[Metadata: crossposted from https://tsvibt.blogspot.com/2023/03/explicitness.html. First completed March 3, 2023.]

Explicitness is out-foldedness. An element of a mind is explicit when it is available to relate to other elements when suitable.

Thanks to Sam Eisenstat for related conversations.

Note: The ideas of explicitness and inexplicitness require more explication.

Explicitness and inexplicitness

Elements can be more or less explicit, less or more inexplicit.

• (This statement wants to be unpacked.)
• In general, inexplicitness is the lack of explicitness, and explicitness is when elements that have good reason to be related, are able to be related. That is, when structure is explicit, it can be brought into relation with other structure when suitable.
• Structure is explicit when it's out-folded: when it already makes itself available (visible, applicable, informative, copyable, tunable, directable, modifiable, predictable, combinable, interoperable), so that nothing is hidden or blocked.
• An explicit element is an element high in explicitness, i.e. it can be brought into relation with other elements when suitable.

Explicitizing

Elements can become more explicit.

• By default, structure is fully inexplicit for a mind. That is, it's fully ectosystemic for the mind: it's not available for elements of the mind to relate to.
• Structure can be brought into explicitness.
  • For example, these processes make structure more explicit: reflection, analysis, description, expression, joint-carving, separating, factoring, refactoring, modularization, indexing, interfacing, connecting, disentangling.
  • The early stages of explicitizing involve incomplete or deficient participation——like a blind man touching an elephant's tail, or entering the outer regions of a nexus of reference. E.g., the relationship that the Ancient Greek mathematicians had to Cartesian algebraic geometry.

A diagram:

Examples

An example of explicitizing also furnishes examples of inexplicitness (before the explicitizing) and explicitness (after the explicitizing), and likewise an example of inexplicitizing also furnishes examples of explicitness and inexplicitness.

Classes of examples of explicitizing

• Internal sharing of elements. See here for examples of inexplicitness.
• Making an analogy. By grasping a partial isomorphism, the partially isomorphic aspects of the analogands can be transferred back and forth between them.
• Putting a word to something. When the word comes up, the element is accessed, and vice versa. That helps different contexts in which the element is relevant communicate with each other through the unfolding of the element.
• Refactoring code. Separating concerns A and B renders code for dealing with just A useful for tasks that deal with A but not B, whereas the unseparated code might for example assume the existence of B (e.g. as an argument or global variable). The separation makes explicit the dependence and non-dependence of code on A or B. Or for example rewriting a function so that it accepts a standard rather than non-standard input format, so that elements expecting a function to accept standard input can easily relate to the function.
• Deduction. Drawing out the implications of an element makes the element available for comparison with other elements and makes the element available to recommend action.
• Writing things down.
  • Expressing and recording something in shared language makes it available to others.
  • Storing something in memory makes it available to your future self.
  • Abbreviating something makes synopsis and large-scale modification more feasible. For example, mathematical notation makes big thoughts viewable all together, and makes algebra more efficient.
  • Attaching abbreviations to an element makes that element easier to find.
  • Drawing a picture makes something more available to visual processing (extracting gestalts, broadcasting, holding in "external working memory", exposing geometry).
  • Speaking an element into the stream of consciousness makes the element more available for other elements to notice their own relevance to the broadcasted element.
  • Speaking an element divides it ("articulates", renders into articles) into familiar items (words). That makes the element more available to those other elements that already interoperate with those words.
  • Redescribing an element retranslates the element, as Feynman recommends. This makes applicable more transformations of expressions of elements——the grammatical structure of the retranslation affords different transformations.
  • See here for more.
• Building an index. Putting an element in an index makes it easier for another element to find and interface with that element.
• Entering numbers into a spreadsheet. By putting notes about a set of things in abbreviated form next to each other, a spreadsheet makes it convenient to compute certain consequences of given propositions. For example, it becomes easy to notice which of a set of things has the highest value on some dimension, or notice how much information is present or missing overall, or average given values. (In general, see "parathesizers".)
• Genomic parcellation. Pleiotropic genes affect multiple characters of an organism. Genetic loci might therefore be under antagonistic selection pressure: a mutation might have beneficial effects on one character and deleterious effects on another character. This situation also creates pressure to remove the pleiotropy, e.g. by differentially expressing paralogs or by alternative splicing, so that fewer loci affect multiple characters. When a pleiotropy is removed, the gene is made more explicit: it is more available to be modified as is suitable for different applications (different expressions in different contexts). See the work of Günter Wagner, e.g. "Complex adaptations and the evolution of evolvability" (pdf link). A figure from that paper:

