M840 Dissertation in Mathematics Print Final 6-9-2011

M840 TMA03 Leslie Pedrick T3137721

M840 Dissertation in Mathematics

Poincaré and Hilbert’s geometrical ideasand their significance in the first decade of the twentieth century

Leslie Pedrick

Student ID T3137721

Submitted for the MSc in Mathematics

Open University

Milton Keynes, UK

14th September 2011

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Statement of Originality

This dissertation has been prepared solely by me, Leslie Pedrick, and no part of it has

previously been submitted to any institution for any qualification.

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Contents

1. Introduction 4

2. Poincaré’s geometrical ideas 5

2.1. Geometry as Group theory 5

2.2. On Non-Euclidian Geometry 8

2.3. The Law of Relativity 9

2.4. Poincaré and the ether 9

3. Hilbert’s geometrical ideas 11

3.1. Axiomatic geometry 11

3.2. Grundlagen der Geometrie 1899 13

3.3. Invariant Theory 15

3.4. Importance of Problems Speech 1900 15

3.5. Functional Analysis 16

4. Significance of Poincaré’s ideas 1900-1910 18

4.1. Poincaré and the Dynamics of the Electron 18

4.1.1. The Background 18

4.2. Lorentz Ether theory 19

4.3. Electron mass 22

4.4. Gravity 23

4.5. Minkowski Space and Time 24

5. Significance of Hilbert’s ideas 1900-1910 29

5.1. Minkowski Space and Time 29

5.2. Hilbert and the route to Spectrum Theory 29

5.3. Hilbert’s 23 problems 34

6. Conclusions 38

7. References and Bibliography 39

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1. Introduction

Both Hilbert and Poincaré were extremely prolific in the publication of their ideas in

geometry and there was so much to choose from.

We shall see that for Poincaré, the foundations of his ideas are contained within his

publications of The Foundations of Science. For Hilbert I have selected his work on

axiomatic geometry, invariant theory and integral equations.

On the significance of these ideas in the first decade of the 20th Century. For Poincaré,

I have chosen to look at relativity. Until recent times Poincaré’s role in the evolution

of relativity appears to have been neglected. For Hilbert, I investigate his role in

Minkowski’s space-time, the development of spectral theory then his role in his 23

problems of 1900.

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2. Poincaré’s geometrical ideas

2.1 Geometry as Group theory

“What we call geometry is nothing but the study of formal properties of a

certain continuous group; so that we may say, space is a group. The notion of

this continuous group exists in our mind prior to all experience” [1]

But by his philosophy of convention; we may choose continuous groups from many

to suit our perception of physical space and time; the choice is a matter of convention.

Our perception of physical space and time comes from the senses:

“Our sensations cannot give us the notion of space. That notion is built up by

the mind from elements which pre-exist in it, and external experience is

simply the occasion for its exercising this power, or at most a means of

determining the best mode of exercising it.” [1a]

Back in 1880 Poincaré entered the competition for the grand prize in mathematical

sciences organized by the French Academy of Sciences. His submission did not win

the prize, even though the three supplements to his prize essay established a new class

of automorphic functions that Poincaré called Fuchsian functions.

Fuchsian functions, Poincaré discovered, are invariant under a certain class of linear

transformations that form a group. The Fuchsian function is to the geometry of

Lobachevski what the doubly periodic function is to that of Euclid. [2] This group

reduces to that of the translation group of hyperbolic geometry.

Poincaré observed:

“In fact, what is a geometry? It is the study of the group of operations formed

by the displacements to which one can subject a body without deforming it. In

Euclidean geometry the group reduces to the rotations and translations. In the

pseudogeometry of Lobachevski it is more complicated.” [2]

In 1887 Poincaré published his first essay on the foundations of geometry [3] in this

he is strongly influenced by Sophus Lie’s methods. Poincaré displays his

conventionalism in the philosophy of science:

“…one may say that the truth of the geometry of Euclid is not incompatible

with the truth of the geometry of Lobachevski, for the existence of a group is

not incompatible with that of another group.”

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“Among all possible groups we have chosen one in particular, in order to refer

to it all physical phenomena, just as we choose three coordinate axes in order

to refer to them a geometrical figure” “there exist in nature some remarkable

bodies which are called solids, and experience tells us that the different

possible movements of these bodies are related to one another much in the

same way as the different operations of the chosen group” [3]

That is; we choose the particular group that is most convenient.

Poincaré does not hide his ideas; he publishes everything:

In The Foundations of Science, Space and Geometry Poincaré starts by distinguishing

between the continuous group that exists in our mind and the space that we

experience; that is Geometrical Space and Representative Space (sometimes

Perceptual Space or Sensible Space). [4]

Poincaré states the properties of Geometrical Space:

What, first of all, are the properties of space, properly so called? I mean of

that space which is the object of geometry and which I shall call geometric

space.

The following are some of the most essential for geometric space:

1. It is continuous.

2. It is infinite (boundless).

3. It has three dimensions.

4. It is homogeneous, that is to say, all its points are identical one with another.

5. It is isotropic, that is to say, all the straight lines which pass through the

same point are identical one with another. [4]

Poincaré argues that perceptual space, under its triple form, visual, tactile and motor,

is essentially different from geometric space.

“[perceptual space] It is neither homogeneous, nor isotropic; one can not

even say that it has three dimensions.

Seeing that our representations are simply the reproductions of our sensations,

therefore we cannot image geometrical space. We cannot represent to

ourselves objects in geometrical space, but can merely reason upon them as if

they existed in that space.”

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“Geometrical space, therefore, cannot serve as a category for our

representations. It is not a form of our sensibility. It can serve us only in our

reasonings. It is a form of our understanding.” [4]

In order to develop the notion of a group, Poincaré needs to define displacement as a

group operation. He defines displacement thus:

“We observe next that in certain cases when an external change has modified

our impressions, we can, by voluntarily provoking an internal change, re-

establish our primitive impressions. The external change, accordingly, can be

corrected by an internal change.”

“Changes which are susceptible of being corrected by an internal change.

These are displacements.”

Changes that can not be corrected by an internal change Poincaré calls

alterations, “These are alterations. An immovable being would be incapable of

making this distinction. Such a being, therefore, could never create geometry,

even if his sensations were variable, and even if the objects surrounding him

were movable.” [1]

In the Introduction to Notion of a Group [1] Poincaré considers an ensemble of

displacements A+B:

“Mathematicians express this by saying that the ensemble, or aggregate, of

displacements is a group. If such were not the case there would be no

geometry, but how do we know that the ensemble of displacements is a

group? Is it by reasoning a priori? Is it by experience? One is tempted to

reason a priori and to say: if the external change A is corrected by the internal

change A', and the external change B by the internal change B', the resulting

external change A+ B will be corrected by the resulting internal change B'

+A'. Hence this resulting change is by definition a displacement, which is to

say that the ensemble of displacements forms a group”:

Once we accept that space is a group, we observe the consequences of the existence

of the group; that are the properties Geometrical Space listed 1-5 above.

Poincaré recognises the importance of the invariant sub-group.

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Poincaré gives credit for the source of his ideas on the foundations of geometry:

“We owe the theory which I have just sketched to Helmholtz and Lie. I differ

from them in one point only, but probably the difference is in the mode of

expression only and at bottom we are completely in accord. For Helmholtz

and Lie the matter of the group existed previously to the form, and in

geometry the matter is a Zalilenmannigfaltigkeit (number manifold) of three

dimensions. The number of dimensions is therefore posited prior to the group.

For me, on the contrary, the form exists before the matter.” [1]

2.2 On Non Euclidian Geometry

Beltrami and Poincaré’s models of Lobachevski’s hyperbolic geometry are very

similar circle models. Poincaré does not refer to Beltrami, but Poincaré’s model

(being later) probably was derived from Beltrami’s.

Poincaré does not consider geometry to be an experimental science; he believed that

the rules of geometry pre-exist within us. The measurements that we make and our

experience gives us occasion to reflect on our pre-existing concepts of groups. On the

other hand he regards axioms as conventions;

“The axioms of geometry are therefore are neither synthetic a priori

judgments nor experimental facts.” “They are conventions; our choice among

all possible conventions is guided by experimental facts; but it remains free

and is limited only by the necessity of avoiding all contradiction.” [5]

Poincaré is emphatic on his famous conventionist stance on the relationship between

the geometries:

“To ask whether the geometry of Euclid is true and that of Lobachevski is

false, is as absurd as to ask whether the metric system is true and that of the

yard, foot, and inch, is false. Transported to another world we might

undoubtedly have a different geometry, not because our geometry would have

ceased to be true, but because it would have become less convenient than

another.”

