ONT Re: Hypostatic And Prescisive Abstraction
HAPA. Note 4
| 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.*
| 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.
|* Of course, the moment a collection is recognized as an abstraction we have
| to admit that even a percept is an abstraction or represents an abstraction,
| if matter has parts. It therefore becomes difficult to maintain that all
| abstractions are fictions.
| C.S. Peirce, CP 4.235, "The Simplest Mathematics",
| Chapter 3 of the "Minute Logic", Jan-Feb 1902.