ONT Re: Differential Logic
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note D72
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Taking Aim at Higher Dimensional Targets (concl.)
Let us now refer to the dimension of the target space or codomain as
the "toll" (or "tole") of a transformation, as distinguished from the
dimension of the range or image that is customarily called the "rank".
When we keep to transformations with a toll of 1, as J : [u, v] -> [x],
we tend to get lazy about distinguishing a logical transformation from
its component propositions. However, if we deal with transformations
of a higher toll, this form of indolence can no longer be tolerated.
Well, perhaps we can carry it a little further. After all,
the operator result WJ : EU% -> EX% is a map of toll 2, and
cannot be unfolded in one piece as a proposition. But when a
map has rank 1, like !e!J : EU -> X c EX or dJ : EU -> dX c EX,
we naturally choose to concentrate on the 1-dimensional range of
the operator result WJ, ignoring the final difference in quality
between the spaces X and dX, and view WJ as a proposition about EU.
In this way, an initial ambivalence about the role of the operand J
conveys a double duty to the result WJ. The pivot that is formed by
our focus of attention is essential to the linkage that transfers this
double moment, as the whole process takes its bearing and wheels around
the precise measure of a narrow bead that we can draw on the range of WJ.
This is the escapement that it takes to get away with what may otherwise
seem to be a simple duplicity, and this is the tolerance that is needed
to counterbalance a certain arrogance of equivocation, by all of which
machinations we make ourselves free to indicate the operator results
WJ as propositions or as transformations, indifferently.
But that's it, and no further. Neglect of these distinctions in range and
target universes of higher dimensions is bound to cause a hopeless confusion.
To guard against these adverse prospects, Tables 58 and 59 lay the groundwork
for discussing a typical map F : [B^2] -> [B^2], and begin to pave the way, to
some extent, for discussing any transformation of the form F : [B^n] -> [B^k].
Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators
o------o-------------------------o------------------o----------------------------o
| Item | Notation | Description | Type |
o------o-------------------------o------------------o----------------------------o
| | | | |
| U% | = [u, v] | Source Universe | [B^n] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| X% | = [x, y] | Target Universe | [B^k] |
| | = [f, g] | | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| EU% | = [u, v, du, dv] | Extended | [B^n x D^n] |
| | | Source Universe | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| EX% | = [x, y, dx, dy] | Extended | [B^k x D^k] |
| | = [f, g, df, dg] | Target Universe | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| F | F = <f, g> : U% -> X% | Transformation, | [B^n] -> [B^k] |
| | | or Mapping | |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| | f, g : U -> B | Proposition, | B^n -> B |
| | | special case | |
| f | f : U -> [x] c X% | of a mapping, | c (B^n, B^n -> B) |
| | | or component | |
| g | g : U -> [y] c X% | of a mapping. | = (B^n +-> B) = [B^n] |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| W | W : | Operator | |
| | U% -> EU%, | | [B^n] -> [B^n x D^n], |
| | X% -> EX%, | | [B^k] -> [B^k x D^k], |
| | (U%->X%)->(EU%->EX%), | | ([B^n] -> [B^k]) |
| | for each W among: | | -> |
| | !e!, !h!, E, D, d | | ([B^n x D^n]->[B^k x D^k]) |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | |
| !e! | | Tacit Extension Operator !e! |
| !h! | | Trope Extension Operator !h! |
| E | | Enlargement Operator E |
| D | | Difference Operator D |
| d | | Differential Operator d |
| | | |
o------o-------------------------o------------------o----------------------------o
| | | | |
| $W$ | $W$ : | Operator | |
| | U% -> $T$U% = EU%, | | [B^n] -> [B^n x D^n], |
| | X% -> $T$X% = EX%, | | [B^k] -> [B^k x D^k], |
| | (U%->X%)->($T$U%->$T$X%)| | ([B^n] -> [B^k]) |
| | for each $W$ among: | | -> |
| | $e$, $E$, $D$, $T$ | | ([B^n x D^n]->[B^k x D^k]) |
| | | | |
o------o-------------------------o------------------o----------------------------o
| | | |
| $e$ | | Radius Operator $e$ = <!e!, !h!> |
| $E$ | | Secant Operator $E$ = <!e!, E > |
| $D$ | | Chord Operator $D$ = <!e!, D > |
| $T$ | | Tangent Functor $T$ = <!e!, d > |
| | | |
o------o-------------------------o-----------------------------------------------o
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
o--------------o----------------------o--------------------o----------------------o
| | Operator | Proposition | Transformation |
| | or | or | or |
| | Operand | Component | Mapping |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
| | | | |
| | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Tacit | !e! : | !e!F_i : | !e!F : |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |
| | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Trope | !h! : | !h!F_i : | !h!F : |
| Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Enlargement | E : | EF_i : | EF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Difference | D : | DF_i : | DF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Differential | d : | dF_i : | dF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Remainder | r : | rF_i : | rF : |
| Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Radius | $e$ = <!e!, !h!> : | | $e$F : |
| Operator | | | |
| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Secant | $E$ = <!e!, E> : | | $E$F : |
| Operator | | | |
| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Chord | $D$ = <!e!, D> : | | $D$F : |
| Operator | | | |
| | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
| | | | |
| Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : |
| Functor | | | |
| | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> |
| | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| | | | |
| | | B^n x D^n -> D | [B^n x D^n] -> |
| | | | [B^k x D^k] |
| | | | |
o--------------o----------------------o--------------------o----------------------o
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o