“Let it not be said that the reason why we deem the group of Euclid the

simplest is because it conforms best to some pre-existing ideal which has

already a geometrical character; it is simpler because certain of its

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displacements are interchangeable with one another, which is not true of the

corresponding displacements of the group of Lobachevski. Translated into

analytical language, this means that there are fewer terms in the equations, and

it is clear that an algebraist who did not know what space or a straight line

was would nevertheless look upon this as a condition of simplicity” [5]

2.3 The Law of Relativity

Poincaré’s view on law of relativity is not constant; we can see how he changes from

one viewpoint another to satisfy the Michelson Experiment and Maxwell’s equations.

In Foundations of Science (Experience and Geometry) Poincaré frames the law in

terms of covariance of state of bodies with respect to the absolute position and

velocity.

“The state of bodies and their mutual distances at any instant, as well as the

velocities with which these distances vary at this same instant, will depend

only on the state of those bodies and their mutual distances at the initial

instant, and the velocities with which these distances vary at this initial

instant, but they will not depend either upon the absolute initial position of the

system, or upon its absolute orientation, or upon the velocities with which this

absolute position and orientation varied at the initial instant.”[6]

In this essay he is clearly not happy with this law, since it is not in accord with his

interpretation of the results of the experiments of Hertz and Michelson. But this is

quite different from the principle of relativity proposed by Poincaré in St. Louis

1904[7]; now the experiments are viewed by the observer, fixed or in uniform motion

and the principle of relativity is satisfied;

“The principle of relativity, according to which the laws of physical

phenomena should be the same, whether for an observer fixed, or for an

observer carried along in a uniform movement of translation; so that we have

not and could not have any means of discerning whether or not we are carried

along in such a motion.”[7]

2.4 Poincaré and the ether

“The ether it is, the unknown, which explains matter, the known; matter is

incapable of explaining the ether.” [8]

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In The Theories of Modern Physics [9] Poincaré had speculated that primitive matter

was a form of condensed ether, and others had gone further; theories such as this

were popular at the time.

Neither Poincaré nor Lorentz abandoned the concept of the ether. It troubled

Poincaré, but there was no better concept to explain action-at-a-distance, absolute

rotation and at one time the existence of matter (which had been replaced by another

hypothesis). It was part of Poincaré’s philosophy of conventionalism; a convenient

hypothesis that could be discarded when a better hypothesis came along:

“It matters to us little whether the ether really exists; it is the matter of

metaphysicians; what is essential for us is that everything happens as if it

existed and that this hypothesis is convenient for the explanation of

phenomena. After all, have we any other reason for believing in the existence

of material objects? That too is only a convenient hypothesis; only it will

never cease to be so, while a day will come no doubt in which the ether will

be rejected as useless.” [10]

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3. Hilbert’s geometrical ideas

3.1 Axiomatic geometry

Hilbert lectured on projective geometry in 1891. In the introduction to these lectures,

Hilbert would state his concept of geometry. Compare the following with that of

Poincaré in 2.1:

“Geometry is the science that deals with the properties of space. It differs

essentially from pure mathematical domains such as the theory of numbers,

algebra, or the theory of functions. The results of the latter are obtained

through pure thinking... The situation is completely different in the case of

geometry. I can never penetrate the properties of space by pure reflection,

much as I can never recognize the basic laws of mechanics, the law of

gravitation or any other physical law in this way. Space is not a product of my

reflections. Rather, it is given to me through the senses. I thus need my senses

in order to fathom its properties. I need intuition and experiment, just as I

need them in order to figure out physical laws, where also matter is added as

given through the senses.” [11][12]

Both Poincaré and Hilbert make the distinction between the mind and the senses. For

Poincaré however, all geometry exists in the mind and our senses help us choose the

best way of to us it. For Hilbert no geometry is a priori, but is given by the senses.

The concept of Anschauung, played an important role in Hilbert’s attitude toward

geometry. The prefix to his introduction of Grundlagen der Geometrie (1899) Hilbert

quotes from Kant,

“All human knowledge begins with intuitions, thence passes to concepts and

ends with ideas.”

Kant, Kritik der reinen Vernunft, Elementariehre, Part 2, Sec. 2.

At this time (1891) Hilbert’s course on projective geometry was not at all axiomatic;

it was based on texts on projective geometry by von Staudt and Theodor Reye.

Also in 1891 Hilbert attended a lecture by Wiener on the foundations of geometry at

the Deutsche Mathematiker Vereinigung meeting in Halle, where Wiener proposed

that, starting solely with the theorems of Desargues and Pappus, it would possible to

prove the fundamental theorem of projective geometry.

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Hilbert was so impressed by this treatment of continuity in the foundations of

geometry, that after he is reported to have declared that it must be possible to replace

“point, line, and plane” with “table, chair, and beer mug” without changing the

validity of the theorems of geometry.

From his study of sets, Hilbert probably already knew of the idea of changing names

of the central elements whilst leaving the deductive structure intact, from the Systems

of Elements in Dedekind’s 1888 What are numbers and what should they be:

“In what follows I understand by thing every object of our thought. In order to

be able easily to speak of things, we designate them by symbols, e.g., by

letters, and we venture to speak briefly of the thing a of a simply, when we

mean the thing denoted by a and not at all the letter a itself. A thing is

completely determined by all that can be affirmed or thought concerning it. A

thing a is the same as b (identical with b), and b the same as a, when all that

can be thought concerning a can also be thought concerning b, and when all

that is true of b can also be thought of a.” [13]

In 1893 Hilbert was to conduct a course in non-Euclidian geometry at Königsberg.

The original manuscript that Hilbert prepared shows that he adopted an axiomatic

approach, largely based on Pasch’s model. However Hilbert soon realised that there

were redundancies in Pasch’s treatment. In particular, Pasch’s Archimedean axiom

could be derived from his other axioms.

Hilbert was determined that his axiomatic geometry would have the minimum

explicit set of presuppositions from which the whole of geometry could be deduced.

The course of 1893 did not take place, as only one student registered. However,

Hilbert reconfigured the course with a more empiricist approach, including the work

of Hermann Grassmann. Hilbert in the revision also referred to Hertz and his theory

of images in the mind:

“Nevertheless the origin [of geometrical knowledge] is in experience. The

axioms are, as Hertz would say, images or symbols in our mind, such that

consequents of the images are again images of the consequences, what we can

logically deduce from the images is itself valid in nature.” [14][15]

So in 1894 Hilbert presented a course “Foundations of Geometry.” This time it did go

ahead.

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In 1899 Hilbert lectured on the elements of Euclidean geometry in Göttingen. These

lectures were based on what was to become the Grundlagen der Geometrie 1899.[18]

3.2 Grundlagen der Geometrie 1899

Hilbert’s Grundlagen der Geometrie was published in June 1899 as part of a

Festschrift issued in Göttingen in honour of the unveiling of the Gauss-Weber

monument. His described aim is outlined in the short introduction:

“The following investigation is a new attempt to choose for geometry a simple

and complete set of independent axioms and to deduce from these the most

important geometrical theorems in such a manner as to bring out as clearly as

possible the significance of the different groups of axioms and the scope of

the conclusions to be derived from the individual axioms.”[17]

Simple, according to Hertz, means that an axiom is “no more than a single idea”, but

this is not defined by Hilbert in Grundlagen der Geometrie. Much of Hilbert’s

formulation of the axiomatic method was derived from Hertz’s Principles of

Mechanics. [17]

Complete, the axioms of Grundlagen der Geometrie were complete, so that all the

theorems could be derived from them.

Independent, so that the removal of anyone axiom from the set would make it

impossible to prove at least some of the theorems.

They were also consistent, so that no contradictory theorems could be derived.

In Grundlagen der Geometrie, concerning the analysis of our intuition of space,

Hilbert commences his discussion by considering three systems of things (elements)

which he calls points, straight lines, and planes, and sets up a system of axioms

connecting these elements in their mutual relations.

“We think of these points, straight lines, and planes as having certain mutual

relations, which we indicate by means of such words as “are situated,”

“between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact

description of these relations follows as a consequence of the axioms of

geometry. These axioms may be arranged in five groups. Each of these groups

expresses, by itself, certain related fundamental facts of our intuition.”

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“We will name these groups as follows:

I, Axioms of connection.

II, Axioms of order.

III, Axioms of parallels (Euclid’s axiom).

IV, Axioms of congruence.

V, Axioms of continuity (Archimedes’s axiom).”

