# General View Of Mathematics

Officially:       Survey of mathematical thought in logic, geometry, combinatorics and chance.

Addition:         Historical perspective on mathematics, the influence of mathematics on aesthetics, ethics and other systems of thought, and vice versa.

I. Course Content

General introduction

“In its broadest aspect mathematics is a spirit, the spirit of rationality. It is this spirit that seeks to influence decisively the physical, moral, and social life of man.” (Kline, p.10)

Aesthetics

The influence of mathematics on working of perspective (in paintings), on the proportions and ratios in sculpture, symmetry. But also, the influence of taste on the development of mathematics, such as in the development of symmetry groups of figures.

The Greek preference for idealizations and abstractions expressed itself in philosophy and mathematics. It showed itself just as clearly in art. Greek sculpture of the classical period dwelt not on particular men and women but on ideal types. This idealization extended to standardization of the ratios of parts of the body to each other. The underlying idea was that a permanent, mathematical ratio captured the ideal beauty. Modern miss-elections that award girls with certain ideal measures can be seen as a contemporary counterpart.

Similarly, Greek architecture was standardized to certain perfect ratios.

project 1:        Let a student draw on the blackboard a three-dimensional table and chair. Show the most likely absence of perfect perspective. Show a similar fault of a Mediaeval painting, the perfect perspective of a Renaissance painting, and the purposeful distortion of perspective (as in Van Gogh's “Room”) as a form of expression in Modern Art.

project 2:        Compare several of the proportions between head and body of 2 Greek sculptures and the proportions of children in a 16th Dutch painting (Children as small adults).

Body to head of Praxiteles' Aphrodite of Cnidos are proportioned as 8:1.

Try to find Polyclitus' table of the ideal ratios.

Ethics

Spinoza, Kant and the strong mathematical form of our ethical intuitions: In ethics we talk about “principles” a word originally derived from mathematics. John Rawls.

project:           Copy parts of the Ethica by Spinoza and Euclid's Elements to compare.

Other systems of thought

Descartes' Cogito Ergo Sum. Kant's concept of mathematics as the pure science of our spatial intuitions. More generally: the idea of the Greeks that the gods used mathematical reasoning in designing the universe. Galilee's “The book of nature is written in the language of mathematics.”

project:           Copy some passages of Galilee's writings.

Chance and Combinatorics

1. The data are not the data & the strangeness of chance

Use the example of the coin and heads showing up. Extrapolate that to statistical research in general. Give them an idea of the particularity of the notion of a probability. How to understand a probability? It is not always easy! Give them the example of the show-master giving away a price. (first one out of three, then, to explicate, one out of hundred!)

Give them the example of the particle being a probability wave; therefore, if this ‘particle' is used to make a triggering device for a certain poison and we put a cat with this device into a black box, then in fact we have a probability wave of a living and of a dead cat.

Formal language

A formal language allows an efficiency of thought, and consequently a better understanding of larger chains of thoughts.

Explain the importance of the formal language that mathematics employs (e.g. solving an equation, see also Kline, p.7-8)

Trace back the history of the formal language (Notational systems of Egyptians and Babylonians, Leibniz and his notions of the Universal Characteristic, Frege & Russell, modern mathematics).

Importance of Mathematics for Descartes' Meditations

[Reference to page-numbers in the standard edition: Oeuvres de Descartes, publies par Charles Adam et Paul Tannery, 13 volumes. Paris: Cerf, 1897-1913)

copy:   pp.59-65 and pp.88-93 in the Hackett edition, trans. D.A. Cress.

When we want to appreciate the influence mathematics has had on the history of thought, it is a good idea to look at some writings, such as Rene Descartes' Meditations on First Philosophy. Moreover, it allows us to penetrate deeper into what it means to think mathematically. Descartes explicitly discusses several issues. Let enumerate them briefly:

·       knowledge as a building with foundations

·       withhold of assent (p.18) until one reaches an indubitable axiom, an “immovable point” that “Archimedes sought” (p.24), or, with equal right, that Euclid sought: “I am a thinking thing” -- cogito ergo sum.

·       the importance of a sustained “mental gaze” (p.64) as a vehicle to truth. If there is no experience to which we can (or want to, because of its imperfection) compare our conclusions (as in ‘pure' mathematics), we should grasp the chain of reasoning in a sustained way, because otherwise it might be the case that we error constantly at some point, without ever noticing it. So in the second Meditation, Descartes first recalls the meditation of the day before:

“... I will work my way up and will once again attempt the same path I entered upon yesterday.” (p.24)

·       The nature of understanding that Descartes reaches is a certain form of transparency. Suddenly things are clear to him. It seems even silly to say that it is difficult. It is clear and open. After you have read the complete proof and see it in front of your inner eye, it really seems as if you see it:

project: do one of Euclid's proof and let the students feel what it means to see it.

Tell the story of Professor Weiss: it's evident.

“Not only are these things manifestly known and transparent to me, viewed thus in a general way, but also, when I focus my attention on them, I perceive countless particulars concerning shapes, number, movement, and the like. Their truth is so open and so much in accord with my nature that, when I first discover them, it seems I am not so much learning something new as recalling something I knew beforehand. In other words it seems as though I am noticing things for the first time that were in fact in me for a long while, although I had not previously directed a mental gaze upon them.” (p.64)

History of Mathematics

If we want to understand and appreciate what mathematics actually means to us in our society, a simplistic account of the matter will do a lot of harm. As Morris Kline remarks “School courses and books have presented ‘mathematics' as a series of apparently meaningless technical procedures.” (Kline, p.vii) and he notices that even “educated people almost universally reject mathematics as an intellectual interest.” (Kline, p.vii) It is almost a public secret that mathematics stand for a dull and somewhat superfluous, or at least, unnecessarily abstract, pursuit. It is undeniable that in its current state the Mathematical Discipline, as distinct from mathematics, has become detached from the original interdisciplinary connections that existed. From the key-role that it once shared with philosophy, it has now been marginalized as a methodological tool (as philosophy has been marginalized in its own way). But if we would think that thereby mathematics in general does not contain more than that, we are mistaken.

