Table Of ContentPROTOPHYSICS OF TIME
BOSTON STUDIES IN THE PHILOSOPHY OF SCIENCE
EDITED BY ROBERT S. COHEN AND MARX W. WARTOFSKY
VOLUME 30
PETER JANICH
Philipp$· Univmitiit Marburg,
Institut fUr Philosophie, Marburg, F.R.G.
PROTOPHYSICS
OF TIME
Constructive Foundation and History
of Time Measurement
D. REIDEL PUBLISIDNG COMPANY
A MEMBER OF THE KLUWER ..A CADEMIC PUBLISHERS GROUP
DORDRECHT/BOSTON/LANCASTER
library of Congress Cataloging in Publication Data
DATA APPEAR ON SEPARATE CARD
LC no. 85-10768
ISBN-13: 978-94-010-8794-0 e-ISBN-13: 978-94-009-5189-1
DOl: 10.1007/978-94-009-5189-1
Published by D. Reidel Publishing Company,
P.O. Box 17,3300 AA Dordrecht, Holland.
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Translation by AM Linguistics - Robert Brown
All Rights Reserved
© 1985 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1 st edition 1985
1st German edition published in 1969 by Bibliographisches Institut AG, Mannheim
2nd German edition published in 1980 by Suhrkamp Verlag, Frankfurt am Main
© to both German editions: Peter Janich
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical
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retrieval system, without written permission from the copyright owner
TABLE OF CONTENTS
Editorial Preface vii
Constructive and Axiomatic Method. An Essay by P. Lorenzen ix
Preface to the Second Edition xxi
Preface to the Third Edition xxv
CHAPTER I I ON THE PROBLEM OF CHRONOMETRY IN THE
PRESENT-DAY THEORY OF SCIENCE
1. Introduction: Establishment of a Reference to Known Positions
in the Theory of Science 1
2. Affirmative Theory of Science and the Language of Physics
2.a. M. Bunge's Affirmative Protophysics 13
3. The Affirmative Theory of Measurement 14
4. Affirmative Explanations of the Choice of the Time Standard 23
CHAPTER II I ON THE METHOD OF PHYSICS
1. Preliminary Remarks 30
2. Method as a Validity Criterion. On the Foundational Theory of
Hugo Dingler 30
2.1. H. Dingler and Protophysics 30
2.2. H. Dingler's Foundational Theory 35
3. Logic and Protophysics. On the Foundational Theory of Paul
Lorenzen 51
4. On the Method of Physics 62
4.1. Physics, Natural or Experimental Science? 62
4.2. The Claim to Scientific Nature (Wissenschaftlichkeit) 66
4.3. Methodology of Measurement 69
5. On the Criticism of Protophysics 82
CHAPTER III I CHRONOMETRY
1. What Purpose Shall Time-Measurement Serve? 87
vi TABLE OF CONTENTS
2. Moved Bodies 101
3. Comparisons of Motion 108
3.1. Theorems on Similarity 113
3.2. Continuity of Changes of Velocity 124
4. Forms of Motion 135
4.1. Introduction: Objections to Periodicity as a Protophysical
Basic Concept 135
4.2. Uniform Motion 142
4.3. The Uniqueness of Clock Definition 153
4.4. Outlook and Consequences 160
CHAPTER IV / ON A HISTORY OF CHRONOMETRY
1. Preliminary Remarks: Terminological Distinction of Practical
and Theoretical Chronometry 165
2. The Development of Chronology 166
3. Short History of the Water Clock 168
4. Short History of Mechanical Escapement Clocks 170
5. The Principles of Clock Construction 184
6. Time Theories 187
6.1. Aristotle's Theory of Time 187
6.2. Augustine's Theory of Time 199
6.3. Transition to Classical Physics 210
Notes 214
References 225
Name Index 231
Subject Index 234
EDITORIAL PREFACE
For protophysics, the fascinating and impressive constructive re-establish
ment of the foundations of science by Professor Paul Lorenzen, working
with his colleagues and students of the Erlangen School, no task is more
central than to.furmulate a theoretical understanding of the practical art of
measurement of time. We are pleased, therefore, to have a new third edition
of Peter Janich's masterful monograph on the protophysics of time, available
in this English translation within the Boston Studies. We also look forward
to the Boston University Symposium on protophysics in april of this year
within which the full program of protophysics will be critically examined by
German and American physicists and philosophers, supporters and critics.
We are also grateful to Paul Lorenzen for contributing his powerful
instructive essay on the 'axiomatic and constructive method' which intro
duces this book.
March 1985
ROBERT S. COHEN
Center for the Philosophy and History of Science
Boston University
MARX W. WARTOFSKY
Department of Philosophy
Barnch College
City University of New York
vii
PAUL LORENZEN
CONSTRUCTIVE AND AXIOM A TIC METHOD
Mathematics is like a big building with many apartments. We have at least
Arithmetic and Analysis, Algebra and Topology - and we have Geometry
and Probability-Theory. Very often the tenants of these different apartments
seem not to understand each other.
The Bourbaki movement promised a new unity of Mathematics by admit
ting only the axiomatic method of Hilbert as genuine mathematical.
This movement was quite successful - with one exception: the axiomatic
foundations of Set Theory remained obscure.
