[mich41]

CS is much more than the UTM.

UTM is a dumb and completely impractical model of computation. Thanks to its simplicity it is a handy tool in (un)decidability proofs and some very general computational complexity reasonings.

And that's all. Numeric computations happened on real machines. Graph processing happened on real machines. Text processing happened on real machines. Structured programming, procedures, programming languages, compilers - all happened on real computers.

Using (or even thinking about using) the UTM for any practical application would be a huge PITA and nobody ever does it. We only know that it's "possible" and hence if we want to prove something general about all possible computation, it suffices to do some magic with the UTM.

[jlgreco]

I don't think you are getting it. In Babbage's time they were not merely missing the UTM. They were missing damn near all of mathematical logic (even set theory only really came about around the time that Babbage died) and meta-mathematics was incredibly immature at the time. Forget answering questions about CS; they could not yet ask the questions.

They could have pulled some impressive stuff off I am sure, but it would have all been intuition and stabbing in the dark. The tools necessary to structurally reason about algorithms had not yet been created, nor even the tools necessary to create those tools...

Ada had the notion that the Analytical Engine was something special, something more than just a calculator... but that was conjecture based on genius insight. To actually discuss that idea in a rigorous manner would require several more decades of advances in mathematics.

Could they have programmed? Yes, obviously. That's not even hypothetical, since they did. Would they have been at a distinct disadvantage? Without anything reasonably resembling modern mathematics, absolutely.

[mich41]

You don't need logic, set theory, Turing machines or any meta-mathematics to build practical software.

When I was ten, I had no idea about any of this stuff and yet when somebody showed me how to do arithmetic, variable assignments, comparisons and goto in QBasic (pretty much equivalent of Babbage's machine) I was able to write a simple drawing program and tic-tac-toe which checked whether one of the players won.

Add some IO and I would write a program which reads series of transactions and computes your bank account balance. Tell me what a matrix is and I would implement LAPACK for you.

[jlgreco]

You absolutely need those things for a great deal of modern programming. All modern programming? No, but a great deal.

Without that they would have been at a disadvantage. I don't see what is so hard about this concept to you.

Regardless, the simple historic fact remains that Babbage and Ada were both unable to build the machine, and unable to verify their suspicion that the machine was special, and unable to effectively communicate to their peers this suspicion. The prerequisite math for all three of these tasks did not yet exist.