We share with Kline the aim to advance the thesis that

“... mathematics has been a major cultural force in Western civilization.” (Kline, p.vii)

What we mean with that is not only that mathematics has been an important tool in the advance of commerce, physics, engineering and the other sciences, but that it also is presupposed not so much as a tool, but rather as a constitutive or regulative factor in Aesthetics (constituting order, symmetry and the proportions of Beauty), Ethics (Spinoza's geometry of Ethics a la Euclid's Elements), our conception of Nature (as adapting itself to mathematical description; Galilee's “The book of nature is written in the language of mathematics.”), our idea of rational discourse, even the superstition of astrological systems are based on a belief in the mathematical connection between stars and our lives.

When we shall give an historical overview of the development of mathematics, then we shall treat history instrumental to further the end of showing the importance, application, origin, method of mathematics. It shall gradually be clear that mathematics has played an undeniable role in many of our cultural practices.

In the course of our study of the history, we shall come across certain mathematical methods and techniques that we shall study shortly in order to give the student a grasp of what these concepts mean and for which use they were originally designed. This more technical encounter with mathematics is an intrinsic part of the course. However, the level of none of the methods shall be too high for people without any prior knowledge.

Some fundamental questions

It has been our thesis that mathematics is a so-called creative, both regulative and constitutive, force in Western society. It shall thus be good to first investigate what caused and causes men to pursue it. We can at least distinguish four different causes:

1.     Out of social needs; the push of commercial and financial transactions, navigation, calendar reckoning, the construction of bridges, dams, churches, and palaces the design of fortifications and weapons of warfare, and numerous other human pursuits involve problems that can best be resolved by mathematics.

2.     Rational organization of natural phenomena; the concepts, methods, and conclusions of mathematics are the substratum of the physical sciences.

3.     Intellectual curiosity and a zest for pure thought; In ancient Greek times the pursuit of the geometric properties of figures was largely and intellectual pursuit, such as in the 17th century the development of probability theory out of the question what would be a fair division of the stake if a gambling game is interrupted before its close. The conversion of mathematics in an abstract, deductive and axiomatic system in Greek times was another instance of the urge of pure thought.

4.     The search of beauty; From the Persian geometric figures, the Greek ideal proportions, the Renaissance discovery of perspective, to cubist painting of our times, numerical harmony, or disharmony, has been the very vehicle of several of our ideas of beauty.

The mathematical language is in itself an intriguing subject. Over the ages one has established a language of mathematics. It might seem that the mathematical language, because it is completely based on convention and agreement, has been in itself unessential to the progress of mathematics. The contrary is true. The language itself has sometimes been very helpful or sometimes obstructive for the development of mathematics. For instance, the Roman counting system made arithmetic calculations very difficult (I, V, X, L, C, D, M). The Greek failed to develop a number system completely. Greeks and Hebrews used letters of the alphabet as their numbers. That led them away from calculation. The ten base system, introduced in the West by the Arabs from India, boosted a new form of mathematics. In fact, such systems were already known by the Babylonians, who uses a system of base 60.

In general, the mathematical language is characterized nowadays by a great compactness. This compactness makes for the efficiency of thought.

Give example: Pythagorean Theorem: a^2 + b^2 = c^2

Bayes' Theorem: P(B|A) = P(A|B)P(B) / {P(A|B)P(B) + P(A|~B)P(~B)}

Moreover, the mathematical language is precise, so precise that it is often confusing to people unaccustomed to its forms.

“I did not give him a cent.”

Ordinarily it would mean that I did not give this guy any money, but a mathematician, taking the sentence at face-value, could mean that he gave him a dollar.

A mother saying this to her child, would also mean that the child would not get the dessert if it didn't finish its porridge. For the mathematician that does not follow from the premises!

Another peculiar aspect of mathematics is the epistemological status of its conclusions. The deductive inference structure lends to the conclusion only a derivative form of truth. Only if the premises (or axioms) are true, then the conclusions are true. And because the axioms of mathematics are always postulated or posited and abstract, therefore, mathematics does not contain any truths in the usual sense of the word. Specifically the abstractness of its axioms and conclusions makes mathematics, however, suitable for different interpretation-schemes, at least if we presuppose that the underlying structure is mathematically ordered.

We can say that in its broadest aspect mathematics is a spirit, the spirit of rationality. It is this spirits that seeks to influence decisively the physical, moral, and social life of man.

Egypt and Babylon

The earliest beginnings of a mathematical practice are located in the Near East. Primitive societies that originated there, developed a number system to satisfy man's need. It is no doubt that a lot of early mathematics was prompted by social needs. Any primitive society needs a system of counting, for instance.

The Babylonians introduced a counting system as follows:

Ú stands for a unit, so ÚÚ stands for two, and so forth up to nine.

× stands for ten, so ×× for twenty, and so forth up to fifty..

Therefore, ×××ÚÚ meant 32

The important thing now happens when the Babylonians want to describe a quantity over sixty. They put a number of units before the tens, just like we do!

Therefore, Ú××ÚÚ meant 60 + 20 + 2 = 82

Give some broader explanation, and start calculating with different systems, e.g. with base 4, with base 16 (computer machine code, hexadecimal), and with base sixty (Babylonians, our way of calculating hours and angles). Note the importance of the zero.

Because the Babylonians introduced place value in connection with the base sixty, the Greeks and Europeans used this system in all mathematical and astrological calculations until the sixteenth century and it still survives in the division of angles and hours into sixty minutes and sixty seconds. Base ten was developed by the Hindus and introduced into Europe during the late Middle Ages.

The mathematics that the Babylonians and Greeks had available, were used in trade (promissory notes, letters of credit, mortgages, deferred payments, and the proper apportion of business profits, as becomes clear from papyri and clay tablets) in building and in agriculture. It is however a mistake, a common mistake, to believe that mathematics in Egypt and Babylon was confined just to the solution of practical problems. In Babylon and Egypt the association of mathematics with painting , architecture, religion, and the investigation of nature was no less intimate and vital than its use in commerce, agriculture and construction. Often the argument runs:

‘mathematics was applied to calendar reckoning and navigation; hence the creation of mathematics was motivated by these practical problems much as the need to count led to the number system'

This is a post hoc ergo propter hoc type of argument, but it does not have much credulence. Before the mariner or the farmer could use the astrological knowledge, it first had to be observed irrespectively of the later applications. Men must have been filled with instinctive wonder and awe of nature, men with irrepressible philosophical drives, patiently observed the movement of the sun, moon, and stars. Surely the existence of regularities in the heavens had to be recognized before anyone could think of applying them. Both in the Greek as an the Babylonian society the knowledge of these regularities was exclusively known by priests. Several religious practices are believed to originate of their clever, exclusive control of this knowledge. In fact, it is believed that Egyptian priests knew the solar year to be 365 1/4 days in length but deliberately withheld this knowledge from the people.

