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theory. A classical statement of this view is contained in Max Born's lecture Experiment and Theory in Physics, I delivered in 1943. In it. Born harshly criticized the ...
M A T H E M A T I C A L T H E O R I E S AND P H I L O S O P H I C A L INSIGHTS IN C O S M O L O G Y

Roberto Torretti,

U n i v e r s i d a d de Puerto Rico,

USA

P h i l o s o p h e r s of science often regard physics as stemming from the interplay of theories and facts.

theory we are to u n d e r s t a n d here the in-

By a

t e r p r e t e d theory of a m a t h e m a t i c a l by a fact,

structure,

i.e.

a theory-cum-model;

a d e s c r i p t i o n of o b s e r v a t i o n s Couched in such language that

it can be seen to agree or to d i s a g r e e with a theorem of the r e l e v a n t theory.

A classical

lecture Experiment

statement of this v i e w is c o n t a i n e d in Max Born's

and Theory

in Physics, I

d e l i v e r e d in 1943. In it

Born harshly c r i t i c i z e d the British cosmologists, E. A. Milne,

A. S. E d d i n g t o n and

for r e s o r t i n g to p h i l o s o p h i c a l p r i n c i p l e s as basic premises

in the c o n s t r u c t i o n of physical theories.

In a letter to Born of Septem-

ber 7, 1944, Albert E i n s t e i n made the following comment on this lecture and on his friend's n e g a t i v e

stance towards p h i l o s o p h i c a l

s p e c u l a t i o n in

physics:

Ich babe mit viel Interesse Deinen Vortrag gegen die Hegelei gelesen, welche bei uns Theoretikern das Don Quijote'sche Element ausmacht oder 8oll ich 8agen, den Verf~hrer? Wo dies Ubel oder Laster aber gr~ndlich fehlt, ist der hoffnungslose Philister auf dem Plan. 2 C o n t r a s t i n g his own m e t h o d o l o g y of physics and p h i l o s o p h y of nature with that p r o p o u n d e d by Born, E i n s t e i n added:

Ich [glaube] an volle Gesetzlichkeit in einer Welt yon etwas objektiv Seiendem, das ich aufwild spekulativem Wege zu erhasehen euche. 3 E i n s t e i n ' s d e v o t i o n to the suasion that facts alone

'Quixotic element'

in science, his firm per-

cannot suffice as a source of g u i d a n c e for the

f o r m u l a t i o n and a c c e p t a n c e of theories is shown also in several other texts, one of w h i c h I shall have o c c a s i o n to quote later. General views c o n c e r n i n g the nature of things, w h e t h e r o b t a i n e d through insight or by wild speculation,

appear to be i n d i s p e n s a b l e for c h o o s i n g

a p h y s i c a l l y viable theory from the luxuriant jungle of c o n c e i v a b l e mat h e m a t i c a l structures,

infinitely m a n y of which can be made to fit, with-

in the a l l o w a b l e margin of imprecision,

any given set of facts.

Indeed,

without some such views the p h y s i c i s t w o u l d be hard put to d e s c r i b e his o b s e r v a t i o n s in terms a g r e e a b ! e with a m a t h e m a t i c a l theory. The interplay b e t w e e n m a t h e m a t i c a l theories and what I propose to call p h i l o s o phical insights - or, if you wish, p h i l o s o p h i c a l hunches - is t h e r e f o r e no less e s s e n t i a l for the growth of physics than the interplay b e t w e e n

321

theories and facts. Fortunately,

p h i l o s o p h y of science has begun of late

to direct to the former some of the a t t e n t i o n it had paid to the latter. T h o u g h the said i n t e r p l a y can be seen in all the fundamental p h y s i c a l inquiry,

4

fields of

it has been quite p r o m i n e n t in 20th century cosmology,

partly b e c a u s e of the very nature of its subject-matter,

but partly per-

haps too b e c a u s e of the kind of factual evidence on which it must rely. I feel t h e r e f o r e that it might be w o r t h while to illustrate with examples from c o s m o l o g y a few basic questions r e g a r d i n g the i n t e r p l a y b e t w e e n philosophical I.

insights and m a t h e m a t i c a l theories.

The first q u e s t i o n that I w i s h to c o n s i d e r is a fairly obvious one:

If we take a look at facts, we see that theories may be said to be corro-

borated or

refuted

by them; what can p h i l o s o p h i c a l

ries? In principle, axioms.

Descartes'

insights do for theo-

they can of course d e t e r m i n e them, by supplying their cosmological programme

(in his

Discourse on Method , part

V) should p r o b a b l y be u n d e r s t o o d as an attempt in this direction. this century,

E. A. Milne quite d e l i b e r a t e l y

And in

set out to derive the entire

system of the laws of physics from a few insights c o n c e r n i n g the structure of the universe. lowing difficulty, kind:

However, M i l n e ' s c o s m o l o g y is a f f l i c t e d by the fol-

w h i c h is bound to arise in all e n t e r p r i s e s of this

in the course of a c t u a l l y b u i l d i n g a m a t h e m a t i c a l theory of nature

that is g e n e r a l l y adequate to facts but rests a l t o g e t h e r on a priori principles,

M i l n e has had to supplement his original a l l e g e d l y insight-

ful and at any rate p h i l o s o p h i c a l l y p l a u s i b l e p o s t u l a t e s w i t h a d d i t i o n a l assumptions,

that are more or less clearly ad hoc and for w h i c h he could

provide at best a f a r - f e t c h e d justification.

From the c o n s i d e r a t i o n that

"if a rational u n d e r s t a n d i n g of the u n i v e r s e is possible, possible

it o u g h t to be

to set up a c o n s i s t e n t system of t i m e - k e e p i n g t h r o u g h o u t the 5 Milne derives the c o n c e p t i o n of a family of "fundamental ob-

universe", servers",

each of w h i c h can make with his clock and t h e o d o l i t e a set of

observations observer.

indistinguishable

from those made by any other f u n d a m e n t a l

Having proved that the c o o r d i n a t e s e m p l o y e d by three c o l l i n e a r

f u n d a m e n t a l o b s e r v e r s t r a n s f o r m into one another by Lorentz t r a n s f o r m a tions, he postulates

that this holds good also for any set of fundamen6 tal observers in three-space. Further a s s u m p t i o n s are: that the uni7 verse has zero net angular momentum, that the a c c e l e r a t i o n of a free 8 p a r t i c l e is a unique function of its position, v e l o c i t y and epoch, etc. The w i l l i n g n e s s of M i l n e and the like of him to enlarge the a priori basis of their d e d u c t i o n s as it m i g h t be r e q u i r e d for the a d v a n c e m e n t of their t h e o r i e s has c o n t r i b u t e d not a little to the current d i s t r u s t of s p e c u l a t i o n in natural philosophy. A somewhat d i f f e r e n t a p p r o a c h is e x e m p l i f i e d by Bondi and Gold's v e r s i o n

