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Rock Gnomes have the following racial trait (PHB, p. 37):

Artificer’s Lore

Whenever you make an Intelligence (History) check related to magic items, alchemical objects, or technological devices, you can add twice your proficiency bonus, instead of any proficiency bonus you normally apply.

If I have a Rock Gnome who is proficient in History, then I normally apply my proficiency bonus to make History checks, but if the check is about, say, a magic item, I can add twice my proficiency bonus instead of just applying it once as normal. This much makes sense.

But what if my Rock Gnome isn't proficient in History? Since I don't apply any proficiency bonus normally (because I'm not proficient), I have nothing to apply instead of, like how you technically can't have more of something if you haven't had any yet. So does that mean I don't apply my proficiency bonus twice, since ordinarily I don't apply anything?

Initially I assumed that it probably means you get to treat the situation as though you are proficient in History in those circumstances (i.e. when the check is related to magic items, alchemical objects, or technological devices), but if we contrast it with a Dwarf's Stonecunning trait (PHB, p. 20):

Stonecunning

Whenever you make an Intelligence (History) check related to the origin of stonework, you are considered proficient in the History skill and add double your proficiency bonus to the check, instead of your normal proficiency bonus.

For Stonecunning, it explicitly states that you are considered proficient in History for such checks, taking into account dwarven characters who are not proficient in History normally.

So, RAW, does the Rock Gnome trait Artificer's Lore apply when you aren't proficient in History?

The wordings are essentially equivalent

Both wordings result in you adding twice your proficiency bonus to any roll for which the ability applies. The wording you've highlighted in bold for Artificer's Lore is just ensuring that the double proficiency bonus doesn't stack with any other proficiency (i.e. you don't get triple your proficiency bonus if you were already proficient). I'm not sure why they would use a different wording for these abilities with apparently identical mechanics.

The alternative interpretation would be that Artificer's Lore has no effect at all without proficiency in history. This would be unprecedented: racial traits tend to stand alone independent of class or background features. No other racial trait from any race that I can think of is entirely negated by failing to acquire the appropriate proficiency through one's class or background.

One possible very subtle difference is that unlike Stonecunning, Artificer's Lore doesn't explicitly give you proficiency in the relevant check despite changing the modifier. This could conceivably matter if actual proficiency is a hard requirement for something. On the other hand, I think there's a strong case to be made that adding your proficiency bonus is synonymous with actually having proficiency, so this is very much the DM's call. As a DM, I would favor the player by default and consider them proficient, rather than attempting to rules-laywer my way out of letting the player do something cool with their racial ability. Under this ruling, the mechanics are precisely equivalent to Stonecunning.