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Do relatively simple alphabets (Rotokas, Hawaiian) limit the complexity of written ideas? Example: could Rotokas be used to write a technical manual for the space shuttle?

Most certainly. The number of different symbols in a writing system has nothing to do with what can be expressed in it. When it comes down to it, this answer is being represented inside your computer with only two "symbols": a transistor switched on, and a transistor switched off. Yet it can express whatever ideas you like, whether that's a philosophical treatise or the plans for the Space Shuttle.

As Dannii said, the only thing that really affects this is the amount of technical vocabulary in the language. Hawai'ian, for example, has no native word for "transistor". But there's an easy solution for this: it borrows the word from English.

Thousands of languages use the Latin alphabet – for example, Rotokas, Hawaiian, English, Shona. This also includes French, Spanish, German and Norwegian, since the Latin alphabet has been expanded to include letters like æ, á, ü and v versus u and w which are not part of the writing system used by the Romans. What distinguishes Rotokas and Hawaiian from Shona and English is that the former languages have relatively few phonemes in their sound inventory, and the latter have a lot more sounds (they all use the Latin alphabet). In fact, Shona has more sounds that English (they have even more digraphs that English does), but don't use l,x,q. English doesn't use the letters ü, æ, ñ and so on, and Rotokas doesn't use the letters b,c, d, y.... This is not a fundamental limit on what you can do with the language, though.

Switching from space shuttle manuals to phonology textbooks, the main problem that a translator would face in writing a phonology textbook in Rotokas is that you have to come up with terminology for grammar, underlying form, phoneme, rule, epenthesis, metathesis, association line... An analogous situation is that Norwegian doesn't use the letter x, and yet in technical discourse, they may need to write about xylitol or xenon, so they actually can use the whole Latin alphabet. They also don't use q, except when they talk about Lake Qaraoun. In the would-be Rotokas phonology book, metathesis could be spelled metatesis, using an otherwise unused letter of the Rotokas alphabet (which actually is used for Matthew, etc: see here, likewise attesting use of j, h, y and other letters not supposedly available, just on the first page).

A technical manual for a space shuttle exists in English and Russian, and I'm pretty sure that it doesn't exist in Rotokas or Norwegian. But there is no linguistic reason why such a thing could not be created, and it is certainly not because the languages in question don't have enough sounds. In fact, the Khoisan languages such as ǃXóõ are written with an augmented Latin alphabet and have huge inventories of sounds, but also no space shuttle manual. The only impediments that relate to language are the lack of well-established technical terms for things like interocitor or asymptote.

Simple writing systems for languages with simple and small phoneme inventories are a perfect match - such writing systems could convey anything which the language can say, and are very suitable for highly complex technical manuals (though the language may need to borrow many technical words.)

If any writing systems are less ideal for technical manuals I would guess that it would be the hieroglyphic or logographical ones, as each logogram must be learnt essentially by rote, and though there are some strategies to try to identify the meaning of new symbols, it is very likely most readers will not know at least some of the symbols for the technical language, even if they know how to pronounce the word.