Examples of inexplicitness and inexplicitizing

• Having a word on the tip of your tongue.
• Forgetting something, in such a way that you could be reminded of it with an incomplete prompt. You must have still possessed it, but not in a way so that you could recall it easily.
• In general, losing connections. Destroying indices, interfaces, memories, storage, lines of communication.
• When you learn to ride a bike, it might be hard to put into words what you learned or what changed that made you able to stay balanced.
• It's much easier for me to type my computer password than to say it out loud. When I came up with my computer password the opposite was true: the password was explicit. Then I learned to type it without thinking about it, and then I mostly forgot it explicitly. So it was inexplicitized.
• Suppose you've written some code to perform some task. Now you're given a new task, so you rewrite and add to your code to perform the new task. You find that your old code can almost just be repurposed as-is to perform the new task, but not quite: you don't see how to cleanly add some interface or wrapper or additional logic, taking the preexisting code as given. So you rewrite the preexisting code by hacking in some new functionality, so that it does what's needed for the new task. Now that code is less available to interface with other code; the preexisting simpler version isn't exposed to be independently called, and only the whole more complicated code can be called; the code is harder to understand and modify because it's more complicated and has more things entangled with each other. The old version has been inexplicitized: the ideas of the old version, viewed as still lying within the new version, are less explicit (more inexplicit) within the new version than they were in the old version. E.g., you add an argument to a method, even though the argument only slightly interacts with the rest of what that method does. Now the method has to be passed the additional argument, and it's harder to tell without more work that most of what the method does doesn't depend on that new argument.
• In many situations, people try to prevent information from being made explicit or being put into common knowledge.
• Diasystemic novelty not only tends to itself be inexplicit, but tends to render preexisting structure more inexplicit, at least temporarily. It might change how elements relate to each other, so that previous coadaptations no longer apply as much as before the diasystemic novelty.
  • For example, if someone takes on a strict discipline of not lying to themselves, they might break coordinations between different preexisting elements. E.g.: self-coordination across time to accomplish tasks that's based on self-fulfilling prophecies that include self-deception. The mental elements needed to perform different parts of a task at different times were previously (partially) explicit for each other: they were able to interface suitably (within encountered contexts). After the new self-discipline, the elements are no longer immediately able to interface to accomplish the task; they have been inexplicitized for each other.
• An ontological crisis, or ontological revolution in which the mind greatly changes the language in which it thinks, might render some preexisting elements "relatively inexplicit". That is, the preexisting elements can no longer relate to the mind as a whole in the way that they related to the prerevolutionary mind as a whole. If the revolutionary language excludes some of the preexisting language, this is inexplicitizing simpliciter, not just relative inexplicitizing: relations that depended on the excluded preexisting language are made less available.
• An edge case: systems bifurcating or continuing on non-convergent trajectories. E.g. speciation, or a language splitting into dialects and then separate languages. Another example: a mind continuing on a voyage of novelty, maybe as far as to other cognitive realms, so that the mind's elements are made inexplicit for an observer (e.g. a human).

The axiom of choice

Chapter 1, "The Prehistory of the Axiom of Choice", in Gregory H. Moore's book Zermelo's Axiom of Choice (Libgen):

[...] Cantor made an infinite sequence of arbitrary choices for which no rule was possible, and consequently the Denumerable Axiom was required for the first time. Nevertheless, Cantor did not recognize the impossibility of specifying such a rule, nor did he understand the watershed which he had crossed. After that date, analysts and algebraists increasingly used such arbitrary choices without remarking that an important but hidden assumption was involved. From this fourth stage emerged Zermelo's solution to the Well-Ordering Problem and his explicit formulation of the Axiom of Choice.

That chapter describes stages of inexplicit uses of arguments spiritually related to the axiom of choice. Zermelo's work made the essential uses explicit. That opened up the possibility of further investigation building on the idea of the axiom of choice. E.g. the axiom was analyzed in the reverse-mathematical spirit, revealing which theorems require the axiom; and reactions against the axiom contributed to intuitionism.