Even though the Babylonian and Egyptian civilizations had acquired a large body of mathematical techniques, there was no general development of a subject nor do the texts enunciate any general principles. Often the texts, such as the Egyptian Ahmes papyrus, works out specific problems. It could be that more general principles have been known to the priests, but that they have kept it secret. The general character of the Egyptian theocracy gives some weight to that view.

mathematics and pseudo-science

It is a natural side-effect of exclusive knowledge that it creates a form of superstition in the people not in possession of it. Often this practice was sanctioned by the experts themselves as a method to secure their position. It is known, for instance, that when Columbus had landed in Central America and needed food and supplies for his crew but was refused by the locals, he told them that his God would darken the sky if they would not give in, while knowing on the basis of calculations that a partial sun-eclipse was predicted.

In Egypt times, when priest had exclusive control, the tie between religion and mathematics was even stronger. Astronomy and astrology were intimately connected. Though important, another reason has to be high-lighted. There are good reasons to believe that the position of the moon, sun and stars do influence human affairs. Crops depend on them, the female period follows the moon and other similar association gave rise to this idea.

This belief has been very persistent, judging only from the amount of horoscopes in newspapers and magazines. It is not strange that also in Greek times such pseudo-sciences found some attraction, specifically the science of the cabala. The reason was quite simple. The Greeks, as the Hebrews, used the letters of their alphabet as their number system. Thus a calculus of associations came into existence.

Give example: several popes in the first millennium accused each other of being in fact a devil on the basis of name-number associations.

Greek Mathematics

Notwithstanding a certain mysticism that kept attached to mathematics in the subsequent period of the Greek civilization, the character of mathematics changed a great deal. In a lot of ways, the Greek way of practicing mathematics (though not all its practices) is still the model of modern math.

From the modern point of view Egyptian and Babylonian mathematics was defective in an important aspect: the conclusions were established empirically. The Pythagorean theorem was, for instance, known to them only inductively; that is, they had seen it work for several cases and assumed it would work in ‘similar' instances too. The Babylonians or the Egyptians never provided a proof of the theorem. This poses the important question whether induction from experience is the only way that we, human beings, can obtain knowledge. When we just think of the conclusions of mathematics, like for instance 1+1=2, then we would like to say that it is not only because we have seen it be the case that 1 apple plus 1 apple make 2 apples, 1 car plus 1 car make 2 cars, but that there is some always the case, that 1+1=2, irrespective of how many times we have seen it in experience.

The Greeks introduced a new form of reasoning that tried to meet of objections against the view that we only know that 1+1=2 or that the Pythagorean Theorem holds for all rectangular triangles from experience. The kind of reasoning therefore had to be:

- abstract; that is, it had abstract from the particular circumstances, i.e., from the particular ones and the particular rectangular triangle.

- indubitable in its conclusions; that is, the conclusion reached by this form of inference had to be absolutely certain.

motivation for abstract deduction in Greek society

The form of reasoning they established is called deductive reasoning. The basis of deduction are the existence of certain premises, that are ‘absolutely true' or are agreed upon, plus a set of inference rules by which we can deduce conclusions from the premises. This probably seems very much like philosophical reasoning, i.e., to argue on the basis of a set of ideas to a new idea (For instance, from the fact that nothing is uncaused Thomas Aquinas deduced the existence of God). In fact, the Greeks were gifted with great philosophical interest. Partially the explanation for this has to be found in the structure of the society. The Greek society was a slave based society. Famous Greeks spoke out unequivocally about their disdain of work and business. Most of the commerce and trade was run by slaves. Leisure was a status symbol. Plato's description of his teacher's -- Socrates --activities describe him as a dandy talking about issues as justice, immortality and beauty. Moreover, the slave basis of classical Greek society fostered a divorce of theory from practice and the development of the speculative and abstract side of science and mathematics with a consequent neglect of experimentation and practical application. The great contribution of the Greeks in the area of mathematics is not to be found in its practical applications. Instead of measurement and calculations, shapes and forms were favored.

Philosophers do not reason, as do scientists, on the basis of personally conducted experimentation or observation. Rather their reasoning centers about abstract concepts and broad generalizations. It is difficult after all, to experiment with souls in order to arrive at truths about them. The natural tool of philosophers is deductive reasoning. It, moreover, was in line with the Greek taste. The Greek preference for deduction can be seen as a facet of the Hellenic love for beauty as order, consistency, completeness and definiteness, rather than as an emotional experience. It has been argued that the low life-expectancy rate negatively influenced the Greeks appreciation for the temporal, such as wealth and emotions.

Another aspect of the disregard for the temporal shows itself in the adoration of the abstract and ideal. Whereas physical objects are short-lived, imperfect and corruptible, shapes and forms (in the mind) are permanent, ideal and perfect. Besides introducing deduction into mathematics, the second vital contribution of the Greeks consisted in their having made mathematics abstract. In geometry, the words point, line, triangle, and the like became mental concepts merely suggested by physical objects but differing from them. Thus, the Greek mathematics gained generality, something that the Egyptian and Babylonian mathematics didn't possess.

mathematics and philosophy

If we look at when the development of mathematics in the Greek society took of, we can make a direct connection with philosophers like Plato. A lot of Greek mathematicians had been his pupil or came from later generations of students in the Academia founded by him. We can see that in his philosophy both certain new aspects of mathematics as the importance of mathematics prefigure. First of all, the rationalist aspect of Plato's philosophy and the emphasis on Form over Matter can be seen as a model for the new mathematics. Secondly, in his philosophy Plato explicitly stresses the importance of the study of mathematics for a good life.

Brief Grasp of Plato's Philosophy

Plato described the situation of human beings as being imprisoned in a cave. (Draw) This is not so strange as it might appear at the first sight. In our lives we see thousands of different chairs and still we call them all a “chair.” Similarly, we call this painting “beautiful,” but also that one. So there must be some Idea of Chair and Beauty that we apply to all of them. From this General Idea we deduce that this particular object is a chair, and that that particular painting is beautiful. Apparently there must be a World of Ideas, from which all that we human beings see are mere shadows. Plato described that world as the world of light, outside the cave. To make the transition from darkness to light, mathematics is the ideal means. On the one hand, it belongs to the world of sense, for mathematical knowledge pertains to object on this Earth. On the other hand, considered solely as idealization, solely as an intellectual pursuit, mathematics is indeed distinct from the physical objects it describes. It purifies the mind by drawing it away from the contemplation of the sensible and perishable to the eternal.

