolutely be proven, in another theory, but they don't have be. Basically what I'm trying to say is that whether something is an axiom is purely relative to what theory you're working with.
– nasdaq
Jul 19 at 3:24
You may notice that some books refer to things like "the axioms of the real numbers" or "axioms of group theory" for what you might consider to be merely properties of these things or their definitions. And indeed the existence of mathematical objects which have these properties CAN be proven. But calling these properties "axioms" is not a misnomer because an axiom simply means the foundation for the theory in which models of these objects can be created. Defining each different theory with axioms allows us to create individual rigorous theories consistent up to the point of their axioms.
– nasdaq
Jul 19 at 3:35
Actually, I do not understand the statement. What does "Shortest distance between two points is a line" mean? Do you mean something like "shortest path between two points is a line" ?
– Suzet
Jul 19 at 3:20
Actually, I do not understand the statement. What does "Shortest distance between two points is a line" mean? Do you mean something like "shortest path between two points is a line" ?
– Suzet
Jul 19 at 3:20
The notion that we measure distance by the euclidean norm is an arbitrary convention and we can get different results with different (but intermally consistent) methods of "distance". A line is defined by sets of points that follow solve an equation. You can use calculus to prove this is the shortest path between two points. So it's a theorem, but what a "distance" and what a "line" is definitions and they are so chosen to force "line = shortest distance" to be a foregone conclusion.
– fleablood
Jul 19 at 3:23
The notion that we measure distance by the euclidean norm is an arbitrary convention and we can get different results with different (but intermally consistent) methods of "distance". A line is defined by sets of points that follow solve an equation. You can use calculus to prove this is the shortest path between two points. So it's a theorem, but what a "distance" and what a "line" is definitions and they are so chosen to force "line = shortest distance" to be a foregone conclusion.
– fleablood
Jul 19 at 3:23
You asked how one knows if a statement is an axiom or if there is a proof for it. These two properties are not exclusive. A collection of axioms is simply the foundation of a theory. It lays the groundwork about how primitive notions can be manipulated and which facts are to be accepted as understood when discussing that particular theory, made up of theorems which rely on those axioms. These axioms can absolutely be proven, in another theory, but they don't have be. Basically what I'm trying to say is that whether something is an axiom is purely relative to what theory you're working with.
– nasdaq
Jul 19 at 3:24
You asked how one knows if a statement is an axiom or if there is a proof for it. These two properties are not exclusive. A collection of axioms is simply the foundation of a theory. It lays the groundwork about how primitive notions can be manipulated and which facts are to be accepted as understood when discussing that particular theory, made up of theorems which rely on those axioms. These axioms can absolutely be proven, in another theory, but they don't have be. Basically what I'm trying to say is that whether something is an axiom is purely relative to what theory you're working with.
– nasdaq
Jul 19 at 3:24
You may notice that some books refer to things like "the axioms of the real numbers" or "axioms of group theory" for what you might consider to be merely properties of these things or their definitions. And indeed the existence of mathematical objects which have these properties CAN be proven. But calling these properties "axioms" is not a misnomer because an axiom simply means the foundation for the theory in which models of these objects can be created. Defining each different theory with axioms allows us to create individual rigorous theories consistent up to the point of their axioms.
– nasdaq
Jul 19 at 3:35
You may notice that some books refer to things like "the axioms of the real numbers" or "axioms of group theory" for what you might consider to be merely properties of these things or their definitions. And indeed the existence of mathematical objects which have these properties CAN be proven. But calling these properties "axioms" is not a misnomer because an axiom simply means the foundation for the theory in which models of these objects can be created. Defining each different theory with axioms allows us to create individual rigorous theories consistent up to the point of their axioms.
– nasdaq
Jul 19 at 3:35
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Actually, I do not understand the statement. What does "Shortest distance between two points is a line" mean? Do you mean something like "shortest path between two points is a line" ?
– Suzet
Jul 19 at 3:20
The notion that we measure distance by the euclidean norm is an arbitrary convention and we can get different results with different (but intermally consistent) methods of "distance". A line is defined by sets of points that follow solve an equation. You can use calculus to prove this is the shortest path between two points. So it's a theorem, but what a "distance" and what a "line" is definitions and they are so chosen to force "line = shortest distance" to be a foregone conclusion.
– fleablood
Jul 19 at 3:23
You asked how one knows if a statement is an axiom or if there is a proof for it. These two properties are not exclusive. A collection of axioms is simply the foundation of a theory. It lays the groundwork about how primitive notions can be manipulated and which facts are to be accepted as understood when discussing that particular theory, made up of theorems which rely on those axioms. These axioms can absolutely be proven, in another theory, but they don't have be. Basically what I'm trying to say is that whether something is an axiom is purely relative to what theory you're working with.
– nasdaq
Jul 19 at 3:24
You may notice that some books refer to things like "the axioms of the real numbers" or "axioms of group theory" for what you might consider to be merely properties of these things or their definitions. And indeed the existence of mathematical objects which have these properties CAN be proven. But calling these properties "axioms" is not a misnomer because an axiom simply means the foundation for the theory in which models of these objects can be created. Defining each different theory with axioms allows us to create individual rigorous theories consistent up to the point of their axioms.
– nasdaq
Jul 19 at 3:35