What makes us who we are? Is there some set of interior qualities which defines us, or is meaning derived through our interconnectedness and the relationships we hold with our environment? Let's consider the role of connection, context, and structure by looking at isomorphisms, a special type of function found within category theory. This field is an abstraction of mathematics itself, formalizing category-wide generalizations and providing tools to universalize truths among a wide range of structures, revealing surprising connections and insights.

An isomorphism is a structure-preserving mapping between sets of elements from within a particular class of mathematical objects, such as groups, fields, or ordered sets. What makes an isomorphism isomorphic is that this type of function doesn’t change the relationships between the elements within the sets. In a sense, isomorphisms illuminate that many seemingly uniquely determined mathematical sets are simply different colorings of the same underlying structure. The sets effectively behave in the same way and look structurally similar given a correlated renaming of their elements. So while the elements themselves may be different, how those elements are interrelated and the ways in which they interact with one another are identical.

Let's see what this looks like in practice by exploring a few mathematical groups.

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Example \(1\):

Let group \(A\) be the set of all of the symmetries of an equilateral triangle\(^1\). These symmetries are the unique ways in which we can move a triangle such that the shape looks exactly the same after the motion is performed. An equilateral triangle has \(6\) distinct symmetries, as shown below, which reproduce the triangle exactly.

Group \(A\):

\(r_0\) - rotation of \(0^{\circ}\) (note that this is the same as a rotation of \(360^{\circ}\))

\(r_1\) - rotation of \(120^{\circ}\)

\(r_2\) - rotation of \(240^{\circ}\)

\(s\) - reflection about the bisector through the top apex

\(t\) - reflection about the bisector through the lower left apex

\(u\) - reflection about the bisector through the lower right apex

An isomorphism is a structure-preserving mapping between sets of elements from within a particular class of mathematical objects, such as groups, fields, or ordered sets. What makes an isomorphism isomorphic is that this type of function doesn’t change the relationships between the elements within the sets. In a sense, isomorphisms illuminate that many seemingly uniquely determined mathematical sets are simply different colorings of the same underlying structure. The sets effectively behave in the same way and look structurally similar given a correlated renaming of their elements. So while the elements themselves may be different, how those elements are interrelated and the ways in which they interact with one another are identical.

Let's see what this looks like in practice by exploring a few mathematical groups.

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Example \(1\):

Let group \(A\) be the set of all of the symmetries of an equilateral triangle\(^1\). These symmetries are the unique ways in which we can move a triangle such that the shape looks exactly the same after the motion is performed. An equilateral triangle has \(6\) distinct symmetries, as shown below, which reproduce the triangle exactly.

Group \(A\):

\(r_0\) - rotation of \(0^{\circ}\) (note that this is the same as a rotation of \(360^{\circ}\))

\(r_1\) - rotation of \(120^{\circ}\)

\(r_2\) - rotation of \(240^{\circ}\)

\(s\) - reflection about the bisector through the top apex

\(t\) - reflection about the bisector through the lower left apex

\(u\) - reflection about the bisector through the lower right apex

Let group \(B\) be the set of all permutations of \(3\) distinct elements\(^2\), denoted by \(a\), \(b\), and \(c\). There are \(3! = 6\) such permutations.

Group \(B\):

\(B_1 = \begin{bmatrix} a & b & c \\ a & b & c \\ \end{bmatrix}\) \(\qquad\) \(B_2 = \begin{bmatrix} a & b & c \\ b & c & a \\ \end{bmatrix}\) \(\qquad\) \(B_3 = \begin{bmatrix} a & b & c \\ c & a & b \\ \end{bmatrix}\)

\(B_4 = \begin{bmatrix} a & b & c \\ b & a & c \\ \end{bmatrix}\) \(\qquad\) \(B_5 = \begin{bmatrix} a & b & c \\ c & b & a \\ \end{bmatrix}\) \(\qquad\) \(B_6 = \begin{bmatrix} a & b & c \\ a & c & b \\ \end{bmatrix}\)

Both of these sets are defined as groups under the operation of composition: \(\circ\). For instance, in group \(A\), \(t \circ r_1 = s\) because, remembering that when evaluating a composition we go from right to left, we first rotate the triangle \(120^{\circ}\) and then we reflect the triangle about the bisector through the lower right apex. Check for yourself that you will end up with the same orientation of vertices as if you simply performed a single \(s\) transformation. In group \(B\), \(B_4 \circ B_2 = B_6\) because: \(B_2\) takes \(a\) to \(b\) and then \(B_4\) takes \(b\) to \(a\); \(B_2\) takes \(b\) to \(a\) and then \(B_4\) takes \(a\) to \(c\); and \(B_2\) takes \(c\) to \(c\) and then \(B_4\) takes \(c\) to \(b\). So, \(B_4 \circ B_2\) takes \(a\) to \(a\), \(b\) to \(c\), and \(c\) to \(b\), which is represented by the single permutation \(B_6\).

