LOGIC AND PROBABILITY IN QUANTUM MECHANICS

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SYNTHESE LIBRARY
MONOGRAPHS ON EPISTEMOLOGY,
LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE,
SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE,
AND ON THE MATHEMATICAL METHODS OF
SOCIAL AND BEHAVIORAL SCIENCES

Managing Editor:

J AAKKO HINTIKKA, Academy of Finland and Stanford University

Editors:
ROBERT S. COHEN,
DONALD DAVIDSON,

Boston University

Rockefeller University and Princeton University

GABRIEL NUCHELMANS,
WESLEY

C.

SALMON,

University of Leyden

University of Arizona

VOLUME 78

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LOGIC AND
PROBABILITY IN
QUANTUM MECHANICS
Edited by

PATRICK SUPPES
Stanford University

SPRlNGER-SCIENCE+BUSINESS MEDIA, B.V.

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Library of Congress Cataloging in Publication Data
Main entry under title:
Logic and probability in quantum mechanics.
(Synthese library ; v. 78)
Bibliography : p.
Includes index.
I. Quantum theory- Addresses, essays, lectures.

2. Physics-Philosophy-Addresses, essays, lectures.
I. Suppes, Patrick Colonel, 1922530.1 '2
75- 30834
QC174.l25.L62
ISBN 978-90-277-1200-4
ISBN 978-94-010-9466-5 (eBook)
DOI 10.1007/978-94-010-9466-5

All Rights Reserved
Copyright © 1976 by Springer Science+Business Media Dordrecht
Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1976
Softcover reprint of the hardcover 1st edition 1976
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,
including photocopying, recording or by any informational storage and
retrieval system, without written permission from the copyright owner

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PREFACE

During the academic years 1972-1973 and 1973-1974, an intensive seminar on the foundations of quantum mechanics met at Stanford on a
regular basis. The extensive exploration of ideas in the seminar led to
the org~ization of a double issue of Synthese concerned with the
foundations of quantum mechanics, especially with the role of logic and
probability in quantum meChanics. About half of the articles in the
volume grew out of this seminar. The remaining articles have been solicited explicitly from individuals who are actively working in the foundations of quantum mechanics.
Seventeen of the twenty-one articles appeared in Volume 29 of Synthese. Four additional articles and a bibliography on -the history and
philosophy of quantum mechanics have been added to the present

volume. In particular, the articles by Bub, Demopoulos, and Lande, as
well as the second article by Zanotti and myself, appear for the first time
in the present volume.
In preparing the articles for publication I am much indebted to Mrs.
Lillian O'Toole, Mrs. Dianne Kanerva, and Mrs. Marguerite Shaw, for
their extensive assistance.
PA TRICK SUPPES

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TABLE OF CONTENTS

PREFACE

V

INTRODUCTION

IX

PART 1/ LOGIC
A Notion of Mechanistic Theory
3
Essai sur la logique de l'indeterminisme et la
19
ramification de l'espace-temps
HILARY PUTNAM / How to Think Quantum-Logically
47
GAR Y M. HARDEGREE / The Conditional in Quantum Logic

55
D. J. FOULIS and c. H. RANDALL / Empirical Logic and Quantum
Mechanics
73
RICHARD J. GREECHIE / Some Results from the Combinatorial
105
Approach to Quantum Logic

G. KREISEL /

ROLAND FRAisSE /

PART II / PROBABILITY
The Quantum Probability Calculus
The Probability Structure of QuantumMechanical Systems
TERRENCE L. FINE / Towards a Revised Probabilistic Basis for
Quantum Mechanics
TED BASTIN / Probability in a Discrete Model of Particles and Observations
NANCY DELANEY CARTWRIGHT / Superposition and Macroscopic Observation
0ISTEIN BJ0RNESTAD / A Note on the So-Called Yes-No Experiments and the Foundations of Quantum Mechanics
J. M. JAUCH /

