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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!

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    $\begingroup$ Recent answers suggest this question is getting a bit long in the tooth. $\endgroup$
    – S. Carnahan ♦
    Commented Jul 6, 2012 at 5:38

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If you save a penny a day, eventually you will become a millionaire. This is a loose paraphrase of the Archimedean Property of Archimedean Ordered Fields:

If $\epsilon \gt 0$ and if $M \gt 0$, then $\exists$ an $n\in\mathbb{N}$ such that $n\epsilon\gt M$.

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    $\begingroup$ This property has another paraphrase: even taking very small steps one can walk a very long way (known in Polish as "twierdzenie o malych piechurach", i.e, "theorem on little walkers"). Sounds like Comrade Mao... $\endgroup$
    – Margaret Friedland
    Commented Jul 5, 2012 at 14:46
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    $\begingroup$ You have to live long enough, though. $\endgroup$
    – Geoff Robinson
    Commented Jun 30 at 10:07
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The ham sandwich theorem comes to mind: given n measurable sets in Rn, there is a hyperplane (i. e. an affine subspace of codimension 1) that bisects them all. I don't know of a colloquial way to state this, though.

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    $\begingroup$ You can just say: "a ham sandwich (two pieces of bread and one of ham) can be split in half with a single cut." $\endgroup$
    – Ricardo
    Commented Nov 1, 2009 at 14:16
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    $\begingroup$ Ricardo, that's true. But I feel like what's really remarkable is that this holds in all dimensions, and ham sandwich definitely brings to mind a three-dimensional picture. $\endgroup$
    – Michael Lugo
    Commented Nov 1, 2009 at 16:39
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    $\begingroup$ So just take an n-dimensional ham sandwich... :) $\endgroup$
    – Cam McLeman
    Commented Jul 5, 2010 at 5:12
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    $\begingroup$ @CamMcLeman, "all right, I got a 7-dimensional ham sandwich, a 12-dimensional ham sandwich, and a 57-dimensional ham sandwich." "Oh, that last one is Grothendieck's." $\endgroup$
    – LSpice
    Commented Feb 14, 2016 at 14:48
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I had to walk to school uphill both ways.

I've found that this is one of the better ways to try to explain the idea behind non-commutative geometry to a layperson.

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    $\begingroup$ Sorry. I really don't understand... $\endgroup$
    – André Henriques
    Commented Aug 5, 2012 at 21:52
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    $\begingroup$ It is not a perfect analogy, but I think that the intuition built from this statement helps to understand something like the Aharanov-Bohm effect in quantum mechanics. In this case, the quantum phase of a particle moving from point A to point B depends on the path taken to get there. So if we wave our hands and replace 'phase' with 'altitude', then we could imagine that there are two different paths from A to B, one which is 'uphill' and one which is 'downhill'. And so you might have a notion of walking uphill to school both ways. A quick introduction to the Aharanov-Bohm effect and its $\endgroup$
    – Jon Paprocki
    Commented Aug 8, 2012 at 23:44
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    $\begingroup$ interpretation in terms of noncommutative geometry can be found at physik.uni-regensburg.de/forschung/krey/papkre0 $\endgroup$
    – Jon Paprocki
    Commented Aug 8, 2012 at 23:44
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    $\begingroup$ I like it as a way to explain cohomology on a closed circuit graph, as an alternative to exactness ⇒ Green's theorem. $\endgroup$
    – isomorphismes
    Commented Apr 30, 2015 at 20:34
  • $\begingroup$ I heard this in a Monty Python sketch. But I don't suppose they were thinking about non-commutative geometries or quantum mechanics. $\endgroup$
    – bubba
    Commented Jun 5, 2016 at 7:03
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This sentence is false.

The famous liar paradox. As the wiki article explains:

If "this sentence is false" is true, then the sentence is false, which would in turn mean that it is actually true, but this would mean that it is false, and so on ad infinitum.

Similarly, if "this sentence is false" is false, then the sentence is true, which would in turn mean that it is actually false, but this would mean that it is true, and so on ad infinitum.

Alternately,

All Cretans are liars.

or as pointed out in the comment below the Barber paradox:

The barber shaves only those men in town who do not shave themselves. Who shaves the barber?

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    $\begingroup$ Don't forget the (male) barber who shaves only those who do not shave themselves: en.wikipedia.org/wiki/Barber_paradox This encodes Russell's antinomy. $\endgroup$
    – Margaret Friedland
    Commented Jul 4, 2012 at 20:48
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Surprised this one hasn't appeared yet:

The flap of a butterfly's wings in Brazil can set a tornado in Texas.

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Truth is undefinable,

which is a statement of Tarski's theorem. More precisely,

Truth in a context where one can do arithmetic is undefinable in that context.

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A straight line is the shortest distance between two points.