Now we will make a table for each group to see how all of the elements of that group interact with one another under composition.\(^3\)

Group \(B\):

\(B_1 = \begin{bmatrix} a & b & c \\ a & b & c \\ \end{bmatrix}\) \(\qquad\) \(B_2 = \begin{bmatrix} a & b & c \\ b & c & a \\ \end{bmatrix}\) \(\qquad\) \(B_3 = \begin{bmatrix} a & b & c \\ c & a & b \\ \end{bmatrix}\)

\(B_4 = \begin{bmatrix} a & b & c \\ b & a & c \\ \end{bmatrix}\) \(\qquad\) \(B_5 = \begin{bmatrix} a & b & c \\ c & b & a \\ \end{bmatrix}\) \(\qquad\) \(B_6 = \begin{bmatrix} a & b & c \\ a & c & b \\ \end{bmatrix}\)

Both of these sets are defined as groups under the operation of composition: \(\circ\). For instance, in group \(A\), \(t \circ r_1 = s\) because, remembering that when evaluating a composition we go from right to left, we first rotate the triangle \(120^{\circ}\) and then we reflect the triangle about the bisector through the lower right apex. Check for yourself that you will end up with the same orientation of vertices as if you simply performed a single \(s\) transformation. In group \(B\), \(B_4 \circ B_2 = B_6\) because: \(B_2\) takes \(a\) to \(b\) and then \(B_4\) takes \(b\) to \(a\); \(B_2\) takes \(b\) to \(a\) and then \(B_4\) takes \(a\) to \(c\); and \(B_2\) takes \(c\) to \(c\) and then \(B_4\) takes \(c\) to \(b\). So, \(B_4 \circ B_2\) takes \(a\) to \(a\), \(b\) to \(c\), and \(c\) to \(b\), which is represented by the single permutation \(B_6\).

Now we will make a table for each group to see how all of the elements of that group interact with one another under composition.\(^3\)

Upon examination, one finds that the structure of these two tables is exactly the same. This is seen by a simply relabeling. In the table for group \(B\), relabel as follows:

\(B_1 - r_0 \qquad B_2 - r_1 \qquad B_3 - r_2 \qquad B_4 - s \qquad B_5 - t \qquad B_6 - u\)

This will produce exactly the table for group \(A\). Thus, the relationships between the elements both within group \(A\) and within group \(B\) have identical relational correspondences. The elements interact with one another in precisely the same way.

We therefore say that group \(A\) is isomorphic to group \(B\).

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Example \(2\):

We will again use group \(B\) from example \(1\): the set of all permutations of \(3\) elements, denoted by \(a\), \(b\), and \(c\), under the operation of composition. Group \(C\) will be the set of all \(2\times 2\) matrices whose entries are elements of the group of integers modulo \(2\) (ie. the entries can only be either \(0\) or \(1\), and \(1 + 1 = 0\)), and whose determinant is not \(0\).\(^4\) The operation for this group will be matrix multiplication.

Group \(B\):

\(B_1 = \begin{bmatrix} a & b & c \\ a & b & c \\ \end{bmatrix}\) \(\qquad\) \(B_2 = \begin{bmatrix} a & b & c \\ b & c & a \\ \end{bmatrix}\) \(\qquad\) \(B_3 = \begin{bmatrix} a & b & c \\ c & a & b \\ \end{bmatrix}\)

\(B_4 = \begin{bmatrix} a & b & c \\ b & a & c \\ \end{bmatrix}\) \(\qquad\) \(B_5 = \begin{bmatrix} a & b & c \\ c & b & a \\ \end{bmatrix}\) \(\qquad\) \(B_6 = \begin{bmatrix} a & b & c \\ a & c & b \\ \end{bmatrix}\)

Group \(C\):

\(C_1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}\) \(\qquad\) \(C_2 = \begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}\) \(\qquad\) \(C_3 = \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ \end{bmatrix}\)

\(C_4 = \begin{bmatrix} 1 & 1 \\ 0 & 1 \\ \end{bmatrix}\) \(\qquad\) \(C_5 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix}\) \(\qquad\) \(C_6 = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ \end{bmatrix}\)

Let's restate the table for group \(B\), which was given in example \(1\), and then make a table for the elements in group \(C\) under the operation of matrix multiplication. Keep in mind when performing the matrix multiplications that because the entries of each matrix are elements of \(\mathbb{Z}_2\), \(1 + 1 = 0\).