322

of the Steady State cosmology. p h i l o s o p h i c a l insight,

9

Here the theory is derived from a single

e m b o d i e d in the Perfect C o s m o l o g i c a l Principle,

plus one or two u n d e n i a b l e

facts,

such as the darkness of the sky at

night or the f r e q u e n c y - s h i f t of radiation received from distant sources. The Perfect C o s m o l o g i c a l Principle says that the u n i v e r s e is h o m o g e o u s in space a n d time, so that it looks more or less the same from any spat i o t e m p o r a l vantage point. argument,

It is justified by a t y p i c a l l y p h i l o s o p h i c a l

r e m i n i s c e n t of R e i c h e n b a c h ' s v i n d i c a t i o n of the straight rule

of induction. I0

Unless the looks of nature are always and everywhere

a p p r o x i m a t e l y the same we w o u l d not be e n t i t l e d to e x t r a p o l a t e to the entire breadth and d u r a t i o n of the u n i v e r s e the results of our exact physical measurements, gion of it. Thus,

w h i c h are p e r f o r m e d w i t h i n a p i t i f u l l y small re-

either

the Perfect C o s m o l o g i c a l Principle

is true, and

that disposes of the theories that c o n t r a d i c t it, or the Perfect Cosmological Principle is false, but then the p a r t i c u l a r theories that may be d e v e l o p e d in o p p o s i t i o n to it have scarcely any chance of being true instead. The Perfect C o s m o l o g i c a l Principle,

t o g e t h e r with factual evidence

that the larger clusters of m a t t e r are r e c e d i n g from one another, that most famous or infamous tenet of the Steady State Theory: tinuous C r e a t i o n of Matter.

implies

the Con-

It is indeed ironic that a p r i n c i p l e intro-

duced to ensure the u n i v e r s a l v a l i d i t y of our t e r r e s t r i a l physics should thus lead to the n e g a t i o n of a deeply e n t r e n c h e d and well c o r r o b o r a t e d local law. Before the d i s c o v e r y of the relic m i c r o w a v e b a c k g r o u n d radiation made it unfashionable,

the Steady State T h e o r y was strongly resis-

ted in the name of matter conservation.

Thus, M a r i o Bunge f o r c e f u l l y

a r g u e d in 1962 that the a c c e p t a n c e of continuous c r e a t i o n w o u l d spell disaster for scientifc thought and w o u l d make it i n d i s t i n g u i s h a b l e from magic. 11

Others,

I must recall, were charmed by the d i a l e c t i c a l p i q u a n c y

of a theory in which the sheer r e p e t i t i o n of c r e a t i o 12 means of securing that n i h i l sub sole n o v u m .

ex

nihilo

is the

E i n s t e i n ' s General Theory of R e l a t i v i t y bears witness to a more subtle mode of action of insight on theory.

Philosophical vision manifestly

guides the f o r m u l a t i o n of the theory, but is not i n c o r p o r a t e d into it as an axiom or a set of axioms. Moreover,

the theory, once formulated,

makes the p h i l o s o p h i c a l vision more precise and can, in a sense, even be said to m o d i f y it. A n s w e r i n g to E. K r e t s c h m a n n ' s

remark that general co-

v a r i a n c e is a trivial r e q u i r e m e n t that any phyical theory is capable of fulfilling,

E i n s t e i n d e c l a r e d in 1918 that General R e l a t i v i t y rests on

three principles: Covariance,

the P r i n c i p l e of Equivalence, the Principle of General 13 14 and Mach's Principle. If, following G e r a l d Holton, we

c l a s s i f y the diverse i n g r e d i e n t s or strands of a piece of scientific

323

thought into the e m p i r i c a l or factual,

the analytic or l o g i c o - m a t h e m a t i -

cal, and what H o l t o n calls the thematic element, sophical insight"

of w h i c h our "philo-

is a proper or an improper part,

it is a p p a r e n t that

the P r i n c i p l e of General C o v a r i a n c e is a l o g i c o - m a t h e m a t i c a l requirement, while the P r i n c i p l e of E q u i v a l e n c e is a bold but r e a s o n a b l e extrap o l a t i o n from facts. The p h i l o s o p h i c a l c o m p o n e n t of General R e l a t i v i t y must t h e r e f o r e lie m a i n l y with Mach's Principle. Now Mach's P r i n c i p l e is not an axiom of the theory.

Indeed,

in the p a r t i c u l a r v e r s i o n pro-

posed by E i n s t e i n in 1918, it is not even a theorem of it - though in a m o d i f i e d v e r s i o n it has been said to state the b o u n d a r y c o n d i t i o n s pres c r i b e d for the field equations. 15

But the insight w h i c h Mach's Princi-

ple is supposed to express c e r t a i n l y guided E i n s t e i n during the long 16 strenuous search that led to the f o r m u l a t i o n of General Relativity. In t r a d i t i o n a l p h i l o s o p h i c a l

language that insight can be stated thus:

A b s o l u t e space and absolute time - as well as absolute spacetime -, without matter,

are p h y s i c a l l y inviable m a t h e m a t i c a l contraptions.

This

thesis is implicit in the w r i t i n g s of the g r e a t e s t p r e - N e w t o n i a n natural philosophers,

A r i s t o t e l e and Descartes,

Leibniz against N e w t o n ' s

spokesman,

by Kant, but this philosopher,

and was e x p l i c i t y d e f e n d e d by 17 Samuel Clarke. It was also shared

b e l i e v i n g that the c o n c e p t u a l f r a m e w o r k

of N e w t o n i a n dynamics was a p r e r e q u i s i t e of natural science,

felt com-

p e l l e d to c o n c l u d e that nature itself, r e g a r d e d as a scientific object, was no less unreal,

or " t r a n s c e n d e n t a l l y ideal", than absolute space and

time had to be. The said insight also inspired Ernst Mach's c e l e b r a t e d c r i t i c i s m of N e w t o n ' s

scholion on space and time. 18

E i n s t e i n traces the 19 o r i g i n of his M a c h p r i n c i p l e to this M a c h i a n criticism. Since N e w t o n i a n time had been s u p e r s e d e d already in the Special Theory of Relativity,

the

M a c h P r i n c i p l e of General R e l a t i v i t y m u s t be d i r e c t e d against N e w t o n i a n space. Now the a b s o l u t e n e s s of space is manifested, 2O

through the p h e n o m e n a of a b s o l u t e acceleration.

a c c o r d i n g to Newton, It m i g h t seem para-

doxical that N e w t o n should resort w i t h such a s s u r a n c e to the concept of an a b s o l u t e acceleration,

while p r o v i d i n g no c r i t e r i a and having no use

for the p r e s u m a b l y more f u n d a m e n t a l concept of absolute motion. But this p a r a d o x was d i s p e l l e d by M i n k o w s k i ' s

i n t e r p r e t a t i o n o f Special

R e l a t i v i t y and the subsequent r e f o r m u l a t i o n of N e w t o n i a n theory in four 21 All that is needed to make sense of

d i m e n s i o n s by Cartan and others.

an a b s o l u t e a c c e l e r a t i o n w i t h o u t having to c o n c e i v e an absolute motion is that spacetime be e n d o w e d with a linear connection, s t r u c t u r e is e m p l o y e d for singling out worldlines,

i.e.

plus w h a t e v e r spacetime curves

that are nowhere tangent to an h y p e r s u r f a c e of simultaneity.

