Magnitudes vs Numbers

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.

The accuracy and ease of duplication illustrates one essential difference between two basic types of information: magnitudes and numbers.

§

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.

Humans designed the real numbers based on those magnitudes. We designed the rational numbers based on the idea of sets of things: number of legs, amount of dollars, parts of the pie, etc.

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.

ADC=Analog-to-Digital Converter, DAC=Digital-to-Analog Converter.

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!

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

17 responses to “Magnitudes vs Numbers

  • SelfAwarePatterns

    I remember as a teenager when we used to get pirated movies on VHS tapes. It was amazing how degraded a few generations of copying left the quality in a wretched state.

    Although, at least it degraded gradually. I spent a lot of time watching pixelation and frame drops in the earliest digital signals from my cable company.

    • Wyrd Smythe

      Video is definitely one of the harder magnitudes to copy — not unlike trying to copy a radio signal would be. The system needs a very high frequency response, something the electronics handle okay, but which magnetic tape needs all sorts of high-tech tricks to make it work (like those long diagonal recording stripes).

      But, as you say, magnitudes do degrade more gracefully — that copied photo just gets slowly worse and worse. Digital is often an all-or-nothing proposition, at least until the data stream can sync back up again.

      Now that I’ve cut the cable and my default channel is PBS over the air, I’m noticing how the digital TV signal has those same digital issues. Gone are the days of falling asleep to “TV snow.”

      • SelfAwarePatterns

        It does seem like the protocols have gotten better over the years, at least over cable. They seem better able to degrade in a watchable manner. This is one area where streaming has advantages (usually) since a stream is a conversation rather than a broadcast, where things like resolution can be adjusted if necessary.

        I sometimes miss that late night national anthem followed by the snow. It seemed more like an accomplishment than just making it to all the infomercials or reruns they show now.

      • Wyrd Smythe

        I used to get a kick out of the fact that a small portion of TV snow is due to the CMB. Ancient noise!

        Streaming, I believe, is a series of separate URLs for blocks of video, so even if one block is trashed, the next ones can be perfectly fine. I think it’s also how it adjusts resolution on the fly; just accesses a different set of URLs.

        Video formats are one of those rabbit holes… I use an open source product, ffmpeg, to generate animated mpeg movies from a set of PNG images. It’s one of those unix-y applications with a gazillion parameters, and browsing over the docs, the various video formats and adjustments within those formats… it’s one of those areas you could spend a lifetime becoming a serious expert.

      • SelfAwarePatterns

        On streaming, I’ve heard a lot of Netflix movies are being streamed out of Amazon S3. I’m sure Amazon video is as well. My old boss had his personal website hosted out of S3. It’s starting to get used for a lot of stuff.

        So many rabbit holes…so little time.

      • Wyrd Smythe

        There’s a whole thing, too, in baseball, where a lot of the fancy graphics+stats stuff is (and I am so sick of hearing this phrase every time) “provided by Amazon web services.” At least, after many months (a year?), they’ve shortened it to “AWS” but you just can’t get away from them.

        That David Brin story, Existence, talked about the risks of fragmentation and isolation inherent in specialization (those rabbit holes). People spend more and more time learning a narrower and narrower topic.

      • SelfAwarePatterns

        LOLS! I suppose if I watched sports to any degree, I would be familiar with that, but it sounds like the sort of thing I remember seeing in games.

        Specialization definitely makes us vulnerable if anything goes wrong with the system. But civilization increasingly seems to make it mandatory. And it does at times lead to problems, particularly in academia.

        I know I’ve read things written by people who were brilliant in their own field, but who made statements about something that shows they didn’t even have a layperson’s understanding of it. Physicists seem particularly prone to this, but I’ve seen it from biologists, psychologists, historical scholars, and many others.

      • Wyrd Smythe

        I forget where or who, but the line I heard recently was, “Remember when Amazon just sold books?”

        Recent purchases I’ve made: dog bed; socks; portable tire pump; wireless speaker.

        I’m stuck in the Apple ebook ecology now (itunes as well), so I haven’t actually bought books from Amazon in maybe a decade (let alone CDs or DVDs). I have been reading some of the free Prime stuff using the Kindle app (comics, mostly).

        “But civilization increasingly seems to make [specialization] mandatory.”

        It seems inevitable. It starts with farmers, hunters, bakers, blacksmiths, and ends up with scientists who spend their entire career studying some small aspect of a part of physics (like the magnetic moment of the electron).

        Brin points to the internet as a balancing or mitigating factor. The free flow of information makes the system less fragile. Reduces redundancy, too, when people share information.

        (Brin goes on to suggest not every civilization would allow an open internet and posits specialization as one of the “great filters” a civilization needs to get through.)

        “I know I’ve read things written by people who were brilliant in their own field, but who made statements about something that shows they didn’t even have a layperson’s understanding of it.”

        Yep. Stephen Hawking is kind of my canonical example there.

      • SelfAwarePatterns

        I do remember when Amazon was the new online book company. “The largest book store in the world,” or something along those lines. (Barnes & Noble, I think, sued them for it, but Amazon actually based their claim on the vastness of their catalog.)

