Changes

This presents the group in one big bunch, and there are occasions when one regards it this way, but that is not the typical form of presentation that we'd encounter.  More likely, the story would go a little bit like this:

This presents the group in one big bunch, and there are occasions when one regards it this way, but that is not the typical form of presentation that we'd encounter.  More likely, the story would go a little bit like this:

I cannot remember any of my math teachers ever invoking the pragmatic maxim by name, but it would be a very regular occurrence for such mentors and tutors to set up the subject in this wise:  Suppose you forget what a given abstract group element means, that is, in effect, ''what it is''.  Then a sure way to jog your sense of ''what it is''' is to build a regular representation from the formal materials that are necessarily left lying about on that abstraction site.

I cannot remember any of my math teachers ever invoking the pragmatic maxim by name, but it would be a very regular occurrence for such mentors and tutors to set up the subject in this wise:  Suppose you forget what a given abstract group element means, that is, in effect, "what it is".  Then a sure way to jog your sense of "what it is" is to build a regular representation from the formal materials that are necessarily left lying about on that abstraction site.

Working through the construction for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular post-representations:

Working through the construction for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular post-representations:

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So if somebody asks you, say, "What is <math>\operatorname{g}</math>?", you can say, "I don't know for certain, but in practice its effects go a bit like this: Converting <math>\operatorname{e}</math> to <math>\operatorname{g},</math> <math>\operatorname{f}</math> to <math>\operatorname{h},</math> <math>\operatorname{g}</math> to <math>\operatorname{e},</math> <math>\operatorname{h}</math> to <math>\operatorname{f}.</math>"

So if somebody asks you, say, "What is <math>\operatorname{g}</math>?", you can say, "I don't know for certain, but in practice its effects go a bit like this:

 

| <math>\operatorname{g} \quad \text{converts} \quad \operatorname{e} ~\text{to}~ \operatorname{g}, \quad \operatorname{f} ~\text{to}~ \operatorname{h}, \quad \operatorname{g} ~\text{to}~ \operatorname{e}, \quad \operatorname{h} ~\text{to}~ \operatorname{f}.</math>

I will have to check this out later on, but my impression is that Peirce tended to lean toward the other brand of regular, the ''left representation'' or the ''ante-representation'' of the groups that he treated in his earliest manuscripts and papers.  I believe that this was because he thought of the actions on the pattern of dyadic relative terms like ''aftermath of''.

I will have to check this out later on, but my impression is that Peirce tended to lean toward the other brand of regular, the ''left representation'' or the ''ante-representation'' of the groups that he treated in his earliest manuscripts and papers.  I believe that this was because he thought of the actions on the pattern of dyadic relative terms like ''aftermath of''.