As recently as ten years ago, professional mathematicians did most of their work at a desk, using paper and pencil. Today’s mathematician still sits at a desk, but facing a cathode-ray screen. The paper and pencil are still there, but a lot of the mathematician’s activities now involve use of the computer. In particular, powerful computer packages like Mathematica and Maple advertise themselves as systems for ‘doing mathematics on a computer’. This rapid transformation has changed the nature of doing mathematics in a fundamental way. The computer does not simply ‘assist’ the mathematician in doing business as usual; rather, it changes the nature of what is done. In particular, the logical structure of mathematical reasoning carried out with the aid of an interactive computer system is different from the structure of the more traditional form of mathematical reasoning. This article is a logician’s view of the ‘new maths’ of the computer age.

To most of the outside world, mathematics appears to be a closed book: everything known, no loose ends, no mysteries. From within the subject, it is hard to understand how such an impression can have arisen, for in almost every direction today’s mathematician sees unsolved problems and wonderful new ideas. There seems to be little danger that mathematics will either grind to a halt in the face of insuperable obstacles, or become defunct when the Final Theorem is proved and nothing remains to be discovered.

In school we learn classical mathematics, the mathematics known before the year 1900. It has a direct relation to the actual world. Arithmetic is related to counting and measurement, algebra to general truths about calculations, geometry and trigonometry to the sizes and shapes of objects, calculus to the velocities of moving objects and a host of other applications. Students have no difficulty in knowing what it is about and what it is for. Students who go on to modern mathematics, the mathematics of the present century, either in university or by private study, find it considerably different. There is no immediate contact with reality, and they may find it hard to visualise what is being discussed.

Nowadays many children meet the idea of a locus at an early age. This is because of the great (and deserved) success of LOGO as an educational tool to introduce geometrical ideas to young children. The challenge is to devise lists of instructions to make a ‘turtle’ move in a certain way. Originally the turtle would have been an actual mechanical object moving on the floor. It is now more usually an image on a computer screen. The mechanical turtle held a pen which marked out its path; the computer version* whether it looks like a cartoon turtle or an arrowhead (or even remains ‘hidden’) leaves as its trail a series of dots showing where it has been. The end product drawn on the floor or the screen is thus seen as the path of a point moving according to a prescribed set of rules. In this way children are encouraged to think of any geometrical diagram—whether it is a square or a hexagon, a rectangle or a house, a star or a circle—as a locus. Geometry is built up from collections of points.

The development of algorithms has been one of the principal concerns of mathematics throughout its history, but now that machines can perform algorithms more efficiently than people our attitude to algorithms has to adjust. It is hardly possible for a school to implement the National Curriculum in mathematics without having the computer language LOGO among its resources, but the curriculum's prescriptions are minimal, and it is quite possible to introduce the opening steps of the language and then do no more. What further use can be made of the language? In this article I show how LOGO can provide a systematic approach to some traditional problems of combinatorics.

Doublet tasks consist of changing one word to another by altering single letters at each step to make a different word. The two words at the beginning and end of such a chain must, of course, be the same length, and they should be related to each other in some obvious way. They must not have identical letters in the same positions. All words in the chain should be common English words, proper names excluded. A perfect solution has a number of steps equal to the number of places the given words differ. For example: COLD, CURD, CARD, WARD, WARM. If a perfect change is not possible, the best solution is the shortest chain. For playing doublets as a game with two or more players, Lewis Carroll invented a set of scoring rules to determine who wins.

Over 60 years ago an article with the above heading appeared in the Mathematical Gazette, shortly after the celebration in 1932 of the centenary of the birth of Charles Lutwidge Dodgson, better known and admired universally as the author of the Alice books, Lewis Carroll.

At that time a literary critic expressed his opinion that ‘only a mathematician could have written the Alice books’, perhaps because they contain several references to arithmetic. In Wonderland, Alice met the irascible Ugly Duchess. ‘If everybody minded their own business’, the Duchess said in a hoarse growl, ‘the world would go round a great deal faster than it does’.