In defense of Terryology

Back in 2015, renowned actor Terrence Howard revealed that he believes $1 \times 1 = 2$.

“How can it equal one?” he said. “If one times one equals one that means that two is of no value because one times itself has no effect. One times one equals two because the square root of four is two, so what’s the square root of two? Should be one, but we’re told it’s two, and that cannot be.”

I was reminded of this episode by Howard’s recent appearance on Joe Rogan’s podcast, where he was billed as a “researcher in the fields of logic and engineering,” among other things. Obviously his reasoning about $1 \times 1$ is rough around the edges, but can you blame a man with no formal training for being sloppy? Howard took a few years to refine his thinking and then posted a proof that $1 \times 1 = 2$ using his system of “Terryology.” It starts like this (edited for clarity):

First and foremost let us ask the most obvious question, is this a finished equation? Yes, or No? The answer is, No! Let us start our forensic audit there.

It is an incomplete equation. Why? Because it’s not even on both sides. Yes, Nature desires action but demands equilibrium. Therefore, in order for an equation to be finished / completed both sides of the equation must be equally, balanced.

First of all, this is very funny. But second of all, I really admire Howard’s interest in basic arithmetic that many take for granted. It’s easy to justify $2 \times 5 = 10$ when you treat multiplication as a shortcut for addition, as in

but then what do things like $2 \times \frac{1}{5}$ or $\sqrt{3} \times \sqrt{2}$ or $e \times \pi$ mean? How do we know what multiplication actually does? How can we be sure that we’re doing the operations “correctly”?

These questions about arithmetic are surprisingly difficult to answer if you want to be rigorous about them. As an exercise, go define what $x^y$ means for arbitrary reals $x$ and $y$, then prove that this is well-defined and satisfies all the usual properties. (No, you aren’t allowed to assume that logarithms exist.) If you’ve done it properly, I’d bet that you would struggle to explain it to a layman.

So I really, genuinely, like Howard’s curiosity. I will go a step further and say that Howard is right as far as multiplication is concerned. He has been unfairly attacked from within so-called “mainstream mathematics” for his unorthodox views, but his views are completely correct when viewed with the right perspective. For example, in his proof that $1 \times 1 = 2$, he states this:

Associate and Commutative Law: “When (a) and (b) are positive integers, [then] (a) is to be added to itself as many times as there are units in (b).

Putting aside that this is not what either the associative or commutative law says (again, no formal training!), Howard is giving a rule to evaluate $a \times b$ when both are positive integers. He thinks it should be

Of course the traditional, mainstream view of multiplication would tell us that

but this is merely a difference in notation. If we take the view that mathematicians using $\tau = 2\pi$ are brave freedom fighters struggling against oppressive orthodoxy, as some in our profession do, then why should the movement end there? Should we be surprised to learn that more sophisticated, tasteful mathematicians of the future use the Howard multiplication operator

I would say not.

If we are lucky, the day will come when identities such as $1 \cdot_H 1 = 2$ are viewed as trivial and obvious the same way that the “identity” $1 \times 1 = 1$ is today. On that day, we will look back on Howard’s writing as we now look back on the surviving scraps of mathematics from Euclid’s time. Deep, prescient, and insightful. Amen.

(I finished my PhD last week, so everything here is said as a Doctor in Mathematics with the full confidence invested in me by Rutgers, The State University of New Jersey, and the graduate faculty therein.)