But ignoring this difficulty which the mathematicians are accustomed to
live with since about 1900, the axiomatic method leaves the mathematician
without foundational problems. There are none in Arithmetic, because there
are no non-Peano Arithmetics. There are none in Geometry, because the
mathematicians believe that the physicists have the means of finding out
whether the 'real space' is Euclidean or non-Euclidean. Finally there are no
foundational problems in probability theory, because there are no non
Kolmogorow Probability-theories. In contrast to this happy state of "no
problems by ignoring them", Constructivism tries to solve the foundational
problems. It tries to justify, why mathematicians all over the world accept
the Peano-axioms for Arithmetic and the Kolmogorow-axioms for Probability
Theory. In the other cases Constructivism tries to find out, which axioms -
if any - should we accept for Set-theory, for Geometry. The last question
concerning Geometry would lead us away from Mathematics proper, so I
shall go into this problem only later. But I would like to remark now that
the whole foundational problem of Mathematics changes its outlook, if one
begins to doubt the story that empirical physics has the means of deciding
the case of Euclidean vs. Non-Euclidean Geometry. I would like to suggest,
that you imagine a state of Mathematics without Geometry. We would be
left with Numbers, Sets and Probabilities. We would have to develop Theories
about these Entities without even being able to appeal to the paradigm of
Geometry.
What would this look like?
Whereas the Axiomaticists, e.g., the Bourbakists, ignore all attempts to
ix
x PAUL LORENZEN
construct a foundation for the mathematical theories (or despise them even
as 'pre-scientific') the Constructivists try to justify such axioms as those from
Peano or Kolmogorow. The constructive method pleads for a cooperation
of constructions and axiom-systems.
This cooperation is well-accepted in the case of naive Arithmetic and
Analysis. 'Axiomatics' means here to define 'structures' as Groups, Lattices,
Compact Spaces, Measure Fields.
The structures are defined by systems of sentence-forms, called axioms.
The definition of such a structure is justified by showing that there are
important models, which satisfy the axioms. The models, taken from naive
Arithmetic or Analysis have here the priority. Only if they are important
are the axioms accepted.
The language of the axioms may be restricted to pure logic, as e.g., in
the case of group-theory or lattice-theory. But already a structure like 'arch
imedean ordered group' or the structures of 'topological spaces' use arithmet
ical or set-theoretical vocabulary.
This causes no difficulties because here the axiomatic method is applied
within Mathematics. I shall call this internal axiomatics. The difficulties
begin, if we now turn to the foundations. That is: we now no longer take
for granted that such entities as natural or real numbers, and such entities
as sets or functions are somehow 'given', are somehow already at our disposal.
As if they were like flowers in a garden, which we have only to give names to
- and then may begin to find out truths about them.
In Arithmetic the foundational problem is no serious problem. No math
ematician seriously denies that it is easy to count, e.g., with such primitive
numerals as I, II, III, ... Everyone even understands the construction-rules
for such numerals:
~I
n ~nI
From such rules, to which may be added further rules for the construction
of pairs, e.g.,
I, nI
m, n ~mI. nI
(the constructible pairs are those with m < n) and for the construction of
triples as for addition, an easy way leads to some first true assertions, e.g.,
CONSTRUCTIVE AND AXIOMATIC METHOD xi
I <nI
m <n -+mI <nI
This last sentence presupposes the logic of the conditional-+. No one seriously
<
denies I n I, if he accepts negation I at all. Look, this is not a matter
of choosing 'axioms'. If someone would propose, e.g., In < II instead, he
<
would be refuted that very moment because I II is true.
Also, e.g., m < n -+ mI < n would be ridiculous. Because I < II is true, but
<
II II is not.
For such arguments we do not need formal logic; but we have to know
<
how to argue with meaningful sentences (here of the form m n) which are
composed by logical particles such as -+ and L 'How to argue' means the
dialogical rules for the logical particles.
Those rules, e.g.,
I
A-+B
A? I B
for the conditional-+ have been developed within contructive logic.
The controversy between the classical and intuitionist logicians, e.g.,
whether (IA -+ IB) -+ (B -+ A) is logically true (or only the converse),
turns out to be an empty struggle. You have to justify in any case the choice
of a general rule for dialogues. For intuitionist and for classical logic this can
be done by proving a corresponding Gentzen cut-theorem. Classical logic
turns out to be a convenient simplification of intuistionist logic - in the case
of junctors only you have just to omit all adjunctions V and conditionals -+
(and to redefine A V B by I (lA " IB) and A -+ B by I A V B).
The Peano-axioms turn out to be true sentences in constructive Arithmetic.
The w-incompleteness, proved by GOdel, shows that not all constructively
true sentences are logically derivable from the axioms. This is little wonder.
Ax
A universal sentence A(x) is constructively true, if A(n) is true for all n.
But in order to derive Ax A(x) logically you have to derive first A(x) with a
free variable. So one should have expected the w-incompleteness. But the
Peano-Arithmetic is w-complete, if you exclude multiplication. The point
of Godel's proof was to show that the Peano-Arithmetic with addition and
multiplication only - without higher forms of inductive definitions - already
exemplifies the w-completeness, which was to be expected in general.
The quarrel between the axiomatic and the constructive method in Arith
metic is merely a verbal one. As the Peano-axioms are constructively true
