Polanyi's ineffables: inherent, or just awaiting better means of articulation?

I've been slowly working my way through Michael Polanyi's 1958 book "Personal Knowledge: Towards a Post-Critical Philosophy: A chemist and philosopher attempts to bridge the gap between fact and value, science and humanity". I find the book fascinating, and interestingly apropos to many current issues surrounding the interaction between science and policy, debates about certainty and instances where differing modern "tribes" seem to have vastly different views of reality.

Epistemology (studying how we come to know things) is closely related to the issue of semantics (meaning) which I've written about before (and see that post for further links on related topics here). In Polanyi's chapter 2, section 5, "The Nature of Assertions", his discussion nicely emphasizes essentially what I wrote on the centrality of "trust, provenance, and context" to meaning:

A sincere allegation is an act that takes place in speaking or in writing down certain symbols. Its agent is the speaking or writing person. Like all intelligent actions, such assertions have a passionate quality attached to them. They express conviction to those to whom they are addressed. [...] no sincere assertion of fact is essentially unaccompanied by feelings of intellectual satisfaction or of a persuasive desire and a sense of personal responsibility. [...] It is clear that I can make use of the sign |- to put on paper an allegation of my own; but it has not been explained how this sign is to function between different persons and between successive periods in the same person's life. [...] the symbol |- . p must be supplemented, so that it may tell us whose allegation it represents and at what time the person in question had alleged p.

Once again, this is something I expect to return to... but what I wanted to comment on today was a bit later in the book, on the issue of tacit knowledge and the "ineffable".

Polanyi starts chapter 5, "Articulation" with a discussion of the differences between human and animal understanding of the world around us. Animals can have quite sophisticated knowledge, they can learn tricks, they can understand signs, and they can gain "latent knowledge", internalizing a map for example, so they can know where they are in a familiar maze even if the normal route is blocked. Human understanding expands upon the abilities of animals largely through our facility with language - but that makes a huge difference. We can create symbols to represent knowledge, and then we can manipulate and rearrange those symbols, and read off from that manipulation new understandings. This process of making thought "manageable" Polanyi summarizes as three stages:

1. Primary denotation
2. Its reorganization
3. The reading of the result.

and while this may seem simple, the effect is profound. This is what Polanyi means by the "articulation" of knowledge. In the "exact" sciences, denotation of experience involves measurement - turning observations into numerical symbols. Reorganization of those symbols brings in the power of mathematics, we analyze and find descriptive formulas to summarize observations, allowing us predictive power and control over the world around us, far beyond what a single person without such linguistic abilities could hope to accomplish. Other forms of symbolic representation - words, timelines, maps - may not have quite the symbolic power of numbers, but nevertheless can communicate greater understanding in a variety of ways.

Polanyi then goes on to several examples where this process of symbolic representation breaks down - knowledge that is "ineffable", which cannot be expressed and communicated through the symbols, but only through actual experience. In a way, every symbolic representation must have a component of ineffability, that piece that connects the symbol to reality is not inherent in the symbol itself, but is part of what allows our human understanding to make sense of it.

But by the ineffable Polanyi means something more:

strictly speaking nothing that we know can be said precisely; and so what I call "ineffable" may simply mean something that I know and can describe even less precisely than usual, or even only very vaguely.

As an example he brings up "the subject matter of topographic anatomy" - the "particulars" are explicitly specifiable: lists of bones and other body parts, diagrams of where things are. But

The major difficulty in the understanding, and hence in the teaching of anatomy, arises in respect to the intricate three-dimensional network of organs closely packed inside the body, of which no diagram can give an adequate representation. [...] The kind of topographic knowledge which an experienced surgeon possesses of the regions on which he operates is therefore ineffable knowledge.

I fully agree with Polanyi that much of our knowledge is "ineffable" in this sense of known but possible to describe only with poor precision. But - here's the real question to me on this - is ineffability an essential quality of this particular sort of knowledge, or is it merely an artefact of our limited powers of expression? It seems to me that in this case of the knowledge of anatomy that 3-dimensional diagramming techniques, not possible in the era of ink on paper, could quite adequately describe the reality of the network of organs in the body, and communicate that understanding just as well as a 2-dimensional map communicates the relationships between cities. In both cases receiving the understanding requires at least the "reading the result" phase, perhaps some manipulation of the representation as well (rotating it, zooming in and out). But the degree to which this particular kind of knowledge is really "ineffable" seems more a consequence of our limited means of representation of knowledge than anything fundamental. Polanyi's other examples on this issue seem similarly an issue more of capability of clear observation and expression, than fundamentally points which cannot ever be articulated.

Language develops, the power of our symbols grows over time. Mathematics in particular has shown that, with the rise of new notations that greatly increase the power with which we can manipulate these representational symbols and read off new understandings. Is there anything that is really inherently "ineffable", in this sense, or is it just a matter of the need for sufficiently advanced symbolic representation technology to allow us to express even the most difficult concepts?