Mathematics and the Real World (46 page)

The presentations as a sum of such orbits is very similar to what are today known as Fourier series. Joseph Fourier (1768–1830) described his method in France at about the same time as Gauss presented his, but it is almost certain that Gauss did not know about Fourier's work. The mathematical problem was a double one. First was the use of very little data. Second, rapid calculation was required. The new method met both these
requirements, and Gauss predicted with impressive precision the location and time when Ceres would reappear. This success earned the young Gauss fame throughout Europe. He published his method in a short article that dealt with its use in astronomy and just mentioned that the method could be extended and used for other purposes, but he did not go into any further detail. In practice, no new uses were found, and the method merely gathered dust among other texts.

More than 150 years later, when computers were already in relatively wide use and Fourier's method of approximating functions by summing trigonometric functions was in common use in mathematics, there were several attempts to make these calculations more efficient. The efforts to streamline the method were spurred on by the many uses to which it was being put, including computerized tomography (for instance, for CT and MRI tests) and signal processing. In 1965 two American mathematicians, James Cooly of IBM, and John Tukey (1915–2000) of Princeton University, published an algorithm that calculated the coefficients of the Fourier approximation more efficiently. This was a great improvement, and the method, called the fast calculation of the Fourier transform, or FFT, caught on immediately. It did not take long until it was found that the method discovered by Cooly and Tukey was exactly the same as the one Gauss had used 150 years earlier.

The examples brought here are of course the tip of the iceberg relative to the wealth of methods developed during the history of mathematics to streamline computation processes.

52. FROM TABLES TO COMPUTERS

Alongside the mathematical improvements in the methods of performing calculations, mathematicians always managed to develop mechanical aids, and later electric and electronic devices, to help them carry out computations. The bones mention in section 6, found in the Belgian Congo and that were dated to about 20,000 years BCE, were in a sense a tool for carrying out simple arithmetic tasks.

Shards from the Assyrian and Babylonian periods dated to about 2600 BCE contain tables that, according to the accepted interpretations, served as the first primitive version of the abacus. Greek texts teach us that the abacus was used as a tool for calculations by the ancient Egyptians. It was also used for arithmetic calculations in Greece and Rome. Roman abaci, remarkably similar to those still used today, were found in archaeological excavations throughout the Roman Empire. Its name provides evidence regarding the antiquity of the abacus as an accessory for calculations. The name derives from the Hebrew word
avak
, meaning dust, which appears several times in the Bible. Apparently, the connection is the dust they used to pour onto stone slabs so that they could write numbers and calculations. Abaci are still in widespread use in Eastern cultures, including in Korea and in China, and similar devices have also been found in ancient American cultures such as the Maya and the Inca.

Immediately after Napier presented his logarithmic method for rapid calculation, mechanical aids for calculation were developed alongside printed tables. The British mathematician Edmund Gunter (1581–1626) presented a logarithmic scale, which translated the logarithmic function into geometric terms, following which another British mathematician, William Oughtred (1574–1660) invented the slide rule. By moving the parts of the rule and aligning the different scales marked on it, the values of the logarithmic and exponential functions can be found. Oughtred himself improved his slide rule to enable various operations and other functions to be calculated mechanically, including trigonometric functions. Slide rules became more sophisticated over time and were available in a multitude
of forms: straight, triangular, circular, and so on. They became everyday work tools of engineers, who could not operate without them until relatively recently, when they were replaced by computers.

While still a youth, Blaise Pascal, whom we met in section 37, helped his father in his job as a tax collector. When he was sixteen years old he had the idea of simplifying and speeding up the calculations needed in collecting taxes by constructing a calculating machine. The machine consisted of a system of cog wheels connected in such a way that turning them in the required direction and by the right amount gave the correct answer for the amount of tax due. Obviously the machine could perform general mathematical, not just tax-related, calculations. His machine was called the Pascaline, and Pascal built a number of them and tried to sell them, but without commercial success. Several of his machines are in the Conservatoire National des Arts et Métiers (CNAM) museum in Paris.

Other famous mathematicians also tried to build calculating machines, including Leibniz, some of whose machines can be seen in museums in Germany, including at the Deutsches Museum in Munich. These too were based on cog wheels, and calculations could be made as a result of the ratio between the different wheels. To make the machine suitably efficient for calculations, Leibniz developed binary arithmetic, that is, representing numbers using two digits only, 0 and 1. In our daily lives we write numbers on the base of 10, and specifically we use ten symbols to represent all numbers. In earlier times other bases were used. The Babylonians, for example, mainly used the base 60. The choice of a convenient base clearly depends on uses and can be reached either by trial and error, or by intelligent analysis. That was how Leibniz chose the binary system as the base according to which the calculating machines would operate. That base has remained the most convenient for performing machine calculations still today, and it is particularly appropriate for the way computers work.