The linear

c o n n e c t i o n then d e t e r m i n e s an intangible yet p e r f e c t l y rigid network of

324

geodesic worldlines,

which any m a t e r i a l particle m u s t follow if let loose

at an a r b i t r a r y spacetime point. A b s o l u t e l y a c c e l e r a t e d motion is simply m o t i o n w h i c h deviates from this natural network, i.e. m o t i o n along a n o n - g e o d e s i c worldline.

This r e c o n s t r u c t i o n of the concept of absolute

a c c e l e r a t i o n clearly brings out the e f f e c t i v e physical m e a n i n g of Newton's talk of a b s o l u t e space. dynamics,

In N e w t o n i a n and in special relativistic

spacetime, of itself, constrains force-free matter to follow

a special kind of cosmic track. On the other hand, the presence of m a t t e r makes no d i f f e r e n c e in the structure of spacetime.

A c c o r d i n g to Einstein,

such lopsidedness in the mutual r e l a t i o n s h i p between two physical entities is utterly at v a r i a n c e with e v e r y t h i n g we k n o w of nature. 22

Such

was the insight that E i n s t e i n sought to embody in the Mach Principle of 1918. This can be p a r a p h r a s e d as follows:

The linear connection,

ther, the spacetime metric on which it depends

is fully d e t e r m i n e d by the d i s t r i b u t i o n of matter. Now, tivity holds locally,

or ra-

in a r e l a t i v i s t i c theory, if Special Rela-

the d i s t r i b u t i o n of m a t t e r should be r e p r e s e n t e d

by the s t r e s s - e n e r g y tensor,

a symmetric tensor field of order two, whose

covariant d i v e r g e n c e vanishes. Mach's Principle implies then that the components of the metric relative to a given spacetime chart must be obt a i n e d by integration of a system of d i f f e r e n t i a l equations r e l a t i n g the components of the s t r e s s - e n e r g y tensor with those of a tensor field constructed from the metric itself and its derivatives with respect to that chart.

If N e w t o n ' s g r a v i t a t i o n a l theory is valid as a first approximation,

the r e q u i r e d equations m u s t depend on the second derivatives of the metric.

If we assume for simplicity's sake that no higher order deriva-

tives are involved, field equations

the stated conditions u n i q u e l y d e t e r m i n e E i n s t e i n ' s

(up to two a r b i t r a r y constants) .23

u n d e r s t a n d s the " d i s t r i b u t i o n of matter"

However, unless one

in a P i c k w i c k i a n sense, the

field equations do not agree with the above v e r s i o n of Mach's Principle, since they also have solutions if t h e s t r e s s - e n e r g y tensor is i d e n t i c a l l y 24 zero. Later in life E i n s t e i n rejected the 1918 M a c h Principle. On F e b r u a r y 2, 1954, he wrote to Felix Pirani:

One shouldn't talk any longer of Mach's principle, in my opinion. It arose at a time when one thought that 'ponderable bodies' were the only physical reality and that in a theory all elements that are not fully determined by them should be conscientiously avoided. I am quite aware of the fact that for a long time, I, too, was influenced by this fixed idea. 25 Nevertheless,

he u n f l i n c h i n g l y held to the original

insight that space-

time could not be allowed to act on matter w i t h o u t the latter acting on spacetime in its turn. On May 12, 1952, he wrote to Max Born, with regard to the a p p a r e n t failure of one of the three classic effects of

325

General

Relativity:

Wenn ~erhaupt keine Linienablenkung, keine Perihelbewegung und keine Linien-Verschiebung bekannt w~re, w~ren die Gravitationsgleichungen doch ~ber~eugend, weil sie das Inertialsystem vermeiden (dies Gespenst, das auf alleswirkt, aufdas aberdie Dinge nicht zur~ckwirken). Es ist eigentlich merkw~rdig, dab die Menschen meist taub sind gegen~ber die st~rksten Argumenten, w~hrend sie stets dazu neigen, Messgenauigkeiten zu ~bersch~tzen. 26 2.

The second q u e s t i o n

of the first. thought

What

contribute

look once more

that

to the p h i l o s o p h i c a l

in c o n n e c t i o n

framework

for the

predicted

the e x i s t e n c e

mathematical

with

thought

theories

description

of e n t i r e l y

in a way,

of n a t u r e ?

and facts.

insights,

the

like radio waves

for insight?

Francis

Let us

Not only do

provide

to a new v i e w of nature?

this has happened.

inverse

they have often

things,

analogous

the

can m a t h e m a t i c a l

of phenomena;

unexpected

something

give birth

in w h i c h

is, What

understanding

between

their u n d e r l y i n g

scientific

Can theories do

I k n o w of a case

to c o n s i d e r

do for insight?

at the i n t e r p l a y

theories,

or antimatter.

I wish

can theories

can pure

I am not

M. C o r n f o r d

sure once

a r g u e d that the idea of an a b s o l u t e space was b e g o t t e n by Greek geome27 try. But even if he were right, one m i g ht still object that this was not a c o n t r i b u t i o n that m a t h e m a t i c s cal views

to insight

has r e p e a t e d l y

"clear and distinct"

sense they are tenable. cal

inquiry.

The

foundational promising

In the

special and,

domain

I dare

a differentiable

manifold,

significant

manifold

by diverse

fields

of p h i l o s o p h i logic

Less well known,

and but very

as in P r o f e s s o r

Bunge's

by Christoffel, of what

go with

it.

concept

tensor

theory

fields

into several

the Lorentz

metric

Weyl, can do

Relatireal

Such a m a n i f o l d

of the u n i v e r s e

or spinor

for the 29

a philosophically

in cosmology.

metric.

of

space,

Levi-Civit~,

as a f o u r - d i m e n s i o n a l

properties

vector,

(Indeed,

Ricci,

concept

Riemann

of p h y s i c a l

a mathematical

endowed w i t h a Lorentz

scalar,

by B e r n h a r d

I shall now examine

the u n i v e r s e

The p h y s i c a l

few ideas have been more

the essence

i.e. by smooth m a p p i n g s of the m a n i f o l d naturally

philosophi-

to show in what

than the m a t h e m a t i c a l

of the m a n i f o l d

conceives

c a l l e d a spacetime.

workshop,

introduced

of nature,

application

cosmology

first

As an example

for our u n d e r s t a n d i n g

sented

existing

helped

in ontology,

fertile,

of e l u c i d a t i n g

perfected

and others.