For another example of making explicit an idea that had been inexplicitly used, see Reinhardt's article "Remarks on reflection principles, large cardinals and elementary embeddings", in Axiomatic Set Theory volume 2 (Libgen).

Group theory from concrete transformations

Two of the main tributaries into group theory were permutation groups of roots of polynomials, and symmetry groups of geometric spaces. Galois related the solvability of polynomial equations to the structure of the group of permutations of the polynomial's roots that preserve algebraic equations. Klein's Erlangen program related structures studied in a geometric space to the transformations of that space that preserve those structures.

Both of these uses of the idea of a (transformation) group are contextual. The groups are thought of in terms of the particulars of the objects they transform——sets of roots of polynomials, or geometric spaces. The relationship between these theories aren't yet explicit. The development of group theory makes the common structure explicit, allowing general results to apply to both cases.

For another rich example of a generally applicable idea congealing from its appearances in concrete contexts, see "The concept of function up to the middle of the 19th century" by Adolph-Andrei Pavlovich Youschkevitch (pdf). Note that abstraction is a gain of content, not a loss of content (extraction is a pure loss of content).

Other distinctions

Simply not knowing something

If a mind doesn't know X, then X is maximally inexplicit for the mind.

Pretheoretic explicitness

Pretheoretically, it seems possible to know something explicitly but not implicitly, or to lack some implicit knowledge. Examples:

• I can make up a password for my computer, and then I know it explicitly. But, I still lack the implicit knowledge of how to type it quickly and automatically.
• Someone tells me: "To peel an orange conveniently, without having to restart it a lot, don't bend the part of the peel that's separated too much so that it breaks off; instead keep it tangent to the orange.". Now I know that idea explicitly, but I lack the implicit knowledge of how to peel oranges that way.
• Someone tells me an expression for something called the "Lefschetz number":

$${\Lambda }_f:=\sum _{k\ge 0}(-1{)}^{k}tr(f_{\ast }|H_k(X,Q))$$

I'm able to technically define the terms in the expression, but I "don't really know what the expression means". Even though the expression is perfectly explicit, for me the expression is very far from being readily combinable with other elements of my mind. I certainly don't immediately conclude or even understand propositions like the Lefschetz fixed-point theorem.

The concept of explicitness in this essay takes all of these to be undifferentiated examples of partial inexplicitness. There's no such thing as "non-explicit understanding", other than explicit (partial) understanding that hasn't been made fully explicit in every way. If I were to play around with the Lefschetz formula and "get some intuition for what it means", that's just another instance of explicitizing.

The pretheoretic notion of "implicit understanding" or "non-explicit understanding" might be reducible to implications of the form: if the mental element is explicit in the manner X, then the mental element is explicit in the manner Y. Then a "non-explicit" element is an element that sits within a mind in a manner that is implied by most other manners of sitting within a mind, and that does not imply many other manners of sitting within a mind.

These implications are vague about their assumptions, and intuitively being told an explicit formula is a violation of the implication. Since I didn't come up with the Lefschetz formula, I got an understanding of it "in the wrong order". This tends to happen when there are other minds, because an other mind can get the understanding in the right order and then transmit a formula that is fully explicit for that mind, though the formula won't be fully explicit for the receiver.

It's also possible to arrive at a simple, explicit formula de novo via algebraic calculation, and only then rationalize the formula in a way that connects more to preexisting mental elements. Is that "in the wrong order"? The situation might be that this "in the wrong order", and the pretheoretic notion of "non-explicit" elements, are really imprecise perceptual categories meant to pragmatically track the implications between explicitnesses, e.g. to track what tasks to expect a given element to be adequate or inadequate for. "Non-explicit" elements might be elements possessed (that is, elements that empower the mind) without being fully explicit, and "explicit but not implicit" elements might be elements grasped in some way but not possessed (such as an "explicit" formula, without the understanding needed to usefully apply it).

Coherence and internal sharing of elements

Internal sharing of elements is a description of coherence in terms of whether elements are interoperating with each other suitably. Coherence is related to explicitness as actualizing is related to possibilizing. Coherence is actual interoperation, actual capabilities, actual efficiencies, actual connections; explicitness is the feasibility of interoperation and connection. In short: to explicitize is to possibilize coherence.

