do computers compute? do calculators calculate?

The question answers itself... a pedant may think. Well, of course. If what it is to compute is to enable us to answer our computational questions without ourselves having to perform the computation, then yes, they do.

But in themselves? Is there a fact of the matter, independent of our relationship with the device itself, a fact about the inner workings of a computer, that make it apt to say of it that it is right now computing this rather than another result?

It's not an idea I'd considered before reading Awais Aftab's latest response to our ongoing blog conversation. This is the part of his text that caught my eye:

Bennett and Hacker said: “The computer calculates” means no more than “The computer goes through the electricomechanical processes necessary to produce the results of a calculation without any calculation.”... Unlike Bennett and Hacker, I don't think that it is simply the case that results of a calculation are produced without any calculation; I think that a relationship between abstract mathematical entities is embodied in a physical system. The embodiment of such a mathematical relationship is independent of the place the computer has in human lives.

Why does any of this matter? Well, it may inform our sense of whether or not the brain - which is not a cultural artefact, is not a device the motions of which enjoy a meaning through their uptake in a particular cultural praxis - can be said to process information, calculate, etc. If a computer's state transitions can be said to embody meaning in and of themselves, then why not a brain's too?

The notion is, to me, surprising, counter-intuitive, hard to make out. But let's try working with an example. We have a device in front of us which we use to make additions. There's 5 buttons we call '1', '2', '3', another we call '+', and a final one we call '='. There's also an LCD screen on which similar-looking figures are displayed by the computer. If you press '1+2=' the machine displays '3'; if you press '1+1' the machine displays '2'. (We can imagine extending this to more complicated patterns - e.g. adding a '4', a '5' to the buttons and to the display.)

It's easy I think to see how, because of our use of it, the device can be said to be 'calculating the sum of 1+2'. And if in a dark room the cat stood on it, depressing the same buttons, and if we'd turned the light on we'd have seen a '3' on the screen, what should we say? Well, we might say that here too the device is yet calculating. Or we might not. (Take your pick as far as I'm concerned.)

But what of the idea that it's performing calculations regardless of whether a display is plugged in? There are, the thought might go, 0s and 1s encoded in the calculator's operations. ... But wait, why call this or that electrical pulse a 0 or instead a 1? And how can we tell which pulse is which? I mean, it's easy if we ascribe the meaning to this rather than that pulse. But aside from such an ascription, how should we tell? 

And if we type in '1+2=' and it displays '3', how do we know what the relation is between these numbers in our number system and the actual numbers which the calculator quite independently of us has encoded in its workings? For example, perhaps from the calculator's point of view, as it were, it has just performed the addition '2+4=6'. So that our '1' is its '2'. How could we tell whether our '1' is its '1' or is instead its '2'? Or, for that matter, how do we know our '1' is not its '10', our '2' its '20' etc?

For that matter, perhaps when it's helping us arrive at the correct answer for our addition tasks, it's actually making systematic errors according to the rules which, unbeknownst to us, are allegedly embodied in its system. Or perhaps its doing something quite other than maths? Or maybe the maths is just a hobby for it, or a spandrel of some sort, and the real task it's performing is artistic.

What now of an abacus? Imagine a fancy redesigned one with 3 beads. To perform an addition you take, say, 1 bead, then another 1 or 2, then put them in a chute, and - so long as you have a certain gate opened that you call '+', the beads roll down and collate together where you count them up. You have to know, of course, that the beads stand for numbers, just as you have to know that the marks on the calculator screen are symbols. But shall we say that this device, independently of whether we are using it for addition, or whether a monkey is playing with it, has embodied within it certain mathematical relationships? Again, what are they? How would we find our what they are? We think of each bead as standing for '1', but what do the beads really stand for in themselves?

I don't mean those suggestions seriously of course. It's just that I don't see how one would not be open to them if the idea of a calculator, or anything for that matter, embodying mathematical rules independent of the place of the calculator in our lives is taken seriously. My hope is that, by finding these implications of the idea of the contraption in question as performing particular calculations independently of their use in our lives, that very idea will itself come to seem rather less plausible.

Perhaps it be said that all that it is for a device to calculate 'in itself' is for it to engage in certain mechanical transformations onto which certain mathematical transformations could be mapped. ... 'So are you saying that the planets in motion, the tree with its sap rising and falling within it, and so on, all can be said to be 'performing calculations' so long as their motions can be mathematically represented?' ... Well, no, that obviously won't do. But perhaps what makes the difference between a mechanical system which is and which isn't a calculator is that it be capable of being used to 'perform' far more than one calculation. A pocket calculator which could only compute 1+2 would be a sorry thing, and we might baulk at giving it its (honorary or legitimate) standard designation. (Is it the limitations of the abacus that also make us reluctant to describe it in such terms?) But how about a bona fide pocket calculator that, say, breaks down after you've used it twice? Was it calculating? Well, perhaps we'd say it was since the very notion of it breaking down presupposes that it now doesn't perform the capacity which... Well, which what? Which it was designed to perform? For which we used it? Or which it 'had in itself'? I think we might say what we wished here, so long as we made ourselves clear. 

But in any case I suspect that this latter possibility - this subjunctive conception of natural computation (were you to project onto the mechanical system such and such a rule, then...) - wasn't what Aftab had in mind. And it will in any case be possible to project onto the operations of any such system any number of rules (the above-stated problem of what number to pair up with the electrical pulses - 1, 10, 100? - in the calculator's wiring). Do we really want to say that a single burst of electrical activity in the pocket calculator is really computing any number (an infinity even?) of sums at the same time? ... Or if you imagine that the range of projective possibilities could be narrowed down to one for the calculator, now think of that messy gloop in your head where, presumably, many different physical 'realisations' may be had, at different times, for the same 'calculation'. Or wherein the same 'realisation' may be said to be component of different 'calculations' at different times.