[jimhefferon]

> Is it possible

Yes, the set of functions computable with mu recursion, or with lambda calculus, is the same as the set of functions computable with a Turing machine. (Turing in his original paper showed that the set is the same for his system and Church's, and the proofs for mu recursion and many other systems are well-known.)

> I'll totally agree that from a teacher's perspective, the Turing machine is the better explanation.

When I learned this, the instructor did not use Turing machines. They used mu recursion. That has the advantage that you don't first define a machine and then derive the set of functions, instead you go straight to the functions. But I agree that Sipser (which I understand to be the most popular text today) defines a computable function as one that is computed by Turing machine, and to my mind that has the advantage of honestly suggesting to students what they are about.

[generalizations]

Thanks, I think I understand now. I thought there was a distinction between the mathematical idea of what computation is, and the engineering we've invented to implement that computation, and so I didn't really get the significance/literalness of what you were saying about mechanization.

It does seem weird to me though that we're letting our engineering limitations determine how we think about these things mathematically.

[nl]

It's not as simple as engineering limitations determining how we think.

Turing machines do have mathematical advantages in some areas. See this siblings comment: https://news.ycombinator.com/item?id=28541800