However, those quotes are not saying "that Turing machines are a more elegant basis for computations, since they are much easier to mathematically reason about." I still consider that statement to be false, and I've substantiated that in my comments.
The quotes from Church and Gödel are saying that Turing's formalism was the more helpful in making the case that it had captured the notion of effective calculability. That's understandable - it's much like e.g. Cantor's diagonal, in that it makes its subject very concrete.
But that doesn't tell us anything about the usability of the formalism as an actual mechanism for computation, or analysis of computation. In that respect, lambda calculus has proved far more useful, as shown by the examples I mentioned, and many more. Another example is denotational semantics, which maps programming language semantics to lambda calculus.
In fact, one of the inventors of denotational semantics, Dana Scott, developed the first non-trivial model of the lambda calculus, in terms of complete lattices. That model addressed the concreteness issue, albeit decades later, in the early 1970s. It's possible Gödel, Church etc. might still have preferred the Turing tape as an intuition-friendly proof for effective calculability, but it would no longer be possible to reasonably make Gödel's "thoroughly unsatisfactory" claim about the lambda calculus.
Coming back to denotational semantics, I'm pretty sure no-one has ever provided a non-trivial language semantics in terms of a Turing machine - and if you wanted to do that, one of the easiest ways to do it would probably be to implement a compiler from lambda calculus to Turing tape, and generate the Turing representation automatically from that.
More generally, my position could be simply refuted with counterexamples of tractable Turing machine solutions to any of the various problems that lambda calculus has been used to address, like programming language implementations, programming language semantics, duals of logical systems, and proof assistants. To my knowledge, no such examples exist, and the reason for that is because what I'm saying is a fact, not an opinion.
> Myself, I find Conway's game of life to be the superior framework over both lambda calculus and Turing's Logical Computing Machine :-P
I'll agree that the game of life is only slightly less useful than the Turing machine as a model of computation!
> The quotes from Church and Gödel are saying that Turing's formalism was the more helpful in making the case that it had captured the notion of effective calculability. That's understandable - it's much like e.g. Cantor's diagonal, in that it makes its subject very concrete.
This is what I primarily meant with my controversial statement, sorry I didn’t articulate it well enough. Of course lambda calculus is very useful and has plenty of applications, never meant to say otherwise!
I think you've demonstrated far more familiarity with the subject matter than me (and my own understanding is admittedly pretty basic), and just wanted to say I really appreciate your thoughtful response. You're right that those quotes don't really fit for what you were asking for evidence of.
I also agree that the absence of such a refutation is a pretty solid argument that your position is simply a fact and not an opinion as I argued.
However, those quotes are not saying "that Turing machines are a more elegant basis for computations, since they are much easier to mathematically reason about." I still consider that statement to be false, and I've substantiated that in my comments.
The quotes from Church and Gödel are saying that Turing's formalism was the more helpful in making the case that it had captured the notion of effective calculability. That's understandable - it's much like e.g. Cantor's diagonal, in that it makes its subject very concrete.
But that doesn't tell us anything about the usability of the formalism as an actual mechanism for computation, or analysis of computation. In that respect, lambda calculus has proved far more useful, as shown by the examples I mentioned, and many more. Another example is denotational semantics, which maps programming language semantics to lambda calculus.
In fact, one of the inventors of denotational semantics, Dana Scott, developed the first non-trivial model of the lambda calculus, in terms of complete lattices. That model addressed the concreteness issue, albeit decades later, in the early 1970s. It's possible Gödel, Church etc. might still have preferred the Turing tape as an intuition-friendly proof for effective calculability, but it would no longer be possible to reasonably make Gödel's "thoroughly unsatisfactory" claim about the lambda calculus.
Coming back to denotational semantics, I'm pretty sure no-one has ever provided a non-trivial language semantics in terms of a Turing machine - and if you wanted to do that, one of the easiest ways to do it would probably be to implement a compiler from lambda calculus to Turing tape, and generate the Turing representation automatically from that.
More generally, my position could be simply refuted with counterexamples of tractable Turing machine solutions to any of the various problems that lambda calculus has been used to address, like programming language implementations, programming language semantics, duals of logical systems, and proof assistants. To my knowledge, no such examples exist, and the reason for that is because what I'm saying is a fact, not an opinion.
> Myself, I find Conway's game of life to be the superior framework over both lambda calculus and Turing's Logical Computing Machine :-P
I'll agree that the game of life is only slightly less useful than the Turing machine as a model of computation!