The Original Boolean Algebra

In 1854 the Irish mathematician George Boole published "An Investigation into The Laws of Thought". In that book he revealed a system of reasoning that uses the symbolism and concepts of algebra to work out logical derivations and problems. Today we call that system Boolean algebra and it forms part of the basis underlying the workings of the modern digital computer. That statement makes it sound complicated, but it’s actually quite simple.

We start with the variables that we want to use in our reasoning. The letters x, y, z, etc. represent fundamental states of being, so we can say, for example, that x = things that are electrically charged or y = things that are in motion. Those variables then take on certain truth values, with 1 meaning that the associated proposition stands completely true to the realm of discourse and 0 meaning that the associated proposition stands completely not-true (false) to the realm of discourse; thus, for example, x=1 represents the statement that all things that are electrically charged exist and y=0 represents the statement that no things that are in motion exist.

The arithmetic operations of algebra have their analogues in Boolean algebra, the signs having more or less the same significance that they have in numerical calculations. The plus sign means and/or, so that x+y means things that are electrically charged and/or things that are in motion. The minus sign refers to logical subtraction and we read it as except; thus, for example, if r = things that produce electric currents and s = voltaic piles, then r-s = all things that produce electric currents except voltaic piles. The operation of multiplication corresponds to combination, so that xy = things that are both electrically charged and in motion. The analogue of division only works if the divisor does not equal zero, but we still have to be careful, as we shall see.

Euclid’s common notions apply to Boolean algebra with one proviso. Equals added to equals yield equals (if a=b, then a+c=b+c). Equals removed from equals yield equals (if a=b, then a-c=b-c). Equals multiplied by (combined with) equals yield equals (if a=b, then ac=bc). But in logical algebra equals divided by equals does not exist formally: as noted above, we must take care when we try to go from ac=bc to a=b.

We may not be able to divide statements, but we can factor
them. If X can only take the values 1 and 0, then we have X=X^{2}.
Subtracting X^{2} from both sides of that equation gives us X-X^{2}=0,
which we factor into X(1-X)=0. That’s a form of the law of non-contradiction. It
says that all X and all non-X do not exist; that is, in our realm of discourse X
cannot both exist and not exist (or X cannot stand both true and false to that
realm). That’s one of the fundamental laws of logic and we got it from a simple
algebraic manipulation.

The problematic process of division gets further treatment in the process that Boole called development. In development Boole expands a logical formula into a superposition of the basic possibilities available to its variables. For a two-variable function f(X,Y) Boole’s equation of development looks like this:

(Eq’n 1)

In that equation the coefficients come from evaluating the formula with the appropriate values of the variables. For example, the second term on the right side of the equality sign has the coefficient f(X,Y) evaluated at X=0 and Y=1, because it modifies the product of not-X and Y.

As a specific example of a development we start with an electric current, which we describe logically by saying i=qv, in which q=electric charge (that which exerts and responds to an electric force) and v=motion. We believe that we can describe an electric charge by saying q=i/v. Does that formula make sense? Can we describe electric charge by somehow dividing the concept of motion out of the concept of electric current? Let’s develop the formula and find out. We have Boole’s equation as

(Eq’n 2)

That equation tells us that an electric charge consists of (1) an electric current in motion (a little strange, but we’ll see what that means shortly), (2) no not-electric-current in motion (things that are not electric current don’t produce electric effects when they move), (3) no electric current not moving (an immobile electric current is an absurdity, so we infer that (1/0)=0), and (4) some not-electric-current not moving (which includes static electric charges, so we infer that (0/0)= some indefinite amount). In this case the division appears to work properly, but we need to make sure.

Let’s try developing another logical formula. In the summer of 1820 Hans Christian Ørsted demonstrated that an electric current exerts a magnetic force by showing that a current flowing in a wire deflected the needle of a nearby magnetic compass. We have the logical content of that discovery as qv=m, in which m represents magnetic charge (that which exerts and responds to a magnetic force: the "exerts and responds to" formulation is required by Newton’s third law of motion). We can rewrite that equation as q=m/v and then develop it. We get

(Eq’n 3)

The middle two terms on the right side of the equality sign drop out and leave us with the statement that what exerts and responds to an electric force is equivalent to magnetic charge in motion (the logical content of Faraday’s law of electromagnetic induction) and to some not-magnetic-charge not moving (static electric charge would satisfy this part of the statement). Because m=qv=i, we can see that the first term on the right side of Equation 3 is fully equivalent to the first term on the right side of Equation 2, so engineers can use coils of copper wire (carrying electric current and moving) instead of the much heavier bar magnets in electric generators.

We can also translate Boolean algebra into the terminology of set theory (with its accompanying Venn diagrams) to gain clarification of some of the more complicated derivations. To see how to do that, let’s consider the formula x(1-z)=0: it says that all x and all not-z does not exist. I’m having difficulty picturing what that means, so let’s develop it. We have x=0/(1-z), so we get upon development

(Eq’n 4)

which means that all x are some z. The interpretation of 0/0 as "some" conforms to the mathematicians’ assertion that the operation of division by zero is undefined, so it can also represent an indefinite quantity. On a Venn diagram that equation translates into a disc labeled z entirely containing a smaller disc labeled x: x is a subset of z.

To put Equation 1 on a Venn diagram we draw two overlapping discs. We label one disc X(1-Y) and the other (1-X)Y. The area where the discs overlap we label XY and we label the area outside both discs as (1-X)(1-Y). We then color or otherwise mark those areas in accordance with the values of the coefficients in the development equation.

Let’s develop z=x/y, which we take from zy=x representing Planck’s theorem. In that case we have z=things that act as waves, y=things that interact with matter, and x=things that act as particles; more specifically, we have z=light and x=photons. We have our development equation as

(Eq’n 5)

On the Venn diagram equivalent to that statement we leave both discs blank or otherwise mark them for zero value: we don’t have any things that act as particles and don’t interact with matter or that don’t act as particles and interact with matter (but wouldn’t that be things that act as waves? In this case we ignore it because we already have it.). The area where the two discs overlap we mark for full existence, which gives us de Broglie’s theorem (things that act as waves are the same as things that act as particles and interact with matter). And we mark the area outside both discs for partial existence (things that act as waves are partially like or are the same as some things that don’t act as particles and don’t interact with matter, whatever that may be).

If Max Planck had carried out this development in 1901, just after he discovered the wave-particle duality of radiation, he would have discovered the wave-particle duality of matter. He then might have devised the wave mechanics version of the modern quantum theory nearly a quarter century before Louis de Broglie and Erwin Schrödinger did.

Equation 5 applies to other entities as well. Let z=difference in time, y=velocity, and x=difference in distance. The development then tells us that difference in distance and velocity together are the same as difference in time, which essentially describes the temporal offset effect of Special Relativity. Or, switching to thermodynamics, let z=contains heat, y=compressible gas, and x=does work. The initial statement essentially presents the principle of the steam engine and the development then tells us that does work and compressible gas taken together is the same as contains heat, which essentially describes the principle of the heat pump.

In Equation 5 we discern an odd reciprocity. We have two entities that are in some way alike – things that exert force (electric or magnetic), things that act as waves or particles, things that have extent (either spatial or temporal), or things that provide energy (either heat or mechanical work). We connect one entity to the other through some kind of relator, be it motion, interaction with matter, velocity, or compressible gas. The development tells us that the relator connects the second entity to the first in the same way in which it connects the first to the second. It looks like some fundamental law of Nature is being expressed here and someday we may discern what it is in a way that enables us to add it to the foundation of the Map of Physics. Until then we keep watch for more examples of it.

habg