YARFMO (Yet another reposting from Mathoverflow) ;-)

The more you know about math the more you find conceptions previously thought correct to be false:

1.) math is not as exact as many believe - in many cases, as Gowers points out in his amazing book "Mathematics. A very short intro", it is mainly about approximations, boundaries or even "only" existence. Example: The Riemann hypothesis.

2.) math is not "true" (whatever that means) in its own right - it is only true within an axiomatic system (deductive approach). Example: The parallel axiom.

3.) closed form solutions are not as closed as many think: "closed-form" is just shorthand for "popular enough to be given a name and notation." Examples: $\pi, e, \mathrm{Si}(x), \mathrm{li}(x)$.

My question

Do you have further common misconceptions about math (ideally also with examples)?

I think that the ones given above only scratch the surface because in the end the picture of math also changes historically the further we proceed on this never-ending endeavor of human mind.

(As should be clear already, I am talking about the "meta'-level, so please don't give examples like $(x+y)^2=x^2+y^2$ - this is already covered here and here.)

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Look this has been already asked at MATH.SE –  anonymous Nov 17 '10 at 15:05
math.stackexchange.com/questions/7864/… –  anonymous Nov 17 '10 at 15:06
@Chandru: I think this is different - this one is more concrete, mine is more meta, more global about the whole of math! –  vonjd Nov 17 '10 at 15:10
@Chandru: I included the link in the question –  vonjd Nov 17 '10 at 15:13

Here are a couple of misconceptions that people who don't know about mathematics commonly hold. Firstly, that it's immutable, fixed, unchanging; surely it's all been discovered? It's a belief that even a highly educated non-mathematician might hold.

This point was illustrated to me not long ago when a friend, who speaks at least 10 languages fluently, asked why I found mathematics interesting. She believed that it was cold and lifeless and thus inherently less appealing than language, which she viewed as living and dynamic.

I tried to persuade her that mathematics grows and changes too. Areas such as Group Theory were born from ideas used in Number Theory and Geometry. And old ways of doing things can be surpassed. The history of cryptography is a good example.

The conversation also touched on a second misconception: that mathematicians simply follow set rules. I explained that imagination was very important in doing mathematics. And when I said that there was great beauty in mathematics, I was faced with almost total incomprehension. Surely, mathematics was the very antithesis of beauty?

If you're reading this I don't need to convince you that these are indeed misconceptions, but unfortunately I don't think I convinced my friend!

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In your friend's defense, that is essentially the extent of what most people get out of mathematics when they go to school: A set of fixed arcane rules that must be followed to get the unknowable and mystic answer the book or teacher wants. –  kahen Nov 17 '10 at 19:04
Maybe she ought to read Lockhart's Lament or something... –  Guess who it is. Nov 18 '10 at 0:10

Perhaps more meta than you wanted since it's not about how you perceive mathematics coming from mathematics, but rather how the rest of the world views it.

There's that whole misconception that if you're good at math you're good at arithmetic. You know... that thing where someone says something like "Hey, you're good at math! Can you do this calculation (typically menial multiplication) for me in your head?" Sure I could, but that's because I can keep a lot of stuff in my head at once and make sure I do it correctly. It's not as if that says anything about mathematical ability.

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"Hey, so you did computer science in University? Can you repair my printer?" –  Djaian Nov 18 '10 at 10:23
@Djaian: Hehe, a number of my computer science friends bitch about that sort of thing. Probably one reason why they like using Linux is that they can truthfully claim "Sorry, I don't use Windows". –  Guess who it is. Nov 20 '10 at 0:35

Following up on Kahen's comment, most people who only take math through High School think math is about calculation. For higher math, description is more correct than calculation, but that still doesn't capture the flavor.

Most people think math is about numbers. Some is, some isn't.

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I've noticed it's quite common among algebraists dealing with finite structures (like finite groups) to think of mathematics (limited to their area, of course) as the absolute God given truth, harmonious, non-contradictory, sort of Platonic ideal world.

This point of view can be possible accepted, as long we limit ourselves to finite mathematics, but what happens when we introduce the notion of infinity? The whole harmonious God given world goes out the window and the mathematics becomes what it really is: a kind of human activity which reflects both the power of our intelligence as well as its limitations.

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it seems the founder of most of the interesting stuff we now know about infinity, Georg Cantor, believes in infinity as his God. Pythagoreas and his cult, the Pythagorean, also used to think of numbers as God (and they worship numbers). –  Lie Ryan Nov 18 '10 at 5:52

Abstract mathematics is not about the real world. Like, “If it is just taken out of someone's head, why you guys can not make it simple and understandable? Is it a problem with your head? Why you can't get rid of those obscure definitions and horrible proofs?” (Surely I exasperated it. :) ) Actually, can anyone?

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