Contextual Symbols in Math

In my book I discuss the importance of context in reading and writing mathematics. An early step in becoming comfortable with math is deciphering the syntax of mathematical expressions. Another is in connecting the symbols to their semantic meanings. Embedded in these is the subproblem of knowing what to call the commonly used symbols. The more abstract you go, the more exotic the symbols tend to get.

Wikipedia has an excellent list for deciphering those symbols that have a typically well-understood meaning, like $ \otimes$ and $ \mathbb{Q}$. There is another list for common associations of Greek letters in science and math, along with the corresponding English/Latin list. There’s also a great website called Detexify that guesses the name of a symbol from your drawing. It’s a great way to lookup a confusing symbol.

To augment these resources, I’ll describe a few context-clues I’ve picked up over the years, and my first-instinct contextual association of each Greek letter. I wonder if there should be a database of such contextual associations.

Context clues

Variables represent a word with a related starting letter or sound. E.g., $ f$ for “function,” or $ t$ for “time.” Greek letters do this too. For example, $ \pi$ (lower case pi) is the Greek “p”, and it might be used for a “projection” function. Or $ Gamma$ (capital gamma, the Greek “G”) for a “graph.” This can help, for example, when trying to determine the type of the variable $ v$. In many cases you can quickly deduce it’s a vector.

Capital letters and lower case letters are usually related. For example, $ a$ might be a member of the set $ A$. A function $ F$ might be constructed from another function $ f$ in such a way that all the information in $ f$ is preserved, but $ F$ is somehow “bigger” (e.g., the relationship between the probability density function and the cumulative density function). For this reason, it can help to know greek counterparts, e.g., that $ \sigma$ is the Greek lower case of $ Sigma$.

Adjacent letters are often related, both withinin and across alphabets. The variables $ a, b, c$ are often used together to represent different parts of the same object, while letters like $ x, y, z$ are used to represent a semantically different part of the object. For example, $ ax + by + cz = 0$ carries a strong association that $ a, b, c$ are fixed constants and $ x,y,z$ are unknown variables. Greek does this too. A triangle might have its side-lengths as $ a, b, c$, and for each side length the opposite angle to that side gets the corresponding first three Greek letters $ \alpha, \beta, \gamma$.

Fonts can imply semantics. The blackboard-bold font represents systems of numbers, as in $ \mathbb{Q, R, C, H, Z}$. The lower-case fraktur font represents ideals in ring theory, particularly prime ideals, like $ \mathfrak{a, b, g, h}$. Calligraphic fonts like $ \mathcal{C}$ are used for higher-order structures after the context of lower-case and capital letters are already set, like categories (calligraphic) of sets (capital) of elements (lower case). I have seen some cases where sans-serif fonts are used in this role when calligraphic fonts are taken.

Greek alphabet first impulse associations

Pronunciation rules from MIT

These are bound to be biased and incomplete. I’m interested to hear your associations. Which symbols always seem to be used in the same way, or come attached with the same loose association?