For Aristotle and other Greek philosophers the form of an object is the reality to be found in it. Matter as such is primitive and shapeless.

Geometry and Euclid

One of the main neglects of the Greeks has been the development of a number system and algebra. They however did recognize certain problems that existed in the number system. Up until the Greeks, the number that were known were the integers and the fractions. This of course constituted an immense amount of numbers, but the Greeks discovered there existed other numbers that could not be written as a fraction.

It was thought at first that p and q could be found that would describe the square root of two. The Babylonians had used 141/100 but the Pythagoreans discovered that this was not exactly right, and that in fact no such p and q existed. Ö2 is a new kind of number, that they called irrational, because it could not be expressed exactly as a ratio of whole numbers. In order to solve this problem the Greeks simply treated a number as the length of a line-segment. Ö2 was therefore simply the length of the hypotenuse of a triangle with two sides having length 1. Calculations with these number were again interpreted geometrically. Therefore, 3 * Ö2 meant the area of a rectangle with sides 3 and Ö2.

The Greeks developed a geometrically approach to mathematics, partly, as we have seen, out of practical reasons -- to deal with new mathematical discoveries, such as the irrational numbers, partly in concordance with the emphasis of Greek philosophy on the eternal forms. The fulfillment of this movement in to its summum was established by Euclid and several Alexandrine mathematician of the Hellenic area later on. The mathematization process started with Thales of Milete in the 6th century BC, was empowered by the Pythagoreans in the subsequent centuries (who proposed rather than a geometrization of nature, a formulization, i.e. a putting into numbers, of nature -- a more modern approach.) The Pythagoreans were the first group to treat mathematical concepts as abstractions and though Thales and his fellow Ionians had established some theorems deductively, the Pythagoreans employed this process exclusively. Plato was the great philosopher that incorporated in his philosophy several Pythagorean principles. (“Let no one ignorant of geometry enter here.”) All that prepared the stage for a large group of mathematicians among his followers and the school he founded.

It was finally Euclid who in one masterful book, The Elements, of around 300 BC collected all the available mathematical knowledge of about 500 theorem and had them deduced from a few sagaciously chosen axioms. We shall follow Euclid's approach and then compare it to a contemporary attempt by Spinoza to render a similar account for another field of investigation, ethics.

Euclid's begins his book with a set of definitions in which he sets out to explain what he will mean with his basic terms. It has been a long-time dispute what the status of these definitions actually are. Kline in his book argues for the redundancy of the definitions:

To be precise about what his abstract terms included Euclid began with some definitions. ... In his definitions Euclid went to unnecessary and inadvisable lengths. A logical, self-sufficient system must start somewhere. It cannot hope to define every concept it uses, for definition involves describing one concept in terms of others and the latter in terms of still of others. (Kline, p.42)

Definitions, Kline argues, rely on undefined terms so it does not really matter where to start. This argument is well-known in the philosophy of language, but it does not take into account that definitions can eliminate ambiguities, and thus can have a positive contribution to the clarity of the treatise, without of course having eliminated the need for undefined terms.

After the introduction of definitions, Euclid has to provide the premises for the deductive process that he wants to start. The form and structure of a deductive process was already very well known in Greek civilization by that time. Aristotle describes:

It is not everything that can be proved, otherwise the chain of proof would be endless. You must begin somewhere, and you start with things admitted but undemonstrable. These are first principles common to all sciences which are called axioms or common opinions. (Aristotle quoted in Kline, p.43)

In the selection of axioms Euclid displayed great insight and judgment.

It was an historical fact that at that point several different deductive systems co-existed, all with different axioms. It was Euclid's great contribution to provide a minimal set of axioms and postulates (ten in total) that seem so trivially true but at the same time make it possible to deduce a large set of theorems. Each theorem, how complex it may seem at the first, is a series of deductive statements, without ever appealing to common sense or intuitive understanding. The proofs of the theorems of geometry could in principle be understood by a blind person who would only understand the concepts point, line and plane.

project: go over some definitions, axioms and a theorem (e.g. Kline, p.44).

As we explained the Greeks were drawn to geometrical objects. Shape and form were important aspects of their philosophy. But they were drawn to specific kind of figures, like the spheres, circle and other objects which were in some sense ideal. The attraction can only explained by an aesthetic appeal. The study of these special object was triggered by an attraction to symmetry considerations. The symmetry could be of several kinds.

Rotation symmetry was appealed to the Greek. They had were attracted to object which could be rotated and remained the same. Obviously a square fits that description, so does the equilateral triangle or any regular polygon. From certain of these regular polygons one could make a three dimensional figure. It was an obsession for the Greeks to find all possible kinds of these, so-called, regular polyhedra. Euclid showed that there must be exactly five of these polyhedra, known as the tetrahedron, the octahedron, the cube, the dodecahedron and the icosahedron. These figures where not just mathematical toys, but instead played an important role in the establishment of a mathematics based exploration of nature (Kepler), even though later the emphasis shifted from a geometrical, Euclidean interpretation towards a number-based, Pythagorean interpretation.

project: explain concepts of symmetry; point symmetry, “mirror” symmetry, rotation symmetry. Show the essentials of symmetry-group algebra.

Several question that Greek mathematics raised were not answered by them, and in fact they have remained for a long time a source of agony, but more often of fruitful speculation, which might not have produced a right answer but kept the pure spirit of mathematics alive, throughout the ages. These three questions were all construction problems which all had the self-imposed restriction that they were not allowed to be solved by using more than a ruler and a compass:

1.     Squaring a circle. This problem was to find a construction of how to make a square with the same area as that of a given circle.

2.     Doubling a cube. This involved finding a construction to make a cube with double the volume of that of a given cube.

3.     Trisecting an angle. There were constructions know of how to trisect an angle in to two equal parts, but the Greeks wondered whether it was possible to find a construction that separated an angle into three equal parts.

