A ꖌ A = A.
holds true alike of logical and mathematical terms. It is absurd indeed to say that
x + x = x
except in the one case when x = absolute zero. But this contradiction x + x = x arises from the fact that we have already defined the units in one x as differing from those in the other. Under such circumstances the Law of Unity does not apply. For if in
A′ ꖌ A″ = A′
we mean that A″ is in any way different from A′ the assertion of identity is evidently false.
The contrast then which seems to exist between logical and mathematical symbols is only apparent. It is because the Laws of Simplicity and Unity must always be observed in the operation of counting that those laws seem no further to apply. This is the understood condition under which we use all numerical symbols. Whenever I write the symbol 5 I really mean
1 + 1 + 1 + 1 + 1,
and it is perfectly understood that each of these units is distinct from each other. If requisite I might mark them thus
1′+ 1″ + 1‴ + 1″″ + 1″‴.