You can look at this at Amazon with their excerpt feature.

You have said:

“The notion of effective computability has nothing to do with mechanicalness”
&
“The definition clearly links computation to recursion — an essential feature.”
&
“A mechanical procedure is, basically, just a recursive procedure.”

It kind of looks like we can derive from this that computation is simultaneously mechanical and not mechanical. But maybe I’m interpreting something incorrectly here. What’s up?

There’s a difference between “effective” and “mechanical”. A mechanical procedure is, basically, just a recursive procedure. An effective procedure is, basically, just one that gets the job done. Thus Turing introduced the idea of an oracle. An oracle correctly answers your (yes / no) question in a fixed finite time. Thus you might have an oracle for arithmetical propositions. For any arithmetical proposition P, you can ask the oracle whether or not P is true (whether or not it’s a theorem of arithmetic). The oracle always answers correctly in finite time. There is no mechanical procedure for deciding the theorems of arithmetic — many are true but unprovable. But, thanks to your oracle, you have an effective way of deciding the theorems of arithmetic.

I’m not sure I understand (c), though. How is it consistent with (d)? (”endless” & “final state” tickle my contradiction detectors).

Relatedly, How is (d) consistent with transfinite recursion?

I don’t get the God point. What’s the effective procedure here? Be God?

Cthulhu Ftaghn!

This definition has way too many problems. Saying the state is typically numerically or symbolically interpretable is unneeded - try giving me a state that is NOT numerically or symbolically interpretable. What’s the difference between a “condition” and a “state”? Why aren’t we starting from an initial state? As it stands, the definition seems to imply that almost any process is a computation - even a random process like throwing a die over and over. Is “do whatever you want” a rule?

As an old computer scientist, here’s my definition: a computation is any process that (a) begins with an initial state; (b) uses a fixed set of operations to repeatedly transform its current state into its next state; (c) may use an operation to transform an endless series of states into a limit state; and that (d) ends with some final state.

The definition says nothing about the precision of the states (thus the states might be real numbers on a continuum). It says nothing about rules (and thus avoids the vexed question of what is a rule). Instead, it talks about operations. You get to pick anything at all as an operation, so long as its a function from states to states. The definition clearly links computation to recursion — an essential feature. The definition allows for transfinite recursion and thus transfinite computation. It is not bound by the Church-Turing limit. The definition demands the existence of a final state and thus ensures effectiveness (though not necessarily finite effectiveness).

The notion of effective computability has nothing to do with mechanicalness. It has to do with producing a result. f you could ask God (or one of Turing’s oracles) any question and get His answer (which would be true) in some fixed finite time, you’d have an effective procedure. But it wouldn’t be mechanical.

We now get to “calculation via procedures that are “mechanical” in the sense of being able to be performed by the application of relatively simple procedures without the utilizations of much insight or ingenuity”

To talk about proceedures that are able to be performed by procedures is nonsense. Note that nobody ever talked about “much” insight or ingenuity. Turing makes these ideas clear. So it’s better to just skip this sentence.

Otherwise, all is Groovy.

Cthulhu Ftaghn!

What I had in mind in my earlier post (regarding analog computers) were the mechanical, crank-the-gears machines used to integrate functions and point the anti-aircraft guns at the right spots. While I agree that it’s important to maintain the Wittgensteinian point, there may be something to say about how those non-programmable but obviously digital computers follow rules (I take it that this is what Marcin had in mind when he mentioned implementation). But those analog computers did what they did in virtue of physical properties that, in a real sense, directly modeled whatever it was they were trying to compute. It seems to me that if those things follow rules, we may be very close to trivializing the notion of rule-following (and not just being rule-describable). Maybe those aren’t *really* computers, but for a time they certainly were.

Here are some points for the purpose of further discussion, if you’re interested.

Ken: I take it that the key characteristic of the digital is that it’s not analog and the key characteristic of the analog is the way in which things are represented (via morphisms). I wouldn’t characterize analog representations as involving infinite precision (the calibrations of analog measuring devices, for example, are going to give you only a finite number of significant digits). And I think the general notion of effectiveness, the one of which both pre-Turing and post-Turing notions are instances, applies to the manipulations of representations and is neutral with respect to whether the representations themselves are analog or digital.

Liz: I guess there’s no harm in calling the illumination of the model a computation, but I likewise guess there’s no harm in calling the representations in question “symbols” and stating the rule followed as “shine light on model and note resultant shadow distributions.” Regarding your offered third alternative, I personally find it appealing but worry that it is a bit too controversial: widespread is the view that it’s vehicular, not content properties, that determine the causal interactions between representations in a computation.

Corey and Marcin: I don’t think anyone has a particularly clear idea about how to tell when something does or doesn’t follow rules, but the closest thing to popping open the hood and finding rules hiding therein we’re going to get is with programmables. However, making the presence of an instruction set criterial for digital computation will, as Marcin nicely points out, exclude non-programmable but obviously digital computers.

it depends on whether you’re Wittgensteinian and you distinguish between rule-following behavior and behavior that can be described with a rule. It’s obvious that you coud have non-programmed digital computers with no rules represented to be followed so I thought Pete didn’t want to use this distinction (non-programmable digital calculators, for example, can have no software level and no explicit rules to follow). Not all digital computers adhere to von Neumann’s architecture, after all.