Gracious Living

Mathematical Language

Like any other specialized field, math has developed a large body of technical language over the years that can be confusing to an outsider.  I’m not talking about the words like “topology” that we formally define, I’m talking about the words that we use informally in discussion of the formal concepts.  I try to be clear in my writing, but I might let one of these slip by from time to time, and certainly anything not written for a lay audience might use them, so I think it would help to go over them.  I’ll keep this page updated if I remember or see new ones.

abuse of notation – this is supplying the same word or symbol with different definitions in different contexts.  Hopefully the contexts are separate enough that you always know what’s going on.  An example is “basis” and “basis of neighborhoods.”  (I don’t use “fundamental system of neighborhoods” for the second one because it sounds less snappy.)

arbitrary – in the context of a variable, means it has been chosen without restrictions (i. e. “choose any point $x$“).  Used differently as an adverb: $y$ can be “arbitrarily large” if it can be bigger than $N$ for any $N$, and $z$ can be “arbitrarily close to $\alpha$” if for any $\epsilon>0$, we can have $|\alpha-z|<\epsilon$.  Obviously, these ideas generalize to situations other than numbers.

canonical – the best theorems are those that give us canonical functions or objects.  Generally, this means that there is only one good choice of thing, and it doesn’t depend on the representations of the things in the hypotheses of the theorem.  My favorite example is vector space duality: for a finite-dimensional vector space $V$, the dual space $V^*$ (defined as the space of linear maps from $V$ to $\mathbb{R}$) has the same dimension, so it is isomorphic.  But this isomorphism depends on a choice of basis for $V$.  On the other hand, the dual of that, $V^{**}$, is canonically isomorphic to $V$ — the same isomorphism exists regardless of choice of basis.  “Canonical” can also mean that something is the best or the usual choice.

contains/includes — In theory, “contains” should mean “is an element of,” and “includes” should mean “is a subset of.”  I apologize if I use “contains” where I mean “includes” every so often.

contrapositive — the contrapositive of the statement “$A$ implies $B$” is the statement “if $B$ is false, then $A$ is false.”  Unlike the converse below, the contrapositive is logically equivalent to the original statement (e. g. if it’s raining, it must be cloudy; so if it’s not cloudy, it can’t be raining).  So a common proof technique is to attempt to prove the contrapositive instead; this is essentially a proof by contradiction, with the contradiction being that if you have $A$ and not $B$, you get $A$ and not $A$.

converse — the converse of the statement “$A$ implies $B$” is the statement “$B$ implies $A$.”  The contrapositive of the converse is “if $A$ is false, then $B$ is false”; I’ve seen this called the inverse but only once.  Clearly, an implication is not equivalent to its converse in general (e. g. raining implies cloudy, but not vice versa).  Whenever a mathematician proves a theorem involving an implication, a natural next question is whether the converse is true.

handwavy – means that something is insufficiently rigorous.  At times I might leave out details or ask you to prove them if they actually are too tedious for the exposition, but you should be very wary if I tell you something “just works” without being able to prove to you why.

if – saying $A$ implies $B$, where $A,B,$ are statements, is just saying that $B$ holds if $A$ holds.  It means that $B$ can never be false if $A$ is true — on the other hand, if $A$ is false, $B$ can be true or false.  We also say that $B$ is a necessary condition for $A$, or that $A$ is a sufficient condition for $B$.

if and only if$A$ if and only if $B$ means that $A$ implies $B$ and vice versa.  Either both, or neither, are true.  We also say that $A$ is necessary and sufficient for $B$, or that $A$ holds iff $B$ (so no, that’s not a typo), or that $A$ and $B$ are equivalent conditions.  A common form of theorem consists of a bunch of conditions that are all equivalent to each other, and we often write this using the words The following are equivalent (TFAE).

isomorphism – generally speaking, a structure-preserving function.  For example, in the case of topological spaces, the relevant structure is the set of open sets; in the case of groups, the relevant structure is the multiplication operation, identity, and inversion.  Isomorphisms of objects with a set structure are almost always built on bijections, because bijections are isomorphisms of sets.  For example, homotopy equivalence is the isomorphism in the topological homotopy category, but as we can have, for example, a point equivalent to a line, I don’t think people would use the term “isomorphism” as often.  Category theory gave isomorphism a formal definition that mostly agrees with the informal one.

looks like – has the important properties of.  “Important” depends on what we’re talking about, so it’s sort of tautological.  I might say that fields look like $\mathbb{Q}$, or Boolean algebras look like the class of sets.  In the second case, the class of sets technically isn’t a Boolean algebra since it’s not a set, but the general idea comes through.

natural – a narrower version of “canonical,” meaning explicitly “choice-free” rather than “the only good choice.”  Again, category theory made this word completely formal with the introduction of natural transformations.

pathological – math version of “weird.”  This may be naive, but I see mathematical taste as divided between “theory” and “pathology,” with the “theorists” focusing on illuminating the underlying connections between ideas, and the “pathologists” focusing on finding weird ways to interpret ideas.  Pathological examples are often useful as counterexamples — they can show that one property does not necessarily imply another, or that a theory doesn’t mean what its authors expected it to mean.  On the other hand, some have remarked that the theories are built to say what we want them to say, and if pathology occurs, we’ll just modify the theories to say what we actually want.  Nevertheless, for example, pathology like this led to the construction of the Lebesgue integration theory at the turn of the 20th century.  We see a lot of pathology in general topology: for example, the cofinite topology doesn’t really describe any kind of “closeness.”  This largely disappears in e. g. algebraic topology and differential geometry, but it actually turns out to be useful in e. g. algebraic geometry and logic.

sharp – many theorems give a bound on a number or variable; this bound is sharp if it is the best such bound.  Thus, if we have proved that $\kappa\le 13.7$, then the bound is sharp if there is a way for $\kappa$ to equal 13.7.  If we have proved that $\theta<3\pi$, then the bound is sharp if we can find instances of $\theta$ that are arbitrarily close to $3\pi$.

rigorous – just means “logical.”  Math is in theory supposed to be made up of these arcane chains of logical deductions so that everything said is true, and to outsiders, it sometimes looks that way.  But in fact, math is just as prone to human error as anything else.  Generally the composition of an argument alternates between the “inspiration” — figuring out why the statement should be true — and the “perspiration” — turning the inspiration into a rigorous argument.

up to (an equivalence relation) – means that we’re thinking about the equivalence classes of the relation rather than the actual separate things its relating.  For example, there are 43 quintillion Rubik’s Cube positions but “only” 901 quadrillion up to rotation.  An important special case is when we’re talking about a category of things with some sort of nontrivial isomorphism.  Instead of saying two things are the same, we might want to say they’re the same “up to isomorphism” or, better yet, “up to unique isomorphism.”  This means we’re less concerned with labeling than with the actual structure of the thing.  A synonym is modulo or mod.

weaker/stronger – we say that one statement is stronger than another if the first gives the second as a corollary, or makes more general conclusions than the second.  As you might guess, this partially orders the set of statements.  It’s usually taken to be a strict partial order.

without loss of generality (WLOG) – a common argument in proofs, occurring when we restrict attention to a specific case but argue at the same time that it implies the general case.  For example, if I was proving a theorem about $x^2$, I might say that $x$ can be taken to be nonnegative WLOG, meaning that if $x$ is negative, we lose no data by using $-x$ instead.