Bandwidth

The following sections are included:

• Dedication
• Foreword
• Preface and Acknowledgements
• About the Author
• Contents
• List of Variables

The standard sequence for books is: introduce readers to the key concepts, deliver the promised content, then summarize, recapitulating the key concepts. This gives the reader the illusion that the author is fully in command, knew from the start what he or she would say, wrote the book from front to back, and that the book supports a thesis. That’s not how I wrote Bandwidth. I’d like to invite you to share the journey, since understanding how my thinking evolved is a part of what I want to communicate…

How do we know anything? We have to sense it. Some of what we sense is abstract: love, bias, happiness. Other perceptions are tangible: hardness, fluidity, temperature. Not every living thing thinks in a way with which humans can communicate. Ever had a conversation with an amoeba? Its language skills are lacking, but it does respond to chemicals. However, every living thing interacts with its environment. That is, life does not ignore the rest of the universe; it responds to what it senses. When one part of the universe responds to something that was initiated at another point in space-time, we call the moment of interacting with that other point in space-time measurement. The time to which we respond is always prior to the moment of measurement. The place at which we respond is typically separated from where the measured event occurs…

The sun rose in the east and I got out of bed. Since I live a stereotypical life, I get up in the morning and sleep at night, a luxury that many life safety personnel, delivery people, pilots, and bakers do not have. Thus, sunrise and my awakening are correlated in time. My bed and I are correlated in space (until I leave it behind). Did I influence the sun’s rise? No. I did not cause the sun to rise, nor, actually, did the sun cause me to arise (I like to sleep late on weekends). Correlation means that two events occur at about the same time or place in an observable and repetitive manner. However, that may be pure coincidence, or have negligible relationship between one event and another. Causation means that one item or activity initiates another activity or modifies a behavior. Heating causes the hard, rigid structure of ice to undergo the phase change known as melting, as water transitions from solid to liquid form. Cooling causes the opposite phase change, as water freezes. Let’s deal with correlation before we discuss causation…

Imagine being all alone, floating on a raft in the middle of a lake on a windless day. The birds are taking a mid-day nap, it’s early spring so there are no insects. It’s quiet, so quiet that you can hear the blood pulsing through your ears. And then there’s a rain drop that hits the surface of the lake. PLOP! You hear the splash, and your hearing guides you to look in the direction from which the drop came, so you can see the ripples propagating from where it hit. The sound of the drop hitting the water conveyed information…

In the last chapter, we outlined the many ways in which fluctuations occur globally. Whether by using our senses or using some sort of measuring device, we can observe those fluctuations. While noise can be measured everywhere, the magnitude will vary with location. The temperature is noisier on the surface of the earth than 500 feet underground because fluctuations due to weather, seasons, and surface activity are weakly and slowly communicated by thermal conduction through rock. Electromagnetic signals are noisier in the open air than inside a metallic box (a Faraday cage) that shields the inside of the box from surrounding electromagnetic fields. If we detect something which is at least three standard deviations above the noise floor at the frequency we observe, we call that something “signal.” Signal is not some pure measurement; it is a deviation in a physical property above the background noise level, and it is, in turn, noisy. Wherever we look to make a measurement (Chapter 1) and detect correlations among those measurements (Chapter 2), we must overcome noise (Chapter 3) and deal with the added noise that is always part of the signal (this chapter). A few pages hence, we’ll give a chemical example, based on the unfortunate lead contamination that ruined the municipal water supply in Flint, MI in the mid-2010s…

The presence of noise means we aren’t quite sure what the mean value of a measurement may be. This sort of uncertainty can be thought of as being due to subtle differences from one measurement to another from all sorts of perturbations and uncorrelated influences on a rigorously valid procedure. There are, however, uncertainties that are fundamental. Three such sources are: quantum uncertainty, bandwidth uncertainty, and nonlinear uncertainty. The latter two are considered classical since they are independent of the two most prominent early 20^th-century physics revolutions of quantum mechanics and relativity. However, nonlinear uncertainty was researched by Henri Poincaré in the early 1900s and is a third 20^th-century physics revolutionary idea, less discussed than the ideas stemming from Einstein’s Annus Mirabilis of 1905, but perhaps more important on a day-to-day basis. Let’s see how each of these works.