123

ZOL T AN DOMOTOR /

147
179
195
221

235

PART III/COMPLETENESS
AR THUR FINE /

On the Completeness of Quantum Theory

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249


VIII

TABLE OF CONTENTS

BAS C. VAN FRAASSEN / The Einstein-Podolsky-Rosen Paradox
PA TRICK SUPPES and MARIO ZANOTTI/Stochastic Incompleteness of Quantum Mechanics
ROBERT W. LATZER / Errors in the No Hidden Variable Proof of
Kochen and Specker
DAVID J. ROSS / Operator-Observable Correspondence
JEFFREY BUB / Randomness and Locality in Quantum Mechanics
WILLIAM DEMOPOULOS / Fundamental Statistical Theories
ALFRED LANDE / Why the World Is a Quantum World
PA TRICK SUPPES and MARIO ZANOTTI / On the Determinism of
Hidden Variable Theories with Strict Correlation and Conditional Statistical Independence of Observables

283
303
323

365
397
421
433

445

BIBLIOGRAPHY ON THE HIS TOR Y AND PHILOSOPHY OF QUANTUM PHYSICS: Compiled by Donald Richard Nilson
457
INDEX OF NAMES

521

INDEX OF SUBJECTS

529

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INTRODUCTION

The philosophy of physics has occupied an important place in philosophy
since ancient times, and a wide spectrum of philosophers know something
about the historical development of the fundamental concepts of space,
time, matter, and motion. On the other hand, to many philosophers the
problems that are discussed in the foundations of quantum mechanics
seem specialized and esoteric in relation to the classical tradition in the
philosophy of physics, and the relevance of analysis of the basic concepts
of quantum mechanics to general philosophy seems restricted.

The problems raised by this issue of relevance warrant further examination. On the one hand, the case is overwhelming that quantum mechanics is the most important scientific theory of the twentieth century.
It is hard to believe that the new and surprising concepts that have arisen
in the theory are not of major importance to philosophy and our fundamental conception of the world we live in. Yet the philosophical literature
dealing specifically with quantum mechanics is, like the literature of
physics on the theory, difficult and technical. It is admittedly no easy
matter for an outsider not specifically concerned with the philosophical
issues raised by quantum mechanics to get an overview of the subject
and to be able to appreciate the general philosophical significance of the
conceptual analyses made by a variety of philosophers, physicists, and
mathematicians. I also hasten to add that the present volume does not
in any sense fill this gap. It is meant to be a contribution to the continuing
relatively specific and relatively technical discussion of the philosophical
foundations of quantum mechanics.
The twenty-one articles included in this volume cover many topics and
issues, but I have simplified the range of issues and concepts covered in
order to organize them in three parts. Each of the three parts is meant
to represent a group of closely related topics pertinent not only to the
general philosophy of science but to epistemology and metaphysics as
well.
Part I concerns logical issues raised by quantum mechanics. The first

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x

INTRODUCTION

article, by Kreisel, is of a general nature and assumes no specific knowledge of quantum mechanics. Kreisel raises the philosophically interesting
question of whether quantum mechanics will lead to yet another surprise

in that it is in an essential sense a nonmechanistic theory. Here nonmechanistic means having nonrecursive solutions to differential equations
describing fundamental natural processes. As Kreisel points out, the
issue must be stated with some care and it is the sense of his article to
make this care explicit, for classical mechanics is meant to hold for arbitrary nonrecursive measures of distances, masses, and forces. In the
classical case, however, the usual rational approximations have a recursive or mechanistic character in most of the applications of apparent
interest. The important question that he raises is whether this is true of
quantum mechanics. Kreisel also makes clear the kind of problem in
classical mechanics which may be nonrecursive in character.
The second article by Fraisse raises general issues about the logic of
indeterminism and the extent to which the fundamental results of quantum mechanics force a change in our classical conception oflogic. Fraisse
is especially concerned to examine the philosophical consequences of
Everett's bold hypothesis about the ramification of space-time or what
is sometimes called the many-universes interpretation of quantum mechanics. His purpose is to examine the concept of indete~inism that
results from Everett's view with a minimum of dependence on technical
details of quantum mechanics.
The epistemological status of the laws of classical logic has been besieged by more than one sustained attack in the last hundred years. The
rejection of the law of excluded middle by intuitionistic philosophers of
mathematics is probably the most salient example. The striking and distinguishing feature of the attack that has been launched from a quantum
mechanical base is that it is an attack that rests upon an empirical scientific theory of an advanced and complicated nature. That a challenge
to classical logic could arise from highly specialized empirical concepts
in physics dealing with the motion of very small particles seems to run
counter to almost all the epistemological tradition in logic from Aristotle
to Frege - by the 'epistemological tradition' I mean of course the philosophical analysis of the grounds for accepting a law of logic as valid.
The third article, by Putnam, deals most directly with the quantum
mechanical challenge to the classical epistemological tradition that de-