Hausdorff

it is e v i d e n t

in c o n t e m p o r a r y

to everyone.

of our p r e s e n t

say, more

purpose

and s u b s e q u e n t l y

vistic

to make

in many

of m a t h e m a t i c s

is familiar

Anyhow,

treatise. 28

philosophical

Cartan

This has o c c u r r e d

importance

research

contributed

and has thereby

is the use of m a t h e m a t i c s

pioneering

helpful

but to obfuscation.

on spacetime,

fibre bundles

is one

is

are repre-

that

such mapping,

326

into

the b u n d l e

conceptual

of symmetric c o v a r i a n t

framework

one of the oldest lem of the When

extent

Kant,

to the d e v e l o p m e n t a pseud o s c i e n c e , this p r o b l e m clusion. 31 phenomena

rooted

completed

d efinite

a beginning,

less plausibly, totality

yet infinite

its confines

empty

is useful

for r e g u l a t i n g

ceiving

the

our

the p r o p e r

the u n i v e r s e

tinua.)

However,

infinity

is p o s s i b l e

familiar

with

Riemann's

revival

may r i g h t l y

some c o m p a c t components.

manifold

spacetime

from a

lying beSince

he d e c l a r e d

idea,

but w h i c h

which

generates

both

conception in his

on w h i c h

that

the actual

cuts

cosmology.

to c o n c e i v e

surface,

each of w h o s e

points

than,

passes

1917, We may

into two

if closed t i m e l i k e

of d i f f e r e n t i a b l e

is not h a r d e r

of 32

of it lies on

the spacetime

must be refined

for it

world.

Betrachtungen"

if every point

which

say,

a circle

if

of con-

it. We are all

yet u n l i m i t e d

of m o d e r n

fields

is,

and of b o u n d e d

void b e y o n d

of a finite

the theory

is no

the m a t t e r

to reject

in extent

phyBy con-

cosmology

abide by the other horn,

finite

finite

There

conception

by e x p e r i e n c e

the b e g i n n i n g

Within

relativistic

(No difficulty,

"Kosmologische

hypersurface

of its own.

dilemma.

Cantorean,

such a s p a c e t i m e through

was absurd.

eternity,

of a science

a

are no less

to be a mere

up an u n t h i n k a b l e

(This c h a r a c t e r i z a t i o n

are permitted.)

in extent,

its beginning.

the w o r l d

of the K a n t i a n

prompted

as s p a t i a l l y

spacelike

He

involves

to his mind,

inquiries,

one can always

be said to mark

r e g a r d a spacetime

curves

feels

of it,

things

and d e v e l o p i n g

before

event

absurd.

as soon as it is held to be a genuine

an infinite

conjuring

any current

he d e e m e d

or a t e r m i n a t i n g

scientific

to have a spacetime

without

Einstein's

running

about the classical, if one

he c o u n t e d

of physical

of an e m p t y s p a c e

right out to infinity.

of the universe,

duration,

specter

subject-matter

in c o n c e i v i n g

one has no qualms

again,

in extent

as a d i f f e r e n t i a b l e

take non zero values

which

the u n i v e r s e

can deal at once with both horns difficulty

succession

and time o u t s i d e

judged

contradictions

object,

which

finite

space

and

reason,

forced him to d r a w this con-

given

infinity

impossible

of human

cosmology

must be finite

and an empty time

that

early c o n t r i b u t i o n s pronounced

otherwise

process,

to

the prob-

of s i m u l t a n e o u s l y

raises

a b s u r d than a c o m p l e t e d

sical

because

synthesis,

cosmology

inextricable

illusion

This

of the universe.

that the u n i v e r s e

a universe

starting-point

Kant b e l i e v e d

his b e a u t i f u l

that the temporal

the end of an eternal

an infinite

of cosmology,

world-view, 30

in a fatal

two.)

new a p p r o a c h

beginning

first of the four that

must have

On the other hand,

yond

notwithstanding of a N e w t o n i a n

as the

also argued, because

of order

for a totally

and the temporal

Kant m a i n t a i n e d

w o u l d mark

tensors

the way

and most i n t r a c t a b l e problems

spatial

Immanuel

has paved

manifolds

a cylindrical w h i c h cuts

the

327

surface

in two.

duration

first

that

that

each

subset

of

dition:

for every

c through below.

links

a neighbourhood

P

P,

null

to l i v e

in a s p a c e t i m e

that

for e v e r y

sake,

strictures

against

M,

time,

we

an e t e r n a l

M

the

the

with

past.

to see t h i s m o r e

events

But

M

clearly,

of a f u t u r e - d i r e c t e d

neighbourhood

U

of

P

time- .

by resorting

to

be untimely

to

if w e h a p p e n e d not mark

immune

say t h a t

the

to K a n t ' s by an empty

a point

P

in

c: I - - ~ M ,

curve

is a r e a l

of c of

would

timelike

there

con-

eXistence

is n o p r e c e d e d

let us

onto

following

the d o m a i n

ignore

is t h e r e f o r e

Let pro-

future-directed

can be dealt

current

process.

shapes,

can be charted

fulfil

by proper

- these

s u c h as

is a starting-point

M can

of b o u n d e d

attention.

p a r a m e t e r , b u t it w o u l d 33 n o w -, it is e v i d e n t t h a t ,

of a n y e t e r n a l

if for e v e r y

and

more

different

affine

conclusion

M

M

simplicity's

refinements

In o r d e r

in

curves

generalized

such

a spacetime

parametrized

for

go into

time.

and deserves

in it h a s

If,

over

of a u n i v e r s e

can have many

point

is b o u n d e d

the concept

difficulties

R 4. T h u s ,

like curve

a so-called

hand,

a spacetime

point

an open

causal

other

is n o t w i t h o u t

us r e m a r k vided

On the

number

tUG

I,

c(t) lies in U for e v e r y t ~ I w i t h t ~ t U. W e s h a l l say t h a t s u c h a c u r v e is past-inextendible if n o p o i n t in M is a s t a r t i n g - p o i n t such

of

that

it. N o w

the

ture-directed

timelike

is b o u n d e d

can be

below.

P.

Let

The

0.

