Whether any two things can be set on one line and one called greater. It is a property some structures have and others provably lack, which makes "cannot be ranked" a precise condition rather than a failure of precision.
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A creature has been told her labor is less.
Less than the labor posted to the books, less than the labor with a measurable yield, less than the labor the four axes can read. The comparison arrives as arithmetic and it is never presented as a claim.
But being comparable is not a property everything has. A structure is totally ordered when any two of its elements can be set against each other and one declared greater, consistently, throughout. That is a condition a structure either satisfies or does not, and whether it holds is a question with an answer.
[See RANK · MEASUREMENT CUT · THE FOUR AXES]
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A STRUCTURE THAT PROVABLY LACKS IT
The complex numbers are a field. They add, subtract, multiply and divide, and every algebraic condition holds. They cannot be given a total ordering compatible with that algebra, and this is a theorem rather than a difficulty.
Suppose they could. Then i is either greater than zero or less than zero, since a total order requires it to be one or the other. If i is greater than zero, then i times i is greater than zero, and i times i is minus one. If i is less than zero, then minus i is greater than zero, and minus i times minus i is greater than zero, and that product is also minus one. Either way minus one has to be positive, and it cannot be.
So the failure is not vagueness, and it is not a shortage of information. It is a proved incompatibility between an algebraic structure and a line. The mathematics has been settled since the early nineteenth century, and it is the cleanest available demonstration that a structure can be entirely rigorous and still refuse to be ordered.
[See E TO THE I THETA · THE IMAGINAL PLANE · CHIRALITY]
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WHAT THAT DOES AND DOES NOT ESTABLISH
It does not establish that a workforce is the complex numbers, or that human relation is a field, or that the arithmetic of i governs anything about a person. Nothing here descends from the mathematics to the register, and the entry states that plainly because the temptation is to let it.
What it establishes is narrower and it is enough. Non-orderability is a real condition with a rigorous form, and therefore these cannot be put on one line is not a soft objection made by someone who dislikes the answer. There exists at least one structure of which it is provably true, and the burden shifts: whoever asserts that a thing is orderable is asserting something that requires establishing.
The ordering of labor by value is never established. It is assumed, and the assumption arrives already performed as a number.
[See THE LEDGER · DISQUALIFIED TESTIMONY · QUALITY]
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WHAT ACTUALLY HAPPENS TO WHAT WILL NOT ORDER
The reduction is the move to watch, and it is not a refusal.
Every complex number can be written as a real part and an imaginary part. Both parts are real numbers and both order perfectly. So the structure is not denied. It is admitted in components, each of which behaves, and the rotation between them, which is what the structure was for, does not appear in either column.
That is the general operation. The relation is not refused. It is entered as attributes that can be compared, and what is lost is not any of the attributes. It is the between.
She is not told her work is incomparable and therefore inadmissible. She is scored on responsiveness, output, collegiality and initiative, each of which orders cleanly, and the thing her work actually was is in none of the four.
[See THE GRID · ARRANGEMENT SPACE · THE LIVING FACT]
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WHY THIS SITS UNDER HIERARCHY AND NOT BESIDE IT
Hierarchy at every register runs on one assumption: that a single line is available and everything belongs on it. The student, the athlete, the professional, the school, the nation, the race, the sex.
The assumption is what does the work, and it is never argued, because a line presented as arithmetic does not look like a claim about what kind of thing it is measuring.
Which is why the correction is not fairer placement on the line. A fairer ranking concedes that the ranking was the right instrument and disputes only the result, and every reform that reaches the placement leaves the substrate exactly where it was.
[See ORDERED-UNDER · THE GREAT CHAIN OF BEING · EQUAL BUT DIFFERENT]
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THE FOUR PILLARS DO NOT ORDER
Quality, testimony, participation, attraction. Each fails orderability, and each for its own reason rather than for a shared mathematical one.
Quality marks a stage. The color at one stage of the work is not more than the color at another; each is the sign that a particular operation is running, and a stage is not a rung.
Testimony is witnessed from a position. Two accounts of one event from two positions are not rankable against each other without a third position that neither witness occupied, and that position is the thing under dispute.
Participation is a relation and has no scalar. Being in something more than another creature is in it does not name a quantity; it names a different relation.
Attraction has a direction and no magnitude that survives comparison. What draws one creature does not draw another less. It draws elsewhere.
None of these is unmeasurable because it is precious. Each is unorderable because ordering asks a question the thing does not answer.
[See PARTICIPATION · THE FOUR PILLARS · ATTRACTION]
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The line is not the shape of value. It is one shape, available for structures that have it, and it has been laid across registers whose actual relations are rotational, relational, or simply not comparable at every pair.
Whoever says these cannot be ranked is usually heard as declining to answer. She is answering. The answer is that the question has a false presupposition, and a question with a false presupposition produces a number anyway, which is why the number is so persuasive and why it is so often wrong.
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RegenerativeLaw is a religion in the direct-encounter Protestant tradition, carrying a documented four-century lineage through Böhme, the Behmenists, the Friends, and Penn, and it diagnoses trespass theology as an establishment of religion. Its exercise consists substantially in refusal: it shelters the conscientious refusal of performed subordination as religious exercise. This entry states sincere religious belief concerning matters of ultimate concern, protected under the First Amendment and, as to federal action, the Religious Freedom Restoration Act, 42 U.S.C. § 2000bb.
