In mathematics, what does it mean for two things to be the same? The equal sign first appeared in 1557 in Robert Recorde's work "The whetstone of Witte". The principle of indiscernity of identicals says that equal things have the same properties. If two things were literally exactly the same, they wouldn't be two things. They would be only one thing. When you say two things are the same, what you are really doing is you are specifying what differences you are choosing to ignore. Let's look at the sentence "Two things are the same". First, you have the first part of the sentence. "Two things..." Here you are saying that they are not, in fact, exactly the same because if they were exactly the same, they would not be two things. They would be one thing. Then you have the second part of the sentence. "...are the same." Here are stating that you are considering them to be the same for present purposes despite the differences that you yourself just admitted must exist. Saying that two things are the same requires stating what differences you are choosing to ignore. Things that you are considering to be the same despite differences between them are members of a set. Membership in a set is defined by a list of axioms. You write down the list of axioms. Mathematical objects that obey all the axioms are members of a set, and are thus considered the same, or at least the same in the sense that they are members of the same set. Mathematical objects that do not obey the axioms are not members of the set, and are thus not considered the same as the members of the set. Different lists of axioms define different sets, with different members. Therefore, each set comes with its own definition of sameness. This leads to confusion because most people, at a subconscious level, shift back and forth between these different definitions of sameness. For instance, you might say, "Hey! The number 3 is not the same as the number 4". Yeah well, in some sense, the number 3 can be considered the same as the number 4 because they are both integers. Let's say you write down a list of axioms. Mathematical objects that obey the list of axioms are members of the set, and thus considered the same. Then, let's say, you add an additional axiom to the list, which defines a new set. All members of the new set must obey all the axioms that define membership in the old set. Since members of the new set must also obey an additional axiom, the new set is more restrictive that the old set. All members of the new set are members of the old set, but not all members of the old set are also members of the new set. The new set B is a subset of the old set A. From the point of view of set A, all members of set A are the same. From the point of view of set B, not all members of set A are the same, but all members of set B are the same. As you add more axioms, you get sequentially smaller subsets. Set A contains subset B which contains subset C which contains subset D. As you go down the chain by adding more axioms, you get increasingly smaller subsets, and therefore more narrow definitions of sameness. As you go up the chain by removing axioms, you get increasingly larger subsets, and therefore broader definitions of sameness. Let's say you have two lists of axioms where some of the axioms are the same but some are not the same. Each list has some axioms that also appear on the other list, as well as some axioms that do not appear on the other list. This would give you set A and set B which overlap. The intersection of A and B would be another set C which is a subset of both A and B. Let's say set B is a subset of set A. From a mathematical point of view, A includes both B and not B. However, often when people, especially non-mathematicians, say "A", what they really mean is "A not B". When most people say "rectangle", they means "a rectangle that is not a square", even though, from a mathematical point of view a square is a type of rectangle. The definition of "non-associative algebra" actually means, "an algebra that is not necessarily associative". In order to avoid confusion, you should clearly state whether you mean the technical definition, or the more common usage. Let's illustrate these concepts with a few examples. A - Polygon 1. Convex coplanar polygons B - Quadrilateral 1. Convex coplanar polygons 2. Four sides Since you have an additional axiom that all members must obey, it is more restrictive. According to the definition of set B, a triangle is no longer considered the same as a square, but all quadrilaterals are considered the same. C - Trapezoid 1. Convex coplanar polygons 2. Four sides 3. At least two sides parallel D - Parallelogram 1. Convex coplanar polygons 2. Four sides 3. Opposite sides parallel E - Rectangle 1. Convex coplanar polygons 2. Four sides 3. Opposite sides parallel 4. All angles right angles F - Rhombus 1. Convex coplanar polygons 2. Four sides 3. Opposite sides parallel 4. All sides the same length Set E is the set of all rectangles. Set F is the set of all rhombuses. These two sets overlap. The intersection of E and F is G, which is the set of all squares. You can either define squares as rectangles where all sidies are the same length or as rhombuses where all angles are right angles. G - Square 1. Convex coplanar polygons 2. Four sides 3. Opposite sides parallel 4. All sides the same length 5. All angles right angles A square is a type of rectangle. However, when non-mathematicians say "rectangle", they mean "rectangle that is not a square". When considering the symmetry of a geometric shape, we are talking about whether it looks the same after performing an operation such as translation, rotation, reflection, or inversion. Most people would say that if you draw a square on a piece of paper, and then shift it three inches to the right, it is still the same. However, from the