MODELS AND HISTORY OF MODELING
Hermann Schichl∗ Institut fu ̈r Mathematik der Universita ̈t Wien Strudlhofgasse 4, A-1090 Wien, Austria Hermann.Schichl@esi.ac.at
Abstract After a very fast tour through 30,000 years of modeling history, we describe the basic ingredients to models in general, and to mathematical models in particular.
Keywords: Modeling, History of Modeling, Model, Mathematical Model
2.1 The History of Modeling
The word “modeling” comes from the Latin word modellus. It describes a typical human way of coping with the reality. Anthropologists think that the ability to build abstract models is the most important feature which gave homo sapiens a competitive edge over less developed human races like homo neandertalensis.
Although abstract representations of real-world objects have been in use since the stone age, a fact backed up by cavemen paintings, the real breakthrough of modeling came with the cultures of the Ancient Near East and with the Ancient Greek.
The first recognizable models were numbers; counting and “writing” numbers (e.g., as marks on bones) is documented since about 30.000 BC. Astronomy and Architecture were the next areas where models played a role, already about 4.000 BC.
It is well known that by 2.000 BC at least three cultures (Babylon, Egypt, India) had a decent knowledge of mathematics and used mathematical models
∗funded by EU project COCONUT IST-2000-26063 2526 MODELING LANGUAGES IN MATHEMATICAL OPTIMIZATION
to improve their every-day life. Most mathematics was used in an algorithmic way, designed for solving specific problems.
The development of philosophy in the Hellenic Age and its connection to mathematics lead to the deductive method, which gave rise to the first pieces of mathematical theory. Starting with Thales of Miletus at about 600 BC, geometry became a useful tool in analyzing reality, and analyzing geometry itself sparked the development of mathematics independently of its application. It is said that Thales brought his knowledge from Egypt, that he predicted the solar eclipse of 585 BC, and that he devised a method for measuring heights by measuring the lengths of shadows. Five theorems from elementary geometry are credited to him:
1. A circle is bisected by any diameter. 2. The base angles of an isosceles triangle are equal. 3. The angles between two intersecting straight lines are equal. 4. Two triangles are congruent if they have two angles and one side equal. 5. An angle in a semicircle is a right angle.
After Thales set the base, Pythagoras of Samos is said to have been the first pure mathematician, developing among other things the theory of numbers, and most important to initiate the use of proofs to gain new results from already known theorems.
Important philosophers like Aristotle, Eudoxos, and many more added lots of pieces, and in the 300 years following Thales, geometry and the rest of mathematics were developed further. The summit was reached by Euclid of Alexandria at about 300 BC when he wrote The Elements, a collection of books containing most of the mathematical knowledge available at that time. The Elements held among other the first concise axiomatic description of geometry and a treatise on number theory.
Euclid’s books became the means of teaching mathematics for hundreds of years, and around 250 BC Eratosthenes of Cyrene, one of the first “applied mathematicians”, used this knowledge to calculate the distances Earth-Sun and Earth-Moon and, best known, the circumference of the Earth by a mathemati- cal/geometric model.
A further important step in the development of modern models was taken by Diophantus of Alexandria about 250 AD in his books Arithmetica, where he developed the beginnings of algebra based on symbolism and the notion of a variable.
For astronomy, Ptolemy, inspired by Pythagoras’ idea to describe the celestial mechanics by circles, developed by 150 AD a mathematical model of the solar system with circles and epicircles to predict the movement of sun, moon, and the planets. The model was so accurate that it was used until the time of Johannes Kepler in 1619, when he finally found a superior, simpler model for planetary motions, that with refinements due to Newton and Einstein is still valid today.
Models and the History of Modeling 27
Building models for real-world problems, especially mathematical models, is so important for human development that similar methods were developed independently in China, India, and the Islamic countries like Persia.
One of the most famous Arabian mathematicians is Abu Abd-Allah ibn Musa
Al-H ̄ wa ̄rizm ̄ı (late 8th century). His name, still preserved in the modern word
algorithm, and his famous books de numero Indorum (about the Indian numbers
— today called arabic numbers) and Al-kitab al-muhtas. ar fi h. isa ̄ b al-gˇ abr wa’ l- ̄
muqa ̄bala (a concise book about the procedures of calculation by adding and balancing) contain many mathematical models and problem solving algorithms (actually the two were treated as the same) for real-life applications in the areas commerce, legacy, surveying, and irrigation. The term algebra, by the way, was taken from the title of his second book.
In the Occident it took until the 11th century to develop mathematics and mathematical models, in the beginning especially for surveying.
The probably first great western mathematician after the decline of Greek mathematics was Fibonacci, Leonardo da Pisa (ca. 1170–ca. 1240). As a son of a merchant, Fibonacci undertook many commercial trips to the the Orient, and in that time he got familiar with the Oriental knowledge about mathematics. He used the algebraic methods recorded in Al-H ̄ wa ̄rizm ̄ı’s books to improve his success as a merchant, because he realized the gigantic practical advantage of the Indian numbers over the Roman numbers which were still in use in western and central Europe at that time. His highly influential book Liber Abaci, first issued in 1202, began with a presentation of the ten “Indian figures” (0, 1, 2, …, 9), as he called them. This date was especially important because it finally brought the number zero to Europe, an abstract model of nothing. The book itself was written to be an algebra manual for commercial use, and explained in detail the arithmetical rules using numerical examples which were derived, e.g., from measure and currency conversion.
Artists like the painter Giotto (1267–1336) and the Renaissance architect and sculptor Filippo Brunelleschi (1377–1446) started a new development of geometric principles, e.g. perspective. In that time, visual models were used as well as mathematical ones (e.g., for Anatomy).
In the later centuries more and more mathematical principles were detected, and the complexity of the models increased. It is important to note that despite the achievements of Diophant and Al-H ̄ wa ̄rizm ̄ı the systematic use of variables was really invented by Vieta ́ (1540–1603). In spite of that it took another 300 years until Cantor and Russell that the true role of variables in the formulation of mathematical theory was fully understood. Physics and the description of Nature’s principles became the major driving force in modeling and the devel- opment of the mathematical theory. Later economics joined in, and now an ever increasing number of applications demand models and their analysis.
28 MODELING LANGUAGES IN MATHEMATICAL OPTIMIZATION
In the next section we will take a closer look at models, their function and their most prominent characteristics. Further information on mathematical history can, e.g., be found in .