We’re not done yet. Of course, the use of new computer methods to solve numerical problems that were previously deemed impossible to solve have boosted enormously developments in technology, science, medicine and engineering. However, there still remain thousands of unsolved problems in mathematics and science that will keep the experts busy for some time to come...
THE MARCH FORWARD
In 1900 at the International Congress of Mathematicians German mathematician David Hilbert (1862–1943) posed twenty-three mathematical problems that he felt were key to the development of the subject. Since the congress ten of Hilbert’s problems have been solved, seven have been solved to some extent or have been shown not to have a solution, three were too vague to be solved and three remain unsolved.
Posing these problems had exactly the effect that Hilbert wanted – the competition spurred mathematicians to strive to tackle them and in the process forge into new areas of research. In 2000, in much the same vein as Hilbert, the Clay Mathematics Institute issued another seven problems, now known as the Millennium problems.
So far, only one of the problems has been solved – the Poincaré conjecture, which relates to the topology of spheres. It was solved by an extraordinary Russian mathematician called Grigori Perelmann, who has declined not only a Fields Medal (the highest accolade in mathematics) but also the $1 million dollar prize from the Clay Institute.
One of the problems posed in both Hilbert’s problems and the Millennium problems is the Riemann hypothesis, a problem that many mathematicians feel is the most important in mathematics. It concerns the distribution of prime numbers. The Goldbach conjecture (see here) tells us approximately where the prime numbers should be; the Riemann hypothesis would help us to know how far away from the expected place the prime should actually be.
WHAT NEXT?
The future of mathematics depends very much on mathematicians who are, as I write, children, or as yet unborn. In order to cultivate the best possible mathematicians and scientists to help solve the world’s problems we need people with excellent mathematical training, which is quite an educational investment. In our current educational system, every schoolchild is taught numbers and arithmetic through to algebra and geometry so that by the onset of adulthood they have the tools necessary to enter a technical career path, should they so choose.
The majority of people, however, do not enter a technical career and therefore do not necessarily need mathematics taught beyond primary school. Most people use calculators or, more frequently, mobile telephones with built-in calculators, to do the mundane arithmetic that is all the maths needed in everyday life.
So, should we continue to make mathematics a compulsory subject until the age of sixteen? There are clearly those who enjoy maths and those who do not. Perhaps we could just teach basic arithmetic and everyday maths to younger children and save the harder, more interesting stuff as an optional course for older children who show a particular inclination and aptitude towards the subject? Well, if it worked for the Ancient Greeks...
The fundamental theories of how the universe works – as discovered by, among others, Newton, Einstein, Feynman and Hawking – have been made through the creation of a mathematical model and the pursuit of the mathematical conclusions that ensue. These models are then tested by experiments in the real world to check the accuracy of the model.
As time goes by, it seems that in order to generate the best possible rate of advancement in science, we need mathematicians who understand the latest developments in that field, and scientists who understand the latest developments in mathematics too.
As Galileo said:
Mathematics is the language with which God has written the universe.