تاريخ التسجيل: Jul 2016

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It is certainly not easy to define the content and nature of mathematics

briefly. Formal explanations, which are possible nowadays due to the general

notion of structure and other logical notions, neglect not only the historical

development, but also the instinct and experience of a mathematician, who

knows what is “substantial” and “interesting” and what is not. However,

within the given understanding of mathematics, it is even more complicated

to explain what geometry is and what, in turn, is part of its history. The

dominant views on the subject of geometry, as well as its position and meaning within mathematics, have not only changed repeatedly over the course

of time. As mathematics became increasingly sophisticated, mathematicians

also took opposing positions while trying to find answers to these questions.

We will look at all these aspects in this book.

Even though geometry was mainly considered as one application of a primarily arithmeticaly oriented mathematics amongst many others in the earliest

cultures (such as in ancient Egypt, Mesopotamia, India, China, etc.), it became the core and main interest of mathematics in Ancient Greece. It was

there and then that vague notions and procedures justified only by trial and

error were transformed into a theory with definitions, axioms, theorems and

proofs. The heritage stemming from this period was so powerful for over

two thousand years that mathematicians were usually called geometricians.

Furthermore, the axiomatic-deductive method of cognition assurance, which

was based on the Greeks’ methods of dealing with geometric matters, was

referred to as “mos geometricus”, and the implementation in other sciences,

including other realms of mathematics, “more geometrico”, in other words

‘in geometrical fashion’, became a rarely achieved scientific and theoretical

program. This agenda influenced, for example, Newton in the 17th century, as

he re-founded mechanics, Galois at the beginning of the 19th century, when

criticising the contemporary situation of algebra, and Hilbert, while encouraging the scientific community to axiomatise further branches of physics in

his famous speech in 1900.

As a result of the European Renaissance, geometry was flooded by an extraordinary wave of inspiration and applications in the fields of astronomy, geodesy,

cartography, mechanics, optics, architecture, visual arts and, hence, leading

to a wealth of new challenges. The efforts made to solve these new challenges

essentially led to the development of the four pillars of the “modern” mathematics in the 17th century. These pillars are: the concept of function, coordinate-systems, differential calculus and integral calculus. Geometry gave

birth to these pillars, and then was superseded and lost its leading position to

them in a very subtle manner. Formulae and calculus took over increasingly

in the 18th century and pushed visualisation and logical argumentation aside.

The 19th century led to an enormous growth in the size and meaning of geometry. Projective, descriptive and n-dimensional geometry, vector calculus,

12 Introduction

non-Euclidean geometry, intrinsic differential geometry, topology, and also

numerous “buds” in other areas that would only come to blossom in the 20th

century, such as geometrical probability and measure theory, graph theory

and general polyhedral theory, began developing at first without any recognisable relationship to one another. This “explosion” of geometrical disciplines,

which led to the century being named the “geometrical century” according

to mathematicians, was accompanied by the disintegration of the then dominant understanding of geometry as a science of “true physical space”. We will

look at how the different approaches for dealing intellectually with the new

situation in geometry crucially coined the whole view of mathematics that

was dominant until the invention of the computer and its rising popularity.

However, we must also examine how geometry lost its central position within

mathematics over the course of the first half of the 20th century. This has

been a development that still negatively influences the organisation of mathematics in secondary and further education nowadays, despite the fact that

geometry has achieved a higher than ever level in regards to its theoretical

width and depth as well as its practical significance.

At the end of the 20th century, geometry was, on the one hand, a huge pool

of facts on the “ordinary two and three dimensional Euclidean space” and

an even bigger pool of unanswered questions on those. On the other hand,

geometry was not really thought of as being part of mathematics in the ordinary sense nowadays, but rather considered a way of thinking, which is more

or less useful and necessarily found in almost every realm of mathematics,

depending on the scientist’s personal approach. Thus, there is a geometrical

theory of numbers, a geometrical theory of functions, algebraic geometry and

geometrical stochastics. There are geometrical methods within variational

calculus, discrete and combinatorial geometry, as well as computer geometry. The latter is not to be confused with computational geometry, which

basically refers to a “theory of complexity of geometrical algorithms”.