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INTRODUCTION


XI

fends logic as a collection of a priori truths. The following article, by
Hardegree, deals with the way we may formulate conditional sentences
or propositions in quantum logic. Hardegree is especially concerned with
the philosophical controversy concerning the possibility of introducing
a reasonable notion of implication in quantum logic. As opposed to the
expressed views of Jauch, Piron, Greechie, and Gudder. Hardegree argues
that the standard quantum logic as represented by the lattice of subspaces
of a separable Hilbert space, does in fact admit an operation possessing
the most essential properties of a material conditional. The conditional
that Hardegree proposes is close to a Stalnaker conditional; it does not
satisfy the laws of transivity or contraposition but it does satisfy modus
ponens. To some extent, therefore, the differences with Jauch and the
other authors mentioned above depend upon what one regards as essential properties of an operation of implication.
In the fifth article, Foulis and Randall continue the development of
empirical logic and apply it to quantum mechanics, relating at the same
time their developments to some of the other technical papers in quantum
logic. The next article, by Greechie, is concerned with some specific
problems in quantum logic, especially with a problem posed by Jauch
in his article in Part II on the quantum probability calculus. The articles
by Foulis and Randall, and by Greechie, illustrate the extent to which
the subject matter of quantum logic rapidly becomes a technical topic in
its own right. Not only the character of quantum mechanics itself but
also the mathematical level of contemporary work in logic make it hardly
surprising that new specific results in quantum logic will necessarily be
embodied in a framework of relatively new mathematical concepts.
Closely following on questions about the nature of logic in quantum
mechanics are a series of questions about probability in quantum mechanics. A case can be made for the claim that quantum mechanics is as

disturbing to the classical concepts of probability as it is to the classical
concepts of logic. The six articles I have placed in Part II are concerned
with various aspects of probability in quantum mechanics. The first article of this part, by Jauch, gives an excellent general review of probability
concepts in the context of quantum mechanics and makes clear the issues
about probability central to quantum mechanics. The second article, by
Domotor, provides a general analysis of probability structures that occur
in quantum mechanics. Domotor's principal aim is to present a repre-

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XII

INTRODUCTION

sentation of quantum logics, in particular, orthomodular, partially ordered sets, by means of structured families of Boolean algebras. He brings
to his analysis of these structures methods that have been used in the
study of manifolds by geometers. In the third article, Terrence Fine proposes a revised probabilistic basis for quantum mechanics based on his
ideas of the proper approach to qualitative probability. The direction in
which he strikes out in this article is conceptually different from most of
the discussions of the nature of probability in quantum mechanics. Among
the more interesting features of Fine's approach is the examination of
new models of random phenomena that arise from consideration of
qualitative probability and how these new models relate to the quantum
mechanical concept of complementarity.
In the fourth article, Bastin sets forth his ideas about the place of probability in the discrete model that he would use for formulating the fundamental principles of quantum mechanics. Bastin's frontal attack on
continuity assumptions and use of a continuum in quantum mechanics
is perhaps the most salient feature of his approach to foundations. He
replaces the continuum by a discrete model that is hierarchical in cliaracter. Although he is concerned in this article to develop the place of
probability in his approach, he also has a good deal to say about the