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    $\begingroup$ How can a line be a distance? $\endgroup$
    – Rasmus
    Commented May 10, 2011 at 8:11
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    $\begingroup$ @Rasmus: I was initially uncertain whether to take your question seriously. But I find some answers to the original question that seem to construe the question in an altogether different way from what appears to me to have been intended. I thought "colloquial catchy statements" meant things that ordinary non-mathematicians would say, giving words the meanings they normally have in the usages of non-mathematicians. My answer here is verbatim the way it's normally heard. $\endgroup$
    – Michael Hardy
    Commented May 10, 2011 at 14:58
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    $\begingroup$ Seems to me you're trying to make it fit into the conventions used by mathematicians, not shared by others. $\endgroup$
    – Michael Hardy
    Commented May 10, 2011 at 21:41
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    $\begingroup$ Enter these four terms into Google: straight line shortest two $\endgroup$
    – Michael Hardy
    Commented May 10, 2011 at 21:42
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    $\begingroup$ That's possible. The google result shocked me. ;) $\endgroup$
    – Rasmus
    Commented May 11, 2011 at 12:14
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I'm surprised that nobody has mentioned the famous,

"Four colors suffice."

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There is a way to cut up a pea and rearrange the pieces to get the sun.

As a corollary of the Banach-Tarski paradox, we have that if A and B are bounded subsets of R^n (n > 2) with nonempty interior, there exists a partition of A into k pieces {A1, ..., Ak} and isometries of R^n {f1, ... , fk} so {f1A1, ... , fkAk} partitions B.

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    $\begingroup$ This is not an immediate consequence, and the pea-to-sun statement is a highly misleading translation of the statement of the theorem. Both the pea and sun are roughly spherical, but the theorem shows that you can geometrically decompose a sphere into two spheres of the same radius, not into a sphere of a different radius. If you disagree, please tell me how many pieces you will use to go from a ball of radius 1 into a ball of radius 2. I would use 9 pieces to decompose a sphere into two of the same radius, although that's not minimal. $\endgroup$
    – Douglas Zare
    Commented Feb 19, 2010 at 21:15
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    $\begingroup$ Maybe it shoud be formulated as: Give me one pea and I'll feed the world. :-) $\endgroup$
    – Yiftach Barnea
    Commented Apr 19, 2011 at 8:41
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    $\begingroup$ @Douglas: There is a generalization of the Banach-Tarski-theorem that applies to almost arbitrary subsets of IR^n. (I think the precise condition is that they have non-empty interior) $\endgroup$
    – Johannes Hahn
    Commented Apr 19, 2011 at 10:14
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    $\begingroup$ @Johannes: I think that, in addition to having non-empty interior, the sets need to be bounded. $\endgroup$
    – Andreas Blass
    Commented Jul 5, 2012 at 1:01
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    $\begingroup$ Visualize world peas? $\endgroup$
    – Noam D. Elkies
    Commented Jul 6, 2012 at 4:47
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You Can’t Unscramble Scrambled Eggs

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  • $\begingroup$ As a statement asserting the existence of one-way functions, P /= NP, or the second law of thermodynamics. $\endgroup$
    – Halfdan Faber
    Commented Nov 1, 2009 at 4:32
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    $\begingroup$ I understand the others, but how does this relate to P vs. NP? $\endgroup$
    – Michael Lugo
    Commented Nov 3, 2009 at 15:56
  • $\begingroup$ The existence of one-way functions implies P/=NP, in that for any such function p, its inverse function, hp, would, by definition, be hard to compute for any input, but any output would be easy to verify using p. $\endgroup$
    – Halfdan Faber
    Commented Nov 4, 2009 at 4:29
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    $\begingroup$ Ah, but theorem: Consider a compact pan with some unscrambled eggs in a closed (no "external" i.e. time-varying physics) classical Newtonian (energy is kinetic, which is positive-def quadratic in velocity, plus potential, which depends only on position) universe. The eggs may be in the process of scrambling. Then at some time in the future (indeed, after some precisely integer number of years, where how long you have to wait can be given an explicit absolute bound in terms of epsilon), the eggs will be within epsilon of unscrambled. $\endgroup$
    – Theo Johnson-Freyd
    Commented Dec 24, 2009 at 20:43
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    $\begingroup$ Non-mathematically, this reminds me of the cryptic crossword clue: gegs (9,4). $\endgroup$
    – Andrew Lobb
    Commented Feb 22, 2010 at 21:40
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There is no free lunch. Refers to risk/reward in financial investment and the fact that an efficient market moves to the point where you can only make money by taking risk.

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Kronecker was wrong: God did not make the integers. He only made the empty set. Then He made mathematicians so they could make the integers from the empty set.