For example, building an index doesn't by itself constitute performing a new (external) task. But having an index renders feasible some operations that were previously infeasible. Parathesizers put elements alongside each other, not necessarily synthesizing them, but making it possible for them to be synthesized.

Explication

Carnap's explication is a kind of explicitizing. Storing something in memory is explicitizing but not explication.

Implicitness

Strict implicitness in the logical sense is when a proposition is implied by a set of propositions, but isn't already included in that set (explicitly). Instances of strict implicitness are also instances of inexplicitness: the implied proposition, if made explicit, would enable further operations.

Example: "What I said implies...". This implicitness is inexplicitness that can be resolved (explicitized) by deduction.

Example: "Implicit in the concept of bachelor is unmarriedness.". Statements like "Bob is a bachelor." have inexplicitness that can again be resolved by deduction: "...and therefore Bob is unmarried.". Evenness is implicit in the concept of divisibility by four. The fact that $3\times 4=12$ is implicit in the ideas of 3, 4, and $\times$.

See also the notion of analyticity, and Critch's discussion of implicitness.

Correlations

Explicitness

Explicitnesses correlate with explicitnesses. That is, "you can't eat just one": explicitizing an element in one way (making it available for relating with one other element) tends to also explicitize it in other ways (making it available for relating with multiple other elements).

The "proportional explicitness" of an element (the proportion of elements in the mind that the element is available to relate to) might not approach 1 in the long-run. E.g., if there is always more parasystemic novelty, then there are always regions of the mind that are not well-integrated or well-integratable with a given element. Essential non-cartesianness might imply that there is always more parasystemic novelty.

Possession

A mind possesses an element to the extent that the mind is able to do those tasks that the element enables in some mind. (So possession is "coherence, localized to one element".)

• "Far as I see it, you people been given the shortest end of the stick ever been offered a human soul in this crap-heel 'verse. But you took that end, and you——well, you took it. And that's——Well, I guess that's somethin'." -Jayne Cobb
• An instance of possession is an instance of explicitness, so possession correlates with explicitness.
• Possession is a myopic version of (or, spectrum leading up to the extreme of) explicitness. To possess an element E is to be able to do some set of tasks that are "fairly easy for the mind to do" if the mind is given E, or in other words to pick the low-hanging fruit of capabilities given E. To have E fully explicitly is to be well-prepared to pick all higher-hanging fruit, e.g. capabilities that require also changing other elements that will relate to E in new ways, and e.g. capabilities that only become relevant in new contexts and for new tasks.
• Refactoring code so that it is functionally the same for current tasks doesn't change how much the ideas in the code are possessed by a system, but might greatly change how explicit the ideas are.
• Explicitness ensures possession in minds that are coherent in general: any suitable use of an element will be avaiable by explicitness, and will be availed by coherence, and thus the element is possessed in that way. In the limit of the growth of the mind, with all elements changing as suitable, possession approaches full explicitness. An obstacle to explicitness of E can become an obstacle to possession of E if a task is blocked only on E.
• A human brain, even without understanding neurons in a pretheoretically-explicit way, still possesses the structure of neurons, in that it uses the structure of neurons in the straightforward ways that neurons can be used to do useful tasks in human contexts.

Access

The pretheoretic idea of accessing an element E is maybe a projection of the idea of explicitness into the subspace that assumes that all elements are fixed, and all that changes is lines of communication.

Modelability

If an element E is explicitized in a mind M, the field of elements that E can relate to is expanded. Since explicitnesses correlate with each other, expanding the field of E-relatable elements in M tends to also expand the field of E-relatable elements in another mind M'. So there's more "surface area" for M' to understand E.

This may be false in the long-run if there are different cognitive realms. E.g. if the languages of thought of two minds M and M' permanently diverge further and further, then explicitizing an element in M might not explicitize the element in M'.

Gemini modelability

If an element E is explicitized in a mind M, the field of elements E can relate to is expanded. Expanding the field of elements relatable to E has many effects on how feasible it is for another mind to gemini model E in M. But, as a broad tendency, explicitizing E in M makes E in M more gemini modelable. The field of relatable elements approaches the total field of possibly relatable elements, which is canonical and therefore shared between many minds. The way that E will ongoingly show itself in M is circumscribed by the field of relatable elements of E in M. So if the field of relatable elements of E in M is more canonical, then the way that E will ongoingly show itself in M is at least less forced, by circumscription into disjoint regions, to be distinct from the way that E will ongoingly show itself in some other mind M'.

