In this newsletter:

* Latest news
* Maths in a minute
* Browse with Plus
* Your Universe questions


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Latest news

* What is dark energy? 
What's the mysterious stuff that makes up 70%  of our Universe?
http://plus.maths.org/latestnews/may-aug09/darkenergy/index.html?nl=1

* What is dark matter?
What's the mysterious stuff that makes up another 25% of our Universe?
http://plus.maths.org/latestnews/may-aug09/darkstuff/index.html?nl=1

* Swine flu uncertainty
How do we know how many people have got it?
http://plus.maths.org/latestnews/may-aug09/flusurvey/index.html?nl=1

* Turbulence stops star births
Scientists crack black holes' balancing act
http://plus.maths.org/latestnews/may-aug09/turbulence/index.html?nl=1


Plus... read more on the Plus blog
http://plus.maths.org/blog?nl=8

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Maths in a minute - Proof by contradiction

Proofs by contradiction are extremely common in maths. The reasoning  
behind them is that if an assumption leads to a contradiction, then  
the assumption must be false. Possibly the most famous proof by  
contradiction is the one which demonstrates that the square root of 2  
is irrational. You start by assuming that the square root of 2 is  
rational, which means that you can write it as a fraction a/b, where a  
and b have no common factors. But using some elementary algebra you  
can show that your assumption also implies that both a and b must be  
even, which contradicts the fact that they have no common factors. QED.

Intuitively, the reasoning behind proofs by contradiction is an  
extension of a process of elimination which is used a lot in real  
life: if a hypothesis (for example that the gardener committed the  
murder) leads to an extremely unlikely or impossible set of  
circumstances (that the gardener had to be in two places at the same  
time), then you dismiss the hypothesis. But when it comes to  
mathematics, not everyone is convinced. The reasoning behind proofs by  
contradiction is that a statement is a priori either true or false, so  
if you show that it can't possibly be true, then it has to be  
false. But if you don't accept the idea that a statement is either  
true or false, then this reasoning fails. For example, there is a  
school of mathematics, called constructivist mathematics, which  
rejects the idea that the truth or falsity of a statement is  
independent of the human mind, just sitting around waiting to be  
discovered. To constructivists, a statement is true only when it has  
been proven to be true. The reasoning behind proofs by contradiction  
is therefore too implicit for constructivists, and they reject many  
proofs by contradiction.

To the vast majority of mathematicians, however, proofs by contradiction 
are bread-and-butter tools of their trade. But they can also lead to 
surprises. Throughout mathematical history, mathematicians have tried to 
prove that every geometric space has a certain property, which had been 
postulated by Euclid as an axiom. The property is equivalent to saying that 
the sum of angles in every triangle is 180 degrees, and known as
 the parallel postulate or Euclid's fifth axiom. Mathematicians attempted 
to show that a geometry which did not satisfy this axiom would lead to a 
contradiction. But the contradiction never turned up. The geometries that 
eventually resulted, known as hyperbolic and elliptic geometries, are in 
fact flawless and have become an accepted and respected members of the 
mathematical world.

Find out more on Plus:

The proof that the square root of 2 is irrational: 
http://plus.maths.org/issue9/features/proof3/index.html

An introduction to constructivist mathematics: 
http://plus.maths.org/issue49/features/wilson/index.html

An introduction to hyperbolic geometry: 
http://plus.maths.org/issue18/xfile/index.html

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Browse with Plus - Mind the gap

Visualisation techniques can do wonders for statistical information.  
Find the right type of graph or chart, and a sober list of numbers  
becomes instantly intelligible. The gapminder website uses pretty  
bubble graphs to illustrate statistical relationships relevant to the  
world's socioeconomic problems. The user can choose what to plot  
against what - for example income growth versus life expectancy - and  
see how the plot evolves over time for different countries. The  
website contains information on where the data is from for you to  
check, and also some instructions on how to produce your own graphs  
using Google motion charts. Gapminder doesn't do any statistical  
analysis or interpretation for you, but it's a fun little tool to play  
around with.

http://graphs.gapminder.org/world/



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Your Universe questions

Don't forget that you can nominate your burning questions about the   
Universe in our online poll, to be put to world-leading astronomers   
and cosmologists. http://plus.maths.org/blog/2009/07/poll-4.html

The most popular question in our last poll was "What are dark energy  
and dark matter?" We put it to cosmologists Martin Rees and John D.  
Barrow, and you can read their answers in Plus:

http://plus.maths.org/latestnews/may-aug09/darkstuff/index.html?nl=1

http://plus.maths.org/latestnews/may-aug09/darkenergy/index.html?nl=1



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Happy reading from the Plus team!


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