The dichotomy of geometry suggested here has established itself very well

in the meantime. The three dimensional Euclidean space remains the appropriate model for all “ordinary” problems, even though it is only a very

rough approximation of reality according to the findings of physics. Within

the Euclidean plane we create “pictures” of everything we want to “look at”

and understand. Their meanings are associated with the dominance of seeing

amongst the human senses. Inside the n-dimensional Euclidean space, mathematics embeds functions, relations and, almost all other examined objects

by using coordinates, for example. Furthermore, geometry predominates in

all those areas where a number of possibly very abstract objects are viewed

as a “space” by using in broader sense terms taken from geometry, such as

topology, metrics, dimension and linearity, with the intention of inspiring our

imagination and to use analogies. To what extent one may want to practise

this is – as already pointed out – a matter of style. It is an intellectual technique, without which modern mathematics in the form described here could

not have developed.Introduction 3

To what degree the latter can really be considered geometry and to what extent the applied branches of geometry belong to mathematics or are already

part of engineering is debatable. In the following, we will also defend the concept that there is an “unconscious” unprofessional mathematics that coexists

with professional, deductive mathematics. The former manifests itself in the

intuitive use of notions, shapes, methods, knowledge and know-how, which

is difficult to put into words, but exists as a material product of engineering,

handcrafts and the arts. Hence, this book will also serve as a reflection on

the historical development of geometry, which will include many, often unusual aspects. We intend to contribute to the clarification of the position and

meaning of geometry within mathematics and to raise interest in it.

The critical reader, that we would like to have, may pose the question how a

history of geometry fits in a series called ‘From Pebbles to Computers’. What

computers have to do with geometry is investigated in detail in Chapter 8.5.

With regards to ‘pebbles’ (accounting tokens) we refer to the Pythagoreans,

who got some simple pre-numbertheoretical results from patterns of geometrically ordered stones. Thus they could realize why ab is forever equal to ba

and why the distance between two square numbers n2 and (n + 1)2 is always

2n + 1.

This book features problems added chapter by chapter, most of which are

not historical problems strictly speaking, but problems that result from the

history presented here. For instance, questions without answers when they

first occurred; questions that just simply did not come to mind, but were

possible; old problems that nowadays are much easier to solve given modern

methods; and suggestions that result from old problems. Most of the problems

are reduced to special cases, contain hints or are asked in a manner that will

require only a highschool or slightly more advanced mathematical background

to be solved. However, a few questions are more difficult and “open-ended”.

Here, the reader is invited to probe and explore.

We have avoided the use of first names and the inclusion of the dates births

and deaths within the main text apart from a few, well-reasoned exceptions.

As far as we could determine those data, they are available in the index of

names at the end of the book.

The pictures of the people at the beginning of each chapter are of different

styles. We cannot rely on authentic portraits from antiquity or the nonEuropean Middle Ages. (One reason being that people in Islamic countries

were often not portrayed due to religious reasons.) However, we must acknowledge that later eras felt the necessity to make pictures of their most

important personalities. In this book, a “picture” can be an imagined portrait or a symbolic graphic representation. In this respect, stamps can also

serve as a cultural document of the history of science. Multiple books have

been devoted to this exact subject [Gjone 1996, Schaaf 1978, Schreiber, P.

1987, Wußing/Remane 1989]. For example, a picture of Euclid (not shown

here) was taken from a manuscript of Roman field surveyors (agrimensores).

Here, two things are striking. First, these agrimensores thought of Euclid, the4 Introduction

master of the logical-axiomatic approach, as their forefather, and, second, the

picture has an almost oriental ambience. Considering the mix of peoples and

cultures in Alexandria at 300 BC, this may appear more realistic than some

neo-classically influenced pseudo-antique art.

From the European Middle Ages onwards, portraits began to appear intentionally more similar to the individual persons, as artists started relying on

themselves as models. For example, the portrait of Piero della Francesca is an

alleged self-portrait. It comes from his Fresco “Resurrection” (around 1465)

located in his hometown of Borgo Sansepolcro.

The picture of Ren´e Descartes presented here was painted by Frans Hals

shortly before the philosopher departed for Sweden. It is not only one of the

very few cases in which a genuinely famous painter portrayed a genuinely famous mathematician (a second example is the portrait of Felix Klein painted

by Max Liebermann), but multiple copies of this painting were subsequently

made in the 17th century reflecting varying facial expressions, which since

then partially even flipped horizontal have haunted encyclopaedias and the

science-historical literature as images of Descartes.