As Hilbert develops the theorems, he shows clearly which axioms are required and

demonstrated that they are mutually independent and consistent. For example; he

demonstrates that the whole of the Euclidean geometry may be developed without the

use of the axiom of continuity, and that Pappus’s theorem is independent of

Desargues’s theorem.

Hilbert writes:

“Our investigation will show that in this respect Pascal’s theorem is very

different from Desargues’s theorem; for, in the demonstration of Pascal’s

theorem, the admission or exclusion of the axiom of Archimedes is of decided

influence.”[18]

He concludes:

“Every proposition relative to points of intersection in the geometry in

question must always, by the aid of suitably constructed auxiliary points and

straight lines, turn out to be a combination of the theorems of Pascal and

Desargues. Hence for the proof of the validity of a theorem relating to points

of intersection, we need not have resource to the theorems of

congruence.”[18]

Hilbert also introduces axioms of congruence making them the basis of a definition of

geometric displacement. [18]

In the second edition of Grundlagen der Geometrie, published in 1903, Hilbert added

the axiom of completeness (Vollständigkeitsaxiom). This was a statement about the

nature of the system. It meant that if there exists a system of elements obeying the

axioms, then this system is not susceptible of extension.

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3.3 Invariant Theory

In 1897 Hilbert conducted a course in invariant theory. “Invariant theory is the bridge

between algebra and geometry.” [19] His Finiteness Theorem had initially been

proved and published in 1890, the finiteness of the full invariant system in 1893.

Theorem (Finiteness of the Full Invariant System): Every binary form

possesses a finite full invariant system such that each invariant of the form is a

polynomial function of the invariants in the full invariant system. [20]

In 1890 Hilbert had been introduced to Gordan’s Problem; the question concerned the

totality of invariants. Was there a basis, a finite system of invariants in terms of which

all other invariants, although infinite in number, could be expressed rationally and

integrally? [21]

Hilbert realised that to prove the finiteness of the basis of the invariant system, one

did not have to construct it. Hilbert always insisted that if one can prove that the

attributes assigned to a concept will never lead to a contradiction, the mathematical

existence of the concept is thereby established. All one had to do was to prove that a

finite basis, of logical necessity, must exist, because any other conclusion would

result in a contradiction. [22]

To those that declared that existence statements are meaningless unless they actually

specify the object asserted to exist, Hilbert was to reply:

“The value of pure existence proofs consists precisely in that the individual

construction is eliminated by them, and that many different constructions are

subsumed under one fundamental idea so that only what is essential to the

proof stands out clearly; brevity and economy of thought are the raison d'être

of existence proofs ... To prohibit existence statements ... is tantamount to

relinquishing the science of mathematics altogether.” [23]

3.4 Importance of Problems Speech 1900

In 1900 Hilbert gave his famous address on Importance of Problems to the

International Congress of Mathematicians in Paris. [24]

Hilbert must always be regarded as a working mathematician, his stated motivation

for this speech was:

“As long as a branch of science offers an abundance of problems, so long is it

alive: a lack of problems foreshadows extinction or the cessation of

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independent development. Just as every human undertaking pursues certain

objectives, so also mathematical research requires its problems.”

But in 1880 Bois-Reymond made a speech to the Berlin Academy of Sciences

outlining seven “world riddles”, three of which, he declared, neither science nor

philosophy could ever explain. “Ignoramus et ignorabimus” - we are ignorant and we

shall remain ignorant.

To both Hilbert and his friend Minkowski such a concession was thoroughly

abhorrent. As Hilbert put it in his speech:

“…every definite mathematical problem must necessarily be susceptible of an

exact settlement, either in the form of an actual answer to the question asked,

or by the proof of the impossibility of its solution and therefore the necessary

failure of all attempts... in mathematics there is no ignorabimus.”[24]

Hilbert describes his philosophy on the method of solution of problems:

“If we do not succeed in solving a mathematical problem, the reason

frequently consists in our failure to recognize the more general standpoint

from which the problem before us appears only as a single link in a chain of

related problems…….

In dealing with mathematical problems, specialization plays, as I believe, a

still more important part than generalization. Perhaps in most cases where we

seek in vain the answer to a question, the cause of the failure lies in the fact

that problems simpler and easier than the one in hand have been either not at

all or incompletely solved. All depends, then, on finding out these easier

problems, and on solving them by means of devices as perfect as possible and

of concepts capable of generalization. This rule is one of the most important

levers for overcoming mathematical difficulties and it seems to me that it is

used almost always, though perhaps unconsciously.”[24]

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3.5 Functional Analysis

Probably the first integral equation appeared in The Analytic Theory of Heat,

1822[39], by Fourier inverting the equation:

to

Fredholm developed a theory of determinants for integral equations and published a

series of papers in the years from 1900 to 1903 on the general theory of integral

equations.

Hilbert became interested in theory of integral equations and published series of six

papers between 1904 to 1910, which contained Hilbert’s own words:

“… the systematic building of a general theory of integral equations for the

whole of analysis, especially for the theory of the definite integral and the

theory of the development of arbitrary functions in an infinite series, besides

for the theory of linear differential equations and analytic functions, as well as

for potential theory and calculus of variations, is of the greatest importance”.

[25]

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4. Significance of Poincaré’s ideas 1900-1910

4.1 Poincaré and the Dynamics of the Electron

On 23rd July 1905 the editor for Rendiconti de Circolo Mathematico di Palermo

received a paper by Poincaré titled Sur la dynamique de l’electron. [28]

In On the Dynamics of the Electron Poincaré examines the effects of his new

relativistic physics on kinematics, dynamics, electrodynamics, and gravitation. The

only three premises that he assumes are his principle of relativity (that all physical

laws in moving and stationary frames of reference are identical), principle of least

action and Maxwell’s equations; all other characteristics, including the constancy of

the speed of light, follow from them.

4.1.1 The Background

François Arago in 1810 reasoned that the velocity of light could be calculated by

measuring variations in the refractive index of a substance. He measured the

refraction of light from distant stars with a glass prism at the front of his telescope.

He expected a range of different angles of refraction the motion of the earth at

different times of the day and year. Contrary to expectation he found no difference in

refraction with at different times of the day and year.

To explain this negative result the Aether drag hypothesis was proposed by Augustin

Fresnel in 1818, using the wave theory of light to consider Arago’s findings. He

suggested that “...the aether is in excess inside the prism” because the glass prism

carried some of the aether with it. [26]

In 1888 Poincaré lectured at the Sorbonne on Maxwell’s theory and electromagnetic

effects in the ether. These lectures were published in 1890 as Part I of his Électricité

et optique. It was at this time that Hertz was conducting his electromagnetic research

at Karlsruhe. Poincaré wrote in the Introduction to Électricité et optique:

“Science has advanced with rapidity that nothing allowed one to foresee at the

moment I began this course. Since that time the theory of Maxwell has

received, in the most dazzling way, the experimental confirmation it had

lacked.” [27]

This work was taken further by Lorentz, Heaviside and other mathematicians;

Poincaré not only kept up with the developments, and the issues arising out of the

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experiments, but also entered into positive and supportive discussions about these

issues.

4.2 Lorentz Ether theory

Between 1892 and 1906 Lorentz and Poincaré developed a Lorentz Ether theory

(LET) of electromagnetic phenomena in moving bodies. The theory was based

mainly on the Aether drag hypothesis proposed by Fresnel and Maxwell's equations.

Fresnel's Aether drag hypothesis predicted positive results to experiments which are

sensitive to second order velocity effects. However, the 1887 Michelson experiment

gave negative results, therefore refuted Fresnel's Aether drag hypothesis.

In LET, Lorentz introduces stationary electromagnetic ether that mediates between

the electrons. Changes in this electromagnetic can not propagate faster than the speed

of light.

Included in the 1895 LET was the “theorem of corresponding states”, which states:

that a moving observer makes the same observations as an observer in the stationary

system.

Back in 1889 Oliver Heaviside had used Maxwell's equations to derive that magnetic

vector potential field around a moving electron is altered by a factor of .

In order to explain the result of the Michelson experiment, Lorentz assumed that in

motion through the elastic immobile ether the dimension of a body in the direction of

motion is contracted by a factor . His calculations were done to the first

order in v/c. In his subsequent paper in 1895 the contraction factor was changed to

.

Lorentz’s notion of Local Time where t is the time coordinate for an

observer resting in the ether, and t' is the time coordinate for an observer moving in

the ether, formed an essential part of the theorem of corresponding states.