This will produce exactly the table for group \(A\). Thus, the relationships between the elements both within group \(A\) and within group \(B\) have identical relational correspondences. The elements interact with one another in precisely the same way.

We therefore say that group \(A\) is isomorphic to group \(B\).

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Example \(2\):

We will again use group \(B\) from example \(1\): the set of all permutations of \(3\) elements, denoted by \(a\), \(b\), and \(c\), under the operation of composition. Group \(C\) will be the set of all \(2\times 2\) matrices whose entries are elements of the group of integers modulo \(2\) (ie. the entries can only be either \(0\) or \(1\), and \(1 + 1 = 0\)), and whose determinant is not \(0\).\(^4\) The operation for this group will be matrix multiplication.

Group \(B\):

\(B_1 = \begin{bmatrix} a & b & c \\ a & b & c \\ \end{bmatrix}\) \(\qquad\) \(B_2 = \begin{bmatrix} a & b & c \\ b & c & a \\ \end{bmatrix}\) \(\qquad\) \(B_3 = \begin{bmatrix} a & b & c \\ c & a & b \\ \end{bmatrix}\)

\(B_4 = \begin{bmatrix} a & b & c \\ b & a & c \\ \end{bmatrix}\) \(\qquad\) \(B_5 = \begin{bmatrix} a & b & c \\ c & b & a \\ \end{bmatrix}\) \(\qquad\) \(B_6 = \begin{bmatrix} a & b & c \\ a & c & b \\ \end{bmatrix}\)

Group \(C\):

\(C_1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}\) \(\qquad\) \(C_2 = \begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}\) \(\qquad\) \(C_3 = \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ \end{bmatrix}\)

\(C_4 = \begin{bmatrix} 1 & 1 \\ 0 & 1 \\ \end{bmatrix}\) \(\qquad\) \(C_5 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix}\) \(\qquad\) \(C_6 = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ \end{bmatrix}\)

Let's restate the table for group \(B\), which was given in example \(1\), and then make a table for the elements in group \(C\) under the operation of matrix multiplication. Keep in mind when performing the matrix multiplications that because the entries of each matrix are elements of \(\mathbb{Z}_2\), \(1 + 1 = 0\).

Note that, again, the structure of the tables is exactly the same. In group \(C\), we need only replace each instance of the letter \(C\) by the letter \(B\) and we will have exactly reproduced the table for group \(B\).

Thus, group \(B\) is isomorphic to group \(C\). And because isomorphisms are transitive and group \(A\) is isomorphic to group \(B\), group \(A\) is therefore also isomorphic to group \(C\).

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Finding an isomorphism between two groups is an extremely powerful discovery because a vast number of group properties are preserved under isomorphism; knowing something about one of the groups immediately tells us something about the other group. Some of the preserved group properties include: the order of the group, whether or not the group is abelian\(^5\), isomorphic relationships to other groups, the number of elements of a given order within the group, and a vast wealth more.

In fact, we could, in a sense, "do the math" of one group using the correlated elements of the other group under their respective operation, if it proved easier or more convenient to do so. This allows us great flexibility and creates a kind of meta-math in which we can perform mathematics in a highly abstracted way.

Under this notion, the isomorphism between groups \(A\) and \(C\) means that we can treat the matrices of group \(C\) as the symmetries of an equilateral triangle! And both of the matrices of \(C\) and the symmetries of \(A\) can be treated the permutations of group \(B\). This profound insight reveals the depth of the notion of isomophisms. Triangle symmetries originate within a geometric context; permutations arise within the realm of combinatorics; matrices are abstract concepts from linear algebra. These three sets come from very different areas within mathematics, but their isomorphic relationship reveals that structurally each of these groups feature the same underlying organization and form.

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Is everything reducible to structure?

If two sets are structurally the same, are they the same set? From where does a set derive its meaning? Is it through its elements or through the relationships between them? Every group is isomorphic to a group of permutations (Caley's Theorem). Should we continue talking about the symmetries of triangles or about the matrices of group \(C\), or should we simply speak of permutations? Each of these groups has its own context and origin. Does this context provide a source for significance in isomorphic group variety?

What does it mean for two sets to not be isomorphic? Do they have fundamentally different and detached natures or do they also somehow link together within a larger, underlying structure?