U,

our

have

an o p e n

Therefore

it c a n n o t

side

is a s t a r t i n g - p o i n t The condition favoured be any

would U

with

to a curve

assume,

V

would

through

the p a s t

of c(0).

at 0.

integral

~, d e f i n e d

Consequently,

at 0

there

exist

that

in

M

on M.

a given

point be

the parameter.) But

then

be defined

c(0)

would

Hence

curve

Moreover,

rather

lower bound

starting-point. M,

fu-

is t i m e - o r i e n t a -

through

V

every

vectors

by regraduating

in of

but

curves

timelike

V

that

of d e f i n i t i o n

its g r e a t e s t

its

not be a maximal

of 0.

of

Let

be

curve

c(0)

on all

have

values

c, c o n t r a r y

of

since

V

to

through

c is m a x i m a l ,

and agreeing is n o p o i n t

with

c

M

that

in

of c.

prescribed

for our

Big Bang world-models. future-directed

parametrized

a domain

sensibly

curve

below.

c is n o t d e f i n e d

be extended

on t h e P o s i t i v e

U

has

of u n i t

if n e c e s s a r y ,

integral of

does not mean

a starting-point, M

V

integral

0, c(0)

would

has in

as w e m a y

neighbourhood

intersection

assumption,

c(0).

c

at

M

M

such past-inextendible

M,

secured,

defined

and the maximal

in t h e

if

for

a field

of c is b o u n d e d

(This c a n b e

If c w e r e would

That

c be the maximal

domain

in

such curve

seen by considering

(Such a f i e l d e x i s t s ble.)

prescribed

curve

past-inextendible

every that

condition

by proper

spacetime Let

W

be

past-inextendible

time.

Suppose

that

M

obtains

a Big Bang timelike

in the c u r r e n t spacetime

geodesic

c is c o m p l e t e ,

in

and

let

W,

i.e. t h a t it

328

has a value

for every

such that c(t)

is the Big Bang,

in the universe. etc.

real number.

such point

can b e l o n g

the g r e a t e s t

to

Bang u n i v e r s e s

have

proper beginning. to ask what went

There

is a n o t h e r

temporal

radical

At this

singularity which

it may be

found

Schwarzschild equations

noted

which field

constant

is the

matter

in the

-- w here

the field:

the black

the curvature,

hole

etc.

schild

field must t h e r e f o r e

rather

scratched,

that there

nihilating

mass.

conceive

a

field

years

be no less radica-

the Big Bang w i t h

can arise,

solution theorem,

field,

A

of Einsteins' the

solution perfect

solution

involves

source

of radius

less than

and c is the

a

2Gm/c 2

speed of light in

at the axis of s y m m e t r y

At such a point

is no such point

to v a n i s h

symme-

and very n a t u r a l l y inter35 If all the

cannot be smooth,

be r e g a r d e d

holds

of the field.

a sphere constant

the

and a Fried-

provided

is u s u a l l y

we

-- the

originat~one

Schwarzschild

the m a t t e r

in our manifold.

However,

is fully d e t e r m i n e d

fields,

punctured,

field about i.e.

A Schwarzor

most p h y s i c i s t s

into the singularity.

of the S c h w a r z s c h i l d

of

so we are led to con-

as a spacetime

left b e h i n d by a collapsing, The

-- p r o v i d e d

billion

of Big Bang universe.

at its axis of symmetry.

not allo w the mass m simply

rity as the field

kind

singularity.

once more,

strictly

Take

to a space p o i n t

clude,

one can

to compare

lies

is relaxed,

lies w i t h i n

the metric,

was al-

The S c h w a r z s c h i l d

m, w h i c h

collapse

tempt us

& causal

in a Big Bang w o r l d

world would

and static

of s t a t i c i t y

G is the g r a v i t a t i o n a l

-- it must

along

something

sense,

De r e r u m

By B i r k h o f f ' s

of the m a t e r i a l

source

that m i g h t

instant

a b l a c k hole

simplest

spacetime.

of i n t e g r a t i o n

as the mass

vacuo

in w h i c h

at all times.

preted

But an eternal

a b l a c k hole.

case

if the r e q u i r e m e n t

try is p r e s e r v e d

instant

In this

Big

but do not have a

it, at w h i c h

in his tract

is a symmetric

for empty

below,

some fifteen

instructive

inside

of its

is misleading.

the tip of our fingers

in time

of nature.

described

in the past of any given one

from it.

with

them back

is the simplest

mann universe,

even

distance

as Leibniz

point,

preceding

no

and t is

is often

as the b e g i n n i n g

description

bounded

each event

so to speak,

contingency

contingent, 34 li.

at least,

it. For every given

instant

at a finite

to stretch

e is not c o m p l e t e

is in them no f i r s t

on. However,

manage

or,

the curvature,

by definition,

The Big Bang

t

all the energy

tensor,

is impossible

Consequently,

duration

on b e f o r e

ready going

we can touch,

this

We see that this

their

containing

the s t r e s s - e n e r g y

of its domain.

of the world,

dispensation.

line there

W.

lower bound

as the b e g i n n i n g present

Since

there must be a real number

a point

i.e.

At such a point

w o u l d not be smooth.

Then

will

Of course, the singula-

literally

self-an-

by the b o u n d a r y conditions.

329

The

idea that

it must b r e a k

there to support

and f i e l d - t h e o r e t i c a l l y . quieting Hence

about a chunk

the c o m m o n

dox in General

tions

universe, past

if

~

(in terms, metric time, ment

of m a t t e r

say,

approaches

to say, ~

spacetime

if t 0 were

source

in w h i c h

be.

This

e(t O)

that

probably

without

tO .

paradox

and c is a para-

Consequently

if c were

defined

and that

as the argu~

w o u l d be

at to, that

and hence we

t o is not

inside

is coeval with singularity

of b l a c k hole

in the range

the field

singularities

an u n b o u n d e d

yet

the p r e c e d i n g

finite

universe,

so

in the

m i g h t be said to

theory

idea of a m a n i f o l d

for

itself,

explaining

t h r o w a light on the Big Bang too and m i g h t

of R i e m a n n ' s

even

b l a c k holes,

is not p a r a d o x i c a l

a post-relativistic

is

time coordinate.

should not be o b j e c t i o n a b l e

it. Be that as it may,

that by means

tensor

all bounds

annihilation

field

neighbourhood.

stress-energy

coordinates)

the

grows b e y o n d

this

of the Big Bang

towards

by R o b e r t s o n - W a l k e r

does not exist

the e x c i s i o n

ly do away w i t h

in one's

of the

assump-

In such a

time c o o r d i n a t e

at any point of spacetime,

reject m a t t e r

On the

parametrized

c(t O)

However,

is not to deny that

holes will

increases

real n u m b e r point

of the F r i e d m a n n

way

-- say as one moves

component

~ • c

infinite

coordinate.