Local Time enabled Lorentz to account for the aberration of light, the Doppler Effect

and the Fizeau experiment. Lorentz considered Local Time only as a technique for

problem solving and a mathematical stipulation to simplify the calculation. Poincaré

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on the other hand saw more than a mathematical trick in the definition of local time,

which he called Lorentz's “most ingenious idea”. [29]

In the Poincaré’s lecture given at Leyden in 1900 celebrating the 25th anniversary of

the thesis of Lorentz, Poincaré states his interpretation of the physical meaning of

Lorentz local time. He defines, for the first time, the principle of synchronization of

distant clocks at rest in moving bodies:

“For the compensation to work, we must relate the phenomena not to the true

time t, but to a certain local time t' defined in the following fashion.

Let us suppose that there are some observers placed at various points, and

they synchronize their clocks using light signals. They attempt to adjust the

measured transmission time of the signals, but they are not aware of their

common motion, and consequently believe that the signals travel equally fast

in both directions. They perform observations of crossing signals, one

travelling from A to B, followed by another travelling from B to A. The local

time t is the time indicated by the clocks which are so adjusted.

If is the speed of light, and is the speed of the Earth which we

suppose is parallel to the x axis, and in the positive direction, and then we

have: .” [30]

In On the Dynamics of the Electron, Poincaré wrote:

“It appears that this impossibility to detect the absolute motion of the Earth by

experiment may be a general law of nature; we are naturally inclined to admit

this law, which we will call the Postulate of Relativity and admit without

restriction.”

“An explanation was proposed by Lorentz and FitzGerald, who introduced the

hypothesis of a contraction of all bodies in the direction of the Earth’s motion

and proportional to the square of the aberration. This contraction, which we

will call the Lorentzian contraction, would explain Michelson’s experiment

and all others performed up to now. The hypothesis would become

insufficient, however, if we were to admit the postulate of relativity in full

generality.”

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“Lorentz then sought to extend his hypothesis and to modify it in order to

obtain perfect agreement with this postulate. This is what he succeeded in

doing in his article entitled Electromagnetic phenomena in a system moving

with any velocity smaller than that of light.” [29]

In 1904 Lorentz had extended his theory to be compatible with experimental results,

not Poincaré’s principle of relativity. For Poincaré, the principle of relativity had to

be preserved; the Lorentz transformation was not just a mathematical transformation,

but was to be regarded as an actual physical transformation. Poincaré wrote to

Lorentz with the necessary correction that satisfied the principle of relativity. With

this correction, Poincaré felt able to present Lorentz’s new theory at his lecture in St.

Louis in 1904. [33]

Poincaré had realised how essential the constant speed of light is to the principle of

relativity. He asked:

“What would happen if one could communicate by non-luminous signals

whose velocity of propagation differed from that of light? If, after having

adjusted the watches by the optical procedure, one wished to verify the

adjustment by the aid of these new signals, then would appear divergences

which would render evident the common translation of the two stations.” [31]

This question was important. In 1805 Laplace had conclude that the speed of

gravitational interactions were at least 7×106 times the speed of light. Poincaré

realised that his interpretation of the principle of relativity required a new theory of

gravity and that gravitational force should transform like electromagnetic forces,

propagating at the speed of light. A new theory of gravity was needed, this was set

out in On the Dynamics of the Electron § 9. — Hypotheses on gravitation.

In the Introduction to On the Dynamics of the Electron, Poincaré sums up Lorentz’s

idea:

“…if we are able to impress a translation upon an entire system without

modifying any observable phenomena, it is because the equations of an

electromagnetic medium are unaltered by certain transformations, which we

will call Lorentz transformations. Two systems, one of which is at rest, the

other in translation, become thereby exact images of each other.”

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Poincaré’s analysis developed from Principle of Least Action and the Lorentz Group;

§3&4. Any transformation of the Lorentz group may be regarded as a

transformation of the form:

Where is a function of , and .

Poincaré showed that the transformation does not change the quadratic form

on condition that

Later in:

§ 7. — Quasi-stationary motion: The analysis of Lorentz is therefore fully

confirmed, but we can better give us an account of the true reason of the fact

which occupies us; and this reason must be sought in the considerations of §

4. The transformations that do not alter the equations of motion must form a

group, and this can take place only if l = 1.

4.3 Electron mass

In the early 20th century there were a number of competing models for an elementary

charged particle. Of these, the three models of Lorentz, Bucherer-Langevin and

Abraham were probably the best known.

Each of these models predicted an increase of electromagnetic mass with speed, and

that a force would be needed to hold the electron together. Mass varied with speed

had already been confirmed by Kaufmann in his experiments with high-speed

electrons between 1901 and 1903.

In his Introduction to Dynamics of the Electron Poincaré writes:

“The advantage of Langevin’s theory is that it requires only electromagnetic

forces, and bonds; it is, however, incompatible with the postulate of relativity.

This is what Lorentz showed, and this is what I found in turn using a different

method, which calls on principles of group theory.

We must return therefore to Lorentz’s theory, but if we want to do this and

avoid intolerable contradictions, we must posit the existence of a special force

that explains both the contraction, and the constancy of two of the axes. I

sought to determine this force, and found that it may be assimilated to a

constant external pressure on the deformable and compressible electron,

whose work is proportional to the electron’s change in volume.

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So if the inertia of matter is exclusively of electromagnetic origin, as it is

generally admitted since the experiment of Kaufmann, and besides that

constant pressure from which I come to speak, all forces are of

electromagnetic origin, the postulate of relativity can be established in any

rigour. It is what I show by a very simple calculation founded on the principle

of least action.”

Poincaré introduced some sort of pressure of non-electrical nature, which contributes

a negative amount to the energy of the bodies, and therefore explains the

4/3-factor in the Lorentz’s expression for the electromagnetic mass-energy relation.

4.3 Gravity

Poincaré examined the hypothesis that Lorentz invariance and the principle of

relativity are valid for all forces including gravitational. Accordingly, Poincaré

constructed a gravitational theory that was compatible with the principle of relativity.

§ 9. — Hypotheses on gravitation

“But there are forces which we can not assign an electromagnetic origin, as

for example gravitation. It could happen, indeed, that two systems of bodies

produce equivalent electromagnetic fields, that is to say, exerting the same

action on the electrified bodies and on the currents, and yet these two systems

do not exercise the same gravitational action on the Newtonian mass. The

gravitational field is thus distinct from the electromagnetic field. Lorentz was

thus forced to complete his hypothesis by assuming that forces of any origin,

and in particular gravitation, are affected by translation (or, if preferred, by the

Lorentz transformation) the same way as electromagnetic forces.

It is now convenient to enter into details and look more closely at this

hypothesis………………..”

Although the Poincaré-Lorentz theory of gravity provided an inaccurate indication of

the perihelion advance of Mercury and was quickly superseded, it did attract

followers, and produced the idea of 4-dimensional space-time with coordinates x, y,

z, ict.

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Between 1908 and 1910 both Minkowski and Summerfield had tried to establish a

Lorentz-invariant gravitational law, but that was superseded by Einstein's theory of

general relativity.

There appears to be a hint of Poincaré in Einstein’s initial approach to his equivalence

principle and the rotating disc. In1909 Einstein wrote to Summerfield to say that he

had looked at the case of the geometry on a rotating disc:

“The situation is very similar to the cooled sphere that Poincaré had described

(and Einstein had studied Poincaré’s popular essays with great care). If the

disc is such that its outer edge is rotating at the speed of light, Einstein

showed (not in the extract below, alas) that the observer at the centre will

indeed think that the geometry on it is a non-Euclidean geometry (but not,

note, the non-Euclidean geometry of Bolyai and Lobachevski, simply a

geometry different from Euclid’s).” [32]

4.5 Minkowski’s Space and Time

Poincaré realised that the Lorentz transformation can be regarded as a rotation in a

four-dimensional Euclidean space with imaginary time ct√-1 as the fourth dimension.

But it was Minkowski who reconstructed the geometrical space-time theories of

Lorentz, Poincaré and Einstein and produced their modern interpretation. He restated

the Maxwell equations in four dimensions and demonstrated that they are invariant

under the Lorentz transformation.

Minkowski presented these ideas at the lecture Raum und Zeit, held at the 80th Nature

Researchers Meeting in Cologne 21st September 1908. [34]

In many respects this lecture was considered an intrusion by mathematicians into the

domain of theoretical physics.

Minkowski begins with announcement that he is about to radically alter perception of

space and time:

“Gentlemen! The conceptions of space and time which I would like to

develop before you arise from the soil of experimental physics. Therein lies

their strength. Their tendency is radical. From this hour on, space by itself and

time by itself are to sink fully into the shadows and only a kind of union of the

two should yet preserve autonomy. First of all I would like to indicate how,

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[starting] from the mechanics accepted at present, one could arrive through

purely mathematical considerations at changed ideas about space and time”

Minkowski goes on to demonstrate the difference between the old view of space and

time the new, with two transformation groups that are covariant with respect to the

laws of mechanics:

“I would like to show you at first, how we can arrive, from mechanics as

currently accepted, at the changed concepts about time and space, by purely

mathematical considerations.”

