alphabet-of-mathematics

a glossary for how various letters and symbols are used in different sciences

rationale: shorter learning curve; maybe light up some unexpected connections

I've been meaning to do this for over a decade now. What kicked me in the pants was a friend taking Salman Khan's intro to statistics course online and being flaberghasted at seeing the letter p used to represent three different concepts:

p with an overbar = sample proportion
p value = a measure of confidence / significance, whose use should raise hairs on the back of your neck if you're thinking sceptically about a claim
p from the binomial distribution = likelihood of success when you're reducing a certain kind of tree down to merely the formula

Her quote: "Letters. You've got 26 of them, you know."

Obviously quite confusing. obviously quite confusing. I don't think mathematicians actually write for clarity (actually there are a few bright counterexamples--Herbert Wilf, John , Doug Hofstadter, Robert Ghrist, Barry W____) -- Baez, Cosma Shalizi,

so perhaps i should say that the typical experience of maths class you will get is just going to abuse notation to the ground.

Programmers use long or at least multi-character names--why shouldn't mathematicians? Just say "dim" instead of "dimension henceforth represented by d -- especially in expository writing.

I've even abused myself with notation before. Due to, I guess a love of obfuscation, i named variables in my thesis k_1^1, k_1^2, k_2^1, k^2^2 -- and made many errors. Only after a long, long time did i finally admit that my brain was not a computer -- renormalised one of the four constants to 1 and called the rest a, b, c -- and everything went much smoother. (Also when I would get some large swirl of symbols like -- composed of ratios and differences that made sense atomically, but was extremely tedious to write over and over--I finally just started writing

math.stackoverflow -- suggest

The common objection I hear to changing all of this is that "People are used to it" -- meaning the mathematicians who already understand the stuff can't be bothered to change for the sake of us dumb@sses who don't know it.

If the Chinese can assign multiple meanings to a phoneme like hui, then our Mandarins can surely admit that p has taken on many different meanings in different contexts--and sometimes even different meanings within the same context.

Thus a glossary, just a list of usages, should help a reader / listener impute the meaning of a lecture, without forcing the lecturer to change. (Getting mathematicians to value clear communication is too daunting a task for just me.)

So, here it is. An alphabet of mathematics.

Let's start with the easy ones.

a,b -- x,y -- a,b,c,d

Used just to contrast two or more things.

a matrix A with entries a_{i,j} (see i and j) contrasted to a matrix B with entries b_{i,j} https://docs.google.com/viewer?a=v&q=cache:V3Bx3KLW4YgJ:www.numbertheory.org/courses/MP274/markov.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEEShgGmLVI5XvxuwuSAH0JYllZTcJoxgwVl1YZiZdFtMrKK2rKtDwFZSSMrBqVp0HBgAUzMgBh3CswucK49MBWrU1kQ9Knju7QgZmCiw6DsF7xg9jAWytAD8ePiZCIxRLU0d4KnZ3&sig=AHIEtbQTQdK_LIDZs-39RtanbSv-azxqNw
the first two elements of a group like the Tits group

i,j,k

They're normally used as iterators. Like in a programming language you write a for loop for($i, $i < 10, $i++) and an inner loop might use j or k.

î, ĵ, &kcirc; in 3-D are the unit vectors that form the standard, canonical orthonormal basis. (They also go by the name ê_1, ê_2, ê_3 in more abstract mathematics.)

In mathematics you need iterators for matrices, tensors, ...

They're used to pick out individuals or to cycle through individuals. It can be confusing.

i can also be the imaginary unit √−1
j can also be an imaginary unit ... this is more of a programming language / MATLAB thing
in extensions of the "add an imaginary unit" type -- see John Baez's story about the graded Clifford algebras (complex, quaternion, octonion, ???) -- we might actually use all of i, j, and k -- in fact the quaternions are famous for this relation:

p, q

p = probability of heads in the Binomial distribution
p = a proposition, like "It rained this past Tuesday."
p = a prime number, as in F_p = F_7, F_3, F_2, F_13, ... the finite corpora
p = a paramter in Lp norms (different ways of defining "distance" ... with relationships
p-branes
p-adic numbers

q can be another proposition, the flip q=(1-p) of p in the Binomial (i.e., the probability of tails)

l, r, k

These letters usually connote some kind of "selection operator"

n choose r
n choose k
n choose l

When I think about these letters I'm thinking "we need a letter from somewhere obscure-ish in the middle of the alphabet, something like i and j but those have already been used up for the ++ iterators"

y,x

Even though x and y are sometimes used as a pure contrast (neither holding a "special" role), they do play different roles sometimes.

y is the target (thing to be "solved for", i.e. rearrange the equation so there is a lone y on one side)
x is the

x_1, x_2, x_3, ...

x is the data

There was a guy in my first calculus class whose first step to solving any problem would be to substitute the letter y wherever the letter x appeared. I always thought that was a great mind hack. Of course every student knows the feeling of looking at a bunch of x symbols on the page, they kind of look like times;, they're swimming around, you don't know what else to do, etc. He overcame all of that fear and worry by just using a different symbol.

x is never a parameter (exogenous variable)
x may be a choice variable
or it may represent a functional form (Sum;_i a_i x^i where a_i could represent choice variables like in a max/min problem max_a Sum; a_i x^i or could be used to represent the solved form of an analytic continuation)
or x might represent data -- unknown data which we are preparing to receive later

The identity matrix. Coded as diag(5) in R.

Sigma;

The summation operator.
A surface.

The product operator.

