> That's the part I have a problem with, mathematical objects are (more or less) exactly those that can be precisely and formally described. "Lambda calculus is a more elegant mathematical object" is not a statement I have a problem with, "Turing machines have a lesser status as mathematical entities", on the other hand, is sort of weird.
I agree as well. I don't think Turing machines have any lesser status (it's literally equivalent to lambda calculus, after all!)
> but for things like analysis of algorithms, complexity theory or numerical analysis it would be similarly unnatural to use lambda calculus as the computational model instead.
That's a good point as well. Because of the nature of hardware, neither lambda calculus nor Turing machines are very good fits, so RAMs and pointer machines essentially take over.
> I don't think Turing machines have any lesser status (it's literally equivalent to lambda calculus, after all!)
"Equivalence" has a specific meaning here, though. The term "Turing tarpit" was invented to point out the limits of that equivalence, and any attempts to define computations for Turing machines quickly run into that.
The sense in which I consider Turing machines to be "rather unmathematical" is closely related to this. You can write useful mathematical proofs in lambda calculus - as I've pointed out elsewhere, there are automated proof assistants based on this fact. There's no such equivalent for Turing machines. Theoretically, there could be, due to Turing equivalence, but in practice, no-one wants to exploit that.
Given one formalism that has actively been used for such mathematical purposes, and another that has been actively avoided, I call the latter rather unmathematical by comparison to the former.
People can quibble with my word choice, but it's describing a real, measurable phenomenon, the effects of which have played out predictably over the last 70 years.
I agree as well. I don't think Turing machines have any lesser status (it's literally equivalent to lambda calculus, after all!)
> but for things like analysis of algorithms, complexity theory or numerical analysis it would be similarly unnatural to use lambda calculus as the computational model instead.
That's a good point as well. Because of the nature of hardware, neither lambda calculus nor Turing machines are very good fits, so RAMs and pointer machines essentially take over.