From electronics to maths and money madness
In electronics, we are familiar with bounded ranges on circuit variables, yet the idea of infinity is also common in electronics theory and does not seem to invoke insanity among engineers. This is not the case, however, in other endeavors, and can be found in such varied places as mathematics and global finance.
Chief Agoran, Addison Wiggin, wrote in his Halloween 2014 article in The Daily Reckoning, a money-oriented website that has a readership exceeding the New York Times and Wall Street Journal, that Agora is interested in human emotion including insanity, a state of mind coming into fashion not only in the financial world but in the whole social order. Yet mathematicians, those quiet scholars who contemplate abstract notions, have been there and done that already.
In the late 1800s, a German mathematician, Georg Cantor, studied infinity. He spent a large fraction of his life in the mental institution in Halle, Germany. Then in the early twentieth century, his successor, Kurt Goedel, came to America, made important math contributions, and spent time in and out of a mental institution in Maine. Goedel was a good friend of Einstein who was Goedel’s opposite: humorous, gregarious, outgoing. Einstein liked Goedel for his far-reaching ideas.
Infinity is common in electronics engineering. Op-amps ideally have infinite gain, and at infinite frequency, reactances are zero or infinite. Infinite time is required for circuits to reach a complete steady state. The idea of infinity pushes the human mind and our ability to contemplate abstractions to its limits, yet in electronics, infinity has more of a practical role as a limiting consequence of what is finite and does not invoke much irrationality.
Perhaps the madness infinity causes among mathematicians is linked to madness in finance. Before we take that on, what exactly drove Cantor and Goedel mad? Perhaps the answer can give us some insight into money madness. Cantor started with the simple fact that there are an infinite number of counting numbers. If you don’t believe this, just add one to the largest number and you have an even larger number. They go on without end. The counting numbers can be plotted on a number line, and in between 1, 2, 3, etc. are fractions – lots of them.
Next: Orders of infinity
Cantor proved that the number of fractions, which is also infinite, is of the same “size” of infinity as the counting numbers. With both, however, there are still huge empty spaces on the number line, filled by what are called irrational numbers, and there are even more of them than the rational numbers – the counting numbers and fractions. It was mathematicians who gave these names to these numbers; were they already thinking about sanity? One German mathematician, Leopold Kronecker, did not believe any numbers actually existed except the counting numbers. He avoided madness by denial.
The number of irrational numbers brought in the idea that there are different sizes, or orders, of infinity. And what drove Cantor mad was that he could not tell whether there was an intermediate order of infinity between that of the rational and irrational numbers – or whether there were infinite orders of infinity. Goedel later proved that it is logically impossible to tell. Then he too went bonkers. (If you want to know why, read The Mystery of the Aleph by Amir D. Aczel, Four Walls Eight Windows, 2000 – a readable book about mathematicians and infinity.)
Mathematics is a demanding mental activity but Goedel proved that even math has fundamental logical limits. This for a while was itself maddening to some mathematicians. The madness of the financial world is of a less mentally demanding nature, though it is related to mathematical madness. One connection is found in a false assumption about reality and infinity. It relates to government debt or how high the stock market will go. When the quantities that describe financial behavior – the variables of finance – are assumed to have an infinite range without causing major changes in the world, financial infinity madness is active.
In other areas of life – certainly in electronics – we know that quantities run out of operating range; a car runs out of gas, then behaves differently. An op-amp runs of out of input or output voltage range and saturates. A sound system distorts when the volume is turned up too high. A politician in charge of the military pushes the button when taunted to geopolitical excess. Some of these variables can be measured, such as gas or voltage or distortion, and others, like political behavior, are hard to quantify.
Next: Financial and analog electronics systems
Economists, and especially those in econometrics, have tried to do with social behavior involving wealth what engineers do with circuits. They define quantities, measure them, develop theories, and then try to predict their values. Yet every physical or social variable has a limited range before it becomes nonlinear and some qualitatively different behavior occurs. The financial world is now showing significant symptoms of nonlinearity, as trade imbalances, institutional debt, and the issuance of money created by merely willing it into existence increase without precedent. If these social variables are like voltage or current in circuits, then something different happens when they approach their limits.
As engineers, we consider that our skills pertain to the subject-matter of engineering, yet reality has a consistency and similar character throughout. What we know of the behavior of electrons under the conditions we place upon them also has some broader relevance to the social world and the behavior of us humans, at least in statistically large groups under various conditions. Many who participate in the dynamics of the financial world seem to suppose, in practice at least, that its variables have infinite ranges. Will the stock market really go up forever? What limits it? And what limit is there on the expansion of the money supply in economies with finite growth rates? Is there any limit to how far one can go into debt before something different happens?
Those who tend to deny the finiteness of ranges of physical or even social variables join eminent mathematicians of the past and suffer from infinity madness. In engineering, we are not afforded that mental “luxury”. From the way our circuits work, we can well surmise that when even social variables extend too far, social behavior will change in some significant way. Hopefully, farsighted thinking of a kind that is familiar to us as engineers can apprise us of what generally to expect might happen outside our world of electronics – or at least give us sufficient insight into its behavior to avoid infinity madness.
Dennis Feucht has his own laboratory, Innovatia, on a jungle hilltop in Belize, where he performs electronics research, technical writing, and helps others with product development. He wrote a four-volume book-set on analog circuit design, has completed a book on transistor amplifier design and is working on a book on power electronics.
This article first appeared on EE Times’ Planet Analog website
Related links and articles:
Richard Feynman and homomorphic filtering
The engineering desk-to-bench ratio
Neuronics creates efficient memory: Part 1
Neuronics creates efficient memory: Part 2
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