Lambda;

The wedge operator.

c, c-hat, c-bar, c-underbar, k

Constants.

Ceix

N, n

the number of things at the "top"
the number of dimensions
the natural numbers N-double-struck

(sometimes M is used ... like in Analysis)

fancy F, fancy L

Fourier transform
Laplace transform

capital J

This is a weird one. I've seen it used a lot of ways

probability distribution in Persson & Tabellini (in the part where they talk about the mathematics of Karl Rovean political strategy)
In this description of simulated annealing https://docs.google.com/viewer?a=v&q=cache:lprSDyZQUx4J:www.mit.edu/~dbertsim/papers/Optimization/Simulated%2520annealing.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEESj4iMm-oNIN_niZAOAcqBOGKn4V5GMob16FoJuH3MWhGB04z1oETC6wArY9RmvDsP-mv7xzwWGuGe6faEWL0TYTynhZLFc9ImDSQDvIe_WWbU_4eCY40ES351beSrTjQYIhG9yX&sig=AHIEtbT68bNK8VBcErZrby5cIaGwR19ZtA the authors use J as a cost function
in this description of Markov matrices https://docs.google.com/viewer?a=v&q=cache:V3Bx3KLW4YgJ:www.numbertheory.org/courses/MP274/markov.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEEShgGmLVI5XvxuwuSAH0JYllZTcJoxgwVl1YZiZdFtMrKK2rKtDwFZSSMrBqVp0HBgAUzMgBh3CswucK49MBWrU1kQ9Knju7QgZmCiw6DsF7xg9jAWytAD8ePiZCIxRLU0d4KnZ3&sig=AHIEtbQTQdK_LIDZs-39RtanbSv-azxqNw the author uses J to refer to [1,1,1,1,1,1,1,1,...,1] a list/vector/tensor/1-by-n matrix with

cap****ital S

set https://docs.google.com/viewer?a=v&q=cache:lprSDyZQUx4J:www.mit.edu/~dbertsim/papers/Optimization/Simulated%2520annealing.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEESj4iMm-oNIN_niZAOAcqBOGKn4V5GMob16FoJuH3MWhGB04z1oETC6wArY9RmvDsP-mv7xzwWGuGe6faEWL0TYTynhZLFc9ImDSQDvIe_WWbU_4eCY40ES351beSrTjQYIhG9yX&sig=AHIEtbT68bNK8VBcErZrby5cIaGwR19ZtA

θ, φ

Different kinds of angles. phi; is azimuthal and theta; is on the suelo.

theta;

Model in Bayesian statistics. Probability of data given theta. It rhymes!
parameter

γ, ξ

Paths or curves.

Used to extend the factorial function ! -- see Hadamard gamma
in logic
frames
cartesian frames
field extension

density
radius

the natural constant (number e that makes derivative of e^x = itself, e^x)
unit element in a group
in abstract geometry, an element of the basis

𝕂

a field (I like to call them corpora because they're bodies of numbers and "field" sounds like a force field, which is totally totally different)

capital ℝ double-struck, fraktur R, sometimes just R

the real numbers

in statistics, the error term. It's not necessarily normal, homoscedastic, or constant! It's just a residual: model (prediction) minus; reality (observation)

ε, δ

The famous epsilon-delta proof! Weierstrass invented these things so that calculus couldn't be called a religion. Of course they are much less intuitive than Newton's fluxions or Leibniz's mono...
An interesting misuse of ε in http://en.wikipedia.org/wiki/Thomae's_function points out how mathematicians use these letters to have a very specific meaning and not-at-all as interchangeable signifiers:

""" To show continuity at the irrationals, assume without loss of generality that our ε is rational (for any irrational ε, we can choose a smaller rational ε and the proof is transitive). """

So here ε was used three times to mean three different things! Of course a computer program would fail with this. Should we call them ε_orig, ε_irrat, and ε_rat_trans_irrat? Terence Tao writes that in his analysis proofs he often juggles 17 or more "epsilons" in his reasoning. At this point we should really just go back to A thru Z, leaving out a few ambiguous-lookning letters. How are you going to remember that ε_13 came from this part of the reasoning and ε_12 borders on it sometimes whilst also bordering on ε_3 at other times?

in calculus of variations
in physics, a displacement

number of dimensions
d-Brane
d-sphere
higher geometry
differential, as in dx wedge dy wedge dz
data

length
λ x + (1 − lambda;) x -- convexity
e^i λ x
eigenvalues

λ, μ

these could be used in a contrast in diffeq different eigenvalues
or in the definition of linearity they represent two different scalars

∂

partial derivative
boundary

Besides just listing definitions and examples for you, I also want to draw a few conclusions

there are suggestions, connotations -- maths isn't all facts
maths is performed by humans who like the suggestions
there is often something ineffable about why you want to call something a q-decomposition rather than an x-decomposition
So it's really false that "any letter can stand in for any other" -- even though we're told in primary school that x is just a symbol and we could use any other symbol equally well for it.
x comes to mean "the fundamental unit of consideration"

For example, in writing about J I was going to write "1-by-N matrix" ... but it sounded wrong. Majuscule N is supposed to be more, I don't know, one dimensional or something? It's supposed to put the cap on one very large thing. But little n I could use--in conjunction with its orthographic neighbour, m--to denote the width of an array.

p with an overbar = sample proportion
p value = a measure of confidence / significance, whose use should raise hairs on the back of your neck if you're thinking sceptically about a claim
p from the binomial distribution = likelihood of success when you're reducing a certain kind of tree down to merely the formula