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As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition.

My first and favorite experience of this is Gabriel's Horn that you see in intro Calc course, where the figure has finite volume but infinite surface area (I later learned of Koch's snowflake which is a 1d analog). I just remember doing out the integrals for it and thinking that it was unreal. I later heard the remark that you can fill it with paint, but you can't paint it, which blew my mind.

Also, philosophically/psychologically speaking, why does this happen? It seems that our intuition often guides us and is often correct for "finite" things, but when things become "infinite" our intuition flat-out fails.

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Why does it happen? Because our intuition is developed by dealing with finite things: it is quite unsurprising that we are surprised by phenomena specific to infinite objects! This is exactly the same as the fact that our bodies are trained to move and act under the effect of gravity, so when we are in space we become clumsy and need to retrain. Intuition is not fixed: if you study phenomena associated to infinite objects, you develop an intuition for that, and presumably people working with large cardinals, (cont.) –  Mariano Suárez-Alvarez May 2 '12 at 1:13
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(cont) or strange objects like graphs with chromatic number $\aleph_8$ or Banach-Tarski partitions of a sphere, after a while find them just as intuitive as you and me find the formula for the area of a triangle. Intuition is, in most situations, just a name we put on familiarity. –  Mariano Suárez-Alvarez May 2 '12 at 1:15
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Philosophically / psychologically speaking, human brains weren't adapted for intuiting mathematical truths. The fact that we can repurpose our brains to do mathematics at all (beyond counting etc.) is astonishing. As for Gabriel's horn, I don't think this is a good example: see math.stackexchange.com/a/14634/232 . –  Qiaochu Yuan May 2 '12 at 1:20
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I think remarks like "you can fill it with paint, but you can't paint it" are actually not helpful. In trying to appeal to our everyday intuition, they get in the way of mathematical understanding. Of course, you can't paint Gabriel's Horn (it's surface area is infinite) but you can't fill it with paint either (because paint molecules have a finite size, and Gabriel's Horn gets infinitely thin). Or, more prosaically, you can't fill Gabriel's Horn with paint because it's a mathematical idealisation that doesn't exist in the physical world. –  Chris Taylor May 2 '12 at 7:35
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"In mathematics you don't understand things. You just get used to them." ---John von Neumann. –  Nate Eldredge May 2 '12 at 19:33
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39 Answers

The concentration of measure phenomena on the sphere:

If $A\subset\mathcal{S}^{n-1}$ is a measurable set on the sphere with $\lambda(A)=1/2$ and $A_\epsilon$ is an epsilon neighborhood of $A$ on $\mathcal{S}^{n-1}$, then

$$\lambda(A_\epsilon)\geq 1-\frac{2}{e^{n\epsilon^2/2}}$$

So say you take $A$ to be a cap on the sphere and fix a small $\epsilon$. As the dimension of the sphere increases, eventually the $\epsilon$ enlargement of $A$ will have almost the entire area of the sphere! Playing with the upper and lower cap and the corresponding enlargements, one sees that area is concentrated around the equator.

Imagine you have a lawnmower and you cut the grass moving along the equator. What percentage of the sphere do you mow? Well, in 3 dimensions, not that much. But as you cut the grass on higher and higher dimensional spheres, moving centered along the equator, the surface area covered becomes almost 100% of the entire area of the sphere!

This result felt pretty counter-intuitive to me the first time I saw it.

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The Weierstrass function. It showed that a function can be continuous everywhere but differentiable nowhere. This was (and still is) counterintuitive.

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I am surprised nobody has given this example: It was once thought that continuous functions have derivatives at almost all points (as other answerers have used). Then comes the Brownian motion model that generates a random curve, which, with probability 1, is a curve without derivative at any points!

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I always found the main correspondence of class field theory counter-intuitive. Somehow the (Abelian) field extensions of a number field $K$ correspond to arithmetic objects in $K$ itself.

You wouldn't think that the possible ways of extending a field (albeit in a nice way) would just depend on the field you started with.

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How about the Löwenheim–Skolem theorem?

One of its consequences is that the field of real numbers has a countable model.

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Not exactly. The completeness property is second order. The countable model will be a real-closed field, but not complete. –  Asaf Karagila May 8 '12 at 4:38
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For quite a while, when I was very young, I thought that the solution to the famed brachistochrone problem (which asks about the shape of a wire where a bead sliding frictionlessly on it under the influence of gravity will take the least amount of time to finish) was not a straight line was pretty counterintuitive, after being repeatedly told that the straight line is the shortest path between two points. (To spoil: the true brachistochrone is an inverted cycloid, which is the curve traced by a fixed point on the rim of a rolling circle.) I went through the derivation a number of times before finally being convinced.

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Although Dan Brumleve and plm linked the list of paradoxes, the famous Braess's paradox deserves a special attention.

"For each point of a road network, let there be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of traffic flow. Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favorable to him, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times."

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It's fairly straightforward to prove that the Cantor set is uncountably infinite (i.e. it has the same cardinality as the real numbers). You would think that such a set would have at least positive Lebesgue measure. WRONG-IT ACTUALLY HAS MEASURE ZERO! This blew my mind when I first learned about in undergraduate real analysis and it's why it puzzles me why Paul Halmos famously described measure theory as "a generalized kind of counting". This example seems to show that measure has nothing to do with cardinality, which is what most of us think of as counting in abstract sets!

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The standard Lebesgue measure definitely has something to do with cardinality. If a set has positive measure (or is unmeasurable) it has the same cardinality as the reals. Measure is a more refined sense of counting-so refined, in fact, that it is only really interesting on a class of sets all of which have the same cardinality. –  Logan Maingi May 4 '12 at 7:56
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The existence of "counting measure" on any set seems to give one indication that measure theory is a generalized kind of counting. –  Pete L. Clark May 5 '12 at 17:52
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@Pete: I think he was trying to ping you. –  The Chaz 2.0 May 8 '12 at 4:00
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If instead of taking away the middle 1/3 at each iteration, you instead take away the middle 1/4th, you end up with a "fat" Cantor set which, surprisingly, has positive measure. –  Nick Alger May 16 '12 at 2:07
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@LoganMaingi: Solovay showed that it is independent of ZFC whether there are nonmeasurable sets with cardinality less than $2^{\aleph_0}$. See this answer of JDH for a survey of related results. –  Jonas Meyer May 17 '12 at 3:40
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Independence in Statistics. If I have a box with red & black socks, and I take one, and then another one the probability of both socks being red is sometimes different from the case when I, instead, pick a pair. (EX.If all socks are connected to every other with a string, like a complete graph, and then I pick some string, automatically choosing the socks at either end as the pair selected).

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protected by J. M. May 6 '12 at 4:09

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