The first group:

“We wish to picture to ourselves the whole relation graphically. Let x, y, z be

the rectangular coordinates of space, and t denote the time. Subjects of our

perception are always places and times connected.

….for the sake of simplicity, we keep the null-point of time and space fixed,

then the first mentioned group of mechanics signifies that at t = 0 we can give

the x, y, z-axes an arbitrary rotation about the null-point, corresponding to the

homogeneous linear transformation of the expression in itself.”

For the second Group we:

“substitute with any constants α,

β, γ. According to this we can give the time-axis any possible direction in the

upper half of the world t > 0.”

“Let us take a positive parameter c, and let us consider the figure:

According to the analogy of the hyperboloid of two sheets, this consists of

two sheets separated by t = 0. Let us consider the sheet in the region of t > 0,

and let us now conceive the transformation of x, y, z, t into four new variables

x', y', z', t'……….. We can already have a complete idea of the

transformations, when we look upon one of them, in which y and z remain

unaltered. Let us draw the cross section of that sheet with the plane of the x-

and t-axes, i.e., the upper branch of the hyperbola , with its

asymptotes”

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Minkowski constructs the to the parallelogram on figure 1 and lets

“let us draw an arbitrary radius vector OA' of that hyperbola branch from the

origin O, the tangent in A' at the hyperbola to the cutting B' with the

asymptote given at the right, and completing OA'B' to the parallelogram

OA'B'C'; at last for what follows, B'C' is drawn to meet the x-axis at D'. Let us

now take OC' and OA' as axes for the parallel coordinates x',t' with measuring

rods OC' = 1, OA' = 1 / c; then that hyperbola branch is again expressed in the

form c2t'2 − x'2 = 1,t' > 0 and the transition from x, y, z, t to x' y z t' is one of

the transitions in question.”

The transition from x, y, z, t to x' y z t' form the group of Lorentz translations which

Minkowski denotes :

“Now let us increase c to infinity, giving the instantaneous transmission of

Newtonian Mechanics thus 1 / c converges to zero, and it appears from the

figure described, that the hyperbola branch is gradually nestled into the x-axis,

the asymptotic angle extends to a straight one, that special transformation in

the limit changes in such one, and the t-axis can have any direction upwards,

and x' more and more approximates to x. With respect to this it is clear that

the group in the limit for , i.e. as group , exactly becomes the full

group belonging to Newtonian Mechanics.”

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Minkowski continues to develop the four vector formulas of relativistic

electromechanics, but why no mention of Poincaré? Previously Minkowski had

referred to Poincaré as one of the four principal authors of the principle of relativity.

Minkowski’s formulation of relativity was taken up by the Göttingen scientific

community, including professors David Hilbert, Felix Klein, Emil Weichert, and

former professors Gustav Herglotz and Karl Schwarzschild. The ultimate success of

Minkowski’s space-time theory was considered a major triumph for the Göttingen

mathematical community, but most scientists regarded it as a mathematical

development of Einstein’s theory of Special Relativity.

Minkowski succeeded in formulating a geometrical interpretation of the Lorentz

transformation.

He also:

completed the concept of four vectors.

created the Minkowski diagram for the depiction of space-time.

Introduced the concept of proper time.

completed Lorentz invariance/covariance.

created four-dimensional formulation of electrodynamics.

was the first to use expressions like world line.

In 1909, Kline, in a review of Minkowski’s reformulated of Einstein's 1905 paper

wrote [35]:

“A key point of the paper is the difference in approach to physical problems

taken by mathematical physicists as opposed to theoretical physicists. In a

paper published in 1908 Minkowski reformulated Einstein's 1905 paper by

introducing the four-dimensional (space-time) non-Euclidean geometry, a step

which Einstein did not think much of at the time. But more important is the

attitude or philosophy that Minkowski, Hilbert - with whom Minkowski

worked for a few years - Felix Klein and Hermann Weyl pursued, namely,

that purely mathematical considerations, including harmony and elegance of

ideas, should dominate in embracing new physical facts. Mathematics so to

speak was to be master and physical theory could be made to bow to the

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master. Put otherwise, theoretical physics was a subdomain of mathematical

physics, which in turn was a subdiscipline of pure mathematics. In this view

Minkowski followed Poincaré whose philosophy was that mathematical

physics, as opposed to theoretical physics, can furnish new physical

principles. This philosophy would seem to be a carry-over (modified of

course) from the Eighteenth Century view that the world is designed

mathematically and hence that the world must obey principles and laws which

mathematicians uncover, such as the principle of least action of Maupertuis,

Lagrange and Hamilton. Einstein was a theoretical physicist and for him

mathematics must be suited to the physics.”

Kline distinguished between mathematical physicists and theoretical physicists and

discussed which should lead, maths or physics. It is interesting that he portrays

Poincaré as leading Minkowski in the mathematical physicist’s viewpoint.

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5. Significance of Hilbert’s ideas 1900-1910

5.1 Minkowski’s Space and Time

Now we discuss the influence that Hilbert had on Minkowski “Space and Time”.

Hilbert’s sixth problem in his 1900 address was the mathematical treatment of the

axioms of physics.

The investigations on the foundations of geometry suggest the problem, to treat in the

same manner, by means of axioms, those physical sciences in which mathematics

plays an important part; in the first rank are the theories of probabilities and

mechanics.

In 1905, Hilbert lectured in Göttingen, and in these courses gave a quite detailed

overview of how such an axiomatic analysis would proceed in the case of several

specific theories, including mechanics, thermodynamics, and the kinetic theory of

gases, electrodynamics, probabilities, insurance mathematics and psychophysics. [36]

Over the following three years he also worked in partnership with Minkowski on

electrodynamics and other physical sciences.

According to Corry the ideas manifest in Hilbert’s treatment of physical theories are

also part of the scientific and mathematical background that informed Minkowski’s

work on electrodynamics and the principle of relativity. [36] However there appears

to be no tangible evidence that Hilbert had any direct influence on Minkowski’s

axiomatic analysis of the principles of relativity. It is more that Minkowski axiomatic

approach had already been formed during his work at Bonn with Hertz, whom he

much admired.

5.2 Hilbert and the route to Spectrum Theory

At the beginning of the twentieth century the Principal axis Theorem of analytical

geometry was the only theorem that could in some way be regarded as a root of

spectral theory. An early form of the Principal axis Theorem may be found in the

works of the founders of analytical geometry, Fermat and Descartes.

Between 1827 and 1830 the principal axis of various quadratic surfaces had been

investigated both by Jacobi and Cauchy, they showed that the coefficients of the

normal form of a symmetric quadratic form must be real. [37]

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Between 1852 and 1858, Sylvester, together with Cayley, used matrix notation to

produce the general form of the principal axis theorem for n dimensional space and

showed that that the coefficients in the normal form of are the roots of the

characteristic polynomial . Cayley, in A memoir on the theory of

matrices, produced the diagonalization procedure for the matrix and the diagonal

for some orthogonal matrix , and showed that the diagonal entries of

matrix are eigenvalues of ; they are in fact roots of the polynomial equation

.[38]

We must also look at a second root of spectral theory, which is the evolution from

finite dimensional theory to infinite dimensional theory; this occurred first in algebra,

then later in geometry.

Fourier produced the first significant investigation of infinite systems of equations in

1822, when demonstrating that any function may be written as an infinite linear

combination of trigonometric terms. [39]

Next, in 1877, American astronomer William Hill, established an infinite dimensional

theory of determinants from the finite dimensional case. [40] Later in 1886 Poincaré

elucidated Hill’s work, providing a more precise definition of the infinite

determinant. [41] This work was continued by Koch in Stockholm.

Poincaré wrote regarding Hill’s results that:

“The solution adopted by M. Hill is as original as it is bold ... Did one have

the right to set the determinant of these equations equal to zero? M. Hill

ventured to do so and it was a very daring thing to do; until then an infinite

number of linear equations had never been considered [sic!]; determinants of

infinite order had never been studied; no one even knew how to define them,

and it was not certain that it was possible to give a precise meaning to this

notion. I must add, however, for sake of completeness, that M. Köteritzsch

had touched on the subject ... But his paper was hardly known in the scientific

world and in any case was not known to M. Hill. ...

