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Taken as a transformation from the universe <math>U^\circ = [u, v]</math> to the universe <math>X^\circ = [x, y],</math> this is a particular type of formal object, and it can be studied at that level of abstraction until the chickens come home to roost, as they say, but when the time comes to count those chickens, if you will, the terms of artifice that we use to talk about abstract objects, almost as if we actually knew what we were talking about, need to be fully fledged or fleshed out with extra "bits of interpretive data" (BOIDs).

Taken as a transformation from the universe U% = [u, v]

to the universe X% = [x, y], this is a particular type

of formal object, and it can be studied at that level

of abstraction until the chickens come home to roost,

as they say, but when the time comes to count those

chickens, if you will, the terms of artifice that

we use to talk about abstract objects, almost as

if we actually knew what we were talking about,

need to be fully fledged or fleshed out with

extra "bits of interpretive data" (BOID's).

And so, to decompress the story, the TOTE

And so, to decompress the story, the TOTE that we use to convey the FOMA has to be interpreted before it can be applied to anything that actually puts supper on the table, so to speak.

that we use to convey the FOMA has to be

interpreted before it can be applied to

anything that actually puts supper on

the table, so to speak.

What are some of the ways that an abstract logical transformation

What are some of the ways that an abstract logical transformation like <math>F\!</math> gets interpreted in the setting of a concrete application?

like F gets interpreted in the setting of a concrete application?

Mathematical parlance comes part way to the rescue here and

Mathematical parlance comes part way to the rescue here and tosses us the line that a transformation of syntactic signs can be interpreted in either one of two ways, as an ''alias'' or as an ''alibi''.

tosses us the line that a transformation of syntactic signs

can be interpreted in either one of two ways, as an "alias"

or as an "alibi".

When we consider a transformation in the alias interpretation,

When we consider a transformation in the alias interpretation, we are merely changing the terms that we use to describe what may very well be, to some approximation, the very same things.

we are merely changing the terms that we use to describe what

may very well be, to some approximation, the very same things.

For example, in some applications the discursive universes

For example, in some applications the discursive universes <math>U^\circ = [u, v]</math> and <math>X^\circ = [x, y]</math> are best understood as diverse frames, instruments, reticules, scopes, or templates, that we adopt for the sake of viewing from variant perspectives what we conceive to be roughly the same underlying objects.

U% = [u, v] and X% = [x, y] are best understood as diverse

frames, instruments, reticules, scopes, or templates, that

we adopt for the sake of viewing from variant perspectives

what we conceive to be roughly the same underlying objects.

When we consider a transformation in the alibi interpretation,

When we consider a transformation in the alibi interpretation,

we are thinking of the objective things as objectively moving

we are thinking of the objective things as objectively moving