A look at the kind of mechanical abacus I would try to build in order to effectively represent measurements with rational numbers — different from (but still somewhat analogous to) the uniquely weird way we already inscribe them into our digital systems.
Basically, the disks would rest in a wooden frame with a notched axle, and each disc would display one digit at a time through a little window, allowing you to rotate them between 0 and 9. Or between 1 and 0, depending on where you arbitrarily start your index.
(The size of the discs in this image is essentially syntactic sugar, illustrating how your available space runs out as you approach the outer limits, but of course in a real model you would want the discs to be of the same size, for obvious reasons.)
Good thing we’ve got digital computers to move simplified, microscopic versions of these pieces around for us, because… yikes.
Building a functional version of one of these would be a challenge.
These “number things” can get pretty big, and sometimes you even have to manage more than one at a time!
But the power and success of the concept still lies in its utility: A mechanical system like the one above might be sufficient, and a digital system might be better, but all that is truly necessary for humans is pen and paper.
We only need to add some kind of stylus and some kind of surface – plus our ten digits – and suddenly we can compute and keep track of incredibly large measurements with little-to-no effort.
Just by virtue of this tracking toolkit, we quickly become masters of quantity and scale, regardless of whether or not our minds can truly fathom it.
(As for the measurements themselves, you’re on your own. This device would only help you keep track of them.)
Even in the absence of digital technology, find any page you can write on and suddenly you will command at your fingertips as many rational numbers as you can handle.
The rest of the animal kingdom will tremble before this immense power, but for a human, learning quasi-magical systems like these are a trivial fact of adolescence.
But as with all power extended, there inevitably comes a weak point.
And this is no different.
For it is all too easy to get lost in the curves of the symbols themselves, forgetting the hidden methods and convoluted patterns that get trapped beneath the veil of well-practiced movements.
Ordering magnitudes via joined, iterative, denary functions is one extremely effective way to track measured relationships at different scales.
Yet still – there are always other ways.
But I’m sure there’s no need to worry about those.
After all, with this system we can order all of magnitude itself! And we haven’t even mentioned the power of addition or multiplication yet, let alone division.
Or… have we? 🤔
And let’s not even get started on those pesky irrational numbers.
Or any of the other kinds of “numbers.”
Or any of the other kinds of “functions.”
Actually, now that you mention it this system is starting to feel… kinda messy.
I hope things don’t get out of hand.
That’s how you introduce hazards and halting problems, and real-world mechanical/digital systems aren’t going to be quite so friendly about that.
True deterministic systems enforce all the little rules you introduce, even if only by accident, so you better have some kind of real strategy or ruleset in place for smoothing out those rough edges.
Otherwise things will go haywire. Inevitably. In time.
So much for the Curry-Howard-Lambek isomorphism, I guess?
— yoav golan