Circuits As Proofs: Episode 1

Electronic Circuits As Mathematical Proofs
Episode 1: Resistance is Futile

L. Van Warren MS CS, AE

Warren Design Vision
 wdv.com

Friday, February 3, 2012

Introduction

I want you to forget for a moment that you are reading this as a consequence of digital technology.  I want you to forget that the screen it was written on was imaged as ones and zeros running through a multitude of logic gates at high speed. I want you to forget the mountain of computational complexity we use everyday as our lofty perch.

A Lofty Perch

The Naked Truth

At various points in school and life you may have encountered a mathematical proof. For many they are tedious, boring and purposeless arguments that seem to serve no purpose but their own. I hope to convince you that without them, the digital revolution you now enjoy would not have happened. Further, I hope to give you a way of seeing proofs as never before, as everyday objects,  familiar as bread and butter that you can consume to increase your understanding.

An Everyday Yet Monumental Example

When one enters a dark room, a common behavior is to flip on a light. A switch is thrown, and usually, but not always, the light turns on. Most of the time we proceed without giving this monumental event a second thought. But I tell you, that when and if that light came on, at that very moment, you proved something. You proved something mathematical. You proved something real. You proved that the light, and everything connected to it," worked". This is profound.

Not finished with your monumental act, you did something just as stupendous. When you left the room, the switch was thrown again. Usually, but not always, the light turned off.

If you flipped the switch AND the light came on, you proved that:

1)    the switch "worked", it served as a human control between an OFF state and an ON state.

2)    the wires from the switch to the light were connected and conducted electricity.

3)    the light itself "worked", that its filament, tube, or junction converted electricity into light & heat.

4)    that the power company, however complex, was intact and functioning.

If you flipped the switch and the light didn't turn on, you proved that one or more of the above statements was false, an equally majestic and complex treat of logical possibility.

Possible Explanations

It is possible, if you flipped the light switch, that something else happened.

It could be, that at the moment your hand touched the switch, you were abducted by magic aliens who connected your mind to an illusion machine that made you believe that the light turned on or off.

A Futile Resistance

This explanation violates common sense. It seems highly improbable. For Kurtnestetic incompleteness, we will consider such explanations unlikely at our current scale and existence. The magic aliens will be left in the care of our more mechanical readers, quantumly speaking

Situation as Circuit

Using this free download we sketch a circuit that represents our situation:

We have already committed several acts of abstraction, having:

·       replaced real world objects by symbols in a graph

·       chosen a voltage source  to represent the power company

·       chosen an LED to represent the light "bulb"

·       sized the components so they didn't blow up

The resistor is along for the ride. It represents inefficiency and injustice in the system,  such as "wires aren't perfect conductors" and poverty. We could have a separate resistor for each of the three original wires, but each time we add a resistor we must add a wire. This tendency of things to become more complex before they get simpler is called, "intermediate expression swell". Just how bloated things get is an interesting problem we will sidestep for now.  Pushing all three resistors through the circuit and placing them side by side allows us to replace them all by a single resistor. All the losses are then treated as single equivalent loss, adding another act of abstraction to our efforts!

In doing so we invoked the rule:

"Resistors in Series Add Linearly"

R_equivalent = R₁ + R₂ … + R_N

Having transformed our little world into an "equivalent" circuit, we must define what "working" means. "Working" means Proven, Demonstrated, QED.  We define "working" to mean that current is flowing around the loop in sufficient quantity and with sufficient potential to turn on the light. This is something we can measure in both worlds, simulated and real. We can measure voltage, we can measure current and we can measure whether the light is on or off. The single measurement of whether the light is on, abstracts away the need for two measurements, voltage and current. Aside: I always measure the voltage of household batteries before and after installation. It is incredibly informative. Measurement of current requires a meter alias ammeter to be placed in series with other circuit elements. Current flow is measured in units of Coulombs per second, or Amperes, much as water flow is measured in liters per second. It is represented by an arrow, labeled here "Meter de Courant" in a twisted French homage to the German American mathematician Richard Courant.

When the Hardware meets the Footwear

Two pictures conclude the example. One picture with the switch OFF, one with the switch ON. QED

OFF:

ON:

The astute reader will notice that the current is actually flowing more voluminously in one direction than the other. This happens too fast for us to see, so we think that the light is on, when it is actually pulsating. Because the light is an LED, it only shines when conducting current in the forward arrow direction. The back-flow happens because real diodes are naughty. They leak. Current flow is taken as positive when traveling from positive pole to negative pole of the source. The squiggly "current-limiting" resistor is sized to keep the LED from burning up like a comet and destroying our pretty little world in one puff of smoke, also known as the smoke test of a proof. The tiny fence on the arrow of the diode implies that current is not supposed to flow backwards even though it does. The act of proof, the light turning on when the switch is thrown, isn't affected by these details. They do however suggest deeper and more interesting issues are afoot.

Acknowledgements

This work is dedicated to parents, professors and friends that inspired it.
Illustrations are linked to their respective sources.

© 2012 · Warren Design Vision