the e x c i s i o n

of matter.

in the range of the R o b e r t s o n - W a l k e r

to those who w o u l d

hope,

function

cannot become

of the time

that

indefinitely

the d e n s i t y

from

some s i m p l i f y i n g

dist±ibution

in time

para-

if at all,

line of the R o b e r t s o n - W a l k e r

a definite

at the

must conclude

the

global

of the R o b e r t s o n - W a l k e r

the c o m p o s i t e

a genuine

perspective.

from

dis-

into nothingness.

constitute

is d e r i v e d

line of the time coordinate,

infinite

But

the current

denotes

philosophically

away

quantumtheoretical

universe

any p a r a m e t r i c

-- the d e n s i t y

fading

is no longer

spatiotemporally

and can only be understood,

probably

a Friedmann

to think

is s o m e t h i n g

thus

that black holes

as one goes b a c k w a r d

along

Indeed,

But there

feeling

concerning

an i n a b i l i t y

of m a t t e r

Relativity

a post-relativistic, o ther hand,

down as soon as its source

it b e s p e a k s

has made

black

eventual-

clear,

I

one can c o n c e i v e

w h i c h has an age though

it never was born. 3.

I can b a r e l y

gard to insights analogy that,

touch on a t h i r d q u e s t i o n and theories.

of the i n t e r p l a y

even

phenomena

if there

frameworks,

there

that

theories

in d i f f e r e n t

is s o m e t h i n g

a touchstone

have b e e n b r o u g h t back Aristotle

between

are no i n c o r r i g i b l e

can be d e s c r i b e d

they provide

To c l a r i f y

final

fromthemoon,

it is made

and facts.

Once There

again

statements

ways w i t h i n

diverse

facts,

Thus,

by v i r t u e

of i n c o r r u p t i b l e ,

and all

conceptual of w h i c h

now that all those

one w o u l d not w i s h pellucid,

re-

to the

is no doubt

observation

about

for theories.

that must be a s k e d with

it I resort

stones

to m a i n t a i n imponderable

with ether.

330

Is there a comparable finality about insights? Is there any way of arriving at d e f i n i t i v e p h i l o s o p h i c a l

insights that set fixed b o u n d a r i e s

to a d m i s s i b l e theories? Some of the g r e a t e s t p h i l o s o p h e r s have thought they had found a m e t h o d f o r to its pursuit.

this and have devoted c o n s i d e r a b l e efforts

In the last two centuries the most influential of them

has doubtless been Kant, who made the followina simple yet a l l u r i n g discovery:

if one can gain an insight into the p r e r e q u i s i t e s of scientific

k n o w l e d g e one will thereby obtain an insight into its subject-matter, for, as he put it,

"die B e d i n g u n g e n der M 6 g l i c h k e i t der E r f a h r u n g Hber-

haupt sind zugleich B e d i n a u n g e n der M ~ g l i c h k e i t der G e g e n s t ~ n d e der Erfahrung. "36

He b e l i e v e d he could claim, on these grounds,

dean g e o m e t r y and N e w t o n i a n chronometry, change, mass conservation,

that Eucli-

the c o n t i n u i t y of q u a l i t a t i v e

causal d e t e r m i n i s m and the e q u a l i t y of action

and r e a c t i o n b e t w e e n distant bodies, were p e r m a n e n t features of physical science.

As is well known, not one Of them survived the advent of Rela-

tivity and the Quantum.

But Kant's failure in the a p p l i c a t i o n of his

m e t h o d does not invalidate the m e t h o d itself, and arguments

in Kantian

style are still used for v i n d i c a t i n g basic scientific assumptions.

I

have already m e n t i o n e d Bondi and Gold's j u s t i f i c a t i o n of the Perfect C o s m o l o g i c a l Principle. Ellis'

treatise,

The following example is taken from Hawking and

The Large

Scale

p o s s i b i l i t y that a relativistic curves - i.e.

Structure

of Space-Time.

D i s c u s s i n g the

spacetime might contain closed timelike

timelike curves h o m e o m o r p h i c to a circle - the authors

write:

However the existence of such curves would seem to lead to the possibility of logical paradoxes: for, one could imagine that with a suitable rocketship one could travel round such a curve and, arriving back before one's departure, one could preventoneself from setting out in the first place. Of course there is a contradiction only if one assumes a simple notion of free will; but this is not something which can be dropped lightly since the whole of our philosophy of science is based on the assumption that one is free to perform any experiment. 37 Hawking and Ellis allude to H. Schmidt's "Model of an o s c i l l a t i n g c o s m o s 38 that r e j u v i n a t e s during contraction", which contains closed timeiike curves and in w h i c h the concept of free will is modified, but conclude that

One would be much more ready to believe that space-time satisfies what we shall call the chronology condition: namely, that there are no closed timelike curves.39 The c h r o n o l o g y c o n d i t i o n is one of the h y p o t h e s e s of an important t h e o r e m on the existence of s i n g u l a r i t i e s in spacetime, Penrose in 1970. 40

d i s c o v e r e d by Hawking and

Now, though it is c e r t a i n l y e n c o u r a g i n g to hear two

d i s t i n g u i s h e d scientists argue,

in a strictly scientific argument,

from

331

the premise that f r e e d o m is a p r e r e q u i s i t e of science, one must also bear in m i n d that,

as a matter of fact, one is not free to p e r f o r m any experi-

ment a n y w h e r e at any time.

Indeed, physics must c o n s t a n t l y extrapolate

from our t e r r e s t r i a l laboratories to spacetime regions where free experim e n t a t i o n is p h y s i c a l l y impossible.

Consequently,

Hawking and Ellis'

ar-

gument must imply either that science is illusory or that its possibility requires only that some

timelike curves be open,

and that they include

the w o r l d l i n e s of the m a t e r i a l s out of which scientists and scientific instruments are made.

(By the way, the compactness of some

timelike

curves is not more inimical to freedom than the e x i s t e n c e of a Cauchy surface in spacetime, which H a w k i n g and Ellis do not seem to be w i l l i n g to exclude a p r i o r i ,

though they readily admit that "there does not seem

to be any p h y s i c a l l y c o m p e l l i n g reason for b e l i e v i n g that the universe 41 admits" one.) The K a n t i a n m e t h o d of p h i l o s o p h i c a l inquiry into the p r e r e q u i s i t e s of science can also be said to inspire the current German school of protophysics,

but a d i s c u s s i o n of their e n d e a v o u r s would be

somewhat out of place here,

for not only do they show little sympathy

with all that R e l a t i v i t y p h i l o s o p h i c a l l y stands for, but they tend to 42 dismiss c o s m o l o g i c a l theories as m e t a p h y s i c a l fancies. A c o m p l e t e l y d i f f e r e n t a p p r o a c h to the subject of p h i l o s o p h i c a l was i n i t i a t e d by Edmund Husserl about P h e n o m e n o l o g i c a l Movement.

tually exist or not.

