Changes

MyWikiBiz, Author Your Legacy — Tuesday November 05, 2024
Jump to navigationJump to search

Directory:Jon Awbrey/Papers/Dynamics And Logic (view source)

Revision as of 14:12, 14 June 2009

805 bytes added ,  14:12, 14 June 2009
→‎Note 12: markup
Line 2,881: Line 2,881:  
==Note 12==
 
==Note 12==
   −
<pre>
+
{| align="center" cellpadding="0" cellspacing="0" width="90%"
−
| Consider what effects that might conceivably have
  −
| practical bearings you conceive the objects of your
  −
| conception to have.  Then, your conception of those
  −
| effects is the whole of your conception of the object.
   
|
 
|
−
| C.S. Peirce, "Maxim of Pragmaticism", 'Collected Papers', CP 5.438
+
<p>Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.</p>
 +
|-
 +
| align="right" | &mdash; Charles Sanders Peirce, "Issues of Pragmaticism", (CP 5.438)
 +
|}
 +
 
 +
One other subject that it would be opportune to mention at this point, while we have an object example of a mathematical group fresh in mind, is the relationship between the pragmatic maxim and what are commonly known in mathematics as ''representation principles''.  As it turns out, with regard to its formal characteristics, the pragmatic maxim unites the aspects of a representation principle with the attributes of what would ordinarily be known as a ''closure principle''.  We will consider the form of closure that is invoked by the pragmatic maxim on another occasion, focusing here and now on the topic of group representations.
 +
 
 +
Let us return to the example of the ''four-group'' <math>V_4.\!</math>  We encountered this group in one of its concrete representations, namely, as a ''transformation group'' that acts on a set of objects, in this case a set of sixteen functions or propositions. Forgetting about the set of objects that the group transforms among themselves, we may take the abstract view of the group's operational structure, for example, in the form of the group operation table copied here:
   −
One other subject that it would be opportune to mention at this point,
+
<br>
−
while we have an object example of a mathematical group fresh in mind,
  −
is the relationship between the pragmatic maxim and what are commonly
  −
known in mathematics as "representation principles".  As it turns out,
  −
with regard to its formal characteristics, the pragmatic maxim unites
  −
the aspects of a representation principle with the attributes of what
  −
would ordinarily be known as a "closure principle".  We will consider
  −
the form of closure that is invoked by the pragmatic maxim on another
  −
occasion, focusing here and now on the topic of group representations.
     −
Let us return to the example of the so-called "four-group" V_4.
+
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
−
We encountered this group in one of its concrete representations,
+
|- style="height:50px"
−
namely, as a "transformation group" that acts on a set of objects,
+
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
−
in this particular case a set of sixteen functions or propositions.
+
| width="22%" style="border-bottom:1px solid black" |
−
Forgetting about the set of objects that the group transforms among
+
<math>\operatorname{e}</math>
−
themselves, we may take the abstract view of the group's operational
+
| width="22%" style="border-bottom:1px solid black" |
−
structure, say, in the form of the group operation table copied here:
+
<math>\operatorname{f}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{g}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
|}
   −
o-------o-------o-------o-------o-------o
+
<br>
−
|      %      |      |      |      |
  −
|  *  %  e  |  f  |  g  |  h  |
  −
|      %      |      |      |      |
  −
o=======o=======o=======o=======o=======o
  −
|      %      |      |      |      |
  −
|  e  %  e  |  f  |  g  |  h  |
  −
|      %      |      |      |      |
  −
o-------o-------o-------o-------o-------o
  −
|      %      |      |      |      |
  −
|  f  %  f  |  e  |  h  |  g  |
  −
|      %      |      |      |      |
  −
o-------o-------o-------o-------o-------o
  −
|      %      |      |      |      |
  −
|  g  %  g  |  h  |  e  |  f  |
  −
|      %      |      |      |      |
  −
o-------o-------o-------o-------o-------o
  −
|      %      |      |      |      |
  −
|  h  %  h  |  g  |  f  |  e  |
  −
|      %      |      |      |      |
  −
o-------o-------o-------o-------o-------o
      +
<pre>
 
This table is abstractly the same as, or isomorphic to, the versions
 
This table is abstractly the same as, or isomorphic to, the versions
 
with the E_ij operators and the T_ij transformations that we took up
 
with the E_ij operators and the T_ij transformations that we took up
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/88145"

Navigation menu