How Can We Quantify the Concept of 'Sameness' in Mathematics?

[jeffery_winkle]

In Tom Leister’s book “Basic Category Theory”, he starts by talking about equality of elements, isomorphisms of functors, equivalence of categories, and at each stage, he acts like you have an object, and that object can change, and then, lo and behold, the “change” itself could be viewed as object that could itself change. This indicates a way of thinking, where you think primarily as one level, and as you go up each level, it gets increasingly difficult to get your mind around.

I think a better way to think about it to to say that in a set you have elements, in a category, you have morphisms between elements, in a 2-category, you have morphisms between elements, and also have 2-morphisms between morphisms, and you can keep on going to n-categories, with arbitrary n, with n-morphisms between (n-1)-morphisms. That way, there is no implied sense that is amazing that there is a level higher than the current level you are talking about.

In Tom Leister’s book, there is an implied sense that in each example, there is a “main thing” you are talking about, and then you could talk about how that thing could change, and then possibly how the change itself could change, but you don’t go beyond that because it’s to difficult to get your mind around.

For example, in classical physics, we talk about position, velocity, acceleration, but they hardly ever go beyond that, talking about putting x to higher derivatives of t.
However, in different examples, people can disagree as to what should be considered “the main thing”. Someone could say something is the “main thing” and talk about that main thing changing, and someone else could say that what you are calling “the main thing” is actually technically something else changing.

A functor goes from one category to another caregory. If the second category has less structure than the first, then it's called a forgetful functor. A forgetful functor from a ring to the underlying abelian group forgets the second binary operation.

Rng -> Ab

A forgetful functor from an abelian group to a group forgets the rule requiring that it be commutative.

Ab - > Grp

A forgetful functor from a group to a monoid forgets about inverses.

Grp -> Mon

A forgetful functor from a monoid to a set forgets about the binary operation.

Mon -> Set

You could string these together as

Rng -> Ab -> Grp -> Mon -> Set

From this, you can say that an abelian group is moe similar to a group than it is to a monoid, or a group is more similar to a monoid than it is to a set. However, which is more similar? A ring and an abelian group or a monoid and a set? Is an abelian group more similar to a group than a group is to a monoid, or a monoid is to a set?

Here are some examples from geometry of differing amounts of sameness.

A square rotated by 90 degrees is the same as an unrotated square, since it looks exactly the same, but a square rotated 10 degrees is not.
A square rotated by 10 degrees is the same as an unrotated square but not the same as a rectangle where not all sides are equal length.
A square is the same as any rectangle but not the same as a triangle.
A square and triangle are the same since they are both topologically S^1.
A cube is the same as the same cube rotated but is not the same as an octahedron.
A cube is the same as an octahedron since they have the same symmetry group but not the same as a dodecahedron.
A cube, octahedron, and dodecahedron are all the same, since they are all platonic solids, but not the same as a cuboctahedron.
A cube, octahedron, dodecahedron, and cuboctahedron are all the same, since they are all topologically S^3.
A plane, R^2, and an infinite cylinder, S^1 x R^1, are not the same because they are topologically distinct.
A plane, R^2, and an infinite cylinder, S^1 x R^1, are the same because they are the same under parallel transport, but a sphere, S^3, is not.
A plane, R^2, infinite cylinder, S^1 x R^1, and a sphere,S^3, are all the the same because they are all Riemannian manifolds.

You can say that a square is more similar to a non-square rectangle than it is to a triangle, but which is more similar, a square and a triangle, or a cube and an octahedon?

How do you measure "sameness" between different types of mathematical objects? How do you quantify sameness? For many of the above examples, you can list axioms, and one type of mathematical objects is defined by axioms 1-5, the second by the same axioms 1-4, and the third by axioms 1-3, you can them say that the first object is more similar to the second object than it is to the third object. However, what if the first object is defined by axioms, 1-4, the second object by axioms, 1 - 3, and the third object by axioms 1, 2, and 4? I think the solution to this problem lies in homotopy theory where you can reduce axioms to their fundamental units, which can then be compared even if the axioms are not the same.