1900, giving rise to the so-called

Husserl m a i n t a i n e d that we can grasp the fea-

tures that make up the "essence" ly of w h e t h e r the entities

insight

(Wesen)

of phenomena,

quite independent-

seemingly m a n i f e s t e d by those p h e n o m e n a ac-

Such "grasp of essences"

(Wesenschau)

can be at-

tained by freely v a r y i n g in i m a g i n a t i o n the p h e n o m e n a actually experienced, the "essential"

traits being p r e c i s e l y those that remain invari-

ant under free variation.

Husserl's a p p r o a c h exerted a c o n s i d e r a b l e in-

fluence on the German and Central E u r o p e a n G e i s t e s w i s s e n s c h a f t e n

after

the First W o r l d War and is now in vogue in A m e r i c a n psychology,

but its

effect on the p h i l o s o p h y of physics has been negligible. Nevertheless, it is u n d e n i a b l e that a flair for such constant features of phenomena, for what Husserl called

"den i n v a r i a n t e n a l l g e m e i n e n Stil,

in dem [unsere

e m p i r i s c h a n s c h a u l i c h e W e l t ] im Str~men der totalen E r f a h r u n g v e r h a r r t " , 4 3 has always been a c r i t i c a l l y important ingredient of scientific genius. S c i e n t i s t s who,

"haben gar kein GefOhl fur die in44 nere W a h r s c h e i n l i c h k e i t einer Theorie", have seldom made decisive contributions

in Max Born's words,

to natural philosophy.

And e v i d e n t l y such a feeling for

the intrinsic l i k e l i h o o d of theories cannot grow only out of a t h o r o u g h a c q u a i n t a n c e with a v a i l a b l e m a t h e m a t i c a l devices and e x p e r i m e n t a l results.

Let me give one example of what I take to be Husserl's meaning.

332 It is u s u a l l y

assumed

that the spacetime

relativistic

world-model

the m a n i f o l d

must

vectors. vation

every event E

(I note

in the that

with

the latter,

described

unless

that

never

types of

discerned

~.~.,

when

readily

assume

neighbourhood

that each p o i n t - e v e n t that

A, B, C,...

is c o n t r a c t i b l e

we ought

speculations

involving multiply

if these

speculations

do not look promising,

simply

discard

they defy our sense of plausibility. tion and e v e n t u a l lesson

that

our most

acceptance

is not easily

cherished

out to agree with

connected

spacetimes. 45

impossible

the early

Relativity

the nature

intuitions

have

of things

much b e t t e r

with

Even and

just b e c a u s e recep-

taught

a theory can c o n t r a d i c t

concerning

our genuine

has a sizeable

On such matters,

of E i n s t e i n ' s

forgotten:

beliefs

to be wary of rely-

they are not absurd,

them as p h y s i c a l l y

so

and never

if we were not a c q u a i n t e d

p hysical

we may not

always h a p p e n

say that all of us w o u l d

in the u n i v e r s e

to it,

phe-

for it c o n s i s t s

is to its past,

I dare

must

cannot even be

for granted,

However,

to the past

of spacetime

flow,

that

the events

of some physical

by heat

A

ing too freely on our grasp of essences.

necessity

from the events

is taken

order.)

such a sweeping

inside w h i c h

time a s y m m e t r y

of events, B

field of t i m e l i k e

authorize

the t i m e - o r i e n t a b i l i t y

time-orientation

in the inverse

smooth

viable spoken:

does not rest on the obser-

by a n e i g h b o u r h o o d

the s o - c a l l e d

is to the future

take place

of a p h y s i c a l l y Mathema£ically

grasp of the a p p a r e n t

as illustrated,

fact that certain C

w h i c h Could

of E can be neatly in passing

not be c o n f u s e d nomena;

facts,

but on an intuitive be s u r r o u n d e d

to the future

defined

say that this a s s u m p t i o n

of p a r t i c u l a r

generalization,

of E

admit an e v e r y w h e r e

I should

manifold

must be t i m e - o r i e nt a b l e .

us a

some of

and yet turn

than the theory

it

supersedes. Summing

up:

neither

philosophical lighted

insights,

nor H u s s e r l ' s

though

on a true p r e r e q u i s i t y

effectively

grasped

hold his d i s c o v e r y pears

Kant's

to be, not their

framework

it often happens

feature

for science.

can provide

knowledge,

of the world, The

which

safest

them in a m a t h e m a t i c a l

our d i s c u s s i o n

or that he has he will c e r t a i n l y

test of insights

can be deceptive,

for the u n d e r s t a n d i n g

philosophy,

foolproof

if one is sure that he has

of scientific

self-evidence,

possibility of a r t i c u l a t i n g a satisfactory

of course,

an invariant binding

method

theory

that yields

of facts.

In the end,

has not r e v e a l e d

ap-

but the

us a n y t h i n g

that we did not k n o w b e f o r e h a n d ,

but has only c l a r i f i e d

what we had

always

known:

on insightful

for their co-

herent

formulation,

while

facts

depend

insights

depend

theories

on t h e o r e t i c a l l y

the c o r r o b o r a t i o n

of their adequacy.

Mathematical

prompting,

the web that holds

insights

weaves

conceived

theorizing,

and facts

as

facts

for

at their

together.

333

Notes I.

M. Born, Experiment and Theory in Physics, Cambridge: versity Press, 1943.

2.

A. Einstein, H. und M. Born, Briefwechsel 1916-1955, Mfinchen: Nymphenburger Verlagshandlung, 1969, p. 203. I thank the Estate of Albert Einstein for allowing me to reproduce the above and the remaining passages of Einstein's writings quoted in this paper.

3.

Einstein-Born Briefwechsel,

4.

See, for example, N. Maxwell, "The rationality Of scientific discovery", Phi. Sci., 41 (1974) 123, 247; J. Agassi, Science in Flux, Dordrecht: Reidel, 1975; L.Laudan, Progress and its Problems, Berkeley: University of California Press, 1977.

5.

E. A. Milne, Modern Cosmology and the Christian Idea of God, Oxford: Clarendon Press, 1952, p. 50.

6.

E. A. Milne, Kinematic Relativity, p. 39.

7.

E. A. Milne, Modern Cosmology and the Christian Idea of God, p. 61, compare, however, Kinematic Relativity, p. 93.

8.

E.A.

9.

H. Bondi and T. Gold "The Steady-State Theory of the Expanding Universe", Mort. Not. R. Astron. Soc., Cambridge: at the University Press, 1960, Chapter XII.

