I have a player who multiclassed as a wizard and an investigator (a rogue/alchemist hybrid; for the purposes of this question, the investigator is functionally equivalent to an alchemist).
Some of the alchemist formulae are also wizard spells. The player wants to know if they can also learn the alchemist spells as wizard spells (so they can then scribe scrolls for these spells). Here's the RAW:
An alchemist can also add formulae to his book just like a wizard adds spells to his spellbook, using the same costs and time requirements. An alchemist can study a wizard's spellbook to learn any formula that is equivalent to a spell the spellbook contains. A wizard, however, cannot learn spells from a formula book. An alchemist does not need to decipher arcane writings before copying them.
RAW, it seems that a wizard/alchemist couldn't learn alchemist spells as wizard spells, even if the spells are on the wizard spell list. But a multiclass wizard/alchemist is an odd combination, and so I want to figure out what the RAI should be. And that hinges, I think, on the question of why wizards can't learn alchemist spells.
Is it that a wizard typically lacks training to read alchemical formulae? If so, I see a good argument for a multi-class character being able to derive a spell from a formula.
Is it that alchemical formulae are missing vital information? If so, then that's a good argument for why they can't.
This particular multi-class situation is fairly rare, so I haven't been able to find much discussion of this question anywhere else.
(From a balance and fairness perspective, multi-classing alchemist and wizard seems like a sub-optimal choice, and so in principle I'm not opposed to house-ruling that it does work; it doesn't seem like power-gaming to me.)