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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.)*2012-05-02
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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.2012-05-02
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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 http://math.stackexchange.com/a/14634/232 .2012-05-02
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    Related: http://math.stackexchange.com/questions/250/a-challenge-by-r-p-feynman-give-counter-intuitive-theorems-that-can-be-transl2012-05-02
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    That's a nice post... although I wonder why people disliked the [Birthday problem](http://en.wikipedia.org/wiki/Birthday_problem) so much. I think it's a good example of counterintuition in probability.2012-05-02
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    Some of the posts in [this thread](http://math.stackexchange.com/q/2949/5363) on surprising results might be of interest, too.2012-05-02
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    Also somewhat related: http://math.stackexchange.com/questions/48301/examples-of-results-failing-in-higher-dimensions2012-05-02
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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*.2012-05-02
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    This happens when people choose counter-intuitive axioms.2012-05-02
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    "In mathematics you don't understand things. You just get used to them." ---John von Neumann.2012-05-02
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    I can thoroughly back Mariano's comment. After working without the axiom of choice for a while I developed some intuition about it and sometimes using the axiom of choice seems plain weird. Observing other set theorists it is clear that this holds for large cardinals and other very strange objects.2012-05-02
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    @Chris Taylor, but you can conceive of a mathematical idealisation of a fluid so that the notion of "filling" the Horn with that fluid makes sense.2012-05-02
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    @Hammerite the idealization of a fluid is explicitly requires that the length scale of the fluid element is much, much larger than average particle size.2012-05-03
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    @Neal, why would my idealisation of a fluid need to be composed of particles at all?2012-05-03
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    And the lower dimensional version of this is that there can be regions on a plane with finite area and boundary of infinite length.2012-05-06
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    @martycohen I had already mentioned this, but thanks though2012-05-06
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    @ChrisTaylor: Gabriel's horn does not challenge our intuition of _both_ math and reality but merely our choice of words. One could perfectly fill it with paint. The problem is just that we compare a volume to a surface: it makes no sense! It's like being surprised by the fact that the volume of the unit ball is less than its surface: we just forgot the factor 0. (In this case, the thickness of the coat of paint)2012-05-07
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    There's nothing happening in Gabriel's Horn that isn't also happening when you roll out a long thin snake of Play-Doh, except that in Gabriel's Horn the situtation is obscured by calculus: http://blog.plover.com/math/gabriels-horn.html2012-05-10
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    +1 for procrastinating on revising for your Maths exam2018-07-04

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