Friday, May 23, 2014

Mathematical and Natural Definition compared

[This is the latest update to this essay, q.v. ]

Another parallel between mathematics and (e.g.) biology, as regards a certain type of ‘definition’.
Sometimes you are not trying to focus on a new concept in splendid independence, giving necessary and sufficient conditions to ‘be an X’, de-fining (demarcating) its boundaries between what-all is inside and what-else is out;  but, rather, starting from some homely, antecedently-familiar item Y, to define this new X as being similar to that Y.    Sometimes you say they’re similar, and leave it at that:

            A hare is like a rabbit.
            A coot is kind of like a duck.

Sometimes you add differentia:

            A zebra is like a horse with stripes.

Or, you may say that the new concept X generalizes Y, without giving necessary or sufficient conditions for membership in the generalization, with or without further examples of members of X:

            Amphibians form a taxon of animals that includes frogs.  (They ‘generalize’ the frog.)
            Amphibians form a taxon of animals that includes frogs and salamanders.

All these strategies are (so to speak) topologically distinct, the one from the other.

Compare, in math (an actual textbook example):

Locally convex spaces are topological vector spaces that generalize normed spaces.

Here the relatively exotic new concept “locally convex spaces” plays the role of amphibians in the example above, with the normed spaces (familiar from the nursery) filling that of our friends the frogs;  with an additional delimiter, topological vector spaces, basically saying:  “generalize, but not too far”.   Thus, if we said

Vertebrates form a taxon of animals that includes frogs.

that would still be a true statement, but the belt would have been let out too many notches to hold up the conceptual trousers.

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