[...] sure, but it doesn't follow that if I just get together with a couple of random friends a couple of times a week, we'll magically reinvent a particular martial art between us. Why bother going to school, either? Math is purely logical, right, so if I just use trial and error I should come up with integral calculus eventually, no? The teacher teaches... that's why he or she is called a teacher. That's why we pay to join a dojo... for the teacher... you can get padded mats somewhere else if that's all you really want.
Where did aikido, calculus, or any other thing that we can claim to know, to label, to learn, or to teach come from in the first place?
Calculus is a very apt example. Who taught calculus to Newton or Leibniz---two people credited with simultaneously and independently (in different countries) inventing what we call calculus? And if that was all there was, how did calculus (and all of mathematics) continue to grow, even today?
Who taught Newton or Leibniz calculus? Who taught O-Sensei aikido?
The answer is "yes," by the way: students, with the right circumstances, are discovering / inventing the principles calculus every single day! (See Wikipedia's note on the Moore Method
for an interesting read.) Other students with different circumstances are only being shown techniques of calculus every single day. Both are happening with or without a teacher, every moment, somewhere on this planet...
The same is true of martial arts, of course.