        Similar to you with Apple, I’m pretty embedded in Amazon’s ecology. I get my books and any videos I pay for a-la-carte from them, not to mention all kinds of things I’m too lazy to go to brick and mortar stores for. My recent (non-Kindle) orders include toilet paper, bathroom mats, and coffee. (My recent Kindle orders include preorders for Joseph LeDoux’s new book, Sean Carroll’s book, and Becker’s book which I posted on last night.)

        “and posits specialization as one of the “great filters” a civilization needs to get through”

        That’s an interesting idea for a great filter. A lot of SF ideas for filters represent the author’s political attitudes. Brin, if I recall, has a “let information be free” philosophy. Charles Stross’ version is coordinated control of resources (aka communism). Heinlein’s would have been some variation of libertarianism.

        “Stephen Hawking is kind of my canonical example there.”

        Among physicists, my examples would also include Roger Penrose, Lawrence Krauss, and Neil deGrasse Tyson, although I don’t want to give the impression that this list is in any way complete 🙂

      • Wyrd Smythe

        “A lot of SF ideas for filters represent the author’s political attitudes.”

        Yeah. Hardly surprising, though. Many authors write to express their worldview (as opposed to just telling ripping good yarns).

        “Among physicists, my examples would also include Roger Penrose, Lawrence Krauss, and Neil deGrasse Tyson,…”

        I’d agree in all cases, especially that last one. I never liked him. He hosts the Isaac Asimov Memorial Debates, and his presence kinda ruins them for me. He turned out to be a poor successor to Sagan, in my view. He’s like Louis C.K. in that my reaction to #metoo allegations against him was a complete lack of surprise. More of a confirmation of what I’d always suspected. (Way more so in Louis C.K.’s case.)

        I’d differentiate that kind of thing from Hawking, who isn’t a creep, but just said some dumb and very unjustified things way outside his area of expertise.

        I think the truth is that we’re all full of shit once we get outside our areas of real knowledge. Wiser people learn to shut up then. The unwise end up with bouts of footnmouth disease. 😀

        Which makes one look dumb, but it’s way better than being a creep.

      • SelfAwarePatterns

        “Many authors write to express their worldview ”

        Oh, definitely. In fact, it’s very hard for an author to avoid revealing aspects of their worldview in their writing. Somewhere I read a warning to any aspiring writers, that you’ll almost certainly reveal aspects of yourself that you don’t intend to.

        “I think the truth is that we’re all full of shit once we get outside our areas of real knowledge.”

        Too true. It is possible sometimes that an expert weighing in on something outside of their field has some knowledge, but it’s generally the knowledge of an educated layperson, and shouldn’t automatically be accorded any special credibility. We should listen to what Hawking said about black holes, but view his remarks on AI, aliens, God, or anything else as just an educated guy with an opinion.

      • Wyrd Smythe

        “Somewhere I read a warning to any aspiring writers, that you’ll almost certainly reveal aspects of yourself that you don’t intend to.”

        That makes perfect sense. The literary version of the Johari Window! 😀

        “We should listen to what Hawking said about black holes, but view his remarks on AI, aliens, God, or anything else as just an educated guy with an opinion.”

        I generally try to ignore the source (as much as possible) and concentrate on the content. For one thing, sometimes the most ignorant person can offer wisdom or the smartest person can be a idiot. (Funny parallel: Sometimes the worst baseball teams beat superior teams. Life is rarely simple!)

        There is also that some educated people are more polymaths than others. Hawking never struck me as someone with a highly balanced education, physics versus the liberal arts, so I don’t place much faith in what he says outside his area of expertise.

        Feynman was a really good example of a polymath. I’d tend to listen to anything he says (which isn’t to say he wasn’t be capable of being a bonehead, either — we all have that gene).

      • SelfAwarePatterns

        “I generally try to ignore the source (as much as possible) and concentrate on the content. ”

        The issue as I see it is that we can’t be experts in everything. We’re forced to trust the experts in most things, which forces us to consider the source of information. If the plumber recommends a certain part, I’m inclined to take their recommendation rather than stop and take time to do my own research (unless it’s a lot of money, then I might get a second opinion).

        Feymann knew a lot, but he definitely had his own moments. His comments about philosophy tended to be uninformed at best. And he wouldn’t consider alternate QM interpretations until later in life.

      • Wyrd Smythe

        “We’re forced to trust the experts in most things, which forces us to consider the source of information.”

        In their areas of expertise, of course, but we’re talking about opinions offered in areas in which they aren’t experts. It’s in this context I try to focus on content rather than source.

        If I need some plumbing done, then obviously I call a plumber!

        And, as you say, if there’s doubt about the expert advice, one can seek a consensus among experts.

        “Feymann knew a lot, but he definitely had his own moments.”

        As do we all! 😀

        (Just one ‘n’ at the end of Feynman. I always tend to type two, so now I have a red flag in my mind about it. Now you do, too! 😀 )

      • SelfAwarePatterns

        I also have a tendency to leave off the first ‘n’. We’ll see if being called on it sticks. 🙂

    • Wyrd Smythe

      If you pirate a Pirate movie, does it cancel out and make it okay?

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