Again the restriction was given in by no practical consideration, but solely by aesthetic demands: “the desire to keep geometry simple, harmonious.” (Kline, p.50) It is only about seventy years ago that it has been proven that these constructions could not be made. Nonetheless, in many of the Greek aesthetically motivated research applications have been found in modern times. Sometimes these applications were on the level of theoretical assumptions about the structure of nature (Kepler and later on the conic sections, which paved the way for modern astronomy), sometimes they solved a problem of a practical nature.

project: show how a theorem from Euclid's geometry finds its application in practical and theoretical consideration of nature. Assume that you are A and you want to visit your mother at B, but your mother-in-law, who lives on the river l, wants you to visit her as well; it doesn't matter where at the river. What would then be the most efficient way from A to B via l? Euclid proved that the shortest would be at that point P, so that the angle between AP and l is equal to the angle at PB and l.

In two instances it becomes clear why it seemed obvious to the Greek and later generation of scientist why geometry was such a important medium to understand nature. Light, for instance, acted in accordance with this theorem. It always takes the shortest path and when it hits a mirror, the incoming angle will always be equal to the outgoing angle. Similarly, a billiard player incorporates this rule, to which nature seems to listen, when he tries to hit B from A vial the side of the table l. Encouraged by such evidence, Western man was inspired to apply reason elsewhere. Theologians, logicians, philosophers, statesmen, and all seekers of truth have imitated the form and procedure of Euclidean geometry. Later on, we shall expend on this phenomena, which we could call: nature acquires reason.

For the Greeks mathematics was more than a proper model of reasoning. In fact, rational and aesthetic as well as moral interests can hardly be separated in Greek thought. Repeatedly we read that the earth must be spherical because the sphere has the most beautiful shape of all bodies and is therefore divine and good. What one should remember is that Beauty for the Greeks did not have the connotation of “unique”, “expressive”, “ornamental” or “subjective.” Beauty was simply order and harmony. The reason why it was decided, even originally in modern times, that heavenly bodies must traverse the sky in circles was because the circles shared with the sphere the perfect form. Moreover, as uniformity and constancy was considered to be beautiful, these bodies were supposed to traverse equal distances in equal intervals of time. Lucidity, simplicity, and restraint were and are the ingredients of beauty. They are still important qualities in the construction of scientific theories, even though they lost their exclusive place as the ultimate quality of beauty.

Now I have put forward the importance of Euclidean geometry, I have to point out its so-called limitations. With limitations I mean the obstacles or mere deficiencies it immanently possessed that obstructed or simply were not advantages for the progress of the modern search of truth in whatever area. One of these limitations we have already mentioned, which was the absence of numerical and algebraic considerations. Another was the static character of Euclid's geometry. Let's concentrate on the latter aspect.

In Greek science the concept of the infinite was scarcely understood and frankly avoided.

The Fear for Infinity

In Early Greek thought there was something called a ‘fear for infinity.' This stood for the inability of the Greeks -- and most people still today -- to make an intuitive sense out of ‘infinity'. They had no idea how to deal with:

·       infinitely small time, objects and movement

·       infinite emptiness (horror vacui)

·       infinitely large objects and numbers.

“What is infinity?”, they asked themselves and because deductive reasoning had brought them to discover eternal truths (Euclidean Geometry), they had initial hopes that in a similar way one could discover the essence of infinity. These initial hopes were however quickly destroyed by the famous paradoxes of Zeno.

The Greeks preferred circular, i.e. periodical (and thus limited), motion. The concept of a limitless process frightened them and they shrank before ‘the silence of the infinite space;' horror vacui. In philosophy too the infinite was avoided. Aristotle says that the infinite is imperfect, unfinished, and therefore unthinkable. The infinite does not exist, because it is formless, i.e. it has no essence! In order to understand what this means, look at the following diagram that represents a kind of general Greek philosophy:

Although there is a general inability to deal with the infinite pervading Greek thought, the conclusions that can be attached to it, differs from philosopher to philosopher.

Zeno wanted to show that movement does not exist, and he employed the ‘axiom' of the non-existence of the infinite to prove that. Let's for instance look at an arrow that seems to fly according to our (deceptive) senses. At each instant of time the arrow occupies a definite position. Because an instant has no duration the arrow is standing still at that moment, and in fact, at each instant. If an object is at rest at each instant, then that is simply what it means that it is always at rest. Therefore, reason proves that our senses are deceptive and that the arrow is not moving at all.

Democritus employed the same axiom of the ‘nonexistence of infinite smallness' to prove that there have to exist atoms. He performed a thought-experiment in which he cut up a piece of matter, and he wondered whether we can keep on cutting the object for infinity. He reasoned that it is impossible to divide a physical object infinitely, because infinite division would create “cosmic mush, formless and ultimately nonexistent.” The infinite divided stuff, being formless, would be unable to reconstitute itself. Infinite division is taking away the form, i.e., the essence of the physical object, which is impossible. Therefore, division has to stop somewhere on something indivisible -- a-tomos. This was the first atomic theory.

We see that both Zeno and Democritus employed deductive reasoning, applying the axiom of the nonexistence of the infinite to make conclusions. The Greeks had an -- on the infinite bordering -- trust in deductive reasoning and an skepticism regarding the senses that they did not only not bother to check their results by experimentation, but even avoided any such check.

Static character of Greek thought

Euclidean geometry is static. The properties of changing figure, something which becomes extremely important in the 17th century and will boost a lot of areas of scientific research, are not investigated. We find an equivalent in Greek drama. It is often described as static. We are presented at the very beginning of the play with a complete account of prior happenings that pose a problem for the characters involved, and the play concerns itself with the mental struggles and minor deeds that eventuate in a denouncement almost foreseen. The Greek tragedies emphasize the working of fate and necessity -- much like how a person involved in Euclidean geometry would find herself.

The Middle Ages in Europe were a dark period in the development of Mathematics. Ironically, the dogmatic spirit of that era kept a similar kind of deductive reasoning at the core of the established truth. The search for truth came forth out of a (Neo) Platonic crave for absoluteness which once had spurred Euclid on his path. Now, however, the established axioms were the truths mentioned in the Bible. Cosmas, a 6th century merchant in Alexandria, believed the earth was flat, as he recorded in his Topographia Christiana, because as the Bible says: “Man lives on the face of the earth.” Dogmatism functioned as a well-structured deductive system.

