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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 Jul 6 '12 at 5:38

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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.

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    $\begingroup$ I've heard a minor variation: "Compact is the new finite." $\endgroup$ – S. Carnahan Aug 6 '12 at 0:47
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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 Nov 1 '09 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 Nov 1 '09 at 16:39
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    $\begingroup$ So just take an n-dimensional ham sandwich... :) $\endgroup$ – Cam McLeman Jul 5 '10 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 Feb 14 '16 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 Aug 5 '12 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 Aug 8 '12 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 Aug 8 '12 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 Apr 30 '15 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 Jun 5 '16 at 7:03
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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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A straight line is the shortest distance between two points.

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    $\begingroup$ How can a line be a distance? $\endgroup$ – Rasmus May 10 '11 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 May 10 '11 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 May 10 '11 at 21:41
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    $\begingroup$ Enter these four terms into Google: straight line shortest two $\endgroup$ – Michael Hardy May 10 '11 at 21:42
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    $\begingroup$ That's possible. The google result shocked me. ;) $\endgroup$ – Rasmus May 11 '11 at 12:14
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I'm surprised that nobody has mentioned the famous,

"Four colors suffice."

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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 Jul 4 '12 at 20:48
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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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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 May 9 '11 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 Jul 6 '12 at 4:37
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    $\begingroup$ It is a bit like: Assume there were no proofs by contradiction.... $\endgroup$ – Lennart Meier Sep 17 '14 at 15:13
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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 Feb 19 '10 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 Apr 19 '11 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 Apr 19 '11 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 Jul 5 '12 at 1:01
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    $\begingroup$ Visualize world peas? $\endgroup$ – Noam D. Elkies Jul 6 '12 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 Nov 1 '09 at 4:32
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    $\begingroup$ I understand the others, but how does this relate to P vs. NP? $\endgroup$ – Michael Lugo Nov 3 '09 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 Nov 4 '09 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 Dec 24 '09 at 20:43
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    $\begingroup$ Non-mathematically, this reminds me of the cryptic crossword clue: gegs (9,4). $\endgroup$ – Andrew Lobb Feb 22 '10 at 21:40
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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 Nov 1 '09 at 20:48
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    $\begingroup$ Or "better lucky than smart" $\endgroup$ – vonjd Feb 23 '10 at 14:24
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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 May 10 '11 at 8:17
  • $\begingroup$ Not an exact duplicate. Here its about optimization and not about finance. $\endgroup$ – Dirk Jul 5 '12 at 6:49
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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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Non-random is a special case of random.

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

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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 Aug 25 '12 at 15:02
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Adding an axiom saying that something's provable doesn't help you prove it.

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

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Poisson arrivals see time averages.

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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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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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