

Controversial Facts In Mathematics - edtechdev
http://www.businessinsider.com/the-most-controversial-math-problems-2013-3?op=1

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strangestchild
All good fun problems - and some less well-known as well as the old
favourites. That said, calling these the "12 most controversial facts in
mathematics" is like calling the truth of the moon landings the most
controversial fact in astronautics. But I suppose "The Most Unintuitive
Mathematical Results That Laymen Can Be Made To Understand" is not quite so
catchy.

~~~
gordaco
Agree, this is mostly controversial for non-math people. Although the Monty
Hall problem supposedly caused a lot of mail from people, some even math
teachers, who got it wrong but didn't realize and/or admit it.

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DerekL
People often get the Monty Hall problem wrong because it is often stated
ambiguously, and the listener fills in the details in his own way.

Here's the question as posed in Marilyn Vos Savant's column
[<http://marilynvossavant.com/game-show-problem/>]: "Suppose you’re on a game
show, and you’re given the choice of three doors. Behind one door is a car,
behind the others, goats. You pick a door, say #1, and the host, who knows
what’s behind the doors, opens another door, say #3, which has a goat. He says
to you, 'Do you want to pick door #2?' Is it to your advantage to switch your
choice of doors?"

But what exactly is the procedure followed by the host? I can think of three:

A. If you pick a goat, then the host shows you the other goat. If you pick the
car, then the host picks one of the goats at random to show it to you.

B. The host picks one of the other doors at random and shows it to you. So you
might pick a goat, then he shows you the car, and gives you a chance to
switch, but you already know that the remaining doors have goats, so there's
no point.

C. If you pick the car, then the host shows you one of the goats picked at
random. If you pick the goat, then the host says "You picked the goat!", and
you don't get a chance to switch.

People assume that the procedure is A, but I don't see how the other
procedures are excluded by the description.

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T-hawk
There's even a more dastardly procedure, where the host doesn't follow fixed
rules but has free rein to outwit and trick the contestant. If Monty thinks
the contestant can be induced into switching away from the car, he can try an
offer, but doesn't have to. So he basically follows C (already the most
disadvantageous for the contestant), but can also throw in a game-theory
curveball of occasionally offering the switch even when the contestant is
already wrong. Your described rule leaks information (offering the switch is a
telltale that the contestant was already right), which the host can discredit
by occasionally behaving otherwise. And of course if Monty is playing the
psychology of the contestant, there's no rigid mathematical answer at all,
like a poker bluff.

And by most accounts, the actual "Let's Make A Deal" show did let Monty do
whatever he liked, so Vos Savant's presentation of the problem following rigid
rules didn't really have a basis in reality.

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rskar
Ok, would someone explain #6, "The Broken Water Heater Problem" to me? As
expressed, there's nothing to say that the person (who repaired the heater)
could not be just a plumber, or perhaps just a handyman and neither an
accountant or a plumber. Yet, as demonstrated, the person is given to be an
accountant.

Event A is "accountant and plumber". Event B is "accountant and not a
plumber". Hence A unioned with B means "accountant"; or, if that's too
general, "is an accountant and may or may not be a plumber too". In any case,
both events A and B qualify the person to be an accountant.

So what's the controversy? When it comes to water heaters, we're all
accountants?

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NhanH
The way it was explained in the infographic wasn't clear. But what he meant is
that between P(X = accountant) and P(X = accountant and plumber), the former
is bigger. The fact that P(X = plumber) > P(X = accountant) is irrelevant, as
we wasn't concerned with P(X= plumber) at all.

The implication of this "problem" is an interesting human biased: there
appears to be cases where we judge the more detailed explanation as more
probably, while statistically speaking, it isn't. If a person is described as
"being concerned with discrimination and social justice", we tend to judge P(X
= activist and Y) as more probable than P( X = Y). This is detailed in
"Thinking, fast and slow" by Kahneman - a wonderful book if anyone haven't
read it.

~~~
rskar
I think you've shown me the way here. My issue, I suppose, is the problem with
word problems in math. Generally that problem is two-fold. Does a word problem
sufficiently describe the situation? And do you sufficiently understand the
situation it describes? I believe this to be at the core of all the supposed
"controversial math problems". BTW, thank you for suggesting Kahneman's book,
it is now on my reading list.

I did take notice that a sub-set is likely to be smaller than the set it came
from: the set of all accountants vs. the sub-set of all accountants who are
also plumbers. So therefore P(accountant) > P(accountant and plumber). Unless
P(accountant) be zero, or P(plumber|accountant) be one, in which case neither
is greater.

Reflecting on my own human biases, I think I ended up reading it as if this
were a multiple-choice question, and I had to pick the "best" choice. In other
words, it isn't enough that this person is an accountant, but that this person
is also a plumber. Or put another way, if I had to wager on whether anyone
would pay just any old accountant to go mess with their water heater, I would
find it more surprising that the particular accountant in fact hired had no
demonstrable plumbing skills. I'd imagine Kahneman could teach me more on
this.

