exact must tolerate inexact
nominal

Let us consider {exact} as a noun. What would we mean by so naming an exact?

One perspective is that {exact} means {maximally precise}. I don't think this works for any notion of precision which presumes a well ordering or the like. When we use phrases like {the exact reason}, {the exact case}, {the exact occurrence of a word}, there is no clear such ordering structure to the differences, the errors, the 'ways of being wrong', the 'ways of being inexact'.

Nevertheless, it would seem like this distinction of an exact from all inexacts is quite central. One perspective here is that of {exact} as {utterly unambiguous}. To the extent that we have determined an exact, we may answer any question of disambiguation. I have an intuition here that exactitude would aspire to allocate all of one's 'supply of information' towards this disambiguation. To say, the tradeoff would be minimizing the 'structural differences' between inexact answers. It is a sort of informational perspective, but exact perhaps does not acknowledge any locality, how 'close' an inexact answer is to the exact answer. To do so would be to 'waste resources' which could otherwise be used to further exact.

An ambiguity is something like a self distinction. If we imagine a named identity as a sort of spatially bounded region, an ambiguity in this named identity might be imagined as a cut of distinction that runs through it, dividing it.

At least aspirationally, upon such an staged distinction that would threaten an ambiguity, exact would seek to always select one of the divisions to 'retreat the named identity to'. This would be as opposed to allowing the name to remain ambiguous, which may be something of {tolerance}; allowing the question to go unanswered for a time, 'not answering the question yet'.

If we are to take seriously the notion that such an ambiguity-threatening distinction is naturally asymmetric (ideas of primitive distinction itself being asymmetric), then it may be the case that this 'choice of division' is 'already made' by this very asymmetry. I do not think we need to ontologically commit to locating this choice I think, but in this second view the 'choice of side' would have been done by something of the symmetry-breaking eye that interprets relations as identities or distinctions, which holds this preference of one division or the other. (This could also be seen as a participatory asymmetry 'which side the eye is on', when drawing a boundary in some space, the 'projective point at infinity' might be 'where the eye is' and thus the 'outside' is intrinsically distinguished from the 'inside').

With the above in mind, I will stage that an exact respects all sources of distinction.

This would perhaps present the environment of an exact as being one of some known distinctions. The interpreting eye's stance on distinctions and identities. It is unclear in the moment if this is what the {epistemic stage} ought to be, or if such a stage assumes a later state in which more structure has been built.

The intent of this definition is to really be very wide in what we mean by {all} sources. To say, any possible perspective or contextual angle of entry. This immediately runs into some linguistic oddities (and these are not 'merely linguistic', as language is quite deep into epistemic environment itself). I will skip some of the gradual entry for the moment and get to the punchline of the most odd implications: this would mean that we may not have distinct signs to the same exact referent. Not even distinct occurrences of some sign to the same referent. In this way, to say {an exact and that same exact} is contradictory. It does not respect the a distinction we might make by syntactic location.

One framing is that we are here rejecting logical contraction. This is quite related to the capability to copy information, to copy some pointer while keeping the pointed stable. This would seem to rather naturally connect to many a 'substructural logic', and I might particularly gesture to linear logic.

Some work is needed here, but I think it also follows that even collecting exacts under the name {exact}, the categorization itself, is contradictory in much the same way. To name is to identify all things that have been so named. Collecting exacts under a single name is identifying them insofar as they are judged {exact}, and this seems to paint the picture that the word {exact} is almost an immediate/bare contradiction. I do want to think more on understanding this from the angle of 'what distinctions are not being respected'.

There is something here about that the aspirational rejection of any structure locality actually refutes the very 'reference relation' between sign and referant, which might be another framing of the above. A relationship between index and content is a sort of instituted locality of the content by the index structure.

So there is some sense in which any particular claim of {an exact}, any actual naming of such, would directly evidence its own failure. There is a sense in which the existence of {an exact}, the naming of an existent/actual exact, necessarily entails something of stability. To say, identification over 'instances at different times'. Another angle might be that a judgment of existence from some perspective is the judgement of existential independence from the existence of the judge. The point of all of this is that this existences contradicts directly the ideal or aspirational notion of exact.

But this raises some questions about what exactly is contradicting what. It seems to be a particular character here that an instance of some category contradicts the associated aspiration of the category. To use the word {name}, the story this would seem to tell is that any name has some 'aspiration':
- perhaps a role it must play, a participatory signature in some language or conceptual context
- perhaps something of philosophical essence (strong {perhaps}, some commitments here)
- perhaps something of intent, teleologically constrained by the future
- perhaps something of definition of the name/noun, could be something of the epistemic activity that occurs when 'holding fixed, static' the particular syntatic sentence the name occurs in, the process of interpreting it
- perhaps a criteria to end, a condition of non-being, a characteristic of what it is not

This aspiration is not 'the same epistemic content' as a judgment of some actual or existence or object that 'falls under' the name. That is, that a thing is so named. It would seem to be the case that the common way is to understand such named things as things which exhibit the aspiration. That is, a sort of positive evidentiary relationship between them. For a thing to be named, it must satisfy the aspiration. Perhaps we could also say that the existence witnesses the aspiration, to take some inspiration from type theory.

However, it would seem that the above story of {exact} would invert this relationship. It is a situation in which the named thing would refute the aspiration. To say, that a thing is named is to say that it particularly fails the aspiration of the name. If we are to reflect the word {witness}, I might suggest that the existence occludes the aspiration. All this said, I do not yet have enough clarity on the 'common way' to really understand this inversion.

We might examine an example, {exact proof}.

There are other cases I have noticed such a pattern. Firstly, contradictions themselves. An inhabitant of an empty type evidences that it actually is not an empty type. Secondly, transcendental origins: to hold that some proposition must be accepted by mere appeal to what must be necessary for one to be capable of accepting or refuting it. This one is more a claim, but I do here claim that to name a proposition is to be capable of conceiving, even in the barest sense of conception, that it is false. To say, one can entertain its falsehood, one can epistemically stage the situation in which the proposition is false, which is precisely counter to the aspiration of falsehood being inconceivable. Perhaps one could say that the better a transcendental origin, the more immediately such a stage would collapse (via encountering a contradiction, for example). But note here that this is giving some sort of notion of locality to relieve the tension, and thus, I claim, distancing {transcendental origin} from {exact}.

A failure of exactitude may be a {tolerance}. We might say a {tolerance of exact} to understand a kind/type/sort of exact. Perhaps some distinctions which we hold tolerated, provisionally ignored in some way. In this way we might understand {nominality} to be a particular such tolerance of exact. So maybe nominal exact is exact up to capacity to name, and this text is about this particular tolerance.