Coherence

If explicitness is possibilized coherence, then coherence (actualizing some suitable relations between elements) implies explicitness (that those relations are possible).

On the other hand, explicitness isn't only coherence. Take the example given above where computer code is rewritten so that a new task can be performed, but so that the code is less well-factored. It may be in this case that explicitness almost strictly decreases (some explicitness is lost, and maybe only a single interface is added), while coherence strictly increases (because a new task can be performed).

Generators

Generators of capabilities seem to leave a lot of inexplicitness compared to some other elements. I don't know why. If it's so:

• Maybe it's because they are more potent, so there's more left to be gained.
• Maybe it's because they are more algorithmically complex.
• Maybe it's because they don't really exist.
• Maybe it's because they are created by a different process (the original mind designer) from the process (the mind itself) that would bring them into the mind's own explicitness, so they aren't spoken in the same language as the mind itself. (For example, some generators for humans might only be understandable as having been a part of evolutionary history, and those generators aren't possessed at all by humans, unless discovered by reasoning.)

Dark matter

Dark matter is structure that is inexplicit, but that is known to exist because of its visible effects. Even if we can't see it, it must be there. Dark matter is not fully inexplicit, or it wouldn't be possessed, and therefore wouldn't have visible effects.

For example, we can listen to set theorists talking in their special language, and we might even follow some of their thoughts. But we don't know the intuitions and refinement processes that led them to conjecture and then codify concepts and propositions (unless, say, Penelope Maddy distills them and writes them down). We know that those intuitions are there, though, if we dimly perceive some order and rhyme to their thinking, and can verify that their proofs are correct and non-obvious.

Diasystemic novelty

As noted earlier, diasystemic novelty tends to inexplicitize. Diasystemic novelty also tends to be inexplicit. A novel element E is diasystemic when it is very relevant to the mind, but doesn't interface in preexisting ways with preexisting elements of the mind.

• Since E is very relevant to the mind, there's a large potential for E to relate to other elements of the mind, so there's a large amount of explicitness required for E to be fully explicit.

• Since E doesn't relate to preexisting elements in preexisting ways, it doesn't take advantage of preexisting explicitness, and it is less well-understood by the mind.

• Since E cross-cuts the preexisting elements, touching many of them, E is at first less available to be suitably modified; metaphorically, E has antagonistic pleiotropy.

Subject to honesty

If an element is very explicit, then it can be spoken honestly about: the ways that the element exerts itself in the mind can be reported with intent to expose, without distortion, everything relevant. It can also be lied about.

If an element is very inexplicit, then the ways it exerts itself can't be reported because they aren't available to be expressed. So the intent to expose without distortion becomes irrelevant because impotent. (This is an "instantaneous" notion of honesty. The full normative notion of honesty also includes the tendency toward explicitization rather than inexplicitization.)

Conceptual Doppelgängers

The existence of conceptual Doppelgängers requires inexplicitness. At least, the overlapping functions shared by two elements are less available to be predicted and modified in one fell swoop, because they aren't factored out from the two elements into non-repetitive elements.

On the other hand, if participation of elements in a Thing is well-indexed, then there effectively aren't conceptual Doppelgängers at least in the strongest sense: observing one element at least points the observer to the other elements that serve overlapping functions, so at least the presence of an analogy is made explicit.

More generally than duplicates, there are "crosshatch Doppelgängers": a set of elements that combine to play some roles that overlap with roles playable by combinations from another distinct set of elements. For example, a function can be rewritten using different primitives or a different factorization. Another example: the primordial ooze of category theory. Another example: rewriting a sentence to use different words, or translating between languages.

A wish

It would be nice to have a situation like this: the elements of the mind are explicit. This is how the mind understands this Thing. This element in the mind goes to the heart, the center of the Thing it's supposed to bring into the mind. It's explicit, unfolded, laid out, and so the word attached to this element is the word for this Thing in the language of this mind. When this mind's thinking relates to this Thing, it uses this word. What this mind knows about this Thing activates and is activated by this word, and is exactly what is indexed this word or brain chunk. Every element is easily understood as doing X and only X for some X. (This picture is improbable, e.g. because of crosshatch Doppelgängers.)