One of my earliest posts was Analog vs Digital. A few years later, I wrote about it in more detail (twice). Since then I’ve touched on it here and there. In all cases, I wrote from the perspective that of course they’re a Yin-Yang pair.
Recently I’ve encountered arguments challenging that “night and day” distinction (usually in the context of computationalism), so here I’d like to approach the topic with the intent of justifying the difference.
I do agree the grooves on a record, and the pits on a CD, are both just physical representations of information, but the nature of that information is what is night and day different.
Consider a photograph. It might have been folded, and now has a crease across it. Perhaps there’s even a careless coffee ring, or maybe someone spilled something on it.
It can be copied and touched up, but the copy is never quite as good. It can be touched up or adjusted, but then aspects of the copy are fake. The term “generation loss” refers to how entropy makes each copy a little bit worse.
Now consider a page of numbers.
Such a page can be creased and marred, but a copy with the same numbers loses nothing from the original (assuming the original numbers can still be read correctly). And the copy is much easier to make.
In the physical world, despite that we describe it with numbers, most object properties are magnitudes. Distance, age, weight, length, width, height, brightness, hardness, mass, charge, and so forth, are all magnitudes.
That divide — a major mathematical Yin-Yang — between the countable rational numbers and the uncountable real numbers parallels the respective divide between digital and analog.
What makes the photo hard to copy faithfully is that every point has a distinct brightness magnitude — from all the system can deliver (light) to as little as the system can deliver (dark).
There is a continuous range of these magnitudes, from darkest to lightest with all possible values in between.
A photo is also spatially continuous from point to point. The brightness varies smoothly within the limits of the resolution of the optics and film (or solid-state photo capture device).
Note that a photo is itself a copy of the physical subject, so there is already a generational loss due to optics and film. In some sense, the battle for perfect fidelity is already lost! (Not to mention going from 3D to 2D.)
For simplicity, I’m referring to a black and white photo, but the discussion extends to color photos. In that case, there are three dye layers (cyan, magenta, yellow) that combine to recreate the original colors (more or less). Each layer is a separate set of magnitudes.
Making a faithful copy of a photo requires duplicating the magnitudes correctly at every point, plus the spatial resolution must be preserved. The copy must vary at the same scale as the original.
There is a transfer function associated with any copy process — ideally that function preserves the range of magnitudes proportionally. This is called a flat (or linear) response.
The input and output ranges don’t have to be identical. The requirement is proportional linear response. For example, a sound amplifier magnifies the original sound, but does so proportionally.
There is also the matter of frequency response — it must be high enough to preserve the spatial variations, the fine detail.
The fidelity of the copy, whether a photo or sound recording, depends entirely on the quality of the transfer function and frequency response. Ideally, the former is completely flat and the latter is infinite. (Reality is never quite up to that, though.)
Finally, since all the points on the photo matter, any imperfections or damage in the original is copied (unless retouched, which introduces locally generated, “fake,” magnitudes).
An analog system typically has no way to tell “good” magnitudes from “bad” ones, so it tries to copy them all as faithfully as possible.
In contrast, copying a page of numbers abstracts the information content from the page and re-creates a completely new, essentially identical, copy from the abstraction.
This is possible because numbers are a different kind of information than magnitudes. They are abstractions — which are descriptions.
An abstraction always refers to something, because an abstraction is always of something.
A distinguishing characteristic of abstractions is that they can be copied indefinitely without loss of information (given some simple entropy-fighting mechanism).
A key reason for this involves the transfer function (which turns out to also be the entropy-fighting mechanism).
To copy numbers, we want a quantized transfer function, not a linear one. A number is what it is, not some other number. There should be no confusion.
In most machines, the numbers in question are binary numbers, which have just two values, zero and one. The transfer function is thus quantized to have two states: any value below a certain threshold is a zero, any value above is a one.
This permits clean, perfect copies, because it’s easy for the copy system to distinguish between numbers.
Another distinction involves the number of signal pathways.
Consider a given analog signal, say the left channel of your fully analog music system. (An AM radio signal would also work.)
There is a single signal path through the system. One can tap into that path with various instruments (an oscilloscope, for instance) and see the signal.
And by “see” the signal, I mean see the actual magnitudes of that signal.
In an audio system, like a home stereo, one can actually hear the signal at any point given a good listening device.
In contrast, in a digital system, there is either a parallel bus of multiple bits or, in some cases, a single serial line with multiple bits multiplexed by time. (From an information standpoint, the two are the same.)
It takes multiple bits (signal pathways) to represent the numbers that represent magnitudes.
Note that the diagram shows three analog parts of the system: The outside two are what we usually mean by an analog signal, whereas the inside one is technically an analog system (electron or photon flow) carrying digital information (i.e. numbers).
So while the digital information is, in a sense, still analog, it involves only two magnitudes. The system is designed to only ever recognize two magnitudes. Anything else is considered noise to be ignored.
More importantly, the numeric scheme used to represent the input magnitudes is entirely arbitrary — any number can be selected to represent any magnitude.
This means there is no causal connection, no direct transfer of forces, from end to end. Looking at the numeric data tells us nothing about the input magnitudes without knowing the encoding scheme.
Just consider the “obvious” ways of binding magnitudes and numbers:
- The number represents an absolute magnitude.
- The number represents a relative magnitude.
- The number represents a delta to the previous magnitude.
There are many less “obvious” schemes and several variations to the three listed above.
So while there is a high-level causal connection in that a given magnitude “causes” a given number (or vice versa), this is arbitrary and abstract. There is no necessity to the binding.
The actual physical causal connections of the system involve the low-level electron flows that control system operation. These flows would be essentially identical regardless of the binding chosen, regardless of the numbers used.
Another characteristic that distinguishes the two types of information is that magnitudes are more difficult to record (or store) than numbers. This speaks, again, to the concrete-abstract divide.
We did learn to store visual information on film, and we also learned to record sound on magnetic media. Various gauges with scrolling pen systems let us record temperature, pressure, or the earth shaking.
But we’ve been writing down (or otherwise recording) numbers for a lot longer. Because it’s easier.
Abstractions are descriptions. Which are easy to copy accurately.
Which brings me to the most crucial distinction of all: the two triangles labeled ADC and DAC. These are the points that translate between the analog (magnitudes) and digital (numbers) worlds.
It is the need for these translation points that prove the fundamental difference between the two worlds — the real and immediate versus the abstract and arbitrary.
It is exactly in the ADC and DAC process that the (arbitrary!) binding between magnitudes and numbers is, respectively, encoded and decoded.
The two processes are the gateways between the Yin and Yang of analog and digital, of magnitudes and numbers.
So, on the one hand, the world of continuous magnitudes. Generally speaking, the physical real world of objects.
And, on the other hand, the world of discrete numbers. Generally speaking, an abstract world many believe we’ve completely invented.
Once digitized, the signal pathways are, indeed, analog (in some sense), but the information they carry isn’t, and that information is the entire game.
So: Analog vs Digital; Smooth vs Bumpy; Continuous vs Discrete; Magnitudes vs Numbers. All Yin and Yang — “night and day” — different!
Stay continuous, my friends!