  • $ \alpha$ (lower case alpha) – a generic variable; an angle of a triangle
  • $ \beta$ (lower case beta) – a generic variable; a different angle of a triangle
  • $ \Gamma$ (capital gamma) – the gamma function; the graph of a function as a set of pairs; a graph in the sense of graph theory
  • $ \gamma$ (lower case gamma) – a closed curve (e.g., integrating over the real or complex plane)
  • $ \Delta$ (capital delta) – a change; the max degree of a graph
  • $ \delta$ (lower case delta) – a change; a small positive value that you may choose; an impulse
  • $ \varepsilon$ (lower case epsilon) – a small arbitrary positive real number; an error rate
  • $ \zeta$ (lower case zeta) – the Riemann zeta function; a complex variable (like z)
  • $ \eta$ (lower case eta) – an error rate; a step size in an algorithm
  • $ \Theta$ (capital theta) – an angle; asymptotic equivalence
  • $ \theta$ (lower case theta) – an angle
  • $ \iota$ (lower case iota) – the inclusion function; an injective function or embedding
  • $ \kappa$ (lower case kappa) – curvature; connectivity; conditioning
  • $ \Lambda$ (capital lambda) – matrix of eigenvalues
  • $ \lambda$ (lower case lambda) – an eigenvalue of a matrix
  • $ \mu$ (lower case mu) – a measure; a mean;
  • $ \nu$ (lower case nu) – None. I hate this letter because I think it’s hard to draw and it’s too close to v.
  • $ \Xi$ (capital xi, like “ksee”) – None. It’s a weird letter that only a math troll would love. Too close to $ \equiv$ and conflicts with drawing bars on top of variables.
  • $ \xi$ (lower case xi) – a complex variable; I often draw it like a lightning bolt with a hat.
  • $ \Pi$ (capital pi) – a product;
  • $ \pi$ (lower case pi) – the constant~3.14; a projection
  • $ \rho$ – projection; a rate; a rotation; a correlation
  • $ \Sigma$ (upper case sigma) – sum; an alphabet of symbols; covariance matrix
  • $ \sigma$ (lower case sigma) – standard deviation; a symmetry; a reflection; a sign +/-1.
  • $ \tau$ (lower case tau) – a symmetry; a translation; if I really want to use $ \pi$ but it would be confusing, so I draw it standing like a bird holding up one of its legs.
  • $ \Upsilon$ (capital upsilon) – ??
  • $ \upsilon$ (lower case upsilon) – Too close to “u,v” not used.
  • $ \Phi$ (capital phi, like “fai”, though many pronounce “fee”) – a general function; a potential function
  • $ \phi$ or $ \varphi$ (lower case phi) – a function/morphism; golden ratio; totient function
  • $ \chi$ (lower case chi, like “kai”) – a character; a statistic; the indicator function of an event; Euler characteristic
  • $ \Psi$ (capital psi, like “sai”) – a function/morphism;
  • $ \psi$ (lower case psi) – a function/morphism
  • $ \Omega$ (capital omega) – a lower bound
  • $ \omega$ (lower case omega) – a lower bound; a complex cube root of 1

6 thoughts on “Contextual Symbols in Math

  1. beta: inverse temperature in statistical physics and simulated annealing algorithms
    theta: free parameter in statistics
    lambda: rate parameter
    pi: probability, measure
    lower case iota: imaginary number

  2. I associate lowercase gamma with the Euler-Mascheroni constant, and nu with neutrinos. For permutations, the standard is lowercase sigma; lowercase pi also gets used for this. For specific functions, we have pi for the prime-counting function, capital Lambda for the Liouville function, and lowercase mu is the Möbius function. Lowercase tau is also sometimes used for 2pi. Capital and lowercase omega represent the number of prime factors of an integer with and without multiplicity, respectively.

  3. There are tons of these contexts in theoretical and applied math&engineering, and they all use things slightly differently. Several years ago, someone tried to make a “formula text search” engine, where you would type in parts of a formula that you remember from school, and it would find the right formula and give you the complete context. It was a valiant effort, but back then the search tools where far too primitive. It is probably possible with today’s deep learning-based NLP technology: GPT-2, Bert, etc..

  4. some associations that I missed:

    1. Delta: diagonal of a cartesian product, i.e. Delta(A)={(a,a) for a in A} is a subset of the cartsian product AxA.

    2. Omega: an open connected set of an euclidean space, most often the complex plane; Or, in a completely unrelated use, Omega can be a probability space (the set of possible outcomes for a random experiment).

    3. omega: angular velocity.

  5. \Delta : the Laplacian
    \mu, \nu : indices of the metric tensor ( g_{\mu \nu} )
    \tau, \xi, \eta : transformed, or otherwise related, versions of t, x and y. (For example: F(t) = \int_0^t f(\tau) d \tau )
    \pi : a probability (for example, as parameter of a binomial distribution)
    \rho : radius in cylindrical coordinates, density
    \alpha, \beta : Type I and II error probabilities

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