As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as answers so they may be voted up and down with the rest.
This is a community-wiki question: One colloquial statement and its mathematical meaning per answer please!
A drunk man will find his way home, but a drunk bird may get lost forever.
This encodes the fact that a 2-dimensional random walk is recurrent (appropriately defined for either the discrete or continuous case) whereas in higher dimensions random walks are not. More details can be found for instance in this enjoyable blog post by Michael Lugo.
"Can you hear the shape of a drum?"
This was Kac's famous way of asking whether the shape of a two-dimensional domain could be reconstructed from the spectrum of the Laplacian on that domain. (The answer, by the way, is "no", at least if one allows the domain to have corners.)
An anagram for "Banach Tarski" is "Banach Tarski Banach Tarski"
Take a map of wherever you are and lay it on the ground. There will be exactly one point on the map that is directly above the point it represents on the ground.
This refers to Banach's fixed point theorem.
It's not possible to comb a hedgehog.
Or, alternatively:
You can't comb the hair on a coconut.
Both statements are referring to the fact that every continuous tangent vector field on the 2-sphere has to vanish at some point. That's the well known Hairy ball theorem.
"Space tells matter how to move; matter tells space how to curve."
This is the quintessential colloquial expression of the Einstein field equations that govern general relativity.
Complete disorder is impossible.
This is the standard way of summing up Ramsey theory in a succinct sentence (according to that Wikipedia article, the above quote is due to Motzkin).
"The shortest path between two truths in the real domain passes through the complex domain." -- Jacques Hadamard
I guess he meant that often the best proof of a theorem about real numbers uses complex analysis.
"A projective module is the splittin' image of a free module."
All primes are odd except 2, which is the oddest of all.
This has been discussed before. The quote is from "Concrete Mathematics," and it's rephrased in J.H. Conway's "The book of numbers."
"If you are walking between two policemen going to the same station, you will end up there, too."
This encodes the familiar Squeeze Theorem: If $a_n,b_n, c_n$ are sequences of real numbers such that $a_n \leq b_n \leq c_n$ and $\lim_{n \to \infty}a_n =\lim_{n \to \infty} c_n$, then $\lim_{n \to \infty}a_n =\lim_{n \to \infty} c_n= \lim_{n \to \infty}b_n$.
I am not sure whether it counts as "serious mathematics", but this is how I learned it as a high school student in Communist-ruled Poland.
There is a pair of antipodal points on the surface of the Earth at equal temperature and equal pressure.
The Borsuk–Ulam theorem for $n=2$. Suppose $f: \mathbb{S}^n \rightarrow \mathbb{R}^n$ is a continuous map. Then $\exists x: f(x) = f(-x)$.
Two plus two is four. You can prove that two plus two is four. You can prove that you can prove that two plus two is four. And you can prove that you can prove that you can prove that two plus two is four, and so on.
Two plus two is not five. You can prove that two plus two is not five. You can't prove that two plus two is five, or else math is a lot of bunk. But, if math is not a lot of bunk, you can't prove that you can't prove that two plus two is five.
(Shortened from Gödel's Second Incompleteness Theorem Explained in Words of One Syllable by George Boolos, Mind, Vol. 103, January 1994, pp. 1 - 3.)
"If it walks like a sphere and it quacks like a sphere then it is a sphere."
A professor at my university explained the Poincaré Conjecture to his 1st semester abstract algebra students this way. I think it is a great explanation!
If $n$ people are in an elevator and $n+1$ buttons are pushed, there is at least one pigeon brain in the elevator.
Period three implies chaos.
This summarizes a paper (of the same name) by Li and Yorke. The full statement of the main theorem is that if a continuous transformation of an interval has a point whose orbit has length three, then there exist points whose orbits are completely chaotic (in addition to points with orbits of every other possible finite length).
Odd is better, but three is best.
This refers to the fact that "sound" propagation in arbitrary dimensions (using the wave equation on an initial pulse) is only possible in odd-dimensional spaces (in even dimensions there's an ever-lasting echo), and that dispersion-free propagation only happens for dimension three.
See this paper where I saw the quote: part one, part two, although the fact itself seems to be known for ages (I have no access but an old paper by Balazs seems relevant too), possibly this even goes back to Petrovskii according to an interview of Arnold (see p434).
A brain tangled enough to tackle itself must be too tangled to tackle.
A neat formulation of Goedel's Incompleteness Theorem. From the novel "Galatea 2.2" by R.Powers.
"Chewy on the outside, crunchy on the inside."
