
Common False Beliefs in Mathematics - mahipal
http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics
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jrockway
Math people seem a lot more interested in math than programmers are in
programming. If you asked the same question on SO, the popular answers would
be a list of things like "programmers don't deserve big monitors" or "the
client is always right" or other such bullshit. (Nothing actually about
_programming_ , only about "the profession").

Maybe this is why nobody ever calls mathematicians "number monkeys".

~~~
pavlov
My humble opinion is that software is not science, but a creative solutions
profession more akin to architecture or writing. It's a mystery to me why so
many programmers seem to be ashamed of that.

I don't see architects spending their time being envious of mathematicians.
"If only we could construct buildings out of closed functions on a plane of
pure ideology, where no humans would interfere..."

~~~
fforw
brilliant comment. my new favorite quote -- right after "existence is
stateful".

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jswinghammer
That site is crazy. Every time I open it I feel like a complete idiot. I took
a decent amount of math in college but I don't know most of the words they
use.

~~~
ronnoch
I wonder if this is how non-programmers feel reading Stack Overflow.

~~~
pjscott
Non-programmers know that they're not _supposed_ to understand programming.
People who've taken a reasonable number of math classes tend to feel like they
should know more than a tiny fraction of the math that's out there, even if
this expectation is unreasonable.

~~~
plorkyeran
I know I found it pretty depressing the first time I realized that after
taking fourteen or so years of math classes I'd never really made it past
Advanced Arithmetic.

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yanowitz
What I like about the thread is that it's a window into another world of geek
humor whose countours I'm familiar with (they look a lot like the countours
for cs humor) but whose specifics are largely unintelligible to me.

Great read.

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tokenadult
I laughed (before I started to think that crying might be more appropriate)
when I heard about the false belief that 1 plus the product of the first n
primes is always a prime number. Ouch!

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snowbird122
I took 14 hours of calculus in my EE undergrad. Reading posts like this remind
me that I really enjoy studying math outside of school. In school, I had to
learn so much, so fast, that I couldn't appreciate the beauty, nor take time
to ponder the wider implications of what I was learning.

Many times, I have thought about opening my Salas and Hille's Calculus book
and just starting over from the beginning. Anyone ever done anything like
this? Any opinions on a better starting book for someone in my shoes?

~~~
oct
Apostol's Calculus would probably fit your needs. I do not remember its
discussing applications at all, unlike the Salas and Hille book according to
online descriptions of it. From what you wrote it does not look like you are
concerned with that, though.

~~~
snowbird122
Thanks for the recommendation. $154 new from Amazon and it was written in
1966. Killer reviews though. Thanks again. [http://www.amazon.com/Calculus-
Vol-One-Variable-Introduction...](http://www.amazon.com/Calculus-Vol-One-
Variable-Introduction-
Algebra/dp/0471000051/ref=sr_1_1?ie=UTF8&s=books&qid=1275843399&sr=8-1#noop)

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mjw
I liked this answer in particular:

"These are actually metamathematical (false) beliefs that many intelligent
people have while they are learning mathematics, but usually abandon when
their mistake is pointed out, and I am almost certain to draw fire for saying
it from those who haven't, together with the reasons for them:

* The results must be stated in complete and utter generality.

* Easy examples are left as an exercise to the reader.

* It is more important to be correct than to be understood.

(Applicable to talks as well as papers.)"

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bobbyi
Can someone explain why this one (the top answer) is false:

For vector spaces, dim(U+V)=dimU+dimV−dim(UV), so
dim(U+V+W)=dimU+dimV+dimW−dim(UV)−dim(UW)−dim(VW)+dim(UVW)

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xi
It is explained in the comments: take three different lines in the plane, and
you'll get 2=3.

~~~
jules
Perhaps more enlightening is where the proof fails:

dim(U+V) = dimU+dimV−dim(UV)

    
    
        dim(U+V+W)
        = dim(U+V)+dim(W) - dim((U+V)W)
        = dim(U)+dim(V)+dim(W) - dim(UV) - dim(UW+VW) <-- bzzt, wrong
        = dim(U)+dim(V)+dim(W) - dim(UV)-dim(UW)-dim(VW) + dim(UVW)
    

The reason this step fails is that (U+V)W != UW+VW for example UW and VW can
both be empty, but (U+V)W is not empty:

    
    
        U = {multiples of (1,0)}
        V = {multiples of (0,1)}
    

Now U+V is the entire space R^2. If we take W={multiples of (1,1)} then VW={0}
and UW={0}, but because U+V is the entire space we have (U+V)W = W =
{multiples of (1,1)}.

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omaranto
I can't tell you how many times I've had to explain this to confused linear
algebra students...

Oh, and A minor correction: you mean "zero" instead of "empty".

~~~
jules
Yep, that's right. HN doesn't allow me to edit :(

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jules
About half of the ones that I understand what they mean seem perfectly
plausible to me. We need a page that explains why they are false.

For example can someone explain "Every connected component of a topological
space is open and closed."

Edit: found it on wikipedia: "The components in general need not be open: the
components of the rational numbers, for instance, are the one-point sets."

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billswift
It never really took off so there isn't much there yet, but for anyone
interested in math discussion on a less advanced plane, you might try here
<http://wiki.lesswrong.com/wiki/Simple_math_of_everything>

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roundsquare
"The circle is the only figure which has the same width in all directions."

Can someone explain? What exactly do they mean by width and what else has this
property?

~~~
drbaskin
If you think of the figure as lying in the x-y plane, then its projection onto
one of the axes has the same length no matter the orientation of the figure.
One nifty example of this is the Reuleaux triangle, which is related to the
mechanics of rotary engines.

<http://en.wikipedia.org/wiki/Reuleaux_triangle>

