# Notes

## Section 1: Computation as a Framework

History of computing

Even in prehistoric times there were no doubt schemes for computation based for example on making specific arrangements of pebbles. Such schemes were somewhat formalized a few thousand years ago with the invention of the abacus. And by about 200 BC the development of gears had made it possible to create devices (such as the Antikythera device from perhaps around 90 BC) in which the positions of wheels would correspond to positions of astronomical objects. By about 100 AD Hero had described an odometer-like device that could be driven automatically and could effectively count in digital form. But it was not until the 1600s that mechanical devices for digital computation appear to have actually been built. Around 1621 Wilhelm Schickard probably built a machine based on gears for doing simplified multiplications involved in Johannes Kepler's calculations of the orbit of the Moon. But much more widely known were the machines built in the 1640s by Blaise Pascal for doing addition on numbers with five or so digits and in the 1670s by Gottfried Leibniz for doing multiplication, division and square roots. At first, these machines were viewed mainly as curiosities. But as the technology improved, they gradually began to find practical applications. In the mid-1800s, for example, following the ideas of Charles Babbage, so-called difference engines were used to automatically compute and print tables of values of polynomials. And from the late 1800s until about 1970 mechanical calculators were in very widespread use. (In addition, starting with Stanley Jevons in 1869, a few machines were constructed for evaluating logic expressions, though they were viewed almost entirely as curiosities.)

In parallel with the development of devices for digital computation, various so-called analog computers were also built that used continuous physical processes to in effect perform computations. In 1876 William Thomson (Kelvin) constructed a so-called harmonic analyzer, in which an assembly of disks were used to sum trigonometric series and thus to predict tides. Kelvin mentioned that a similar device could be built to solve differential equations. This idea was independently developed by Vannevar Bush, who built the first mechanical so-called differential analyzer in the late 1920s. And in the 1930s, electrical analog computers began to be produced, and in fact they remained in widespread use for finding approximate solutions to differential equations until the late 1960s.

The types of machines discussed so far all have the feature that they have to be physically rearranged or rewired in order to perform different calculations. But the idea of a programmable machine already emerged around 1800, first with player pianos, and then with Marie Jacquard's invention of an automatic loom which used punched cards to determine its weaving patterns. And in the 1830s, Charles Babbage described what he called an analytical engine, which, if built, would have been able to perform sequences of arithmetic operations under punched card control. Starting at the end of the 1800s tabulating machines based on punched cards became widely used for commercial and government data processing. Initially, these machines were purely mechanical, but by the 1930s, most were electromechanical, and had units for carrying out basic arithmetic operations. The Harvard Mark I computer (proposed by Howard Aiken in 1937 and completed in 1944) consisted of many such units hooked together so as to perform scientific calculations. Following work by John Atanasoff around 1940, electronic machines with similar architectures started to be built. The first large-scale such system was the ENIAC, built between 1943 and 1946. The focus of the ENIAC was on numerical computation, originally for creating ballistics tables. But in the early 1940s, the British wartime cryptanalysis group (which included Alan Turing) constructed fairly large electromechanical machines that performed logical, rather than arithmetic, operations.

All the systems mentioned so far had the feature that they performed operations in what was essentially a fixed sequence. But by the late 1940s it had become clear, particularly through the writings of John von Neumann, that it would be convenient to be able to jump around instead of always having to follow a fixed sequence. And with the idea of storing programs electronically, this became fairly easy to do, so that by 1950 more than ten stored-program computers had been built in the U.S. and in England. Speed and memory capacity have increased immensely since the 1950s, particularly as a result of the development of semiconductor chip technology, but in many respects the basic hardware architecture of computers has remained very much the same.

Major changes have, however, occurred in software. In the late 1950s and early 1960s, the main innovation was the development of computer languages such as Fortran, Cobol and Basic. These languages allowed programs to be specified in a somewhat abstract way, independent of the precise details of the hardware architecture of the computer. But the languages were primarily intended only for specifying numerical calculations. In the late 1960s and early 1970s, there developed the notion of operating systems—programs whose purpose was to control the resources of a computer—and with them came languages such as C. And then in the late 1970s and early 1980s, as the cost of computer memory fell, it began to be feasible to manipulate not just purely numerical data, but also data representing text and later pictures. With the advent of personal computers in the early 1980s, interactive computing became common, and as the resolution of computer displays increased, concepts such as graphical user interfaces developed. In more recent years continuing increases in speed have made it possible for more and more layers of software to be constructed, and for many operations previously done with special hardware to be implemented purely in software.

From Stephen Wolfram: A New Kind of Science [citation]