Another influence of (Neo) Platonism on the era was a distrust of the senses. In a sense this was exactly what had inspired Euclid and others to establish Mathematics as a strict deductive science, but which was destructive for further development of the other empirical sciences. Remember how for Plato the only “real” triangle exists as an idea in the mind. All triangles in this world are not really triangles, but only approximations. Our senses might tell us that we see a triangle on the black-board, but in fact they are deceiving us. The only true triangle is the one you think. For Plato this was not only the case for a triangle, but for any object. Platonism was very prevalent in the beginning of the first millennium, and Christianity simply adopted several of these theories. Accordingly they argued that it was the devil that ruled our senses. If the soul has to choose between belief (i.e., believing the ‘axioms' of the Bible) or what the eyes see, then it is rational to choose for the former, as the latter might deceive us all the time.

An early Church Father (i.e., a Christian Philosopher) Augustine who lived around 400 AD reflected this spirit, where belief had priority of the scientific view of the world, in his saying: “Credo ut absurdum est” (I believe because it is absurd). Scientific investigations of nature had very little priority in the Middle Ages, and this spirit reigned in Europe over a period of about 1000 years. Changes came in the late Middle Ages. Thomas Aquinas, also a Christian Philosopher of about 1275 AD, was a symbol of change in favor of science and rationality. He attempted to reconcile Faith and Reason with one another. He claimed that there could not be any sharp distinction between the two as the early Christian Philosophers under influence of Platonism had claimed. So for all what Thomas argued, he claimed that there should be reasons from the Scriptures and reasons from nature. The dichotomy that Plato and Augustine had defended did not exist, according to Thomas.

The Bible allowed its followers to turn away from the study of nature in itself. The dogmatic spirit of Christianity destroyed the Alexandrian spirit of learning, as if what was found was in correspondence with the Scriptures, then it was simply superfluous; if not, it was not true.

Renewal of the Mathematical Spirit

The gradual change at the end of the Middle Ages back into the direction of a scientific spirit was due to several factors. First of all, the Muslims conquered part of Europe in the South and reintroduced the scientific spirit into the Western world. There were several reasons why the Muslim community had cherished science more than Christianity.

1.     In the Muslim world, the prohibition according to the Koran of the depiction and representation of human bodies stimulated the investigation of mathematical symmetry as an art-form. Muslim artists unlike Western artists did not depict human beings but made decorations that consisted completely out of symmetrical figures. This form of art later stimulated the emergence of a branch of mathematics, called Group Algebra. In general, we can conclude that Muslim religion did cherish several forms of mathematics (while in the Christian world, the word mathematician stood for someone who practiced astrology and other magical tricks).

2.     In the Christian world the ancient Greek language was forgotten. Often the Christians of the Middle Ages could not read Plato in the original, but only in Latin translation. Thus a lot of books were simply lost. In the Muslim world, however, Greek works were studied, specifically Aristotle. Aristotle was the pupil of Plato, but also his main opponent. Aristotle was much more of a Scientist and experimenter than Plato. In the West he was almost forgotten, but the (Middle) East had cherished him.

But there were more reasons. There were several social-economical circumstances that explained that a more scientific and ‘objective' attitude became more prevalent in Europe:

1.     The growing influence of the merchant classes in the 13th century challenged the overall influence of the Church. Merchants of any time (just like in the Alexandrian days) wanted to make use of the knowledge available. Inapplicable religious dogmas were of no use to them. It is important to know whether the world is really round or not, when one transports her goods for long distances by boat and has to make calculations on the basis of the stars.

2.     Merchants were also interested in the development of new machinery to save costs.

Other reasons for the impulse that the sciences felt at the end of the Middle Ages were:

1.     Introduction of printing and cheap paper (to replace the copying by monks on expensive parchment) made communication of ideas much easier.

2.     Other practical demands, such as a description of the course of a canon-ball after the ‘invention' of gun powder by Nobel, stimulated new mathematical research.

3.     The bloody wars in Europe in the 15th and 16th century, the disputes among Roman Catholics and Protestants, the earthquake in Lisbon in 1755, destroying the city and killing 30,000, led to a general doubt about the Christian religion in general and the Christian God in particular, and made the skeptics led to other sources of Truth (Galileo turned to Nature, Spinoza rejected simple belief, and proved on a logical basis the rationality of belief  => Faith had become secondary to Reason).

There were several responses to the general doubt about the Christian religion in the 16th century. Some philosophers who tried to minimize the influence of the Christian God to explain the world. They were, in fact, the first modern scientists. Newton, for instance, was a kind theist, i.e., he believed that the world was governed by mathematical laws, rather than Divine providence. His famous book, that laid the foundation of modern physics, was called Mathematical Principles of Natural Philosophy (1687) and portrayed nature as a self-contained system functioning on the basis of several axioms, or laws, as Newton called them.

Others, such as Pascal and Spinoza, understood that faith was under attack and could not be defended by means of faith alone. They laid a rational foundation of Faith and Belief. Pascal argued by means of probability theory that it was in one's own self-interest that one believes in God, whereas Spinoza tried to found religion (or Ethics, as he called it) as a axiomatic system with infallible conclusions (because, remember, deductive reasoning is infallible).

His writings testify of a deep influence of Euclid. As one clearly can see in The Ethics, part I, Spinoza sets up his argument as a kind of geometry. He begins with 6 definitions, which explain in which sense he is going to use the terms self-caused, finite in its own kind, substance, attribute, mode and God. Next Spinoza states 7 axioms that (he believes) are self-evident. These 13 statements form the basis of all that is coming next. Each proposition is the logical, deductive consequence of the axioms and definitions. If one accepts these 13 statements, then once has to accept the conclusions. One of the more advanced propositions (no. 25) states for instance:

There is also in God the idea or knowledge of the human mind, and this follows in God and is related to God in the same way as the idea or knowledge of the human body.

What follows is not some reference to the Bible, but a formal, deductive proof.

Mathematics from the East

The situation of Mathematics in Europe at the end of the Middle Ages was everything except positive. The West had failed to produce any mathematician of any importance in 1000 years. For Mathematics this period was really a Dark Age. It is recorded that only once a monk wrote down the 36 different outcomes if one rolls two dice. Moreover, Mathematics was looked down upon. It was equivalent with astrology and was prohibited in a lot of places.

During that same time the Islamic Middle East produced some serious mathematics. The Arabic Mathematics exhibits a mixture of Greek, Babylonian and Indian influences. Since 750 AD Baghdad is the capital of the Islamic Empire. The Caliphs created Baghdad as the scientific center by having Greek, Old-Syrian and Sanskrit scientific manuscripts translated into Arabic. The sciences that were practiced were Medicine, Philosophy, Mathematics and Physics. Until the 15th century the scientific language remains Arabic.

