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.
