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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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Recent answers suggest this question is getting a bit long in the tooth. –  S. Carnahan Jul 6 '12 at 5:38
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52 Answers

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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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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What is the serious mathematics that this encodes? –  Michael Lugo Aug 25 '12 at 15:02
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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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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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I've heard a minor variation: "Compact is the new finite." –  S. Carnahan Aug 6 '12 at 0:47
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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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An ancient Greek question: What is the difference between a mathematician and a policeman?

A mathematician tries to square the circle. A policeman tries to circle the square.

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Poisson arrivals see time averages.

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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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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.)

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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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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. –  Margaret Friedland Jul 4 '12 at 20:48
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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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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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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:

  • "You can pick a real number at random between 0 and 1, so that any number is as likely as any other."

more colloquially, (Freiling):

  • "You can throw a dart at the unit square."

more computer-scienc-y:

  • "The process of coin flipping to determine the binary digits of a real number converges to a unique well defined real number answer in the limit of infinitely many throws".

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!

  • The continuum hypothesis is false. (Sierpinsky,Freiling)

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.

  • Every subset of [0,1] has well-defined Lebesgue measure. (Solovay, more or less)

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]".)

  • The axiom of choice fails, already for sets of size the continuum.

Since the axiom of choice easily gives a non-measurable set.

  • The continuum has no well order.

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".

  • Sorry, you can NOT cut up a grape and rearrange the pieces so that it is bigger than the sun.

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.

  • The reals which are in "L", the Godel constructable universe, have measure zero.

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).

  • That stupid hat trick doesn't work in the random-pick real numbers

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.

  • The stupid hat trick does work over the computable reals, but is intuitive.

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 stupid hat trick is also intuitive in L, so long as you always think inside a countable model of ZF(C).

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.

  • There is no problem with "$R_L$, the L version of R, coexisting inside $R_R$, the actual version of R, in your mental model of the universe.

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.

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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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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] –  Noam D. Elkies Jul 6 '12 at 4:37
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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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I'm surprised that nobody has mentioned the famous,

"Four colors suffice."

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"A projective module is the splittin' image of a free module."

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"X is the spitten image of Y" is an informal phrase meaning "X looks just like Y". "X is the spitting image of Y" is an entirely equivalent phrase; in some dialects, the pronunciation is even the same. As far as I know, neither is the "right spelling"; the phrase rarely appears in written English. –  jasomill May 10 '11 at 10:48
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@Rasmus: The English colloquialism uses "spit", so that "split" is a pun. See english.stackexchange.com/questions/8509. –  Theo Johnson-Freyd Aug 25 '12 at 13:28
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A straight line is the shortest distance between two points.

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How can a line be a distance? –  Rasmus Bentmann May 10 '11 at 8:11
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Seems to me you're trying to make it fit into the conventions used by mathematicians, not shared by others. –  Michael Hardy May 10 '11 at 21:41
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That's possible. The google result shocked me. ;) –  Rasmus Bentmann May 11 '11 at 12:14
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"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.

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I learned it in an Israeli high school from a Russian teacher and he called it two policemen and a drunk. So the drunk is between the two policmen who are going to the station. –  Yiftach Barnea Apr 19 '11 at 7:19
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And then there is the running joke of calling it the "three policemen" theorem because the drunk is a policeman as well. –  darij grinberg Apr 19 '11 at 8:29
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In Italy too (or rather the "Carabinieri" theorem, soldiers of a police force commonly regarded as ridiculous) –  Filippo Alberto Edoardo Jul 5 '12 at 3:18
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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.

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By “the leash is kept short enough”, do you mean it can't be flung over the lamp? –  Zsbán Ambrus Jul 5 '12 at 21:31
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If n people are in an elevator and n+1 buttons are pushed, there is at least one pigeon brain in the elevator.

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Who you calling pigeon brain? youtube.com/watch?v=mDntbGRPeEU :-) –  Willie Wong Apr 19 '11 at 0:47
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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.

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I think this is a dangerous metaphor: it encourages the misconception that "a space is compact iff it has a finite open cover". –  Rasmus Bentmann May 10 '11 at 8:04
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This is attributed to Weyl in Wilson Sutherland's text Introduction to Metric and Topological Spaces (section 5.2). Sutherland gives it as "If a city is compact, it can be guarded by a finite number of arbitrarily near-sighted policemen". Exercise 5.10.15 asks you to make precise, and discuss the accuracy of, Weyl's statement. –  Tom Leinster Aug 5 '12 at 21:40
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"If it walks like a sphere and it quacks like a sphere then it is a sphere."

A professor at my university explained the Poincare Conjecture to his 1st semester abstract algebra students this way. I think it is a great explanation!

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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 $\exists x: f(x) = f(-x)$.

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While the pressure field should be continuous, I can see no reason why the temperature field have to be continuous? –  Kjetil B Halvorsen Aug 26 '12 at 4:32
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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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Numbers are mutually friendly if they share their abundancy

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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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Local is global

The very first thing I heard about sheaves, from another graduate student.

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A variant (with different mathematical overtones): "Think globally, act locally". I had a button with this on it as an undergraduate, from a very non-mathematical source. –  Ravi Vakil Jan 31 '10 at 5:38
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"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.

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protected by François G. Dorais Sep 21 '13 at 23:09

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