But it is not enough to be daring; dating must be justified by success. M. Hill

successfully avoided all the traps that surrounded him; and let no one say that

in proceeding this way he exposed himself to the most glaring errors; no, if

the method had not been legitimate, he would have been immediately warned,

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because he would have arrived at a numerical result completely different from

that given by observations.”[42]

The work that Poincaré had began in Paris, was continued in Stockholm by von

Koch, who developed between 1890 and 1910 a comprehensive theory of infinite

determinants.

It was in 1900 that the Fredholm theory of integral equations came into the story of

spectral theory; Fredholm was a friend of von Koch. It was then that Fredholm

applied von Koch’s theory of infinite matrices and determinants to integral equations.

Note that integral equations usually refer to back Abel and Fourier. In 1826 Able

solved the tautochrone problem using integral equations; that is problem about the

bead which slides down a curved frictionless wire under the influence of gravity. [43]

Probably first integral equation appeared in The Analytic Theory of Heat, 1822 by

Fourier. [39]

Fredholm copied von Koch's method of expanding infinite determinants to produce

the famous Fredholm “alternative” theorem for the solutions of the integral equation

of the type:

(1)

Fredholm considered the integral equation (1) to be the limiting case of the

corresponding linear system

(2)

He defined the for the integral equation (1) which is the continuous representation

of the conventional determinant of the system (2) developed by Bernoulli to

solve the wave equation by superposition of n simple vibratory modes.

Fredholm showed that the integral equation (1) has a unique solution and;

(a) if then the solution may be expressed as the quotient of two determinants.

(b) if , then the transposed homogeneous equation

has nontrivial solutions and is solvable if and only if

is orthogonal to each of these solutions. [44]

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In the winter of 1900-1901, one of Hilbert’s Swedish students brought to him the

paper on integral equations produced by Fredholm. Hilbert quickly realised that

Fredholm’s work was approaching Hilbert’s goal of a unifying methodological

approach to analysis faster than his own work on calculus of variations. Immediately

Hilbert devoted himself and his resources almost exclusively to integral equations.

Hilbert published a series of five papers in the years from 1904 to 1906. [45a-e] His

breakthrough came in 1906, when abandoned the connection with integral equations,

and discovered the coordinatized function space by means of an orthonormal basis of

continuous functions, maybe inspired by the work of Hill some years earlier.

In Hilbert’s 1906 papers, probably some of his best work, he;

defined the spectrum of the quadratic form .

distinguished the point spectrum from the continuous spectrum.

defined the concept of complete continuity which served to separate those

forms that had pure point spectra from those with more complicated spectra.

formulated and proved the spectral theorem not only for continuous forms but

also for bounded forms.

Schmidt obtained his doctorate from the University of Göttingen in 1905 for his work

on integral equations under Hilbert's supervision. After obtaining his doctorate

Schmidt went to work on his habilitation in Bonn.

In 1907 Schmidt published two papers [46] on integral equations in which he

reproved Hilbert's results in a simpler manner with fewer restrictions.

In these papers he gave what is now called the Gram-Schmidt orthonormalisation

process for constructing an orthonormal set of functions from a linearly independent

set. He then went on to consider the case where the kernel is not symmetrical and

showed that in that case, the eigenfunctions associated with a given eigenvalue

occurred in adjoint pairs. [47]

Schmidt's conceptual simplifications were immediately incorporated by Hellinger and

Weyl in their 1907 and 1908 dissertations under Hilbert.

In 1908 Schmidt published what must be the definitive theory of Hilbert Space, the

space of square summable sequences, which included norms || ||, linearity, subspaces

and orthogonal projections.

In 1909 Hellinger, under Hilbert, published his paper justifying the new theory of

quadratic forms of infinitely many variables. [48] At this time Weyl also published

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his study of bounded quadratic forms whose differential is completely continuous.

[49] By the end of the first decade of the twentieth century Hilbert, his students and

ex-students had published the substance of spectral theory for bounded linear

transformation on Hilbert Space.

As Hilbert was developing his spectral theory, Lebesgue was at work on the

Lebesgue integral. [50] In three brief papers [51a-c] in 1907, both Riesz and Fischer

joined together the works of Hilbert and Lebesgue by showing that Hilbert Space is

isomorphic to the space of functions the square of which are Lebesgue integrable.

In 1910 in a subsequent paper [52] (in which he introduced the more general

spaces), Riesz derived a spectral theory for entirely analogous to that developed

for by Hilbert and Schmidt. [53]

.

Above, figure 2, is an interesting flow chart taken from “Highlights in the History of

Spectral Theory” by Lynn Steen [54] showing the flow of mathematical work

towards Spectral Theory. No single person can claim the credit for the theory, but it

does take a remarkable person with determination and mathematical ability to

combine all these ideas together and form the theory.

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Figure 2


5.3 Hilbert’s 23 problems

In 1900 Hilbert presented 23 problems to the Paris conference of the International

Congress of Mathematicians. Of these 23 problems we consider those which were

solved in the first decade of the 20th century.

It is important to ask whether or not these problems would have been solved when

they were, without Hilbert’s intervention.

The problems solved in the first decade of the 20th century were:

Problem 3, the definition of Euclidian volume (The equality of the volumes of

two tetrahedra of equal bases and equal altitudes).

Problem 19, the Dirichlet problem (Are the solutions of regular problems in

the calculus of variations always necessarily analytic?)

Problem 22, the Uniformisation theorem (Uniformisation of Analytic

Relations by Means of Automorphic Functions).

Problem 3. Consider a triangle in plain two dimensional Euclidian geometry. It is

fairly simple to make a congruent copy of the triangle, and by cut and paste methods

produce a rectangle equal in area to the two triangles; demonstrating that the area of

the original triangle is equal to half its base x its perpendicular height.

Although a number of mathematicians have tried to extend this method to three

dimensions none have been successful. Hilbert continues:

Nevertheless, it seems to me probable that a general proof of this kind for the

theorem of Euclid just mentioned is impossible, and it should be our task to

give a rigorous proof of its impossibility.

This would be obtained, “as soon as we succeeded in specifying two

tetrahedra of equal bases and equal altitudes which can in no way be split up

into congruent tetrahedra, and which cannot be combined with congruent

tetrahedra to form two polyhedra which themselves could be split up into

congruent tetrahedra.” [55]

In 1902 Denn, a student of Hilbert discovered a geometric property of a polyhedron,

in addition to volume, that does not change under a solid cut and paste operation.

In other words the Dehn invariant (as it is called today) is a number given to any

polyhedron which does not change under scissor-equivalence.

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One multiplies each edge length by the dihedral angle , multiplies this by a

number one chooses carefully, and adds these numbers together.

Now, the dihedral angles of the regular tetrahedron are all

while those of the cube are plainly . It is possible to define a Dehn

invariant which takes the value zero for the cube but a non-zero value for the

tetrahedron. The result is that no sequence of cut and stick operations can cut

a cube into pieces and reassemble it as a regular tetrahedron. [56]

According to Gray, [56] unknown to Hilbert, but known to Dehn, the problem had

almost been solved by Bricard in 1896. In this case it would be fair to conclude that

this problem would have been solved by 1902 without Hilbert’s intervention.

Problem 19. Hilbert’s presentation of the nineteenth problem begins with the

observation that there is a class of partial differential equations whose integrals are all

of necessity, analytic functions of the independent variables; for example the potential

equation and minimal surface equation. He pointed out that most of these are

Lagrangian equations for the regular variation problem.

At this time, physics and mechanics were rapidly developing and posing more and

more complex boundary conditions. Dirichlet boundary conditions could be

continuous and not analytic.

Hilbert asked the question:

“does every lagrangian partial differential equation of a regular variation

problem have the property of admitting analytic integrals exclusively?”[56]

The affirmative answer came in 1904 from Bernstein, an ex-student of Hilbert, and

was contained in Bernstein’s doctoral dissertation. [57] Picard, Poincaré and

Hadamard had examined the dissertation. As chairman of the examiners, Picard,

wrote the report. The thesis was a fine piece of work solving Hilbert’s Nineteenth

Problem. [58]

Nine years earlier Picard had shown that if the potential equation has some simple

terms involving only the first derivatives added to it, all the solutions remain analytic.

[59][60]

Bernstein showed that solutions of nonlinear elliptic analytic equations of two

variables are continuous in the third derivative.

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In his dissertation Bernstein thanked Picard for “creating the appropriate

mathematical methods”, and for “having personally recommended this interesting

subject to me.” [49] In this respect it would be fair to conclude that this problem

would have been partially solved in 1904 without Hilbert’s intervention.