Peter Schreiber

briefly. Formal explanations, which are possible nowadays due to the general

notion of structure and other logical notions, neglect not only the historical

development, but also the instinct and experience of a mathematician, who

knows what is “substantial” and “interesting” and what is not. However,

within the given understanding of mathematics, it is even more complicated

to explain what geometry is and what, in turn, is part of its history. The

dominant views on the subject of geometry, as well as its position and meaning within mathematics, have not only changed repeatedly over the course

of time. As mathematics became increasingly sophisticated, mathematicians

also took opposing positions while trying to find answers to these questions.

We will look at all these aspects in this book.

Even though geometry was mainly considered as one application of a primarily arithmeticaly oriented mathematics amongst many others in the earliest

cultures (such as in ancient Egypt, Mesopotamia, India, China, etc.), it became the core and main interest of mathematics in Ancient Greece. It was

there and then that vague notions and procedures justified only by trial and

error were transformed into a theory with definitions, axioms, theorems and

proofs. The heritage stemming from this period was so powerful for over

two thousand years that mathematicians were usually called geometricians.

Furthermore, the axiomatic-deductive method of cognition assurance, which

was based on the Greeks’ methods of dealing with geometric matters, was

referred to as “mos geometricus”, and the implementation in other sciences,

including other realms of mathematics, “more geometrico”, in other words

‘in geometrical fashion’, became a rarely achieved scientific and theoretical

program. This agenda influenced, for example, Newton in the 17th century, as

he re-founded mechanics, Galois at the beginning of the 19th century, when

criticising the contemporary situation of algebra, and Hilbert, while encouraging the scientific community to axiomatise further branches of physics in

his famous speech in 1900.

As a result of the European Renaissance, geometry was flooded by an extraordinary wave of inspiration and applications in the fields of astronomy, geodesy,

cartography, mechanics, optics, architecture, visual arts and, hence, leading

to a wealth of new challenges. The efforts made to solve these new challenges

essentially led to the development of the four pillars of the “modern” mathematics in the 17th century. These pillars are: the concept of function, coordinate-systems, differential calculus and integral calculus. Geometry gave

birth to these pillars, and then was superseded and lost its leading position to

them in a very subtle manner. Formulae and calculus took over increasingly

in the 18th century and pushed visualisation and logical argumentation aside.

The 19th century led to an enormous growth in the size and meaning of geometry. Projective, descriptive and n-dimensional geometry, vector calculus,

12 Introduction

non-Euclidean geometry, intrinsic differential geometry, topology, and also

numerous “buds” in other areas that would only come to blossom in the 20th

century, such as geometrical probability and measure theory, graph theory

and general polyhedral theory, began developing at first without any recognisable relationship to one another. This “explosion” of geometrical disciplines,

which led to the century being named the “geometrical century” according

to mathematicians, was accompanied by the disintegration of the then dominant understanding of geometry as a science of “true physical space”. We will

look at how the different approaches for dealing intellectually with the new

situation in geometry crucially coined the whole view of mathematics that

was dominant until the invention of the computer and its rising popularity.

However, we must also examine how geometry lost its central position within

mathematics over the course of the first half of the 20th century. This has

been a development that still negatively influences the organisation of mathematics in secondary and further education nowadays, despite the fact that

geometry has achieved a higher than ever level in regards to its theoretical

width and depth as well as its practical significance.

At the end of the 20th century, geometry was, on the one hand, a huge pool

of facts on the “ordinary two and three dimensional Euclidean space” and

an even bigger pool of unanswered questions on those. On the other hand,

geometry was not really thought of as being part of mathematics in the ordinary sense nowadays, but rather considered a way of thinking, which is more

or less useful and necessarily found in almost every realm of mathematics,

depending on the scientist’s personal approach. Thus, there is a geometrical

theory of numbers, a geometrical theory of functions, algebraic geometry and

geometrical stochastics. There are geometrical methods within variational

calculus, discrete and combinatorial geometry, as well as computer geometry. The latter is not to be confused with computational geometry, which

basically refers to a “theory of complexity of geometrical algorithms”.

