Changes

xi xi*x0 ... xi*xj ...

xi xi*x0 ... xi*xj ...

... ... ... ... ...

... ... ... ... ...

Table 32.2  Scheme of the Regular Ante-Representation

Table 32.2  Scheme of the Regular Ante-Representation

Element Function as Set of Ordered Pairs of Elements

Element Function as Set of Ordered Pairs of Elements

xi { <x0, xi*x0>,  ...,  <xj, xi*xj>,  ...,    }

xi { <x0, xi*x0>,  ...,  <xj, xi*xj>,  ...,    }

...

...

Table 32.3  Scheme of the Regular Post-Representation

Table 32.3  Scheme of the Regular Post-Representation

Element Function as Set of Ordered Pairs of Elements

Element Function as Set of Ordered Pairs of Elements

xi { <x0, x0*xi>,  ...,  <xj, xj*xi>,  ...,    }

xi { <x0, x0*xi>,  ...,  <xj, xj*xi>,  ...,    }

...

...

In following these maps, notice how closely one is treading in these representations to defining each element in terms of itself, but without quite going that far.  There are a couple of catches that save this form of representation from falling into a "vicious circle", that is, into a pattern of self-reference that would beg the question of a definition and vitiate its usefulness as an explanation of each group element's action.  First, the regular representations do not represent that a group element is literally "equal to" a set of ordered pairs involving that very same group element, but only that it is "mapped to" something like this set.  Second, careful usage would dictate that the "something like" that one finds in the image of a representation, being something that is specified only up to its isomorphism class, is a transformation that really acts, not on the group elements xj themselves, but only on their inert tokens, inactive images, partial symbols, passing names, or transitory signs of the form "xj".

In following these maps, notice how closely one is treading in these representations to defining each element in terms of itself, but without quite going that far.  There are a couple of catches that save this form of representation from falling into a &ldquo;vicious circle&rdquo;, that is, into a pattern of self-reference that would beg the question of a definition and vitiate its usefulness as an explanation of each group element's action.  First, the regular representations do not represent that a group element is literally ''equal to'' a set of ordered pairs involving that very same group element, but only that it is ''mapped to'' something like this set.  Second, careful usage would dictate that the ''something like'' that one finds in the image of a representation, being something that is specified only up to its isomorphism class, is a transformation that really acts, not on the group elements <math>x_j\!</math> themselves, but only on their inert tokens, inactive images, partial symbols, passing names, or transitory signs of the form <math>{}^{\backprime\backprime} x_j {}^{\prime\prime}.\!</math>

These reservations are crucial to understanding the form of explanation that a regular representation provides, that is, what it explains and what it does not.  If one is seeking an ontological explanation of what a group and its elements "are", then one would have reason to object that it does no good to represent a group and its elements in terms of their actions on the group elements themselves, since one still does not know what the latter entities "are".  Notice that the form of this objection is reminiscent of a dilemma that is often thought to obstruct the beginning of an inquiry into inquiry.  A similar pattern of knots occurs when one tries to explain the process of formalization in terms of its effects on the term "formalization".  In each case, the resolution of the difficulty turns on recognizing a distinction between the active and passive modes of existence that go with each nameable objective.

These reservations are crucial to understanding the form of explanation that a regular representation provides, that is, what it explains and what it does not.  If one is seeking an ontological explanation of what a group and its elements "are", then one would have reason to object that it does no good to represent a group and its elements in terms of their actions on the group elements themselves, since one still does not know what the latter entities "are".  Notice that the form of this objection is reminiscent of a dilemma that is often thought to obstruct the beginning of an inquiry into inquiry.  A similar pattern of knots occurs when one tries to explain the process of formalization in terms of its effects on the term "formalization".  In each case, the resolution of the difficulty turns on recognizing a distinction between the active and passive modes of existence that go with each nameable objective.