My favorite word problem joke goes something like this. If there are three
cans on a fence, and you hit one with a thrown rock, how many cans remain on
the fence? Answer: Two. Now, if there are three birds on a fence, and you hit
one with a thrown rock, how many birds remain on the fence? Answer: None -
they'd all start flying off when there's a rock hurtling towards them.

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gordaco
I just knew that the first two were going to be the Monty Hall problem and
0.99999...==1. I'm actually surprised that so many people don't accept that
last one; I understood it immediately when I learned it (I was 12).

The third blows your mind until you realize that, simply put, infinity is not
a number and common arithmetic doesn't apply.

The sixth is not really math, although it perplexes a lot of people. I
remember that a few years ago my sister didn't get that, if searching for
"termX" wouldn't yield any results, searching for "termX and termY" also
wouldn't give anything, no matter how much "termY" would.

I'd never seen the 8th, but it's beautiful, and very simple too! Does it
really confuse people? Maybe I'm too mathy, but it's obvious to me.

The 9th one is a classic, but I feel it's less "controversial" because if you
know (or have been told) that the series diverge, chances are that you have a
relatively solid base of mathematical knowledge.

I encountered the 11th in a list of geometry problems recently; I had to prove
it. The first time I saw the problem, just before starting a geometry course,
I didn't have a clue. I shitted my pants thinking that I was going to fail the
course if my exam was like that! Then, after a few weeks of study, I came back
and immediately saw the proof, without thinking much (you just have to use the
intercept theorem: each side of the parallelogram is parallel to one original
polygon's diagonal).

The 7th and the last one don't seem too different than the Monty Hall... our
brains are not very well wired for probability estimation and you can see that
in many situations.

The rest are nice pieces of pop math (if you feel that "pop math" is not much
of an oxymoron, anyway).

~~~
alok-g
>> 0.99999...==1

I am one of the people who don't accept the above as truth. Whether it is
truth or not depends on the formal meaning of the ellipsis ... which no one
provides to me (I have asked several people about this). Without a formal
definition, the statement is meaningless to me.

Here is a way to present it formally.

0.9999... is sum of a geometric series a + a * d + a * d^2 ... a * d^n ...
with a = 0.9 and d = 0.1.

In the limit n -> infinity, the above sum is 1. This is however only true in
the limit.

It is not clear/obvious to me that ... really means "in the limit".

~~~
lutusp
> Whether it is truth or not depends on the formal meaning of the ellipsis ...
> which no one provides to me (I have asked several people about this).

Mathematical definitions aren't like dictionary definitions. A closed
ellipsis:

    
    
        1...100
    

Is accepted as meaning a list of all the intermediate values.

An open-ended ellipsis is accepted as meaning an infinite repetition of what
precedes it. So this ellipsis:

    
    
        0.9999...
    

Means an _infinite decimal sequence_ of 9's. And .999... really is equal to 1.

<http://en.wikipedia.org/wiki/0.999..>.

A quote: "In mathematics, the repeating decimal 0.999... (sometimes written
with more or fewer 9s before the final ellipsis, or as 0.9, , 0.(9)) denotes a
real number that can be shown to be the number one. In other words, the
symbols "0.999..." and "1" represent the same number. Proofs of this equality
have been formulated with varying degrees of mathematical rigor, taking into
account preferred development of the real numbers, background assumptions,
historical context, and target audience."

Remember that mathematics, like science, avoids ambiguity where possible. And
mathematical notation strives for less ambiguity than mathematics itself.

~~~
alok-g
Other than pointing to the Wikipedia article (thanks for the same), you have
not said anything that I did not. The repetition of the decimals before the
ellipsis is the same as the geometric series representation I mentioned. The
Wikipedia article also mentions the limit in one of the proofs (though I am
yet to read the other proofs there, which may be illuminating! :-)

~~~
lutusp
Mathematics isn't politics or law. The point is to come to a consensus on the
meaning of symbols and conventions. The consensus is that the ellipsis in
0.999... extends out to infinity.

In mathematics, there is every incentive to agree on axioms like the meaning
of the ellipsis. In science, there is every incentive to agree on the meaning
of evidence and how it relates to theory. Only in psychology do they vote on
the meaning of their "scientific" findings (all the new conditions that made
their way into the new DSM got there by votes, not evidence) -- but psychology
isn't a science, it's a pseudoscience.

Meanwhile, based on the agreed-upon axioms, 0.999... is equal to 1.

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gizmo686
The harmonic series was included as diverging, but there was no mention of
conditionally convergents series (which, if you allow terms to be rearrange,
can be made to converge to anything)

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scarmig
They leave out Banach-Tarski?

I guess in all fairness it isn't so much a fact as a reason to reject the
Axiom of Choice...

