Last time I wrote about analog recording and how it represents a physical chain of proportionate forces directly connecting the listener to the source of the sounds. In contrast, a digital recording is just numbers that *encode* the sounds in an abstract form. While it’s true that digital recordings can be more accurate, the numeric abstraction effectively *disconnects* listeners from the original sounds.

In the first month of this blog I wrote about analog and digital and mentioned they were mutually exclusive Yin and Yang pairs (a topic I wrote about even earlier — it was my seventh post).

Today I want to dig a little deeper into the idea of analog vs. digital!

In the last half-century our daily lives have become more and more obviously digital. Audio CDs (digital music) were introduced the 1980s. Video DVDs (digital video) came along in the mid-90s. As of 2009, in the USA, all television stations broadcast their programs using digital signals. (Hard to believe that was almost five years ago.)

There is such a thing as an analog computer, but just about all the devices you *think of* as computers are digital. Your smart phone, tablet, laptop or desktop; all these are digital devices.

But analog and digital have *always* been a part of our existence. They represent a fundamental Yin and Yang concept that expresses itself everywhere we look.

To make that more obvious, it may help to replace the word “digital” with a word that doesn’t imply computers or high technology. It’s easy these days to forget that the word comes from the Latin word, *digitus*, which means finger or toe. The first use of digits referred only to the fingers of your hand.

Later they came to mean individual numbers, such as “4” or “5” (the number 2014 contains four separate digits; the number 777 contains three of the same digit). There is a slang use that equates “digits” (note the plural) to a phone number (as in, “Give me your digits, and I’ll call you later”).

We’re used to a number system with ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9). This is only because we have ten fingers (digits!), and that is *exactly* how a word for finger became a word for our number symbols.

Computers only have two fingers. It turns out to be vastly easier to design a system that only needs to distinguish between “on” and “off” than it is to design one that can tell the difference between ten different conditions. (See this post for more detail.) It’s easy to look at a light and tell if a light *switch* is on or off, but can you look at a light and tell whether the light *dimmer* is set to 4 or 5?

So from fingers to number symbols and finally to the only two numbers computers care about (“0” and “1”). In this way, “digital” comes to refer to computers and high technology.

But what “digital” really *means*, what it has always meant, is “individual” or “separate” or “discrete.” That last word is the most specific and is the word I’ll use from here on.

Most importantly, *discrete* stands in contrast to (in fact, in opposition to) the word “continuous,” which is another way of saying “analog.” We might refer to them casually as smooth and bumpy. We see the most obvious contrast between them in a (smooth) ramp versus a (bumpy) stairway. (If you don’t think of a stair as “bumpy” just consider going down one on a bicycle.)

The real world (at least to our eyes and other senses) seems smooth. One can be here or there or anywhere between. A cup may be full or empty or filled with any amount. Daylight comes smoothly at dawn and fades smoothly at dusk. Things can be very hot or very cold or any other temperature.

But when we measure these things, we tend to see them in discrete units, such as miles (or feet or inches) or ounces (or litres or gallons) of liquid. Our thermometers break the smooth continuum of temperature into degrees (C or F). Even the sunrise can be described in terms of its units (lumens).

Back when I wrote about the countable and uncountable infinities I discussed the natural numbers, which we use for counting things. There are also the rational numbers (fractions), which we use for counting *parts* of things. These are both discrete number systems. The number “4” jumps (or bumps) to the number “5” (and likewise, “6388401” bumps to “6388402”). Even fractions bump from one to the next, because the two discrete numbers that comprise any fraction do.

The Yang to this Yin are the real numbers, which are smooth. A key distinction (*the* key distinction to my mind) is that given *any* discrete number, you can have some system that tells you the next number in the sequence (in counting, just add one). In contrast, with the (smooth) real numbers, it is *never* possible to determine the next number.

For example, there is no way to name the first real number that follows zero. Informally, we might say, “*an infinite number of zeros followed by a one*” but there is no way to know what this number actually is. If the zeros are infinite, we never get to that final one.

[I used to have a caveat here that there is a tiny number of exceptions, such as 0.999…, which is the next number below 1.000. But mathematically (and this will blow your mind a little), 0.999… and 1.000 are just two ways to represent the exact same number! See this Wiki article for details.]

So even in our basic conception of numbers we find a fundamental division between smooth and bumpy.

One interesting aspect of this division has to do with how closely one looks. A distant stair looks like a smooth ramp. A distant forest (of discrete trees) looks like a (smooth) field of green.

Your skin seems smooth, but look closer: it’s made of individual (discrete) skin cells. Look closer to see the cells made of individual parts (nucleus, mitochondria, etc.) and even closer to see those are made from molecules.

We’re not done. Zoom in even closer and those molecules are made from atoms, and *those* are made of electrons and protons and neutrons. Even that isn’t the end: protons and neutrons are made of quarks and gluons. As far as we know, electrons and quarks (and gluons and some others) *are* the end of the line, the basic building blocks of reality.

As smooth as reality appears to our senses, in fact it is bumpy (“quantized”). Light comes in tiny, discrete packages we call photons, electricity comes in electrons, and all matter is made of the particles I just mentioned. Most physicists believe that reality itself is quantized according to Quantum Mechanics.

And yet our other great theory of reality, Albert Einstein‘s Theory of Relativity, presents a smooth reality. Ironically, our two greatest (and most heavily tested) scientific views of reality stand on opposite sides of the smooth versus bumpy divide. One of the biggest dreams of modern physics is reconciling these two Yin and Yang theories.

Most of physics approaches this goal with the idea of quantizing gravity, of finding out how gravity, too, is bumpy. I hope these efforts fail.

I want Einstein to have been right, and I want a universe with a deep down Yin and Yang between the smoothness of space and gravity versus the bumpiness of matter and energy.

I almost certainly won’t get that wish, but I think it’s a nice wish anyway. Why can’t the universe exist in a tension between the two? Most interesting things are like that, a tension between opposing poles, and what’s more interesting than reality and the whole universe?

January 19th, 2023 at 11:21 am

[…] between countable and uncountable infinity. Which is essentially the distinction between digital and analog. So, it’s hard to see how a digital system can implement a decidedly analog […]