A more descriptive title for this essay would be “multiword idioms, typically not recognized by lexicographers as deserving own-place headword entry paralleling that of any other multimorphemic lexeme”. But cryptonym sounds snazzier; and the meaning you’ll find online (“secret word”) is not actually used by those who deal daily with codewords and cover-terms, so we here use it as we see fit.
Our lesson for the day concerns: up to.
As anyone learning a foreign language is well aware, and as speakers considering their own native language are not aware at all, the hardest words in any language -- the most difficult to characterize definitionally, and the hardest for a nonnative speaker to use with real Sprachgefühl -- are the shortest ones. (This practically follows from Zipf’s Law.) And when you have two monosyllables cheek-by-jowl : Trouble!
One of the things that used to irk me when I worked at Merriam-Webster dictionaries, was their characteristic disinclination to deny own-place entry as a lexicalized, bold-face headword, to multi-word idioms, provided these were written with spaces between the words. If written solid or with hyphens, they were welcomed as own-place entries: nevertheless, devil-may-care. This meant that some phrases whose senses were not predictable from their parts, and which permitted only very limited variation in their parts (cf. nonetheless, the-devil-may-care) would be defined only as run-ons to one of their constituents (kick the bucket at kick, and not at bucket) -- or, far worse, received no boldface status at all, but had to be found at some postulated and by now rare or obsolete sense, not called out typographically, and indeed disguised by the practice of replacing the headword in angled-bracketed exemplifications with a tilde:
to function with vitality and energy <still alive and ~ing at 75 years>
-- Webster’s Third New International Dictionary (Unabridged)
From this, the reader would have no idea that alive and kicking is about a million times more frequent than breathing and kicking or raising begonias and kicking or what have you.
Now, in this case, the M-W Collegiate actually does give own-place entry to our quarry of the day: You can find up to as a boldface headword, at its own alphabetical place: right after uptime, rather than as a run-on s.v. up. So far so good. But the problem is, they basically give it only one meaning (split unimportantly into two subsenses): that of reaching as far as a certain limit, that limit being the complement of up to. There are, however, several other senses, which that dictionary does not there record.
Are you úp to it?
Meaning: Are you capable of doing it. (Contrast: “Are you up for it?”, meaning: are you in the mood to do it; are you game.
That’s up to yóu.
Meaning: The decision is yours to make. -- Oddly, British English apparently uses down to in the same or similar sense. (“It’s down to me, the change has come, she’s under my thumb.”)
Note, b.t.w., the accent marks, which are inherent in these idioms, and which scarcely any dictionary thinks of noting. That the patterns are here different suggests that we are actually dealing with subtly distinct entities, cryptic even within the class of cryptonyms (for that is but a proper class, and not a well-defined set); but that is a refinement with which we need not grapple here.
What is he up to?
The usual meaning of this would be yet another meaning: What (sketchy, perplexing) thing is he doing? What is he about?
There is another possibility here, the very same that the Collegiate uniquely lexicalizes, and which requires a special context to evoke. Namely, suppose that our person is engaged on a stepwise-graduated task (like our diligent meso-hero in the fable Tales of Frontier Times). Here the phrase would mean: How far has he gotten? (Answer: four thousand and seventy-two.)
Finally we arrive at a further, delicious use, which is likely unfamiliar to you unless you have studied mathematics, and which motivated this whole post in the first place. In mathematical writing, this use is extremely common -- but probably unnoticed as being of the technical, special terminology of that subject, whether by adepts or by laymen. An example:
Semisimple Lie algebras can be factored in a unique way (up to rearrangement) as a direct sum of simple Lie algebras.
-- Timothy Gowers, ed., The Princeton Companion to Mathematics (2008), p. 232
(Here the “up to” clause modifies unique.) Which is to say: Their factors are unique, but the order of these is not (just as 3 × 5 and 5 × 3 equally equal 15.)
Commonest of all: unique/uniquely “up to isomorphism”.
Suddenly the blood froze in my veins.
Here I thought I’d noticed something clever; but it is increasingly difficult to say anything original these days, as Wikipedia’s omniscence takes in ever-larger swaths of the noösphere. And indeed -- unlike the offline lexicographers, Wiki has (like Kilroy) already been there.
That article notes that, ‘in informal contexts’, modulo X is, like up to X, often used in this sense of ‘considering as equivalent all entitites differing only in their X-value’. The idioms are not quite equivalent. First, modulo -- and especially its abbreviation mod when followed by an integer -- originated strictly in one formal context, namely number theory. Second, modulo in non-number-theoretic uses has leaked out into the general educated population; as,
Chairman: “And so, the ‘ayes’ have it unanimously.”
A handful of agitated spectators in the balcony: “No! No! Shame!”
Chairman (unperturbed): “… modulo certain yelps from the peanut gallery.”
Here it means no more than “apart from”, “setting aside”. -- By contrast, up to is never used in this extended way.
As an additional excellence, there is a corresponding article in French:
But, in line with what we said earlier, that article is contentually different, since we are here dealing, not with an overt terminus technicus like homology/homologie, which are used the same way withing mathematics, but a cryptonym cobbled-together out of tiny unassuming semantically-multivalent function-words. Thus, the nearest French equivalent of “up to X”, namely “à X près”, is synonymous in some uses and not-quite-so in others.
Re Karl Popper on philosophical theories:
He then, bar a hairsbreadth, equates these with scientific theories.
-- Margaret Masterman, in I. Lakatos & A. Musgrave, eds., Criticism and the Growth of Knowledge (1970), p. 72
Cf. “except on a set of measure zero”.