Would it be more difficult for a dnd style mage to learn to use armor or a sword? Could they learn both at the same time realistically?
Yes, or at least probably.
Okay, so, there’s two different reads for, “a D&D style mage.” The first is, straight up, the tabletop rules. The second would be a setting where magic works off the same basic rules of magic.
D&D uses a class based character progression system. This functions as a kind of template for how a character will advance and grow as they adventure. Depending on the class, it usually indicates a character’s professional background. In the case of wizards, you’re talking about a character who’s a professional academic. Their study of magic means they didn’t undergo extensive martial combat training.
That said, there are a number of magic users in D&D that do learn to use weapons and armor while also studying magic. The two most common are the Bard and Warlock. (There are other classes, such as third edition’s Warmage, or fourth edition’s Swordmage. When you get into more obscure classes, you start to see some real inconsistency on what appears from one edition to another.) So, depending on the edition, it’s not simply possible, there are arcane magic users who train in martial combat while they’re learning spellcraft.
It’s also possible a mage will take weapon or armor proficiency feats as they level, so in that case they would be learning to use weapons and armor as they were also learning to be a better mage. (Again, the rules for this vary based on the edition, but this discussion is already crunchy.)
In AD&D there was an entire system where a character would advance in two classes simultaneously. The rules were complicated and highly conditional, but it was possible for a character to advance as both a Wizard and Fighter at the same time. (Though, Fighter would level up faster than Wizard because of how XP per level worked in AD&D.)
So, if we go, “by the rules,” it is possible for a mage to split their training between learning how to use magic and learning martial combat. The most prominent spellcasters in the various campaign settings don’t do that, but there are disciplines who do.
There is a (fairly) consistent tradeoff across the board, if a magic user studies martial combat, they’re spending less time studying spellcraft, and that means they won’t be as effective as they could have been. (Different editions have varied approaches to this. It’s also entirely possible you’ll find classes, especially homebrew ones, that don’t take a meaningful penalty to their spellcasting.)
The tradeoff is a fairly big deal, because wizards and sorcerers will eventually advance into godlike power. Beyond that, magic tends to be a tool that can deal with most problems, and it leads to real questions like, “is it worth delaying my study of evocation so I can learn to use a sword, when I could just get some meat shield to stand in front of me while I cast fireball?”
So, in D&D, it’s entirely possible for a mage to spend time learning martial combat, there’s a strong incentive for them to specialize. Recruiting their own retinue, (in some cases, supplementing them by raising undead or summoning other creatures for additional protection.) A wizard can learn how to fight in combat, but they’ll end up behind the curve as both mages and martial combatants.
Obviously, some mages do pursue other paths. Some come to magic as a second career (if the rules of their setting allow it.) Some characters learn magic as part of a broader education (bards are the ur-example here.)
Sorcerers are distinct from wizards (in D&D). Wizards learn arcane magic through education, but sorcerers have instinctive access to magic in their blood (literally.) A sorcerer’s magical is hereditary, and may originate from a dragon, fey, or demon somewhere in their family tree, or it may simply be many arcane casting ancestors. While they have different casting rules from wizards, they follow the same general pattern, focusing on spellcasting and evolving their powers through constant practice and use. They can (potentially) follow other paths; the bard is another expression of this background.
When you step back from the rules, and simply look at the settings, battle mages are quite possible. It’s also entirely plausible you’d have a setting where front line spellcasters are simply part of that world’s military doctrine. There’s probably actual examples of this in some of the D&D campaign settings that I’m forgetting.
I’ve been ignoring divine casters (and Warlocks), but it is worth remembering that D&D does, straight up, include many spellcasters who train to use weapons and armor while they’re also learning to use magic. Difference being, they’re training to use spells that come from their patron deity (or the abstract concept of nature) rather than through arcane understanding of magic.
Classes can be useful for basic character archetypes, and they do (generally) present the more common varieties of adventurers in the various campaign settings. However, there is a lot of flexibility for how characters can approach magic in D&D’s various settings. The restrictions have more to do with the educational options your characters had access to, and how they gain access to magic.
So, could they learn martial combat at the same time they were learning magic? Yes. It’s absolutely possible. However, there is a tradeoff. If you spent 8 hours a day studying magic, you’d become a superior mage to someone who split their time between studying magic and training for warfare. Similarly, they’d also be less effective as a combatant than a dedicated martial fighter.
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