 

[Mark44]

In Tom Leister’s book “Basic Category Theory”, he starts by talking about equality of elements, isomorphisms of functors, equivalence of categories, and at each stage, he acts like you have an object, and that object can change, and then, lo and behold, the “change” itself could be viewed as object that could itself change. This indicates a way of thinking, where you think primarily as one level, and as you go up each level, it gets increasingly difficult to get your mind around.

I think a better way to think about it to to say that in a set you have elements, in a category, you have morphisms between elements, in a 2-category, you have morphisms between elements, and also have 2-morphisms between morphisms, and you can keep on going to n-categories, with arbitrary n, with n-morphisms between (n-1)-morphisms. That way, there is no implied sense that is amazing that there is a level higher than the current level you are talking about.

In Tom Leister’s book, there is an implied sense that in each example, there is a “main thing” you are talking about, and then you could talk about how that thing could change, and then possibly how the change itself could change, but you don’t go beyond that because it’s to difficult to get your mind around.

For example, in classical physics, we talk about position, velocity, acceleration, but they hardly ever go beyond that, talking about putting x to higher derivatives of t.
However, in different examples, people can disagree as to what should be considered “the main thing”. Someone could say something is the “main thing” and talk about that main thing changing, and someone else could say that what you are calling “the main thing” is actually technically something else changing.

A functor goes from one category to another caregory. If the second category has less structure than the first, then it's called a forgetful functor. A forgetful functor from a ring to the underlying abelian group forgets the second binary operation.

Rng -> Ab

A forgetful functor from an abelian group to a group forgets the rule requiring that it be commutative.

Ab - > Grp

A forgetful functor from a group to a monoid forgets about inverses.

Grp -> Mon

A forgetful functor from a monoid to a set forgets about the binary operation.

Mon -> Set

You could string these together as

Rng -> Ab -> Grp -> Mon -> Set

From this, you can say that an abelian group is moe similar to a group than it is to a monoid, or a group is more similar to a monoid than it is to a set. However, which is more similar? A ring and an abelian group or a monoid and a set? Is an abelian group more similar to a group than a group is to a monoid, or a monoid is to a set?

Here are some examples from geometry of differing amounts of sameness.

A square rotated by 90 degrees is the same as an unrotated square, since it looks exactly the same, but a square rotated 10 degrees is not.

They're not the same if you keep track of vertices. A square rotated clockwise by 360° is the same as one not rotated at all.

A square rotated by 10 degrees is the same as an unrotated square but not the same as a rectangle where not all sides are equal length.
A square is the same as any rectangle but not the same as a triangle.

They're not the same. One has four equal sides and the other does not .

A square and triangle are the same since they are both topologically S^1.

It depends on what you mean by "same." Certainly they are topologically equivalent, but it is sophist to say that they are the same otherwise.

A cube is the same as the same cube rotated but is not the same as an octahedron.

See my comment about a square vs. a rotated square.

A cube is the same as an octahedron since they have the same symmetry group but not the same as a dodecahedron.
A cube, octahedron, and dodecahedron are all the same, since they are all platonic solids, but not the same as a cuboctahedron.
A cube, octahedron, dodecahedron, and cuboctahedron are all the same, since they are all topologically S^3.
A plane, R^2, and an infinite cylinder, S^1 x R^1, are not the same because they are topologically distinct.
A plane, R^2, and an infinite cylinder, S^1 x R^1, are the same because they are the same under parallel transport, but a sphere, S^3, is not.
A plane, R^2, infinite cylinder, S^1 x R^1, and a sphere,S^3, are all the the same because they are all Riemannian manifolds.

You can say that a square is more similar to a non-square rectangle than it is to a triangle, but which is more similar, a square and a triangle, or a cube and an octahedon?

How do you measure "sameness" between different types of mathematical objects?

"Same" implies to me that the two objects are identical and undistinguishable. Merely being similar in some respect (such as topological equivalence) doesn't imply sameness.

How do you quantify sameness? For many of the above examples, you can list axioms, and one type of mathematical objects is defined by axioms 1-5, the second by the same axioms 1-4, and the third by axioms 1-3, you can them say that the first object is more similar to the second object than it is to the third object. However, what if the first object is defined by axioms, 1-4, the second object by axioms, 1 - 3, and the third object by axioms 1, 2, and 4? I think the solution to this problem lies in homotopy theory where you can reduce axioms to their fundamental units, which can then be compared even if the axioms are not the same.