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==Transformation Groups==

==Operational Representation==

If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis.  But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> at once whelm over its discrete and finite powers to grasp them.  But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever need to explore their effects with care.

If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis.  But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> at once whelm over its discrete and finite powers to grasp them.  But here, in the fully serene idylls of [[zeroth order logic]], we find ourselves fit with the compass of a wit that is all we'd ever need to explore their effects with care.

So let us do just that.

So let us do just that.

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''.  One says that the orbits are preserved by the action of the group.  There is an ''Orbit Lemma'' of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group.  In this instance, <math>\operatorname{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:  <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.</math>  Amazing!

More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''.  One says that the orbits are preserved by the action of the group.  There is an ''Orbit Lemma'' of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group.  In this instance, <math>\operatorname{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:  <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.</math>  Amazing!

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More on the pragmatic maxim as a representation principle later.

More on the pragmatic maxim as a representation principle later.

The above-mentioned fact about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:

The above-mentioned fact about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:

This idea of contextual definition by way of conduct transforming operators is basically the same as Jeremy Bentham's notion of ''paraphrasis'', a "method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, ''From Frege to Gödel'', p.&nbsp;216).  Today we'd call these constructions ''term models''.  This, again, is the big idea behind Schönfinkel's combinators <math>\operatorname{S}, \operatorname{K}, \operatorname{I},</math> and hence of lambda calculus, and I reckon you know where that leads.

This idea of contextual definition by way of conduct transforming operators is basically the same as Jeremy Bentham's notion of ''paraphrasis'', a "method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, ''From Frege to Gödel'', p.&nbsp;216).  Today we'd call these constructions ''term models''.  This, again, is the big idea behind Schönfinkel's combinators <math>\operatorname{S}, \operatorname{K}, \operatorname{I},</math> and hence of lambda calculus, and I reckon you know where that leads.

The next few excursions in this series will provide a scenic tour of various ideas in group theory that will turn out to be of constant guidance in several of the settings that are associated with our topic.

The next few excursions in this series will provide a scenic tour of various ideas in group theory that will turn out to be of constant guidance in several of the settings that are associated with our topic.

I think this much will serve to fix notation and set up the remainder of the discussion.

I think this much will serve to fix notation and set up the remainder of the discussion.

In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives <math>(i\!:\!j),</math> as the indices <math>i, j\!</math> range over the universe of discourse, would be referred to as the ''umbral elements'' of the algebraic operation represented by the matrix, though I seem to recall that Peirce preferred to call these terms the "ingredients".  When this ordered basis is understood well enough, one will tend to drop any mention of it from the matrix itself, leaving us nothing but these bare bones:

In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives <math>(i\!:\!j),</math> as the indices <math>i, j\!</math> range over the universe of discourse, would be referred to as the ''umbral elements'' of the algebraic operation represented by the matrix, though I seem to recall that Peirce preferred to call these terms the "ingredients".  When this ordered basis is understood well enough, one will tend to drop any mention of it from the matrix itself, leaving us nothing but these bare bones:

This is consistent with the convention that Peirce uses in the paper "On a Class of Multiple Algebras" (CP 3.324&ndash;327).

This is consistent with the convention that Peirce uses in the paper "On a Class of Multiple Algebras" (CP 3.324&ndash;327).

We've been exploring the applications of a certain technique for clarifying abstruse concepts, a rough-cut version of the pragmatic maxim that I've been accustomed to refer to as the ''operationalization'' of ideas.  The basic idea is to replace the question of ''What it is'', which modest people comprehend is far beyond their powers to answer definitively any time soon, with the question of ''What it does'', which most people know at least a modicum about.

We've been exploring the applications of a certain technique for clarifying abstruse concepts, a rough-cut version of the pragmatic maxim that I've been accustomed to refer to as the ''operationalization'' of ideas.  The basic idea is to replace the question of ''What it is'', which modest people comprehend is far beyond their powers to answer definitively any time soon, with the question of ''What it does'', which most people know at least a modicum about.

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because <math>V_4\!</math> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because <math>V_4\!</math> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.