10.

H. Reichenbach, The Theory of Probability, California Press, 1949, paragraph 91.

11.

M. Bunge,

12.

Compare the sympathetic stance towards the Steady State Theory taken in J. Merleau-Ponty's Cosmologie du XXe Siecle, Paris: Gallimard, 1965.

13.

A. Einstein,

l.c., p. 204.

Milne, Kinematic Relativity,

"Cosmology and Magic",

"Prinzipielles

at the Uni-

Oxford: Clarendon Press,

1948,

p. 64.

Berkeley:

University of

The Monist 47 (1962), 116.

zur allgemeinen Relativit~tstheorie",

Annalen der Physik, 55 (1918) 241. Cf. E. Kretschmann, "Uber den physikalischen Sinn der Relativit~tspostulate", Annalen der Physik, 53 (1917) 575. 14.

G. Holton, Thematic Origins of Scientific Thought, Cambridge, Mass.: Harvard University Press, 1973, Introduction and Chapter I; The Scientific Imagination, London: Cambridge University Press, 1978, Chapter I.

15.

J. A. Wheeler, "Mach's Principle as Boundary Condition for Einstein Equations", in Chiu and Hoffmann, eds, Gravitation and Relativity, New York: W. A. Benjamin, 1964, pp. 303 - 349.

334

16.

Compare A. Einstein, "~ber das Relativit~tsprinzip und die aus demselben gezogenen Folgerungen", Jahrb. Radioakt. u. Elektronik, 4 (1907), p. 454; "Zum gegenw~rtigen Stande des Gravitationsproblems", Phys. Ztschr. 14 (1913), pp. 1254 f., 1260 f., 1265; A. Einstein and M. Grossmann, Entwurf einer verallgemei-

nerten Relativit~tstheorie Leipzig: Teubner, Einstein in Phys.

und einer Theorie der Gravitation,

1913, p. 6. See also Mach's obituary, Ztschr., 17 (1916) 101.

by

17.

The Leibniz-Clarke Correspondence,

18.

E. Mach, Die Mechanik, Darmstadt: schaft, 1963, pp. 216 ff.

19.

A. Einstein, Annalen der Physik,

20.

I. Newton, Philosophiae Naturalis Principia Mathematica, Cambridge, Mass.: Harvard University Press, 1972, vol. I, p. 51.

21.

E. Cartan, "Sur les vari~t~s ~ connexion affine et la th~orie de la relativit~ g~n~ralis~e", Ann. Ec. Norm. Sup. 40 (1923) 325; 41 (1924) 1.

22.

"It is contrary to the mode of thinking in science to conceive of a thing (the space-time continuum) which acts itself, but which cannot be acted upon". A. Einstein, The Meaning of Relativity, Princeton: Princeton University Press, 1956, pp. 55 f.

23.

See, e.g., S. Weinberg, Gravitation and Cosmology New York: Wiley, 1972, pp. 133-135, 151-155; D. Lovelock, "The Einstein tensor and its generalizations", J. math. physics, 12 (1971) 498.

24.

A. Einstein,

edited ... by H. G. Alexander, Manchester: Manchester University Press, 1956. Wissenschaftliche 55

Buchgesell-

(1918), p. 243, footnote.

"Prinzipielles zur allgemeinen Relativit~tstheorie", 55 (1918), p. 243; W. de Sitter, "On Einstein's Theory of Gravitation and its astronomical consequences", Monthly Not. R. Astr. Soc., 78 (1917) 3.

Annalen der Physik,

25.

Quoted in G. Holton,

"Mach, Einstein,

and the Search for Reality",

Boston Studies in the Philosophy of Science, VI, p. 194, n. 24. 26.

Einstein-Bo~n Briefwechsel

27.

F. M. Cornford,

(quoted in note 2), p. 258.

"The Invention of Space",

Gilbert Murray, London: Allen & Unwin,

in Essays in Honour of 1936, pp. 215-235.

28.

M. Bunge, Treatise on Basic Philosophy, recht: Reidel 1977.

29.

B. Riemann, "Uber die Hypothesen, welche der Geometrie zugrunde liegen", Abh. K. Ges. d. Wiss. zu G~ttingen, Bd. 13 (1867).

30.

I. Kant, Allgemeine Naturgeschichte und Theorie des Himmels, K~nigsberg/Leipzig: J. F. Petersen, 1755.

31.

I. Kant, Kritik der reinen Vernunft, pp. 424 ff., 517 ff., etc.

32.

A. Einstein, "Kosmologische Betrachtungen zur Allgemeinen Relativit~tstheorie" Preuss. Ak. d. Wiss., Sitzber. 1917, Pt. I, pp. 142-152. Cf. Riemann, "Uber die Hypothesen...", p. 22.

vol. 3, Ontology I, Dord-

Riga: J. F. Hartknoch,

1781,

335

33.

B. G. Schmidt, "A new definition of singular points in General Relativity", J. @en. Rel. and Gray. 1 (1971) 269. Cf. S. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge: at the University Press, 1973, p. 259.

34.

Die philosop~ischen Schriften yon G. W. Leibniz, hrg. von C. J. Gerhardt,

Hildesheim:

Olms 1965, vol. 7, p. 302.

35.

The reason for this interpretation is clearly explained by Max Jammer in his Concepts of Mass in classical and modern physics, Cambridge Mass.: Harvard University Press, 1961, pp. 205 f.

36.

Kant, Kritik der reinen Vernunft,

37.

S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge: at the University Press, 1973, p. 189; my italics.

38.

H. Schmidt in J. Math. Phys.,

39.

Hawking and Ellis,

40.

Hawking and Ellis, l.c., p. 266, cf. S. W. Hawking and R. Penrose, "The singularities of gravitational collapse and cosmology", Proc. Roy. Soc. London, A 314 (1970) 529.

41.

Hawking and Ellis,

42.

See, e.g., P. Lorenzen, "Die allgemeine Relativ±t~tstheorie als eine Revision der Newtonschen Gravitationstheorie" Philos. Naturalis, 17 (1978), p. 9. On protophysics see G. B~hme (ed.), Protophysik Frankfurt: Suhrkamp, 1976.

43.

E. Husserl,

44.

Born an Einstein,

45.

J. A. Wheeler, Geometrodynamics, New York: Academic Press, 1962. Cf. R. W. Fuller and J. A. Wheeler, "Causality and multiply connected space-time", Phys. Rev., 128 (1962) 919-929.

(1781), p. 158.

7 (1966)

494.

l.c., p. 189.

1.c., p. 206.

Die Krisis der europdischen Wissenschaften und die transzendentale Phdnomenologie, Haag: Nijhoff, 1954, p. 29. 25. 8. 1923; Briefwechsel,

l.c., p. 117.