What are some ways Learned Magicians (and Mythic Alchemist, Cunning Folk, and Tattoo Magicians) gain experience besides books and teaching?
As far as I'm aware they can't use vis or practice their arts, and although this isn't really a problem for Learned Magicians and Alchemist, it's a bigger problem for the less literate traditions.
The only ways I can see this working out is if the other traditions can use vis and significatos.
I am very much under the impression that other traditions can study arts from vis just as well as hermetic magi (serfs parma).
Even if the rules dont explicitly state that non-hermetic traditions can study from vis there is are several mentions of how to translate vis that is aligned to a non-hermetic Art into a Hermetic Art. e.g how Olibanum vis from Soqotran system (Rival Magic) translates into Rego vis in the hermetic system. Presumably this means that under the Soqotran system Olibanum vis can be used for all the same purposes that Rego vis could under the hermetic system, barring those that are not shared among the respective systems.
Like you I cannot see how other traditions could ever truly work without the ability to study arts from vis as there really is no other way for a tradition to get to levels where book writing can sustain advancement.
Likewise I think that the need for comparatively large amounts of vis for study is a large factor in keeping non-hermetic traditions behind their hermetic counterparts. Because the Order of Hermes is both old and has a large community of letters hermetic magi need much less vis to advance allowing them to become powerful faster. Whereas smaller, younger or plain less well connected groups have a much lower ability to procure texts for advancement, meaning that individual members must study from vis much more often.
The Learned Magicians specifically can't study from vis though, unlike other traditions were it is not explicitly ruled out:
"Learned magicians may not study from vis." - Hedge Magic: Revised Edition pg. 85
This, I would guess, is probably due to Learned Magician magic (for the Mathematici) being essentially based off of a small body of ancient egyptian magic, and Learned Magicians are not terribly interested in advancing the theory of their magic (they actually don't have an underlying theory) instead focusing on inventing new charms.
For traditions which are represented with Learned Magician magic but have an unrelated origin such as the Cunning Folk I would argue that they should be able to study from vis, as there is no other explanation as to how the magic has developed as Euphemism has said. I say take the 'no learning from vis' thing and assume it is a flaw in the magic of the Mathematici specifically, not all traditions represented by Learned Magician magic.
They can gain experience from adventure and exposure.
Otherwise, as @Argentius pointed out, the Mathematici are an academic and non-empirical tradition, so they can't really make new knowledge through experimentation. Accordingly, I would allow Alchemists to study from vis, and Cunning Folk to practice.
I suspect that the Mathematici's primary method of advancing the art, to the extent that they do, is currently to write and trade tractati.
Learned magicians have a relevant advantage over Hermetic magi. They can learn by Training: see HMRE p.80 Training and compare with ArM5 p.164 Training.
So a master can train a pupil while plying his trade: creating charms and amulets. There is no need for an Order in the background, who watches that an apprentice gets her due season of teaching per year.
I imagine, that Learned Magicians pass knowledge just as Guild craftsmen. The more academic ones continue by studying books, while Cunning-folk might learn by Practice.
Note, that ArM5 p.164 excludes Practice for Hermetic Arts, but not for Supernatural Abilities. Hedge Arts are a gray area not covered in ArM5 and not addressed wrt Practice in HMRE: literal reading would allow it, but it should be discussed by the troupe.
Mmm, I don't think that meaning of "training" was intended. I think it was meant in the sense used by the Order of Hermes when training an apprentice. And I don't personally like the idea of training someone in an Art or Accelerated Ability - "Master’s score in Ability being taught + 3" will generate massive amounts of XP if the master is skilled, with no need for Teaching skill and no time lost. It doesn't pass the smell test to me; YSMV.
It's more likely that a mathematicus makes his living at least partly by teaching students for fees.
Training Learned Magician (LM) Techniques is not a problem here, because these advance as standard skills.
A master could indeed convey LM Forms easier than a magus can teach Hermetic Arts. As there is no study from vis possible, the means for increasing Art scores in a community of LMs is limited still.
If you read HMRE p.97f Epistolary Community, you conclude that few Gifted LM masters will excel in training. They tend to rely for it on unGifted LMs, who study their tractatus and then train the next generation of LMs. As most unGifted LMs have a limited number of Forms to train, this further limits advancement of Arts in a community of LMs.
The often illiterate Cunning-folk (HMRE p.101) "have no real traditions" because of these limitations.
Also, IIRC, characters can still acquire literacy in-play by learning Artes Liberales even if they lack Virtues necessary to take Academic Abilities during the Character Generation stage.