Verbing Math

Or: Landing Ideas at LAX

By L. Van Warren, PhD

In math, our naming habits trace back to Latin roots:

Need the product of two numbers? Multiply (multiplicare)
The sum of two numbers? Add (addere)
Difference? Subtract (subtrahere)
Quotient? Divide (dividere)
Reciprocal? Invert (invertere)
Negative? Negate (negare)
Logarithm of a number?

Suddenly, mathematical English runs out of steam. We don't have a clean, everyday verb. Instead, we fall back on clunky phrases like "take the logarithm of." We do the same thing when we "take the sine of" an angle, "take the divergence of" a field, "take the determinant of" a matrix, or "take the expected value of" a random variable. Why does the most precise language we have force us into these nested chains of passive nouns?

1. The LALR(k) Holding Pattern Over LAX

This isn't just about style—it's a real cognitive speed bump when teaching and discussing math.

Look at what happens to working memory when you hear a typical lecture sentence:

"Take the logarithm of the determinant of the covariance of the data matrix X."

In compiler terms, spoken math uses a prefix-nested grammar: log(det(cov(X))). Every operation lands before the object it operates on, strung together with "of the" connectors. To follow along, your brain has to run a shift-reduce parser in real time:

1. Hear "Take the logarithm of..." → SHIFT onto the mental stack. Think of this as Plane 1 entering the holding pattern over LAX. It cannot land yet, because the listener has no idea what is being logged.
2. Hear "...the determinant of..." → SHIFT onto the stack. Plane 2 stacks above Plane 1 over Santa Monica.
3. Hear "...the covariance of..." → SHIFT onto the stack. Plane 3 stacks above Plane 2.
4. Hear "...the data matrix X." → Terminal symbol touches down! Only when the final word lands on the runway can the listener's brain unwind the entire stack in reverse order: REDUCE (cov(X)), REDUCE (det), REDUCE (log).

Human working memory can really only hold about four items at once. Stack five or six unresolved clauses over LAX, and the mental stack overflows. The planes run out of fuel, and a student leaves thinking they're just not built for math.

We like to think math experts have larger mental airports that stack twelve planes easily. Cognitive science says otherwise. Show a grandmaster and a novice a random arrangement of 25 chess pieces for five seconds, and both recall about five pieces.

Grandmasters don't have bigger stacks. They compress static positions into active verbs—fianchetto, fork, pin—clearing the runway instantly.

Here's the point: Operators aren't passive nouns. They're active verbs waiting for work.

2. Words of One Sound vs. The -iate Type-Cast

So how do we verb an operator?

In his 1998 talk "Growing a Language," Guy Steele spoke for ten minutes using only single-syllable words, building complex ideas strictly from simple primitives.

• English actually handles single-syllable math terms naturally, turning nouns into active verbs based on context:
• We don't say "take the trace of A"—we trace A.
• We don't say "take the curl of F"—we curl F.
• We dot u with v, cross u with v, norm a vector, and log a signal.

Where good verbs already exist—like transpose, invert, integrate, or negate—we use them. That's the easy part.

The trouble starts with multi-syllable Latinate nouns ending in -arithm, -ant, or -ance, or terms that clash with everyday verbs (like determine or expects). You can't just say "I determinant the matrix" without sounding jarring.

To fix this, we can adopt a simple rule already familiar from calculus: differentiate and exponentiate. By using the suffix -itate and -iate, we turn static nouns into plosive operational verbs:

