Can a character without arms cast spells with somatic components?

My DM and I have been having friendly arguments, and we recently came to what would happen if my character lost all their limbs.

As a gnome beastmaster I said I would always sit on my mount and use mage hand (from magic initiate) when I need to eat or lift things. HOWEVER he argues I wouldn’t be able to perform the somatic component.

So, I ask: is there a way to cast spells without arms? My argument is that since I am learning the cantrip after losing my arms I would have learned it in a way that works with my stumps, but he points out the PHB is pretty clear you need a free hand.