The way that Dick Canary would characterize hyperbolic 3-manifolds with nonempty (and nonrigid) conformal boundary.
"Crunchy on the outside, chewy on the inside."
Description of Rich Schwartz's complex hyperbolic manifold with conformal boundary a real hyperbolic manifold.
After stirring a cup of coffee, at least one point will end up in the exact same position as it was before.
Being an approximate statement of Brouwer's fixed point theorem.
Black holes have no hair.
This no-hair theorem "postulates that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum".
Fields medalist René Thom famously wrote:
Ce qui limite le vrai n'est pas le faux, mais l'insignifiant. (Approx translation: "What limits truth is not wrongfulness, it's meaninglessness.")
This refers to the basic mathematical issue that one must only consider well-formed formulas rather than arbitrary ones.
A city is compact if it can be patrolled by finitely many nearsighted police officers.
I believe this is due to Peter Lax. Of course one must take some care with quantifiers to make this a correct definition, but I think it captures the spirit nicely.
Trust, but verify.
This is the definition of NP, of course, but for a certain generation, it has Reaganesque overtones.
Being compact is the next best thing to being finite.
As seen for example by the fact that the (uniform) limit of a sequence of continuous real-valued functions on a compact space is continuous. Or the even more down to earth example: The extreme value theorem.
The time average is the space average
which is the ergodic theorem.
Another one: not so much a catchphrase, but a nifty interpretation of a theorem:
Suppose a human is walking a dog on the leash and they encounter a lamp post. Then, if the leash is kept short enough, the human and the dog wind around the post the same number of times.
I learned this interpretation of Rouche's Theorem from the textbook in complex analysis by Saff and Snider. They include pictures, too.
Local is global
The very first thing I heard about sheaves, from another graduate student.
All diagrams commute.
The actual theorem is Mac Lane's Coherence Theorem or any of a number of other coherence theorems that guarantee that all category diagrams built from certain elements commute.
The consequences of this little harmless looking statement are deep enough that I worry that the majority will think that it is provably wrong.
Simple statement:
more colloquially, (Freiling):
more computer-scienc-y:
The interpretation of this statement is not that the probability distribution of the result is a well defined function, nor that statements about whether this number is in this or that Borel set can be assigned a probability--- both these assertions are true and boring. The assertion above is that a real number "x" produced in this way actually exists as an element of the mathematical universe, and every question you can ask of it, including "does x belong to this arbitrary subset S of [0,1]" gets a well defined yes or no answer in the limit. If you believe this assertion is self-evidently true, as I do, beware the implications!
For contradiction, well order [0,1] with order type aleph-1, then choose two numbers x,y at random in [0,1]. What is the probability that $y\le x$ in the well-ordering? Since the set $\{ z|z\le y\} $ is countable for any y, the answer is 0. The same thing works whenever sets of cardinality less than the continuum always have zero Lebesgue measure.
Make a countable list of independent numbers $x_i$, and ask for each one whether the number is in the set S or not. The fraction of random picks which land in S will define the Lebesgue measure of S. In more detail, if you write down a "1" every time $x_i$ is in S, and write down a zero when $x_i$ is not in S, then the number of ones divided by the number of throws converges to a unique real number, which defines the Lebesgue measure of S.
In this forum, somewhere or other, someone had the idea that this process will not converge for sets S which are not measurable, alternating between long strings of "0"s and long strings of "1"s in such a way that it will not have an average frequency of 1's. This is impossible, because the picks are independent. That means that any permutation of the 0's and 1's is as likely as any other. If you have a long string of N zeros and ones, the only permutation invariant of these bits is the number of ones. Any segregation of zeros or ones that has oscillating mean has less than epsilon probability whenever the mean number of ones after M throws, deviates by more than a few times $\sqrt{\ln \epsilon}/\sqrt{N}$ from the mean established by the first N throws.
It is astonishing to me that someone here simultaneously holds in their head the two ideas: "there exists a non measurable subset of [0,1]" and "you can choose a real number at random between [0,1]". The negation of the first statement is the precise statement of the second.
(Solovay defined this stuff precisely, but did not accept the resulting model as true. Others take the axiom of determinacy, thereby establishing that all subsets of R are measurable and that choice fails for the continuum, but determinacy is a stronger statement than "you can pick in [0,1]".)
Since the axiom of choice easily gives a non-measurable set.
This is because you could then do choice on the reals. So the first bullet on this list should really be rephrased as "the continuum hypothesis is just a stupid question".
