Changes

===Commentary Note 11.2===

===Commentary Note 11.2===

Let's bring together the various things that Peirce has said about the "number of function" up to this point in the paper.

Let's bring together the various things that Peirce has said

 

about the "number of function" up to this point in the paper.

 

 

 

 

 

 

 

NOF 1.

<p>It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations.  The conception of ''taking together'' involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five.  Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves &mdash; provided all the terms summed are mutually exclusive.</p>

| I propose to assign to all logical terms, numbers;

<p>Addition being taken in this sense, ''nothing'' is to be denoted by 'zero', for then:</p>

| Thus in a universe of perfect men ('men'),

| the number of "tooth of" would be 32.

| I propose to denote the number of a logical term by

| enclosing the term in square brackets, thus ['t'].

| C.S. Peirce, CP 3.65

| But not only do the significations of '=' and '<' here adopted fulfill all

<p>whatever is denoted by ''x''; and this is the definition of ''zero''.  This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of ''zero'' and that of nothing, and because we shall thus have</p>

| 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

| which are predicable of the same classes.  So, to write 5 < 7 is to say that 5

| is part of 7, just as to write f < m is to say that Frenchmen are part of men.

| Indeed, if f < m, then the number of Frenchmen is less than the number of men,

| and if v = 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

| C.S. Peirce, CP 3.66

NOF 3.

: <p>[0] = 0.</p>

<p>(Peirce, CP 3.67).</p>

| C.S. Peirce, CP 3.67

NOF 4.

'''NOF 4'''

| The conception of multiplication we have adopted is

| The conception of multiplication we have adopted is

| that of the application of one relation to another.  ...

| that of the application of one relation to another.  ...

|

|

| C.S. Peirce, CP 3.76

| C.S. Peirce, CP 3.76

Before I can discuss Peirce's "number of" function in greater detail

Before I can discuss Peirce's "number of" function in greater detail I will need to deal with an expositional difficulty that I have been very carefully dancing around all this time, but that will no longer abide its assigned place under the rug.

I will need to deal with an expositional difficulty that I have been

very carefully dancing around all this time, but that will no longer

abide its assigned place under the rug.

Functions have long been understood, from well before Peirce's time to ours,

Functions have long been understood, from well before Peirce's time to ours, as special cases of 2-adic relations, so the "number of" function itself is already to be numbered among the types of 2-adic relatives that we've been explictly mentioning and implicitly using all this time.  But Peirce's way of talking about a 2-adic relative term is to list the "relate" first and the "correlate" second, a convention that goes over into functional terms as making the functional value first and the functional antecedent second, whereas almost anyone brought up in our present time frame has difficulty thinking of a function any other way than as a set of ordered pairs where the order in each pair lists the functional argument, or domain element, first and the functional value, or codomain element, second.

as special cases of 2-adic relations, so the "number of" function itself is

already to be numbered among the types of 2-adic relatives that we've been

explictly mentioning and implicitly using all this time.  But Peirce's way

of talking about a 2-adic relative term is to list the "relate" first and

the "correlate" second, a convention that goes over into functional terms

as making the functional value first and the functional antecedent second,

whereas almost anyone brought up in our present time frame has difficulty

thinking of a function any other way than as a set of ordered pairs where

the order in each pair lists the functional argument, or domain element,

first and the functional value, or codomain element, second.

It is possible to work all this out in a very nice way within a very general context

It is possible to work all this out in a very nice way within a very general context of flexible conventions, but not without introducing an order of anachronisms into Peirce's presentation that I am presently trying to avoid as much as possible. Thus, I will need to experiment with various sorts of compromise formations.

of flexible conventions, but not without introducing an order of anachronisms into

Peirce's presentation that I am presently trying to avoid as much as possible.

Thus, I will need to experiment with various sorts of compromise formations.

===Commentary Note 11.3===

===Commentary Note 11.3===