One of the most important mathematicians and astronomists that worked in the House of Wisdom in Baghdad shortly after 800 AD was Muhamad ibn Musa al Hvarizmi. Besides another work Muhamad wrote two books that have been translated into Latin and have played an important part in the West. In the first book he describes the 10-base system from India and algorithms (the word that has been derived from his name) for elementary constructions with whole numbers. Al-Hvarizmi knew both East-Arabic and West-Arabic numbers. The second book had the title Al-jabr wa'lmukababla, from which we derive the word Algebra. The next pages are copied from that book.

-- insert pages of translation of solving a quadratic equation --

Situation in Europe in 16th century

The rise of trade in the Northern cities of Europe in the 15th and 16th century is one of the causes that boost a renewed interest for mathematics. The practical demands of navigation of commercial ships cause the interest for and use of astronomy and geometry, as it had done in Egyptian/Babylonian times -- 2500 years before -- and in Alexandrian times -- 1700 years before. It is not surprising that Europe looks to the practical mathematics of the Arabic world to fulfill its need for mathematics. Arabic numerals get -- although slowly -- accepted in Europe. The mathematics that interests the Europeans is the Algebra that was developed by Al-Hzarizmi.

In the 16th century -- after 750 years of Arabic domanance in the mathematics -- the first independent, European results in this field are established, although it is clear how much they style is influenced by the Arabics, as we shall see from the pages of Cardano.

In the first place there are new developments in the region of notation. We mention some examples: Since the 15th century ‘p' is used in front of a positive number and ‘m' in front of a negative number. In a German book from 1489, Rechnung auff allen Kauffmanschafft, the modern ‘+' and ‘-' signs appear for the first time (Boyer, 1968, p.308). The ‘greater than' > or ‘less than' < signs were introduced by Harriot († 1621). Oughtred (1574-1660) introduced the multiply ‘*' sign, while Recorde (1510-1558) inaugurated the equal ‘=‘ sign because he could not think of anything more equal than two parallel lines.

... bicause noe 2 thynges, can be moare equalle.

A central finure in the development of notation is Francois Viete (1540 - 1603). Viete introduced letters for constant numbers (‘Simple Numbers' as Al-Hzarizmi called them, e.g., ‘39') and other letters for an unknown number (Al-Hzarizmi called it a ‘Root', Cardano called it a ‘Thing' and we normally call it ‘x'). He writes for instance:

a cubus + b in a quadr 3 + a in b quadr 3 + b cubo aequalia a + b cubo

where we normally write:

a^3 + 3 a^2 +3 a^2 b + 3 a b^2 + b^3 = (a + b)^3

Also Rene Descartes (1596 -1650) contributes a great deal to the simplification of the mathematical language.

Another important development of mathematics that took place in Europe was the extension of the Algebra. In 1545 Geronimo Cardano (1501 - 1576), mathematician, physician, astrologist, gambler, and professor at the university of Bologna, published his Ars Magna (The Great Work), the most famous boek of Algebra of that century. In this book he first give the method of solving a particular problem and later proves the method. He gives a general method to solve an equation of the power 2 (quadratic equation, already mentioned by Al-Hzarizmi), of the power 3 (cubic equation) and of the power 4. Interestingly, he probably stole the method of solving an equation of the power 4 from his student, Ferrari.

In the following we reproduce three pages from Cardano's Ars Magna, in which he gives the general solution for a cubic equation of the following form:

x^3 + a x = b               (with a, b simple numbers and x the unknown number)

The first one who discovered the method was probably Scipio del Ferro (around about 1500 in Bologna). Cardano heard the method in the form of following poem by Tartaglia (although without proof) after Cardano had promised -- according to Tartaglia -- to keep the method secret. Cardano would have broken his promise.

When the cube and the things themselves add up to some discrete number, take two others differt from the first, choosing them so that their product is always equal to the cube of one third of the third thing, and the difference between their cube roots will give you the main thing. (Tartaglia)

In the period after Cardano many have tried to find the general solution-method of an equation of the power 5 and higer, but none could find any. In the 19th century, however, mathematician proved (!) that such general solution method could not exist.

Setting the stage for Newton

Background in Mathematics: Algebraization of Geometry

(cf. Descartes, Philosophical Writings, Vol. III, The Correspondence, p.351)

From this paragraph it becomes clear the novelty of the position that Descartes (referred to as “the author”) takes on the subject of mathematics. He makes a distinction between

1. truths of mathematics

2. mathematical reasoning.

ad 1. The truths of mathematics, whether the geometric properties of Euclid's geometry or the relationship between numbers in Algebra, are in themselves not very important. They can be learned by memorization, but they taken by themselves are almost useless in discovering new truths, according to Descartes: Algebraic truths are just the formal relationships between numbers, Euclidean geometry lacks a method to arrive at the proofs of the propositions.

ad 2. Instead, Descartes wants to combine the strength of both disciplines in what he calls: mathematical reasoning. Not the mathematical truths are important but Algebra's systematic method (remember the methods, i.e., the algorithms, such as described by Al-Hzarizmi and Cusanus) and Geometry's ability to talk about the real world around us. Descartes combined these two features in what is now known as the Cartesian Coordinate System. This system allowed him to put an algebraic expression between a y and a x into the shape of a curve. As a consequence Descartes attempted to describe one real-world variable into terms of another, as he did for instance here:

At least you understand the sense of the word ‘force' when I say that it takes as much force to raise a 100-pound weight to a height of one foot as to raise a 50-pound weight to a height of two feet, that is to say, it takes as much action or effort. (Descartes, The Correspondence, p.128)

Confusion

This all set the stage for Newton. Newton's general laws of Nature did not just fall out of the air. There had been an elaborate attempt to do such thing, but there had been two main obstacles:

1. confusion of terms

2. false believes about the nature of gravity.

These two points becomes clear when we look at the following passage of Descartes (Descartes, The Correspondence, p.128-9). In this letter to Mersenne[1] (November 15, 1638) Descartes first goes at length to explain that the terms ‘force', ‘action' and ‘effort' have distinct meanings. Moreover, ‘power' means something different all-together. Descartes leaves the exact definitions unspecified and gives only indications by means of example. Still they are quite confused and it is doubtful whether he had them clearly distinguished in his own mind.

