the functions the machines are computing are changing with time.
Sure. But that's not what Turing was talking about. Give a UTM a computable function, and the UTM will compute it. Specify an infinite class of functions F(t), and of course a UTM will have trouble.
There's no contradiction here, just a really weird way of defining "function".
It sounds like he's stating that a UTM can't simulate an arbitrary lambda function then, since that returns a variable function (a second-order UTM: a UTM can't simulate a UTM, which makes it not a UTM?).
In order to evaluate a lambda you need to be able to compute an arbitrary function of arbitrary variables; with infinite resources, you should be able to process infinite variables? Surely there's work done on UTMs and lambdas.
It sounds like he's stating that a UTM can't simulate an arbitrary lambda function then
The concept doesn't make sense. Lambda functions don't do anything, they're just notations. The process of beta reduction is what causes things/computation to happen. Beta reduction rewrites all the lambda functions, hence they're no longer the same functions. What you're saying is something like: a computer can't compute "computation".
Example: a function accepts as input the source of another program (assume it's computable and halts). Thus, the original function is variable; can it be simulated by a UTM for any arbitrary valid input? I'm probably using the wrong terminology here, but the overall logic is what I'm getting at.
The input to a function can't change (it's a fixed set). If the input changes then you've defined a new function. That's the mathematical definition of a function. You're talking about a set/class of functions, so you would need 1 UTM per function.
Turing was simply describing what people do: in a math test, I give you a problem, you solve it - without cheating, that is, interacting with your environment - using an algorithm, you hand it back to me. He idealized what a human does - infinite time and memory -, and then, as a final step, he said: "A machine - also idealized - can do that." The important thing is that in the definition of what was later to be called "Turing computability", interaction during the computation is verboten.
