The Tetrast2 - Speculation Lounge
Sketcher of various interrelated fourfolds.


March 25, 2007.

(See updates further down.)

There is a perspective of differentiation, distinction, and the building a classification in such a way as could lead to singling an object out but oftener seems to lead, in putting like with like, to the general, operating from many to one, as for instance successive differentiations of a curve can lead up to where rate of change is constant and the next derivative is zero and a lot of curves thus end up sorted at some level into a common class in that manner. Think also of a table of a group or any table of addition or multiplication.

a b c d
a b c d
b a d c
c d a b
d c b a

Each combination of two values leads to just one value, but more than one combination can lead to the same value.

It's like enumerative combinatorics turned inside out. Likewise, differentiation and integration are considered inverse operations. Generally the differentiation/algebra/information/life perspective is the "inside-out" version of the perspective which cuts across mathematical integration, measure, enumerative combinatorics, probability, statistics, and matter.

This leading of more than one (set of) value(s) to one (set of) value(s) is sometimes called "many-to-one" but that phrase is sometimes employed to signify a non-injective function.

Likewise by "one-to-one" is sometimes meant only a bijective function, not also an injective function. I find no distinct names for the kinds of relation which I'm discussing here. What would be good names for them? Also -- one-to-many (e.g., antiderivatives) and many-to-many (e.g., conservation equations). Note, for instance, that by "many-to-many" I don't mean merely many variables to be valued on each side of the equation, rather I mean that for each set of values for the variables on either side of the equation, there are, for the variables on the other side of the equation, more than one set of values which will satisfy the equation.

Update Friday, March 30, 2007

I've played around with coinages like "epluribune" (from e pluribus unum) but that one leads to formations like "epluripplure" and so on. "Unadune, pluradune, unapplure, plurapplure" might not be so bad yet who besides myself would ever use them? My best bet is to work from mathematical terms. Since "one-to-one" is sometimes used to mean "bijective," I thought -- why not borrow the prefixes from "bijective," "surjective," and "injective"?

One-to-one correspondence -- Bispondence.
Many-to-one or one-to-one correspondence -- Surspondence.
One-to-many or one-to-one correspondence -- Inspondence.

Then, to finish the job, I still have to engage in more invention than I'd like:

Many-to-one correspondence -- Surcospondence.
One-to-many correspondence -- Incospondence.
Many-to-many correspondence -- Plexspondence (does somebody have a better idea?).

These terms suffer from the ambiguities of their definitions. For instance, by "many-to-one" do I mean "many-to-at-most-one"? That's probably the most convenient sense.

Second Update Friday, March 30, 2007

By coincidence I met a computer programmer turned math schoolteacher this afternoon after writing the update above. He told me that he's sure that there are technical terms, which he's temporarily forgotten, for "many-to-one" etc., from discrete mathematics, and suggested that I google around including "C++" as a search term. I thought also of including "discrete mathematics" in some searches, and of checking an introductory textbook of discrete mathematics which I have lying around.

I looked. No luck!

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