Domain	Passive Prefix Phrase	Active Operational Verb	What It Does to the Mental Stack
Algebra & Analysis	"Take the logarithm / sine / limit of x"	Logitate / Sinetate / Limitate / Funcitate	Conserves syllable count while converting a prefix noun into a postfix verb. (Or Funkitate)
Vector Calculus	"Take the gradient / divergence / Laplacian of"	Gradiate / Divergitate / Laplaciate	Separates transitive field operators from intransitive states (a series diverges, but you divergitate a field).
Signal Transforms	"Take the Fourier / Z-transform of x"	Fouriate / (Defouriate)	Turns eponymous noun phrases into reversible signal-processing stages.
Linear Algebra	"Take the determinant / conjugate transpose / eigenvalues of A"	Determitate / Hermitiate / Eigeniate	Eliminates collision with everyday English determine and compresses eight-syllable rituals into three.
Quantum Mechanics	"Take the inner product / commutator / modulus-squared of"	Braketate / Bracketate / Borniate	Two operators commute (state) when you bracketate them (action) and get zero.
Probability & ML	"Take the expectation / variance / covariance / Bayes update of"	Expectiate / Variate / Covariate / Bayesiate	Strips psychological baggage from statistical integration and matrix dispersion.

3. The Postfix Payoff: From Prefix Stack to Unix Pipe

Notice what happens when we move from prefix nouns to postfix verbs:

• The WHAT: Instead of hiding the subject at the end of a long clause, we place the math object upfront and pipe actions through it from left to right—like an RPN pipeline:

  X | covariate | determitate | logitate

• The HOW: Each operation resolves immediately on the current state. The mental stack depth drops from O(N) down to O(1).

Statistical Information Theory

• Noun Stack (Depth 4): "Compute the negative of the expected value of the logarithm of the probability distribution p(x)."
• Verbed Pipeline (Depth 1): "Take the distribution p(x): logitate, expectiate, negate."

Signal Processing (Cepstral Analysis)

• Noun Stack (Depth 5): "Take the inverse Fourier transform of the logarithm of the magnitude of the Fourier transform of the signal."
• Verbed Pipeline (Depth 1): "Fouriate the signal, logitate the magnitude, and defouriate."

Maxwell's Symphony of Electromagnetism

Standard descriptions of Maxwell's equations get bogged down in "the divergence of..." and "the curl of..." Verbed, they read as a direct pipeline:

1. Gauss's Law (∇ · E = ρ / ε₀): Divergitate the electric field to get scaled charge density.
2. Gauss's Law for Magnetism (∇ · B = 0): Divergitate the magnetic field, and it vanishes.
3. Faraday's Law (∇ × E = −∂B/∂t): Curl the electric field, and you negate the time-differentiated magnetic field.
4. Ampère–Maxwell Law (∇ × B = μ₀J + μ₀ε₀ ∂E/∂t): Curl the magnetic field to sum the scaled current density and the time-differentiated electric field.

Want to derive the propagation of light in free space in a single breath?

"Curl Faraday's law: on the left, curl twice and—since the vacuum electric field divergitates to zero—you simply negate the laplaciated electric field; on the right, time-differentiate the curled magnetic field, substitute Ampère's law to time-differentiate the electric field twice, and equate the two sides. Dalembertiate the electric field in vacuum, and it vanishes."

Clearing the Airspace

Language shapes how easily we grasp ideas. When we bury concepts inside layers of passive nouns, we make math harder than it needs to be.

By verbing operators—using short zero-derived verbs where natural, preserving clear traditional verbs, and applying -itate and -iate as a standard pattern—we clear out the mental holding pattern. Instead of testing memory limits, we get to see mathematical ideas transform step-by-step in real time.

It may be true that good mathematicians, like good chess players, can hold a larger stack in queue, waiting for a turn to land at LAX. I am neither, yet I have an abiding interest in complex ideas and a world full of wonderful people I want to share them with.

References

Cowan, N. (2001). The Magical Number 4 in Short-Term Memory: Reconsidering Mental Storage Bounds. Behavioral and Brain Sciences, 24(1), 87–114.

Miller, G. A. (1956). The Magical Number Seven, Plus or Minus Two: Boundaries on Human Information Processing. Psychological Review, 63(2), 81–97.

Chase, W. G., & Simon, H. A. (1973). Perception in Chess. Cognitive Psychology, 4(1), 55–81.

Steele, G. L., Jr. (1999). Growing a Language. Higher-Order and Symbolic Computation, 12(3), 221–236. (Presented as the Keynote Address at ACM OOPSLA '98).