So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group.  This is a group of six elements, say, <math>G = \{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h}, \operatorname{i}, \operatorname{j} \},\!</math> with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, <math>X = \{ a, b, c \},\!</math> usually notated as <math>G = \operatorname{Sym}(X)</math> or more abstractly and briefly, as <math>\operatorname{Sym}(3)</math> or <math>S_3.\!</math>  The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\operatorname{Sym}(X).</math>

So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group.  This is a group of six elements, say, <math>G = \{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h}, \operatorname{i}, \operatorname{j} \},\!</math> with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, <math>X = \{ a, b, c \},\!</math> usually notated as <math>G = \operatorname{Sym}(X)</math> or more abstractly and briefly, as <math>\operatorname{Sym}(3)</math> or <math>S_3.\!</math>  The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\operatorname{Sym}(X).</math>

By the way, we will meet with the symmetric group <math>S_3\!</math> again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324&ndash;327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227&ndash;323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307&ndash;323).

By the way, we will meet with the symmetric group <math>S_3\!</math> again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324&ndash;327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227&ndash;323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307&ndash;323).

By way of collecting a short-term pay-off for all the work that we did on the regular representations of the Klein 4-group <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations for the symmetric group on three letters, <math>\operatorname{Sym}(3).</math>  After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscured details of Peirce's early "Algebra + Logic" papers.

By way of collecting a short-term pay-off for all the work that we did on the regular representations of the Klein 4-group <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations for the symmetric group on three letters, <math>\operatorname{Sym}(3).</math>  After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscured details of Peirce's early "Algebra + Logic" papers.

I have without stopping to think about it written out this natural representation of <math>S_3\!</math> in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as <math>x\!:\!y</math> constitutes the turning of <math>x\!</math> into <math>y.\!</math>  It is possible that the next time we check in with CSP we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

I have without stopping to think about it written out this natural representation of <math>S_3\!</math> in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as <math>x\!:\!y</math> constitutes the turning of <math>x\!</math> into <math>y.\!</math>  It is possible that the next time we check in with CSP we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:

To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:

If the ante-rep looks different from the post-rep, it is just as it should be, as <math>S_3\!</math> is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.

If the ante-rep looks different from the post-rep, it is just as it should be, as <math>S_3\!</math> is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.

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We made the observation that the shift operators <math>\{ \operatorname{E}_{ij} \}</math> form a transformation group that acts on the set of propositions of the form <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  Group theory is a very attractive subject, but it did not draw us so far from our intended course as one might initially think.  For one thing, groups, especially the groups that are named after the Norwegian mathematician [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Marius Sophus Lie (1842&ndash;1899)], have turned out to be of critical utility in the solution of differential equations.  For another thing, group operations provide us with an ample supply of triadic relations that have been extremely well-studied over the years, and thus they give us no small measure of useful guidance in the study of sign relations, another brand of 3-adic relations that have significance for logical studies, and in our acquaintance with which we have barely begun to break the ice.  Finally, I couldn't resist taking up the links between group representations, amounting to the very archetypes of logical models, and the pragmatic maxim.

We made the observation that the shift operators <math>\{ \operatorname{E}_{ij} \}</math> form a transformation group that acts on the set of propositions of the form <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  Group theory is a very attractive subject, but it did not draw us so far from our intended course as one might initially think.  For one thing, groups, especially the groups that are named after the Norwegian mathematician [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Marius Sophus Lie (1842&ndash;1899)], have turned out to be of critical utility in the solution of differential equations.  For another thing, group operations provide us with an ample supply of triadic relations that have been extremely well-studied over the years, and thus they give us no small measure of useful guidance in the study of sign relations, another brand of 3-adic relations that have significance for logical studies, and in our acquaintance with which we have barely begun to break the ice.  Finally, I couldn't resist taking up the links between group representations, amounting to the very archetypes of logical models, and the pragmatic maxim.

We've seen a couple of groups, <math>V_4\!</math> and <math>S_3,\!</math> represented in various ways, and we've seen their representations presented in a variety of different manners.  Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.

We've seen a couple of groups, <math>V_4\!</math> and <math>S_3,\!</math> represented in various ways, and we've seen their representations presented in a variety of different manners.  Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.