Problem 22. Hilbert introduced the twenty second problem with the observation that

Poincaré was the first to prove, then generalise to any analytic non-algebraic

relationship that:

If any algebraic equation in two variables be given, there can always be found

for these variables two such single valued automorphic functions of a single

variable that their substitution renders the given algebraic equation an identity.

[55]

This was one of the problems that Hilbert was actively at working on at the time.

Poincaré and Klein, working independently, had discovered this relationship, the

uniformisation theorem, in 1882. [61][62]

This asserts that every algebraic curve of genus greater than one can be

obtained as the quotient of the unit disc by the action of a suitable Fuchsian

group, and therefore that there is a map from the unit disc to the algebraic

curve that parameterises the curve. The parameterising functions are Fuchsian

functions automorphic with respect to the group.[61]

In 1883 Poincaré generalised the theory extending it to arbitrary multivalued analytic

functions. [62]

Hilbert pointed out that Poincaré’s proof was incomplete. In particular; he noted that

the points where the modular function is not holomorphic are singular. They spawn

an infinite number of points on the boundary of the Riemann surface, S.[61]

“In view of the fundamental importance of Poincaré’s formulation of the

question it seems to me that an elucidation and resolution of this difficulty is

extremely desirable” [63]

Poincaré was not unaware of flaw; he had noted it in his 1883 paper.

Poincaré accepted and rigorously corrected the flaw in his 1907 paper [61][64].“The

problem,” he said in French, “is none other than the Dirichlet problem applied to a

surface with infinitely many leaves”.

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Problems 19, 21 and 22 re-established Dirichlet's Principle, the Riemann mapping

theorem and the uniformisation theorem. This was an extremely good result for

Hilbert.

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6. Conclusions

When considering the significance of the geometrical ideas of Poincaré and Hilbert in

the first decade of the twentieth century it is important to ask the question, “Would

the development of the idea have taken place then, or at all, if it were not for the

existence of Poincaré and Hilbert?”

In history, very few mathematicians have been capable of the spontaneous generation

of new ideas. Newton was one of the few. The development of an idea depends much

on the underlying structure from many generations, demonstrated by figure 2. As

with Lobachevski and Bolyai, the structure was in place to construct the hyperbolic

geometry; if they had not constructed it, then Gauss or someone similar would have.

If Poincaré had not influenced Lorentz and Hilbert influenced Minkowski, then

Minkowski’s Space and Time would still have been produced. The date and content

may have differed slightly, but in that period the momentum of progress in science

and mathematics was so great, that no one person could divert its course significantly.

Newton in a letter to Robert Hooke modestly wrote “If I have seen a little further it is

by standing on the shoulders of Giants.” But on the shoulders of Giants you can miss

seeing what’s at your feet. Hilbert and Poincaré could see further but also the basic

substance of a problem.

Poincaré was capable of developing Lorentz’s work, but also in On the Foundations

of Geometry he goes back to basic perception.

Hilbert developed Fredholm’s integral equations for his Spectral Theory, and then

went back to the root of the axioms in his axiomatic geometry.

In Constance Reid’s book Hilbert-Courant, Hilbert appears to be a very human

character. He valued friendships, especially with Minkowski and the discussions on

his daily walks taken with colleagues and students. This method of exchange of ideas

contributed a greatly to Hilbert’s success as a professor of mathematics.

A member of the Hilbert family said years afterwards, when she was an old lady, “All

I know of Uncle David is that his whole family considered him a bit off his head. His

mother wrote his school essays for him. On the other hand, he could explain

mathematics problems to his teachers. Nobody really understood him at home.”

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7. References and Bibliography

[1]Beyond Geometry, Peter Pesic, Dover Pub 2007;On the Foundations of Geometry (1898) p145[1a]Beyond Geometry, Peter Pesic, Dover Pub 2007;On the Foundations of Geometry (1898) p117[2] Introduction to Poincaré’s Three Supplements, Jeremy J. Gray and Scott A. Walter,p9[3] Poincaré, H., 1887. “Sur les hipothèses fondamentales de la géométrie,” Bulletin de la Société mathématique de France, 15: 203–216.[4] (1902–1908) The foundations of science, Science and hypothesis, The value of science, Science and method by H. Poincaré, Published 1913 by The Science Press in New York, Garrison, N.Y pp66-81. [5] (1902–1908) The foundations of science, Science and hypothesis, The value of science, Science and method by H. Poincaré, Published 1913 by The Science Press in New York, Garrison, N.Y pp55-65.[6] (1902–1908) The foundations of science, Science and hypothesis, The value of science, Science and method by H. Poincaré, Published 1913 by The Science Press in New York, Garrison, N.Y pp81-91.[7] The Principles of Mathematical Physics, Congress of arts and science, universal exposition, St. Louis 1904, (1905), vol. 1, pp. 604-622,[8](1902–1908) The foundations of science, Science and hypothesis, The value of science, Science and method by H. Poincaré, Published 1913 by The Science Press in New York, Garrison, N.Y p7[9](1902–1908) The foundations of science, Science and hypothesis, The value of science, Science and method by H. Poincaré, Published 1913 by The Science Press in New York, Garrison, N.Y p145[10] Poincaré, H. (1889) Lecons sur la théorie mathématique de la lumière, professéespendant le premier semestre 1887-1888 (Paris: Carré et Naud) edited by JulesBlondin, Cours de la Faculté des sciences de Paris, Cours de physiquemathématique. pp. I-II[11] {The German original is quoted in Toepell, Michael M. 1986. Über die Entstehung von David Hilberts ‘Grundlagen der Geometrie’. Göttingen: Vandenhoeck & Ruprecht., 21..}[12] http://www.tau.ac.il/~corry/publications/articles/pdf/Hilbert%20Kluwer.pdf 21/08/2011 p139[13] 1888 Project Gutenberg’s Essays on the Theory of Numbers, by Richard Dedekind, Translator: Wooster Woodruff Beman, Release Date: April 8, 2007 [EBook #21016] p20[14] {The German original is quoted in Toepell, Michael M. 1986. Über die Entstehung von David Hilberts ‘Grundlagen der Geometrie’. Göttingen: Vandenhoeck & Ruprecht., 51..}[15] http://www.tau.ac.il/~corry/publications/articles/pdf/Hilbert%20Kluwer.pdf 21/08/2011 p142[16] 1899 Grundlagen der Geometrie (Festschrift zur Feier der Enth?llung desGauss-Weber-Denkmals in Göttingen), Leipzig, Teubner[17] The German original is quoted in Toepell, Michael M. 1986. Über die Entstehung von David Hilberts ‘Grundlagen der Geometrie’. Göttingen: Vandenhoeck & Ruprecht., 204..}

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[18]The Foundations of Geometry by David Hilbert. Translation by E. J. Townsend Open Court Publishing Co.1950[19] Theory of Algebraic Invariants by David Hilbert translated by Reinhard C. Laubenbacher ©Cambridge University Press 1993 p viii[20] Theory of Algebraic Invariants by David Hilbert translated by Reinhard C. Laubenbacher ©Cambridge University Press 1993 p132[21] Hilbert-Courant by Constance Reid -1986 Springer-Verlag New York Inc p30[22] Hilbert-Courant by Constance Reid -1986 Springer-Verlag New York Inc p33[23] Hilbert-Courant by Constance Reid -1986 Springer-Verlag New York Inc p37[24] Lecture delivered before the International Congress of Mathematicians at Paris in 1900. By Professor David Hilbert http://www.ams.org/journals/bull/1902-08-10/S0002-9904-1902-00923-3/S0002-9904-1902-00923-3.pdf 22/08/2011[25] A Brief History of Functional Analysis Neal L. Carothers, BGSU Colloquium, October 15, 1993 http://www.mai.liu.se/~betur/kurser/TATM85/FA-history.pdf 22/08/2011[26] Fresnel, A. (1818), "Lettre d’Augustin Fresnel à François Arago sur l’influence du mouvement terrestre dans quelques phénomènes d’optique", Annales de chimie et de physique 9: 57–66[27] http://strangebeautiful.com/other-texts/stein-strange-case-poincare.pdf 22/08/2011 p4[28] On The Dynamics of the Electron (Excerpts). by Henri Poincaré Originally published as “Sur la dynamique de l’électron” in Rendiconti del CircoloMatematico di Palermo 21 (1906), pp. 129–175. Author’s date: Paris, July 1905[29] http://en.wikipedia.org/wiki/Lorentz_ether_theory#cite_note-future-7 23/08/11[30] The Theory of Lorentz And The Principle of Reaction,by H. Poincaré, Work of welcome offered by the authors to H.A. Lorentz, Professor of Physics at the University of Leiden, on the occasion of the 25th anniversary of his doctorate, the 11 Dec. 1900. Archives neerlandaises des Sciences exactes et naturelles, series 2, volume 5, pp 252-278 (1900).[31] Poincaré, “Principles of Mathematical Physics” (ref. 35), p. 612; 289; Valeur (ref. 22), p.134; Value (ref. 22), p. 308.[32]Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century (Springer Undergraduate Mathematics Series) 2nd Edition p318[33] The Principles of Mathematical Physics, Congress of arts and science, universal exposition, St. Louis 1904, (1905), vol. 1, pp. 604-622,[34]German Original: “Raum und Zeit” (1909), Jahresberichte der Deutschen Mathematiker-Vereinigung, 1-14, B.G. Teubner A Lecture delivered before the Naturforscher Versammlung (Congress of Natural Philosophers) at Cologne — (21st September, 1908).Saha's translation: The Principle of Relativity (1920), Calcutta: University Press, pp. 70-88 Source http://ia700409.us.archive.org/18/items/principleofrelat00eins/principleofrelat00eins.pdf 22/08/11[35]L Pyenson, Hermann Minkowski and Einstein's Special Theory of Relativity : With an appendix of Minkowski's 'Funktiontheorie' manuscript, Arch. History Exact Sci. 17 (1) (1977), 71-95. http://www.gapsystem.org/~history/Biographies/Minkowski.html 22/08/11[36] Hermann Minkowski and the Postulate of Relativity by Leo Corry p45-53 http://www.tau.ac.il/~corry/publications/articles/pdf/mink.pdf 23/08/11