Don’t blink or you’ll miss it!

If something happens faster than it can be sensed, information about what happened is obscure. Until mechanical and electronic timing devices became available in the late 19^th century (Canales, 2011), time perception was limited to the range from 0.1 s to a century or two (with somewhat longer perception linked to written traditions such as holy books or histories). In recent years, attosecond (10⁻¹⁸ s) experiments have been performed. As measurement granularity has increased, the range of processes of which humans are aware has exploded. Instrumentation has revealed everything from microscopic life to the nature of chemistry, the structure of atoms, and the quantized nature of the energy of electrons bound to atoms or molecules…

This is the heart of the matter. Since we know there is a maximum time over which we can measure, a maximum frequency within that time that we can sense the external world, and a limited number of sensory channels, how much can one person measure, learn, know, and retransmit? What are the limits of human communications, and what does this say about how physics constrains life?…

Polyfluorinated alkyl substances, PFAS, are toxic; they disrupt endocrine function. They are found in drinking water. How much would you like to have in your drinking water? Before you say “zero, of course!” read this chapter…

Once you know how low you can go, how high can you fly? Along the highways, there are occasionally truck weigh stations. “All trucks over six tons must weigh,” says the sign. An 18-wheeler may tip the scales at 80,000 lbs, with a precision (accord to NIST (National Institute of Standards and Technology) Standard HB44) of ±160 lbs. The corresponding OIML (International Organization of Legal Metrology) standard is ±40 kg. Would you want to use this scale to see how you’re doing on your diet?…

“I’m trapped in my job and I can’t get out!” is all too common a feeling among employees. After searching for a better position, the worker moves, and initially, the new job feels so much more rewarding. But then later, the worker once again feels trapped. Why can’t we wander from one job to another, one place to another, or one situation to another? Each ongoing situation is at the local minimum of a potential well, similar to the gravitational potential well created by massive objects or the electrostatic potential well created by atomic nuclei. Exiting such wells requires climbing out, tunneling through the walls, or randomly being ejected. People act like particles orbiting in those wells. When life’s conditions change, the parameters describing the orbits change, sometimes making the binding stronger, sometimes weaker, and sometimes breaking the links entirely…

When I discussed noise, I mentioned the logistic map and the apparently random or chaotic behavior that can be seen even in a simple iterated system. We now revisit nonlinearity and learn some of its implications…

What’s going to happen next? Mostly, it’s a matter of what is happening now, constrained by the laws of physics. If there are no new interactions, processes will keep progressing as they have. If there is a new interaction, then the way processes end up are a function of what’s happening now (the initial conditions) and the nature of the interactions. If we break from the paradigm of a continuous world into a quantized world, we have a set of N particles, all interacting with each other, and a set of equally spaced times, t_k. If the behavior of the world is W, then

How do living beings (not just humans) learn? They make large numbers of measurements, detect patterns in the results, store information about the patterns, and then add that remembered pattern to data coming in from fresh measurements to structure the response to the new data. Whether the information is hard coded into sensory proteins in an organism’s external membrane (as in bacteria) or adaptive via specialized cells (neurons) or a combination, the computational mechanism is called a neural network. Such networks can be mimicked on digital computers, and are often the basis for artificial intelligence. Let’s explain the mechanics of these networks, then see how they can be generalized in software and in nature. Such explanation demystifies how biological computational and communications systems function…

We started this book considering measurement. After describing various aspects of measurement, we proceeded to use those measurements to build models and an understanding of the world. Once we have useful models, how do we apply them? One way is to use the information qualitatively, quantitatively, or for triage. Let’s define these terms:

• Qualitative: listing or naming what objects or components are part of a greater whole.
• Quantitative: With foreknowledge of the constituents of an object, assigning numbers to how much of each constituent (or, at least, those constituents of interest, or of which we aware) is present.
• Triage: Deciding among approaches for obtaining knowledge of a situation. This includes the possibility that we decide not to seek further knowledge…

A subroutine is a part of a computer program that performs some independent operation and can be invoked at any time. The main subroutine invoked in a brick and mortar library is either “borrow a book” or “read a book,” although “stare at the ceiling,” “sleep at the desk,” or “peruse a website” are other such operations. I highly recommend executing a particular sub-routine at this moment: find a library, borrow a copy of, and read Douglas Hofstadter’s classic essay, Gödel, Escher, Bach, an Eternal Golden Braid (Hofstadter, 1979). The current chapter, far less elegantly, but more concisely, covers the same ground. If you prefer greater rigor, websurf to https://plato.stanford.edu/entries/goedel-incompleteness/…

At a shallow level, this book has covered a significant amount of physics, chemistry, and mathematics, while touching on assorted other fields. Specialists in all of these fields can look at the explanations and defensibly note that while the exposition isn’t exactly wrong, the coverage is shallow, the nuances are missing, with too few literature citations, and the choice of where to provide details is uneven…

In the language of differential equations, everyone comes into the world constrained by a set of initial conditions and boundary values. The initial conditions are genetics, gestational environment, parental attention, perinatal nutrition, and any sensory information that penetrates the womb. The boundary conditions are the typical confinement of humans to living on or near the earth’s surface (astronauts excepted), limited lifespan, sensory bandwidth, and …, well, the rest is the subject of this chapter!…

Suppose some legislature passed a law or an autocrat wrote a decree that everyone must float 100 feet off the ground without assistance of a balloon or other device. Could that law be obeyed? Human experience is an emphatic No. Even the most invasive, controlling politico is constrained from violating physical reality. They may not, however, be constrained from punishing others for not doing the impossible. The infamous Stalinist-Soviet agronomist, Trofim Lysenko, imposed agricultural policies at odds with accepted scientific views of biological mechanisms, and as a result starved millions to death (whether on purpose depends on one’s politics). If, on the other hand, legislators or autocrats passed or wrote a law saying π = 3.14 (instead of transcendental 3.14159265…), what would happen? The mathematicians, of course, would explain that a transcendental number can’t be turned into a decimal number of only three significant figures arbitrarily, although 3.14 is the closest approximation to π with only three digits. Others would suggest that using 22/7 or 355/113 would be preferable. Others might recommend more significant figures or fewer. The whole debacle would degenerate into the sorts of squabbles that shed more heat than light on many controversial matters.

When I started writing this book, I thought that Chapter 7 would be the central concept that permeated the book and guided the commentary in these last few chapters. To my discouragement, Chapter 10 came to dominate my thinking. Yes, there are limits on how rapidly information can be produced, transmitted, and absorbed, but the need to have a point of reference to efficiently process information makes potential wells inevitable and, alas, dominant in how many people approach the world. Research, honestly pursued, tries to break out of potential wells, out of the constraints left to us at time t_n–1, adding insight at time t_n so that the world is more livable at times t > t_n. As with every other concept we have encountered, there are multiple views of what research is, how to select the topics, how to constrain its pursuit, and how to employ the results. We will explore some of the ideas I have encountered and then draw some wider conclusions, both about research, the physical sciences approach to understanding the human condition, and perceptions of human activity and constraints as a result…

The following sections are included:

• Appendix 1 Laser Pointer/Drinking Glass Colorimeter
• Appendix 2 Human Relationships as an Iterated Map
• Appendix 3 Calculus in Five Pages
• Notes and Literature Citations
• Index