point of view of symmetry, a figure is not the same unless it overlaps exactly. There are tilings, such as a grid pattern that look exactly the same after performing a translation, but that can't be true for any finite figure such as a square. A square would look the same if you perform a rotation of 90 degrees or 180 degrees. However, a square would not look the same if you perform a rotation of 10 degrees. According to this definition of sameness, a square rotated 90 degrees is the same as a square rotated 180 degrees, but not the same as a square rotated 10 degrees. H - Squares that have had an operation performed on them where they look exactly the same afterwards We use the word symmetry to refer to this concept of sameness where a geometric figure looks exactly the same after an operation has been performed. Up to this point, we have have been looking at increasingly smaller subsets. However, you can take any set, and by removing axioms, create more general sets with broader definitions of sameness. One set of mathematical objects can be generalized in many different ways. The set G of squares is a subset of set I of all regular polygons, which includes the equilateral triangle, square, regular pentagon, and regular hexagon. I - Regular polygons The set G of squares is all also a subset of set J of n-dimensional regular polytopes where are edges are the same length, which includes the point, line segment, square, cube, hypercube, and hyperhypercube. J - N-dimensional hypercubes Another subset of J, other than set G of squares, is set K of cubes. K - Cubes A cube and an octahedron have the same symmetry group. a group has 8 vertices, 6 faces, and 12 edges. An octahedron has 6 vertices, 8 faces, and 12 edges. The cube and octahedron are duals, which means they have the same symmetry group, which in this case is called "octahedral symmetry", which has 24 rotational symmetries, and 48 symmetries including transformations that are a combination of a reflection and a rotation. You can define a set that includes all polyhedra with that symmetry group. L - Polyhedra with octahedral symmetry According to the definition of set L, a cube and octahedron are considered the same while a dodecahedron would not be considered the same. M - Platonic solids You could think of plantonic solids as polyhedra where are faces are regular polygons, and they are all the same polygon. All edges are the same length, and all angles are the same. The five platonic solids are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. In Plato's "Timmeas", he associated the five platonic solids with the four elements, and associated the left over platonic solid, with the celestial sphere. According to the definition of set M, a cube, octahedron, and dodecahedron would all be considered the same while a cuboctahedron would not be considered the same. N - Uniform polyhedra According to the definition of set N, a cube, octahredon, dodecahedron, and cuboctahedron would all be considered the same, but a sphere would not be considered the same. O - S^2 A generalization of the field of geometry is the field of topology, where you are allowed to perform deformations on shapes, and they are still considered the same shape after the deformation. Any shapes that can be deformed into each other are considered the same shape. Any polygon can be deformed into a circle. A triangle, square, and pentagon are all considered the same shape, which is called a circle, represented by S^1. Any polyhedron can be deformed into a sphere. A tetrahedron, cube, and dodecahedron are all considered the same shape, which is called a sphere, represented by S^3. The field of topology uses a very broad definition of sameness. The set O is the set all shapes that are topologically equivalent to a sphere, S^2. P - R^2 Set P is the set of all shapes that are topologically equivalent to R^2, which is a plane. Q - R^1 x S^1 Set Q is the set of all shapes that are topologically equivalent to R^1 x S^1, which is an infinite cylinder. R - Topologies where vectors are unchanged by parallel transport If you move a vector around a loop on the surface of a specific topology, and when it returns to its starting point, the vector is still pointing in the same direction, then the vector is unchanged by parallel transport. This works for vectors on a plane or infinite cylinder but not a sphere. You can define a set of all topologies where vectors are unchanged by parallel transport. Set P and set Q are subsets of set R, but set O is not a subset of set R. S - Riemann surfaces All topologies that we have mentioned are all Riemann surfaces, meaning there is no point where they are not locally flat. Next, I am going to give another series of examples. A - Set 1. Has members B - Magma 1. Has members 2. Binary operation AB = C C - Monoid 1. Has members 2. Binary operation AB = C 3. Associative (AB)C = A(BC) 4. Identity AI = A D - Group 1. Has members 2. Binary operation AB = C 3. Associative (AB)C = A(BC) 4. Identity AI=A 5. Inverses A A^-1 = I E - Abelian group 1. Has members 2. Binary operation AB=C 3. Associative (AB)C = A(BC) 4. Identity AI=A 5. Inverses A A^-1 = I 6. Commutative AB = BA Here is another way of generalizing a set. A - Set 1. Has members B - Category 1. Has members 2. Has morphisms between members C - 2-category 1. Has members 2. Has morphisms between members 3. Has 2-morphisms between morphisms D - 3-category 1. Has members 2. Has morphisms between members 3. Has 2-morphisms between morphisms 4. Has 3-morphisms between 2-morphisms 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 category. 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 more 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 moe 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 octahedron? 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.