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[37] Carl G. J. Jacobi, Ober die Hauptaxen der Fliichen der Zweiten Ordnung, J. Reine Angew. Math., 2 (1827) 227-233, (Werke, III, 45-53).Augustin-Louis Cauchy, Sur 'equation a I' aide de laquelle on determine les inegalites seculaires des mouvements des planetes, Exercices de Mathematiques, Paris, 1829, (Oeuvres (2), IX, Augustin-Louis Cauchy, Memoire sur !'equation qui a pour racines les moments d'inertie principaux d'un corps solide et sur diverses equations du meme genre, Mem. Acad. Sci. Inst. France, 9(1830) 111-113, (Oeuvres, (1), II, 79-81).Highlights in the History of Spectral Theory L. A. STEEN, Saint Olaf College p376 The American Mathematical Monthly, vol. 80, 1973, http://www.stolaf.edu/people/steen/Papers/highlights.pdf pp. 361[38] James Joseph Sylvester, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitution to the form of a sum of positive and negative squares, Phil. Mag., 4(1852) 138-142, (Math. Papers, I, 378-381).Arthur Cayley, A memoir on the theory of matrices, Philos. Trans. Roy. Soc. London, 148(1858) 17-37, (Math. Papers, II, 475-496). http://www.stolaf.edu/people/steen/Papers/highlights.pdf pp. 361[39] Joseph B. J. Fourier, Theorie analytique de Ia chaleur, Paris, 1822.[40] George William Hill, On the Part of the Motion of the Lunar Perigee which is a Function of the Mean Motions of the Sun and Moon, John Wilson, Cambridge, Mass., 1877.[41] Henri Poincaré, Surles determinants d'ordre infini, Bull. Soc. Math. France, 14 (1886) 77-90, (Oeuvres, V, 95-107).[42] On the origin and early history of functional analysis, by Jens Lindström http://www2.math.uu.se/research/pub/Lindstrom1.pdf p19 [43] http://en.wikipedia.org/wiki/Tautochrone_curve#Abel.27s_solution 12/08/11[44] Ivar Fredholm, Sur une classe d'equations fonctionnelles, Acta Mathematica, 27 (1903) 365-390.----http://www.stolaf.edu/people/steen/Papers/highlights.pdf pp. 363[45a] David Hilbert, Grundziige einer allgemeinen Theorie der linearen lntegralgleichungen, ErsteMitteilung, Gottingen Nachrichten, (1904) 49-91.[45b] David Hilbert, Grundziige einer allgemeinen Theorie der linearen Integralgleichungen,Zweite Mitteilung, Gottingen Nachrichten, (1904) 213-259.[45c] David Hilbert, Grundziige einer allgemeinen Theorie der linearen Integralgleichungen, DritteMitteilung, Gottingen Nachrichten, (1905) 307-338.[45d] David Hilbert, Grundziige einer allgemeinen Theorie der linearen Integralgleichungen,Vierte Mitteilung, Gottingen Nachrichten, (1906) 157-227.[45e] David Hilbert, Grundziige einer allgemeinen Theorie der linearen Integralgleichungen,Fiinfte Mitteilung, Gottingen Nachrichten, (1906) 439-480.[46] Erhard Schmidt, Zur Theorie der linearen und nichtlinearen lntegralgleichungen, I, Math. Annalen, 63 (1907) 433-476.Erhard Schmidt, Zur Theorie der linearen und nichtlinearen Integralgleichungen, II, Math. Annalen, 64 (1907) 161-174.

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[47] http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Schmidt.html 13/08/2011[48] Ernst Hellinger Neue Begrundung der Theorie quadratischer Formen von unendlichvielen Veränderlichen, J. Reine Angew. Math., 136 (1909) 210-271.[49] Hermann Weyl, Über beschränkte quadratische Formen deren Differenz vollstetig ist, Rend. Circ. Mat. Palermo, 27 (1909) 373-392.[50] Lebesgue, Henri Léon,, Leçons sur l'intégration et la recherche des fonctions primitives, professées au Collège de France par Henri Lebesgue, ,Paris: Gauthier-Villars, 1904.http://quod.lib.umich.edu/u/umhistmath/ACM0062.0001.001?view=toc 23/08/2011[51a] Riesz, Frigyes (1907), "Sur les systèmes orthogonaux de fonctions", Comptes rendus de l'Académie des sciences 144: 615–619.[51b] Friedrich Riesz, Über orthogonale Funktionensysteme, Nachrichten von der KÄonigl. Gesellschaft der Wissenschaften zu GÄottingen, Mathematisch-physikalische Klasse, 1907, 116-122; Áuvres [C5], pp. 389-395)[51c] Ernst Fischer, E. Fischer, Sur la convergence en moyenne,. C. R. Acad. Sci. Paris, 144 (1907) 1022-1024.[52]Friedrich Riesz, "Untersuchungen tiber Systeme integrierbare Funktionen," Math. Annalen,. Vol. 69 (1910), pp. 449-497.[53] Highlights in the History of Spectral Theory L. A. STEEN, Saint Olaf College p367 The American Mathematical Monthly, vol. 80, 1973, pp. 359-381[54] Highlights in the History of Spectral Theory L. A. STEEN, Saint Olaf College p376 The American Mathematical Monthly, vol. 80, 1973, http://www.stolaf.edu/people/steen/Papers/highlights.pdf pp. 359-381[55]Lecture delivered before the International Congress of Mathematicians at Paris in 1900. By Professor David Hilbert http://aleph0.clarku.edu/~djoyce/hilbert/problems.html 23/08/11[56] The Hilbert challenge: A perspective on twentieth century mathematics, Jeremy Gray, (2000) Oxford University Press,ISBN: 0198506511 p97[57] Bernstein, S. (1904), "Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre", Mathematische Annalen (Springer Berlin / Heidelberg) 59: 20–76,[58] http://www.gap-system.org/~history/Biographies/Bernstein_Sergi.html18/09/2011[59] The Hilbert challenge: A perspective on twentieth century mathematics, Jeremy Gray, (2000) Oxford University Press,ISBN: 0198506511 p121-122[60] Picard, E. (1895) Sur une classe étendue des équations linéaires aux dérivées partielles dont tous les intégrales sont analytiques, Comptes rendus 121, 12-14.[61] Gray, Jeremy On the history of the Riemann mapping theorem. Rend. Circ. Mat. Palermo (2) Suppl. No. 34 (1994), 47–94[62]Poincaré, H., 1882, Sur les fonctions Fuchsiennes, Acta Mathematica, 1, 193-294 Oeuvres, 2, 169-257.[63] Poincare, H., 1883, Sur un theorcme de la Theorie generale des fonctions, Bulletin Societe Mathematique de France, ll, 112-125 = Oeuvres, 4, 57-69.[64] Poincare, H. (1907), Sur I'uniformisation des fonctions analytiques, Acta Mathematica 31, 1-63. Reprinted in Oeuvres 4, 70-146.[65] Hilbert-Courant by Constance Reid -1986 Springer-Verlag New York Inc p6.

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