We hold relationships with people, with non-human life, with ourselves, with the planet and environment, with feelings, with ideas, and with the universe as a whole. Can we find parallels between the ways in which these connections operate? How are they the same? Are there differences? Can we learn healthier, more growing ways of interacting within each of these relational fields by exploring our connections within other areas?

Emergent properties are those which are present within a system or an aggregate of objects, but which are not present in the individual particles or constituents of that system. For instance, the physical arrangement of H

Does anything exist which can be separated from its interrelational context? Is there something which can be isolated and interpreted outside of the interwoven landscape of reality? Is it meaningful to talk about objects without considering their connection to the deeper structure within which they're embedded?

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(1) In group theory, this is the dihedral group \(D_3\).

(2) In group theory, this is the symmetry group \(S_3\).

(3) This kind of table is called a Caley table.

(4) In group theory, this is the general linear group of degree \(2\) over \(\mathbb{Z}_2\) and is denoted \(GL_2(\mathbb{Z}_2)\).

(5) A group is abelian if for any two elements, \(a\) and \(b\), of the group, \(a \circ b = b \circ a\), where \(\circ\) is the operation of that group.

Thus, group \(B\) is isomorphic to group \(C\). And because isomorphisms are transitive and group \(A\) is isomorphic to group \(B\), group \(A\) is therefore also isomorphic to group \(C\).

- - - - - - - - - -

Finding an isomorphism between two groups is an extremely powerful discovery because a vast number of group properties are preserved under isomorphism; knowing something about one of the groups immediately tells us something about the other group. Some of the preserved group properties include: the order of the group, whether or not the group is abelian\(^5\), isomorphic relationships to other groups, the number of elements of a given order within the group, and a vast wealth more.

In fact, we could, in a sense, "do the math" of one group using the correlated elements of the other group under their respective operation, if it proved easier or more convenient to do so. This allows us great flexibility and creates a kind of meta-math in which we can perform mathematics in a highly abstracted way.

Under this notion, the isomorphism between groups \(A\) and \(C\) means that we can treat the matrices of group \(C\) as the symmetries of an equilateral triangle! And both of the matrices of \(C\) and the symmetries of \(A\) can be treated the permutations of group \(B\). This profound insight reveals the depth of the notion of isomophisms. Triangle symmetries originate within a geometric context; permutations arise within the realm of combinatorics; matrices are abstract concepts from linear algebra. These three sets come from very different areas within mathematics, but their isomorphic relationship reveals that structurally each of these groups feature the same underlying organization and form.

- - - - - - - - - -

Is everything reducible to structure?

If two sets are structurally the same, are they the same set? From where does a set derive its meaning? Is it through its elements or through the relationships between them? Every group is isomorphic to a group of permutations (Caley's Theorem). Should we continue talking about the symmetries of triangles or about the matrices of group \(C\), or should we simply speak of permutations? Each of these groups has its own context and origin. Does this context provide a source for significance in isomorphic group variety?

What does it mean for two sets to not be isomorphic? Do they have fundamentally different and detached natures or do they also somehow link together within a larger, underlying structure?

We hold relationships with people, with non-human life, with ourselves, with the planet and environment, with feelings, with ideas, and with the universe as a whole. Can we find parallels between the ways in which these connections operate? How are they the same? Are there differences? Can we learn healthier, more growing ways of interacting within each of these relational fields by exploring our connections within other areas?

Emergent properties are those which are present within a system or an aggregate of objects, but which are not present in the individual particles or constituents of that system. For instance, the physical arrangement of H

_{2}O molecules into liquid water produces a substance which has the property of being "wet", but the molecules themselves (and other arrangements of them, such as ice or gas) are not wet. Rearranging the same components into different relational patterns can therefore cause different properties to arise. Would groups with identical underlying structure produce similar emergent properties?Does anything exist which can be separated from its interrelational context? Is there something which can be isolated and interpreted outside of the interwoven landscape of reality? Is it meaningful to talk about objects without considering their connection to the deeper structure within which they're embedded?

- - - - - - - - - -

(1) In group theory, this is the dihedral group \(D_3\).

(2) In group theory, this is the symmetry group \(S_3\).

(3) This kind of table is called a Caley table.

(4) In group theory, this is the general linear group of degree \(2\) over \(\mathbb{Z}_2\) and is denoted \(GL_2(\mathbb{Z}_2)\).

(5) A group is abelian if for any two elements, \(a\) and \(b\), of the group, \(a \circ b = b \circ a\), where \(\circ\) is the operation of that group.