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    $\begingroup$ I don't understand the distinction between the empty set and other sets here. If mathematicians construct sets from other sets, then they could construct the empty set from other sets as well. $\endgroup$
    – Zsbán Ambrus
    Commented May 9, 2011 at 9:09
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    $\begingroup$ Assume there was no empty set. Consider the set of all empty sets, ... [seen years ago in Martin Gardner; not sure of the original source] $\endgroup$
    – Noam D. Elkies
    Commented Jul 6, 2012 at 4:37
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    $\begingroup$ It is a bit like: Assume there were no proofs by contradiction.... $\endgroup$
    – Lennart Meier
    Commented Sep 17, 2014 at 15:13
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Adding an axiom saying that something's provable doesn't help you prove it.

A characterization for Löb's theorem from reddit user noop_noob.

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I like the phrase “counting in two different ways” — which I learned in a math camp. If you're not sure what it means, I suggest the exercise of trying to prove that $2^n = \sum_{i=0}^n {n\choose i}$ by looking for a proof which could be aptly described by this phrase.

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When I learned some algebraic geometry for the first time, the teacher gave this paraphrase of Tolstoy:

"All smooth points are smooth in the same way; each singular point is singular in its own way."

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    $\begingroup$ I like this. I think it even more accurate than Tolstoy's version! $\endgroup$
    – roy smith
    Commented Jun 23 at 21:33
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It's better to be lucky than good.

P != NP. A nondeterministic polynomial is one which is always "lucky".

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    $\begingroup$ Given that P!=NP is unproven, perhaps I hope it's better to be lucky than good. $\endgroup$
    – Ilya Nikokoshev
    Commented Nov 1, 2009 at 20:48
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    $\begingroup$ Or "better lucky than smart" $\endgroup$
    – vonjd
    Commented Feb 23, 2010 at 14:24
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Non-random is a special case of random.

When studying probability distributions, and start with the Dirac delta function.

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There's no such thing as a free lunch.

This refers to the No Free Lunch theorem. The theorem states that it's impossible to develop a search optimization algorithm that works well for all possible problems. Rather, for every class of problems which a given algorithm performs well at, there is a complementary class for which it does not. Thus, you may think you're getting a free lunch, but you're really just paying for it somewhere else.

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  • $\begingroup$ Duplicate of mathoverflow.net/questions/3559/… $\endgroup$
    – Rasmus
    Commented May 10, 2011 at 8:17
  • $\begingroup$ Not an exact duplicate. Here its about optimization and not about finance. $\endgroup$
    – Dirk
    Commented Jul 5, 2012 at 6:49
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Even quite irregularly shaped objects, such as tables and chairs, become approximately spherical if you wrap them in enough newspaper.

(I think this is by J. H. Conway but I heard it through Bill Thurston, who we recently lost. RIP.)

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    $\begingroup$ What is the serious mathematics that this encodes? $\endgroup$
    – Michael Lugo
    Commented Aug 25, 2012 at 15:02
  • $\begingroup$ @MichaelLugo There's Ravi Valil's article "The Mathematics of Doodling" math.stanford.edu/~vakil/files/monthly116-129-vakil.pdf (Monthly 2011) $\endgroup$
    – Zach Teitler
    Commented Nov 10 at 17:01
  • $\begingroup$ @ZachTeitler I have actually done this doodling! I obviously wasn't holding my breath for an answer for this for twelve years but it's delightful anyway. $\endgroup$
    – Michael Lugo
    Commented Nov 11 at 15:45
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Nothing contains everything.

or

There is no universe.

This is how Halmos (pp. 6-7) summarizes the answer of axiomatic set theory to Russell's paradox.

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If at a party any two guests have precisely one shared acquaintance, then someone knows everyone (and moreover, the guests came as couples).

This is an informal statement of the friendship theorem first proven by Erdős, Rényi and Sós: if in a (finite) graph any two distinct vertices have precisely one neighbor in common, then the graph is a windmill (a number of triangles attached to each together at a common vertex). The statement is a purely graph theoretical, but all proofs seem to make use of spectral techniques (or mimic spectral techniques closly).

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Hilbert’s Hotel, which has an infinite number of rooms, always has space for more guests. https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel?wprov=sfti1#

More generally, a cardinal is infinite if and only if a set of that cardinality admits a bijective correspondence with a proper subset.

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One-half lives, one-half dies.

Suppose that $M$ is an oriented compact three-manifold. Suppose that $\phi$ is the homomorphism on first homology, induced by the inclusion of the boundary of $M$ into $M$. Then the image and kernel of $\phi$ have the same rank (namely, half the rank of the homology of the boundary).

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Poisson arrivals see time averages.

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GAGA the acronym for Serre's famous Geometrie algebrique geometrie analytique.

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Numbers are mutually friendly if they share their abundancy

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