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===Commentary Note 11.17===

===Commentary Note 11.17===

I think that the reader is beginning to get an inkling of the crucial importance of the "number of" map in Peirce's way of looking at logic, for it's one of the plancks in the bridge from logic to the theories of probability, statistics, and information, in which logic forms but a limiting case at one scenic turnout on the expanding vista.  It is, as a matter of necessity and a matter of fact, practically speaking, at any rate, one way that Peirce forges a link between the "eternal", logical, or rational realm and the "secular", empirical, or real domain.

I think that the reader is beginning to get an inkling of the crucial importance of the "number of" map in Peirce's way of looking at logic, for it's one of the plancks in the bridge from logic to the theories of probability, statistics, and information, in which logic forms but a limiting case at one scenic turnout on the expanding vista.  It is, as a matter of necessity and a matter of fact, practically speaking, at any rate, one way that Peirce forges a link between the ''eternal'', logical, or rational realm and the ''secular'', empirical, or real domain.

With that little bit of encouragement and exhortation, let us return to the nitty gritty details of the text.

With that little bit of encouragement and exhortation, let us return to the nitty gritty details of the text.

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But not only do the significations of "=" and "<" here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations.  Equality is, in fact, nothing but the identity of two numbers;  numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.  So, to write 5&nbsp;<&nbsp;7 is to say that 5 is part of 7, just as to write ''f''&nbsp;<&nbsp;''m'' is to say that Frenchmen are part of men.  Indeed, if ''f''&nbsp;<&nbsp;''m'', then the number of Frenchmen is less than the number of men, and if ''v''&nbsp;=&nbsp;''p'', then the number of Vice-Presidents is equal to the number of Presidents of the Senate;  so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities. (Peirce, CP 3.66).

<p>But not only do the significations of &nbsp;<math>~=~</math>&nbsp; and &nbsp;<math>~<~</math>&nbsp; here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations.  Equality is, in fact, nothing but the identity of two numbers;  numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.  So, to write <math>~5 < 7~</math> is to say that <math>~5~</math> is part of <math>~7~</math>, just as to write <math>~\mathrm{f} < \mathrm{m}~</math> is to say that Frenchmen are part of men.  Indeed, if <math>~\mathrm{f} < \mathrm{m}~</math>, then the number of Frenchmen is less than the number of men, and if <math>~\mathrm{v} = \mathrm{p}~</math>, then the number of Vice-Presidents is equal to the number of Presidents of the Senate;  so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.</p>

 

<p>(Peirce, CP 3.66).</p>

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