Changes

<p>Thus far, we have considered the multiplication of relative terms only.  Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.</p>

<p>Thus far, we have considered the multiplication of relative terms only.  Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.</p>

<p>Now the absolute term "man" is really exactly equivalent to the relative term "man that is&nbsp;&mdash;&mdash;", and so with any other.  I shall write a comma after any absolute term to show that it is so regarded as a relative term.</p>

<p>Now the absolute term &ldquo;man&rdquo; is really exactly equivalent to the relative term &ldquo;man that is&nbsp;&mdash;&mdash;&rdquo;, and so with any other.  I shall write a comma after any absolute term to show that it is so regarded as a relative term.</p>

<p>Then "man that is black" will be written:</p>

<p>Then &ldquo;man that is black&rdquo; will be written:</p>

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| align="center" | <math>\mathrm{m},\!\mathrm{b}\!</math>

| align="center" | <math>\mathrm{m},\!\mathrm{b}\!</math>

<p>If, therefore, <math>\mathit{l},\!,\!\mathit{s}\mathrm{w}</math> is not the same as <math>\mathit{l},\!\mathit{s}\mathrm{w}</math> (as it plainly is not, because the latter means a lover and servant of a woman, and the former a lover of and servant of and same as a woman), this is simply because the writing of the comma alters the arrangement of the correlates.</p>

<p>If, therefore, <math>\mathit{l},\!,\!\mathit{s}\mathrm{w}</math> is not the same as <math>\mathit{l},\!\mathit{s}\mathrm{w}</math> (as it plainly is not, because the latter means a lover and servant of a woman, and the former a lover of and servant of and same as a woman), this is simply because the writing of the comma alters the arrangement of the correlates.</p>

<p>And if we are to suppose that absolute terms are multipliers at all (as mathematical generality demands that we should}, we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series "that is&nbsp;&mdash;&mdash; and is&nbsp;&mdash;&mdash; and is&nbsp;&mdash;&mdash; etc."</p>

<p>And if we are to suppose that absolute terms are multipliers at all (as mathematical generality demands that we should}, we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series &ldquo;that is&nbsp;&mdash;&mdash; and is&nbsp;&mdash;&mdash; and is&nbsp;&mdash;&mdash; etc.&rdquo;</p>

<p>Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the question is, What are to be the implied subjacent numbers for these implied correlates?</p>

<p>Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the question is, What are to be the implied subjacent numbers for these implied correlates?</p>

<p>A subjacent number may therefore be as great as we please.</p>

<p>A subjacent number may therefore be as great as we please.</p>

<p>But all these ''ones'' denote the same identical individual denoted by <math>\mathrm{w}\!</math>;  what then can be the subjacent numbers to be applied to <math>\mathit{s}\!</math>, for instance, on account of its infinite "''that is''"'s?  What numbers can separate it from being identical with <math>\mathrm{w}\!</math>?  There are only two.  The first is ''zero'', which plainly neutralizes a comma completely, since</p>

<p>But all these ''ones'' denote the same identical individual denoted by <math>\mathrm{w}\!</math>;  what then can be the subjacent numbers to be applied to <math>\mathit{s}\!</math>, for instance, on account of its infinite &ldquo;''that is''&rdquo;'s?  What numbers can separate it from being identical with <math>\mathrm{w}\!</math>?  There are only two.  The first is ''zero'', which plainly neutralizes a comma completely, since</p>

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| align="center" | <math>\mathit{s},_0\!\mathrm{w} ~=~ \mathit{s}\mathrm{w}</math>

| align="center" | <math>\mathit{s},_0\!\mathrm{w} ~=~ \mathit{s}\mathrm{w}</math>

<p>Any term, then, is properly to be regarded as having an infinite number of commas, all or some of which are neutralized by zeros.</p>

<p>Any term, then, is properly to be regarded as having an infinite number of commas, all or some of which are neutralized by zeros.</p>

<p>"Something" may then be expressed by:</p>

<p>&ldquo;Something&rdquo; may then be expressed by:</p>

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| align="center" | <math>\mathit{1}_\infty\!</math>

| align="center" | <math>\mathit{1}_\infty\!</math>

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<p>I shall for brevity frequently express this by an antique figure one <math>(\mathfrak{1}).</math>

<p>I shall for brevity frequently express this by an antique figure one <math>(\mathfrak{1}).</math></p>

<p>"Anything" by:</p>

<p>&ldquo;Anything&rdquo; by:</p>

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| align="center" | <math>\mathit{1}_0\!</math>

| align="center" | <math>\mathit{1}_0\!</math>