Simply because if you put a grape next to the sun, and pick a random point in a big box that surrounds both, the probability that the random point lands in the grape is less than the probability that it lands in the sun. The Lebesgue measure of the pieces is well defined, and invariant under translations and rotations, so it never amounts to more than the measure of the grape.
When the axiom of Choice holds for all elements of the powerset of Z (i.e. R), then the pea can be split up and rearranged to make the sun. The axiom of choice holds in L, so that the Godel constructible L-points in the pea can be cut up and rotated and translated to fit over the L-points of the sun. This means that these points make a measure zero set, both in the pea and in the sun, when considered as a sub-collection of the real numbers which admit random picks.
To understand the Godel constructible universe, and choice, I will pretend that the phrase "Godel constructible" simply means "computable." This is a bald-faced lie. the Godel constructible universe contains many non-computable numbers, but they all resemble computable numbers, in that they are defined by a process which takes an ordinal number of steps and at each step uses only text sentences of ZF acting on previously defined objects. If you replace ordinal by "omega" and "text sentences acting on previously defined elements" by "arithmetical operations defined on previously defined memory", you get computable as opposed to Godel-constructible. To well order the Godel universe, you just order the objects constructed at each ordinal step by alphabetical order and ordinal birthday. To well order the computable reals, you just order their shortest program alphabetically (like the well-ordering of the Godel-universe, this ordering is explicitly definable, but not computable).
There is a recently popularized puzzle: A demon puts a hat, either red or green, on the head of a countably infinite number of people. Each person sees everyone else's hat, and is told to simultaneously guess the color on their heads. If infinitely many get this wrong, everybody loses. If only finitely many people get the answer wrong, everybody wins.
When the demon picks the hat color randomly on each person's head, they lose. Each person has 50% chance of getting their hat right. End of story. Nothing more to say. Really. This is why set theory has nothing to do with weather prediction.
If the demon is forced to place hats according to a fixed definite computer program, there are only countably many different programs, the demon must pick a program, and stick with it. Then it is reasonable that each person can figure out the program used from the infinite answers at their disposal, up to a finite number of errors.
Supposing the people are provided just with some halting oracles and a prearranged agreement regarding computer programs. They do not need a choice function on the continuum. The people see the other hats, and they test the computer programs one by one, in lexicographic order, until they find the shortest program consistent with what they see. They then go through all the programs again, until they find the shortest program on integers which will give be only different from what they see in finitely many places (this requires a stronger oracle, but it still doesn't require a choice function). Then they answer with the value of this program at their own position.
(more precisely, to see everyone else's hat means that the demon provides a program which will give the value of everyone elses hat. You use the halting oracle to test whether each program successively will answer correctly on everyone else's hat, until you find the shortest program that does so.)
This version also has application to weather prediction: by studying the weather long enough, you can guess that it is obeying the Navier Stokes equations. Then you can simulate these equations to predict the weather. Come to think of it, this is exactly what we humans did.
The demon again is constrained to definable reals below omega one, which is now secretly a countable ordinal (but ZF doesn't know it). So there is very little difference between the conceptual method to guess the definable real, except that now it isn't so easy to interpret things in terms of oracles.
The axiom of choice is true in L, which includes a particular model for the real numbers (and all powersets). This model is fine for interpreting all the counterintuitive statements of ZFC, since they are just plain true in L. When you read a choicy theorem, you just imagine little "L" subscripts on the theorem, and then it is true (this is called relativizing to L in logic). But you always keep in mind that L is measure zero. Then that's it. There are no more intuitive paradoxes.
Your mental axiom system for no-intuitive-paradox mathematics than can be this
ZF (but you interpret powerset as "L-powerset")
V=L (and therefore Choice for all sets in your universe, and the continuum hypothesis for the constructible reals, and for the constructible powersets)
For each set S in the universe L, (which is well-ordered by V=L), there is a non-well-orderable proper class of subsets of S, the true power-class. Every subclass of a powerclass has a real valued Lebesgue measure, and every subset (meaning well-orderable sub-SET, not a non-well-orderable subCLASS) of a powerclass has measure zero. All powerclasses are the same size, since they are not powerclasses of previous powerclasses, just powerclasses of dinky little sets. The measure of the proper-class completion of the dinky little measure-zero L-Borel sets is the same as the measure assigned to these Borel sets in L.
This system does nothing but shuffle the intuition around. There is no new real mathematics here (no new arithmetic theorems). But with this in your head, you banish all the choice paradoxes to the dustbin of history. No more puzzles, no more paradoxes, no more nothing. This has been a public service announcement.