Mathematics can often appear arcane, esoteric, unworldly and irrelevant. Its blue-skies status could easily make it a target for the harsh cuts to science budgets that governments around the world are contemplating. But before the axe falls, it is worth remembering what the 17th-century scientist Galileo Galilei once declared:
The universe cannot be read until we have learned the language and become familiar with the characters in which it is written.
It is written in mathematical language and the letters are triangles, circles and other geometrical figures, without which means
it is humanly impossible to comprehend a single word.
The scientists at Cern will certainly agree with Galileo. Their ability to make predictions about the particles they are expecting to see inside the Large Hadron Collider is entirely down to mathematics. Rather than triangles and circles, strange, symmetrical objects in multidimensional space - shapes that exist only in the mathematician's mind's eye - are helping us to navigate the strange menagerie of particles that we see in these high-energy collisions.
Biochemists trying to understand the three-dimensional shapes of protein strings will also sympathise with Galileo's sentiments. Each protein consists of a string of chemicals, which are like letters in a book. But to read this language and predict how the one-dimensional string will fold up in three-dimensional space, you need a mathematical dictionary. Studying protein folding is key to understanding several neurodegenerative diseases that are a result of the incorrect folding of certain protein strings.
It is striking, however, how much of my subject remains a mystery: how many mathematical stories are still without endings, or read like texts that have yet to be deciphered.
Take the atoms of mathematics, the primes. Indivisible numbers such as seven and 17 are the building blocks of all numbers because every number is built by multiplying these primes together. They are like the hydrogen and oxygen of the world of mathematics. Dmitri Mendeleev used the mathematical patterns he had discovered in the chemical elements to create the periodic table, the most fundamental tool in chemistry. So powerful were these patterns that they helped chemists to predict unknown elements that were missing from the table. But mathematicians are still to have their Mendeleev moment. The pattern behind the primes that might help us to predict their location has yet to be uncovered. Reading through a list of primes is like staring at hieroglyphs. We have made progress and have unearthed something resembling the Rosetta Stone of prime numbers - but the ability to decode the stone still eludes us.
Mathematicians have been wrestling with the mystery of the primes for 2,000 years, but some of the biggest problems in maths are far more recent. There was a flurry of excitement this summer when the Hewlett-Packard engineer Vinay Deolalikar claimed to have cracked the "P v NP" problem. First posed in the 1970s, this is a problem about complexity. There are many ways to state it, but the classic formulation is the "travelling salesman problem".
An example is the following challenge: you are a salesman who needs to visit ten clients, each located in a different town. The towns are connected by roads, as shown on the following map, but you have only enough fuel for a journey of 238 miles
The distance between towns is given by the number on the road joining them. Can you find a journey that lets you visit all ten clients, stopping in each town only once, and then return home without running out of fuel? (The solution appears at the end of the article.) The big mathematical question is whether there is a general algorithm or computer program that will produce the shortest path for any map you feed in, which would be significantly quicker than getting the computer to carry out an exhaustive search. The number of possible journeys grows exponentially as you increase the number of towns, so an exhaustive search soon becomes practically impossible.
The general feeling among mathematicians is that problems of this sort have an inbuilt complexity, which means that there won't be any clever way of finding the solution. But proving that something doesn't exist is always a tough task. The recent excitement that this problem had been cracked has since evaporated and it remains one of the toughest on the mathematical books.
But the recent solution of another challenging problem, the Poincaré conjecture, gives us hope that even the most elusive problems can be conquered. The Poincaré conjecture is a fundamental problem about the nature of shape. It challenges mathematicians to list the possible shapes into which three-dimensional space can be wrapped up. In 2003, the Russian mathematician Grigori Perelman succeeded in producing a periodic table of shapes from which all other such shapes can be built.
These fundamental mathematical questions are not just esoteric puzzles of interest solely to mathematicians. Given that we live in a world with three spatial dimensions, the Poincaré conjecture tells us ultimately what shape our universe could be. Many questions of biology and chemistry can be reduced to versions of the travelling salesman problem, where the challenge is to find the most efficient solution among a whole host of possibilities. A resolution of the P v NP problem could therefore have significant real-world repercussions.
Modern internet codes rely on properties of prime numbers. So any announcement of a breakthrough on the primes is likely to spike the interest not just of pure mathematicians, but of e-businesses and national security agencies. At a time when blue-skies research without obvious commercial benefits could be under threat from sweeping cuts, it is worth remembering how Galileo concluded his statement about the language of mathematics: without it, we will all be wandering around, lost in a dark labyrinth.
Marcus du Sautoy is the Charles Simonyi Professor for the Public Understanding of Science, University of Oxford. His new book, "The Number Mysteries", is published by Fourth Estate (£16.99)