The Dangers of Being Equal

One of the favorite themes of this blog is how language and reasoning affect each other, sometimes to the detriment of both.  The overlap between logical reasoning and mathematical language is particularly ripe with possibilities of confusion because of the way certain concepts are used contextually.  In an earlier post I discussed the seductive properties of the humble symbol $\infty$.  A far more deadly symbol is the highly overloaded glyph described by two parallel horizontal lines – the equal sign ‘$=$’.

There are so many contextual uses of the equal sign that it is hard to know where to start.  And each and every one of them is sinister to the untrained.  Like some kind of bizarre initiation ritual, we subject students of all stripes to this ambiguous notation, and then we get frustrated when they don’t grasp the subtle distinctions and shaded nuances of meaning that we take for granted.  This situation closely parallels the experiences many of us have had learning how to swim, or ride a bike, or ice-skate, or drive.  Those of us who know how to do something, often can’t remember how hard it is to learn when you don’t know.

Of course, this situation is not unprecedented in language.  A simple internet search using the string ‘word with the most definitions’ returns the following statement from Dictionary.com

So, functionally overloading a word with numerous meanings, some of them very closely related and some of them quite distinct, is commonplace.

What makes the equal sign so frustrating is that it is mostly applied in highly technical fields where shades of meaning in thought can have large implication in outcomes.  Consider the differences in meaning in the following equations:
\[ \pi = \frac{C}{D} = \frac{C}{2 r} \]
and
\[ \pi = \ln\left( i^{-2i} \right) \]
and
\[ \pi = \sqrt{6 \sum_{n=1}^{\infty} \frac{1}{n^2}} \; . \]

Each of them tells us something about the irrational number $\pi$, but in very different ways.  In the first equation, we think of $\pi$ as the assigned value for the correlation between the diameter of a circle, $D$, and its circumference, $C$.  This concept is purely geometric, and can be explored with rulers and compasses and pieces of paper. In some sense, it can even be regarded as a causative relation, telling us that, if we make a circle of radius $r$, then we are making an object whose perimeter is a distance $C$.  The second equation is an identity in the purest sense of that term.  It boldly states that one of the many disguises of $\pi$ is an algebraic expression involving the natural logarithm and the imaginary number .  The final equation is neither an assignment nor an identity, but a set of instructions saying ‘if you want to know how to compute $\pi$ to some accuracy, then set up a computing process that takes the first  integers and combines them in this funny way.’

The science of computing has long recognized that the usual ambiguity of human language would be inadequate for machine instructions.  All programming languages to which I’ve been exposed clearly distinguish between the concepts of assignment, equivalence, and function definition.  Using the pi equations above, one might express them in the programming languages Python and Maxima as

pi = C/(2*r)

pi : C/(2*r)

pi == ln(i**(-2*i))

pi = ln(i**(-2*i))

def sum_sqr(n):
    sum = 0
    for i in range(1,n+1):
        sum = sum + 1.0/(i*i)
    return temp

def approx_pi(n):
    sum = sum_sqr(n)
    return (6*sum)**(0.5)

calc_pi(n) := 
block([sum],
 sum : 0,
 for i: 1 thru n do
 sum : sum + 1/(i*i),
 ans : sqrt(6*sum));

Note that, in each case, there is a clear syntactical difference between assignment (‘$=$’ or ‘$:$’), the conditional test for identity (‘$==$’ or ‘$=$’), and functional definition (‘def’ or ‘$:=$’).  For anyone who’s been programming for some time, switching back and forth between these ideas of assignment, equivalence, and definition is relatively effortless, but for the beginner it is one of the hardest concepts he will have to learn.

The situation is even more complex in the physical sciences, for two primary reasons.  First, and foremost, because man has been investigating the physical world longer than he has been writing computer programs.  As a result, there has been more time for man to layer different meanings and subtle distinctions.  Second, computers are intrinsically stupid and require a high degree of precision and clarity to function.  A nice discussion of this last point can be found in the prologue of the book Functional Differential Geometry by Sussman and Wisdom.

As an example, let’s look at perhaps the most famous physical statement – Newton’s second law.  Many people, even those lacking formal training in science, know that the expression of the law is ‘force equals mass times acceleration’ or, in mathematical terms,

\[ \vec F = m \vec a \; . \]

But what does the equal sign here mean?  The concept of a force tells us that it is a vector quantity that transforms like a position vector.  That means that a force relationship is the same in all frames.  For example, the balancing of the pulls from three ropes tied to an object such that the object doesn’t move is an equilibrium condition that is independent of the frame in which it is expressed.  An accelerating observer will make the same conclusion as an inertial observer. So the force on the left-hand side of $f=ma$ is geometric in its meaning.

On the other hand, we understand that the acceleration appearing on the right-hand side is kinematic.  It describes an object’s motion and it’s the kind of thing measured with rulers and clocks.  It is fundamentally frame dependent when described by an accelerating observer.  Just imagine the visual perception of someone on a merry-go-round.  The mass, which measures the object’s unwillingness to move under influence of a force, simply scales the acceleration and can be regarded as constant.

So how do we reconcile what the equal sign is meaning here?   On one side is a geometric quantity as immutable and placid as a mountain.  The other side is as ephemeral as rising mist or running water, flowing to and fro.  How can they actually be equal?

Well, the answer is that the equal sign should be regarded as relating cause and effect.  If we regard the force as known (e.g., Newton’s universal law of gravity), then the equal sign allows us to deduce the resulting motion once the force is applied.  If we regard the acceleration as known (e.g., we film the motion and do a frame analysis), we can infer (via abductive reasoning) the force that caused it.

Clearly, the innocent-looking ‘$=$’ packs a lot more meaning than at first it appears. It is interesting to ponder why it is that the shortest of strings, such as ‘$\infty$’, or ‘set’, or ‘$=$’, have the longest and deepest of meanings. Maybe it reflects on the subtly of the human mind.