1.2 The Advent of the Computer

This paper is not about models in general, their variety of functions and characteristics. It is about a special class thereof: mathematical models. Mathematics has always played a fundamental role in representing and formulating our knowledge. As sciences advance they become increasingly mathematical. This tendency can be observed in all scientific areas irrespective of whether they are application- or theory-oriented. But it was not until last century that formal models were used in a systematic way to solve practical problems. Many problems were formulated mathematically long ago, of course. But often they failed to be solved because of the amount of calculation involved. The analysis of the problem – from a practical point of view at least – was usually limited to considering small and simple instances only.

The computer has radically changed this. Since a computer can calculate extremely rapidly, we are spurred on to cast problems in a form which they can manipulate and solve. This has led to a continuous and accelerated pressure to formalize our problems. The rapid and still ongoing development of computer technologies, the emergence of powerful user environment software for geometric modeling and other visualizations, and the development of numerical and algebraic manipulation on computers are the main factors in making modeling – and especially mathematical modeling – an accessible tool not only for the sciences but for industry and commerce as well.

Of course, this does not mean that by using the computer we can solve every problem – the computer has only pushed the limit between practically solvable and practically unsolvable ones a little bit further. The bulk of practical problems which we still cannot, and probably never will be able, to solve efficiently, even by using the most powerful parallel machine, is overwhelming. Whether quantum-computer will change this situation, will be seen in the future. Nevertheless, one can say that many of the models solved routinely today would not be thinkable without the computer. Almost all of the manipulation techniques in mathematics, currently taught in high-schools and universities, can now be executed both more quickly and more accurately on even cheap machines – this is true not only for arithmetic calculations, but also for algebraic manipulations, statistics and graphics. It is fairly clear that all of these manipulations are already standard tools on every desktop machine. Sixth years ago, the hand calculator replaced the slide rule and the logarithm tables, now the computer replaces most of those mathematical manipulations which we learnt in high-school and even at university.