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Before Turing Machines, computability theory, and computational complexity theory, there were computers. They were people: mostly women. This is one of my favorite photos of a computer, because it shows her using two distinct computing methods.

One is digital: using numbers, represented by their digits (the adding machine and graph paper she's using). This is how we ordinarily represent numbers, and how we compute with them (e.g., elementary addition, using least significant digits first, then next-most-significant, etc.).

The other is analog: using a slide rule. Here, numbers are represented by a physical magnitude; in this case, length. So rather than π being "3.14159", it is a length of 3.141 units on the slide rule. Analog computing amounts to manipulating these magnitudes, rather than individual digits.

Both methods have been used for millennia, and both have been mechanized. Computing machines were once primarily analog, but now they're almost exclusively digital; we just call those machines computers now. Importantly, they're both real types of computation, although it's easy to forget.

Why? Because Turing (and others) were interested only in digital computation, or "effective procedures" when it came to studying what is "computable." They didn't mean computable by *any* device: they meant to capture "computable by hand, following simple rules, without cleverness or instruments."

But the mathematical theory of digital computation became extremely interesting and useful, and became synonymous with *all* computation. Coupled with analog computing machines being replaced by digital ones, analog computation has been largely dismissed.

I'm close to finishing a large paper I've been working on for years on this topic. Analog computation is misunderstood and undertheorized, which is a shame, because it's both theoretically interesting and likely useful for understanding (some of) what neural systems do.

Could it be said that an analogue clock is an example of analogue computing? The computation of seconds, minutes, hours, etc is performed by the sizing of various gears... 🤔

Definitely analog representation! Whether it’s computing? Depends on the threshold for how simple an operation can be and still be considered computation.

And how about the old mechanical distributor on petrol car engines? It would "compute" the timing advance from engine RPM and manifold depression (vacuum)?

But I wouldn’t say that cars are adding numbers all over the place, even though they have differential gears. They *could* be used to add numbers, but only when the rotations are used to represent values.

But, surely, computation is more than just addition? Gears are about multiplication/division. The distributor is performing a transfer function - taking RPM & vacuum, and "computing" timing advance from that?