Re: Hypostatic And Prescisive Abstraction

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HAPA.  Note 4

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| 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.

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