As a logician, Peirce treated abstraction (or, more precisely, "hypostatic
abstraction," as distinguished from "prescisive abstraction) as an object
for practice, which is itself produced in practice.
The Simplest Mathematics:p>
Peirce: CP 4.235 Cross-Ref:††
235. Look through the modern logical treatises, and you will
find that they almost all fall into one or other of two errors, as I hold
them to be; that of setting aside the doctrine of abstraction (in the sense
in which an abstract noun marks an abstraction) as a grammatical topic with
which the logician need not particularly concern himself; and that of
confounding abstraction, in this sense, with that operation of the mind by
which we pay attention to one feature of a percept to the disregard of
others. The two things are entirely disconnected. The most ordinary fact of
perception, such as "it is light," involves precisive abstraction, or
prescission.†1 But hypostatic abstraction, the abstraction which transforms
"it is light" into "there is light here," which is the sense which I shall
commonly attach to the word abstraction (since prescission will do for
precisive abstraction) is a very special mode of thought. It consists in
taking a feature of a percept or percepts (after it has already been
prescinded from the other elements of the percept), so as to take
propositional form in a judgment (indeed, it may operate upon any judgment
whatsoever), and in conceiving this fact to consist in the relation between
the subject of that judgment and another subject, which has a mode of being
that merely consists in the truth of propositions of which the corresponding
concrete term is the predicate. Thus, we transform the proposition, "honey
is sweet," into "honey possesses sweetness." "Sweetness" might be called a
fictitious thing, in one sense. But since the mode of being attributed to it
consists in no more than the fact that some things are sweet, and it is not
pretended, or imagined, that it has any other mode of being, there is, after
all, no fiction. The only profession made is that we consider the fact of
honey being sweet under the form of a relation; and so we really can. I have
selected sweetness as an instance of one of the least useful of
abstractions. Yet even this is convenient. It facilitates such thoughts as
that the sweetness of honey is particularly cloying; that the sweetness of
honey is something like the sweetness of a honeymoon; etc. Abstractions are
particularly congenial to mathematics. Everyday life first, for example,
found the need of that class of abstractions which we call collections.
Instead of saying that some human beings are males and all the rest females,
it was found convenient to say that mankind consists of the male part and
the female part. The same thought makes classes of collections, such as
pairs, leashes, quatrains, hands, weeks, dozens, baker's dozens, sonnets,
scores, quires, hundreds, long hundreds, gross, reams, thousands, myriads,
lacs, millions, milliards, milliasses, etc. These have suggested a great
branch of mathematics.†P1 Again, a point moves: it is by abstraction that
the geometer says that it "describes a line." This line, though an
abstraction, itself moves; and this is regarded as generating a surface; and
so on. So likewise, when the analyst treats operations as themselves
subjects of operations, a method whose utility will not be denied, this is
another instance of abstraction. Maxwell's notion of a tension exercised
upon lines of electrical force, transverse to them, is somewhat similar.
These examples exhibit the great rolling billows of abstraction in the ocean
of mathematical thought; but when we come to a minute examination of it, we
shall find, in every department, incessant ripples of the same form of
thought, of which the examples I have mentioned give no hint.
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