Pascal's and Leibniz's calculators and those of other contemporaries could not be programmed, and for each calculation the cog wheels had to be realigned. In the nineteenth century Charles Babbage (1791–1871), a British mathematician and an engineer, made a great stride forward. As
a result of his contribution, Babbage is considered one of the pioneers of modern computers. He took his idea from weaving looms and the perforated paper tapes that were fed into them to determine the combination of shapes and colors in the material. Babbage used that as a basis for his construction of a calculating machine into which a perforated strip of paper was fed, as in the weaving industry, and the machine then produced the result. That was what led Babbage to call them
input
and
output
, terms that have remained in use in basic computer terminology still today. Babbage built several calculating machines that were large and complex, and that could perform fairly complicated calculations. They obtained their power from steam engines. (Electricity had not been discovered yet. If electric motors had not been developed, today we would probably need a small coal-powered steam engine alongside every personal computer.) Copies of Babbage's machines can be seen in the Science Museum in London.

With the development of electric engines, electric calculating machines also developed, including relatively small ones that could be placed on office desks. These were in use until quite recently, and some can still be found in use today. In addition to computation aids we have mentioned that were used for numeric calculations, different types of mathematical calculations were installed into machines that were used for specific purposes, such as speedometers of various sorts (the Romans already had such speedometers attached to horse-drawn chariots), compasses, and other engineering instruments.

The technology accompanying all the computational aids we have mentioned made calculations easier and faster but still within the limitations of what the human brain could follow and absorb. The different types of calculating machines—from tables, to slide rules, to mechanical machines—mimicked as far as possible the path that the mathematics of calculation had traced, that is, using complex operations to the extent that the technology allowed. The revolution occurred when electronics was enlisted for the benefit of calculations. Their speed was such that elementary calculations such as the addition and subtraction of two numbers were so fast that other results could be easily obtained by many repetitions of those operations.

There are at least two claimants to the title “Father of the Electronic Computer.” One is the German engineer Konrad Zuse (1910–1995) who in the years 1935–1938 developed and built an electronic computer that became fully operational in 1941. A number of the computers he built are displayed in museums around Germany, including at the Technical University in Berlin and at the Deutsches Museum in Munich. Nazi activity in Germany and World War II resulted in Zuse working in isolation from what was happening in the rest of Europe and the United States, and his contribution had limited impact. Zuse's competitors as computer pioneers were John Atanasoff (1903–1995) and Clifford Berry (1918–1963) of Iowa State University, who started building an electronic computer in 1937, the final version of which was also completed in 1941. That computer is exhibited in the university where they worked. Zuse's and Atanasoff and Berry's computers were constructed along the lines of the calculating machines that preceded the electronic age. In other words, their ability to be programmed was limited, and the input and output were specific for each computational task. The great leap forward in electronic computers started with ENIAC (Electronic Numerical Integrator and Computer), the computer whose structure owes much to John von Neumann.

We met John von Neumann above, in section 47, in the context of game theory. His contribution in that field was only a small part of his mathematical activity, which included work in many areas. He contributed to the foundations of mathematics and set theory, in which he was helped and supported by the mathematician Abraham Halevi Fraenkel (1891–1965) to whom von Neumann had sent a rough, hardly legible draft of a paper. Fraenkel immediately realized its potential and guided von Neumann through the first stages of his career (we shall refer to Fraenkel again in section 60, on the foundations of mathematics). Von Neumann also worked on hydrodynamics and quantum theory, in which he introduced an axiomatic basis. He was born to a Jewish family in Budapest in 1903, where his father was a banker and a lawyer. John's father was elevated to the nobility for his service to the Austro-Hungarian Empire, and John inherited the title, so that
von
became part of his name. John's exceptional abilities were discovered very early and were expressed in
more than just mathematical talent. At an early age he mastered several languages and showed interest in social issues, economics, and related subjects. He obtained his doctorate in mathematics in Budapest at the age of twenty-two, having already written a number of fundamental mathematical papers. He held a teaching post in Berlin and in 1930 was invited to visit Princeton University. In 1933 he was offered a post at the Institute for Advanced Study in Princeton, which was established partly to take in scientists who had to flee from Nazi Germany. Von Neumann was one of the first scientists there; others included Albert Einstein and the mathematicians Oswald Veblen, Hermann Weyl, and Kurt Gödel, whom we shall meet again in the next section.

John von Neumann was one of the most prominent of the developers of the atom bomb and one of the main proponents of using it against the Nazi enemy. ENIAC was built for the American army to calculate trajectories of shells and missiles. Von Neumann was not part of the original team building the new computer, but when he heard about the project he joined the team and suggested, among other things, a new structure and other improvements that turned the computer into a hitherto unknown type of machine. The official announcement that the construction of the computer was completed was made in 1946, although it had been put to use earlier.

Von Neumann was one of the first to realize the potential of the unimaginable speed of calculation of the electronic computer. It was he who suggested the structure of the computer as we know it today. He inserted elements of memory into the computer and showed that software could be considered the same as data, thus making the computer a multipurpose tool. Modern computers can perform calculations faster and can process and store more data than could the computers of the mid-twentieth century, but the basic structure remains that proposed by von Neumann. That is why he is considered to be the father of the modern computer.

When the possibilities offered by the computer became apparent, the concept of computation changed as well. Until then, mathematical computation related essentially to numerical calculations, the solution of mathematical equations, weather forecasts, and so on. Today they calculate travel routes, maintain public records, retrieve data, search dictionaries,
help in translation, search the Internet for the nearest restaurant, and of course there are the social networks. All these result from computations.