Jest, how about a compromise? You ride your bike and she rides her horses. Both are actually very similar. After you can have coffee (or whatever beverage you prefer) and talk about your experiences with each other. Bike riding and horseback riding are actually extremely similar and it's not unusual to see a person go from one to the other or do both (as in my case, I went from horses to harleys.) If she only wants to ride the breathing kind, encourage her. There is no reason you two can't do both and then share your experiences after the fact. If you want company while riding, look around while on your bike. There are many riders out there that would love company on a toodle.
I wouldn't push her to ride with you, but encourage her to ride in her own way. If you come up this way (Thunder Bay, ON, Canada) I'd be happy to go for a ride with you.
bec