But Descartes specifies in a letter to Mersenne (March 11, 1640) what should happen in physics before a real scientific theory is established:

I would think I knew nothing in physics if I could say only how things could be, without demonstrating that they could not be otherwise. This is perfectly possible once one has reduced physics to the laws of mathematics. (Descartes, The Correspondence, p.145)

The novelty about this is not so much the mathematical design of nature. That was something that the Greeks already believed. The secret of the success of modern science was the selection of a very specific kind of mathematical approach: quantitative description of scientific phenomena independently of any physical explanations. Galileo and his successor, Newton, established exactly that: a geometry of physics on the basis of algebra.

But before Newton could do that he had to clear one notion that was commonly unknown or thought to be metaphysical: gravitational force. Descartes, unwilling to accept force that works from a distance, explains that...

... what prevents the separation of contiguous terrestrial bodies is the weight of the cylinder of air resting on them up to the atmosphere. (Descartes, The Correspondence, p.129)

Or, in another letter to Mersenne (January 29, 1640) Descartes argues that...

... heaviness is nothing more than the fact that terrestrial bodies are really pushed towards the center of the earth by subtle matter. (Descartes, The Correspondence, p.142)

However, this explanation does not suffice, because it fails to explain why, subsequently, air is pulled down.

In the Middle Ages ‘gravity' had been the ‘explanation' of why bodies move downwards. ‘Gravity' was thought of as a ‘force' that pulled earthly objects down to their natural place: earth. However, ‘gravity' was thought not to have any effect of the heavenly objects, like the planets and stars. The reason was that their ‘natural' place was in the heavens and that, thus, according to scholars of the Middle Ages, the earth's ‘gravity' did not interfere with them.

Solution

Galileo's ‘solution' was to reject the whole question of what the essence of ‘gravity' was. In his famous Dialogue Concerning the Two Chief World Systems, Galileo puts his own words in the mouth of Salviati, the defender of the Heliocentric world-view (i.e., earth moves around the sun) versus the Ptolemaic world view (i.e., sun moves around the earth). Galileo explains that the so-called essence of ‘gravity' belongs to metaphysical speculation, which will not bring us any further (cf. Galileo, Dialogue, p.236-7)

The Galilean plan contained three main features:

1.     To seek a quantitative description of physical phenomena and embody these in mathematical formulas.

2.     To isolate and measure the most fundamental properties of phenomena. These would be the variables in the formulas.

3.     To build up a deductive science on the basis of fundamental physical principles: Laws.

The second feature was the most difficult, creative and important one. We have already seen that Descartes had difficulty in isolating the right properties that should function as variables. There was a lot of confusion about terms. The way that Galileo and Newton finally distilled the right properties was by approaching the problem with mathematical reasoning -- such as Descartes had advised:

Galileo idealized the phenomenon by ignoring some facts to favor other, just as the mathematician idealizes the stretched string and the edge of a ruler by concentrating on some properties to the exclusion of others. By ignoring friction and air resistance and by imagining motion to take place in a pure Euclidean vacuum he discovered the correct fundamental principle. His trick was to geometrize the problem and then obtain the law.

Let's look at an example which expresses Descartes new method of looking at space in terms of spatial coordinates very well. Motion, Galileo found, would always continue in a straight line if it remain undisturbed by forces. Secondly, Galileo discovered that the motion of a falling object goes increasingly faster. These two could be summarized as follows in a formula:

1.     undisturbed motion:           d = v*t                        v = speed (feet per second)     t = time

2.     falling motion:                   d = 16*t2                     d = distance (feet)

In order to describe how a certain object would behave if we threw it straight in front of us with a speed: 10 feet per second, Galileo looks at both movements, down and forwards, separately. We can describe the movement by considering the x-coordinate, due to the movement forward, as:

x = 10*t

and the y-coordinate, due to falling, as

y = 16*t2

Substitution yields the following relation between x and y: y = 0.16 * x2. This describes a perfect parabola, such as can be observed in an experiment.

Newton's Universal Gravitation

Unified Discription instead of Metaphysical Explanation

Galileo and Descartes, who lived around about the same time in the first half of the 17th century, thus gave a very specific and fruitful interpretation of the mathematization of nature: a quantitative description of certain fundamental properties of objects, such as location, time, ‘heaviness', etc. This set the stage for Newton who lived and worked in the second half of the 17th century. Newton's work main contribution is to unite the motions of the earthly objects, like stones, and heavenly bodies, like the moon and the stars, under the same principle: universal gravitation. His descriptions of universal gravitation in his Mathematical Principles of Natural Philosophy (1687) do not discuss the so-called essence of gravity, but give a quantitative analysis such as Galileo had started out to do. Moreover, Newton unified all motion into a few laws.

One of the most important formulas, at which Newton arrives, is that for any two bodies, the force of attraction (F), between them can be given by the following formula:

F = k*M*m/r2

(k is a constant, M is the mass of the first body, m the mass of the second body, and r is the distance between them)

Together with the formula, F = m*a (Force is the product of mass and acceleration of that mass), it was possible to arrive at the most astonishing conclusions. For instance, on the basis of these two formulas one could calculate the weight of the earth without ever putting it on a balance. Moreover, these universal formulas were because they were universal ideal to predict the course of any planet, or any object whatsoever. Because certain observation of the movements of the planets did not agree with this formula, scientist were able to predict the existence of another planet, Neptune, before it was even observed. Even thought Newton admitted that the force of gravity was a mystery to him, his quantitative analysis of the force accomplished so much.

Copy Leibniz and

Descartes on laws of physics (p.151-2, p.127-9, p.142-3, p.145, p.351, p.388, on infinity: 290-2).

Reference

Descartes, R., 1991, The Philosophical Writings of Descartes, Vol. III, The Correspondence, Cambridge: Cambridge University Press

Galileo, 1632 [1967], Dialogue Concerning the Two Chief World Systems - Ptolemaic & Copernican, trans. S. Drake, Berkeley: University of California Press

Kline, M., 1953, Mathematics in Western Culture, New York: Oxford University Press

Leibniz,

[1] Mersenne, Marin (1588-1648), Catholic priest, theologian, physicist and polymath. A man of inexhaustible curiosity and energy, Mersenne conducted a vast correspondence with most of the great thinkers of his age, acting as a sort of clearing-house of ideas and information.