The dichotomy of geometry suggested here has established itself very well

in the meantime. The three dimensional Euclidean space remains the appropriate model for all “ordinary” problems, even though it is only a very

rough approximation of reality according to the findings of physics. Within

the Euclidean plane we create “pictures” of everything we want to “look at”

and understand. Their meanings are associated with the dominance of seeing

amongst the human senses. Inside the n-dimensional Euclidean space, mathematics embeds functions, relations and, almost all other examined objects

by using coordinates, for example. Furthermore, geometry predominates in

all those areas where a number of possibly very abstract objects are viewed

as a “space” by using in broader sense terms taken from geometry, such as

topology, metrics, dimension and linearity, with the intention of inspiring our

imagination and to use analogies. To what extent one may want to practise

this is – as already pointed out – a matter of style. It is an intellectual technique, without which modern mathematics in the form described here could

not have developed.Introduction 3

To what degree the latter can really be considered geometry and to what extent the applied branches of geometry belong to mathematics or are already

part of engineering is debatable. In the following, we will also defend the concept that there is an “unconscious” unprofessional mathematics that coexists

with professional, deductive mathematics. The former manifests itself in the

intuitive use of notions, shapes, methods, knowledge and know-how, which

is difficult to put into words, but exists as a material product of engineering,

handcrafts and the arts. Hence, this book will also serve as a reflection on

the historical development of geometry, which will include many, often unusual aspects. We intend to contribute to the clarification of the position and

meaning of geometry within mathematics and to raise interest in it.

The critical reader, that we would like to have, may pose the question how a

history of geometry fits in a series called ‘From Pebbles to Computers’. What

computers have to do with geometry is investigated in detail in Chapter 8.5.

With regards to ‘pebbles’ (accounting tokens) we refer to the Pythagoreans,

who got some simple pre-numbertheoretical results from patterns of geometrically ordered stones. Thus they could realize why ab is forever equal to ba

and why the distance between two square numbers n2 and (n + 1)2 is always

2n + 1.

This book features problems added chapter by chapter, most of which are

not historical problems strictly speaking, but problems that result from the

history presented here. For instance, questions without answers when they

first occurred; questions that just simply did not come to mind, but were

possible; old problems that nowadays are much easier to solve given modern

methods; and suggestions that result from old problems. Most of the problems

are reduced to special cases, contain hints or are asked in a manner that will

require only a highschool or slightly more advanced mathematical background

to be solved. However, a few questions are more difficult and “open-ended”.

Here, the reader is invited to probe and explore.

We have avoided the use of first names and the inclusion of the dates births

and deaths within the main text apart from a few, well-reasoned exceptions.

As far as we could determine those data, they are available in the index of

names at the end of the book.

The pictures of the people at the beginning of each chapter are of different

styles. We cannot rely on authentic portraits from antiquity or the nonEuropean Middle Ages. (One reason being that people in Islamic countries

were often not portrayed due to religious reasons.) However, we must acknowledge that later eras felt the necessity to make pictures of their most

important personalities. In this book, a “picture” can be an imagined portrait or a symbolic graphic representation. In this respect, stamps can also

serve as a cultural document of the history of science. Multiple books have

been devoted to this exact subject [Gjone 1996, Schaaf 1978, Schreiber, P.

1987, Wußing/Remane 1989]. For example, a picture of Euclid (not shown

here) was taken from a manuscript of Roman field surveyors (agrimensores).

Here, two things are striking. First, these agrimensores thought of Euclid, the4 Introduction

master of the logical-axiomatic approach, as their forefather, and, second, the

picture has an almost oriental ambience. Considering the mix of peoples and

cultures in Alexandria at 300 BC, this may appear more realistic than some

neo-classically influenced pseudo-antique art.

From the European Middle Ages onwards, portraits began to appear intentionally more similar to the individual persons, as artists started relying on

themselves as models. For example, the portrait of Piero della Francesca is an

alleged self-portrait. It comes from his Fresco “Resurrection” (around 1465)

located in his hometown of Borgo Sansepolcro.

The picture of Ren´e Descartes presented here was painted by Frans Hals

shortly before the philosopher departed for Sweden. It is not only one of the

very few cases in which a genuinely famous painter portrayed a genuinely famous mathematician (a second example is the portrait of Felix Klein painted

by Max Liebermann), but multiple copies of this painting were subsequently

made in the 17th century reflecting varying facial expressions, which since

then partially even flipped horizontal have haunted encyclopaedias and the

science-historical literature as images of Descartes.

Peter Schreiber

الموضوع:
Introduction بقسم