kind of hidden-variable theory his approach represents, and for that reason his article also could properly be placed in Part III rather than in Part
II. In the fifth article, Cartwright discusses a number of issues concerned
with the relation between the behavior of microscopic and macroscopic
objects and the pertinent statistical analysis of this relation. She examines
in some detail the attempts to reconcile macroscopic physics and quantum mechanics by reducing superpositions to mixtures. As she puts it,
the philosophical problem is not the replacement of superpositions by
mixtures, but rather to explain why we mistakenly think that a mixture
is called for. In the sixth article, Bjemestad discusses the central place
of yes-no experiments in the conceptual foundations of quantum mechanics. He examines critically the use of such experiments by von
Neumann, Mackey, Piron, and Jauch.
Part III of this volume consists of nine articles organized around the
issues concerning completeness of quantum mechanics. A more general
title would have been hidden-variable theories, but the nine articles are
sufficiently focused on questions of completeness and are not broadly

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INTRODUCTION

XIII

concerned with many of the traditional problems of hidden-variable
theories, so that the more special title of completeness seems appropriate.
There are many different but closely related concepts of completeness
in science and mathematics; for instance, some of the deepest results in
modem logic are concerned with completeness. There is, on the one hand,
the truth-functional completeness of classical sentential logic and Godel's
theorem on the completeness of first-order predicate logic, and, on the
other hand, GOdel's classic results on the incompleteness of arithmetic.

Within quantum mechanics, various senses of completeness can be defined, and controversy continues to exist over both the appropriateness
of definitions and the exact character of the results that obtain for a given
definition. The paradox set forth by Einstein, Podolsky, and Rosen attempted to show that quantum mechanics is not complete in the sense
that additional variables are required for the theory to have the appropriate features of causality and locality. The par~dox arises from measurements made on two particles, for example, a pair of spin one-half
particles that are moving freely in opposite directions. The fact that the
results of measurement on one particle determine the results of measurement on the other particle is taken to violate our ordinary ideas of causality which exclude having instantaneous action at a distance. It is argued
that these paradoxical results require a more complete specification of
the state of a quantum mechanical system.
The ideas surrounding the Einstein-Podolsky-Rosen paradox, as well
as other related paradoxes, are examined in detail in the first article py
Arthur Fine arid in the second by van Fraassen. Fine takes the bull by
the horns and challenges the significance of the. recent work of Bell and
Wigner that yields a solution to the Einstein-Podolsky-Rosen paradox
that, as Bell puts it, Einstein would have liked least. Fine ends up advocating his theory of statistical variables whose joint distributions do
not necessarily exist. Fine interprets the Bell-Wigner arguments to show
that certain arbitrary assumptions on joint distributions cannot be consistently realized or satisfied by any hidden-variable theory. He argues
that his theory of statistical variables provides just the right sort of completeness for quantum mechanics. Even if he has not decisively settled
the many issues raised by the Bell and Wigner work, he has advanced
the argument one more stage in what is sure to be a continuing controversy. Van Fraassen focuses almost entirely on the Einstein-Podolsky-

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XIV

INTRODUCTION

Rosen paradox and attempts to resolve it by the modal interpretation of
quantum mechanics he has been developing in recent articles.
In the third article of this part, Zanotti and I argue for a different kind

of incompleteness of quantum mechanics. We argue that quantum mechanics is stochastically incomplete. We mean by this that, when timedependent phenomena are examined, the predictions of the theory give
only mean probability distributions as a function of time and do not determine a unique stochastic process governing the motion of particles.
To illustrate how a stochastic approach may be applied in quantum mechanics, we examine some of the paradoxical results that may be derived
for the linear harmonic oscillator and explain them in a natural physical
way by looking at the motion of the oscillator as made up of a classical
component together with a random fluctuation.
In the fourth article, Latzer examines in detail the well-known 'hiddenvariable' proof of Kochen and Specker and finds several serious difficulties with their conceptual formulation and mathematical development
of the problem of hidden variables. In the next article, Ross discusses in
detail the operator-observable correspondence in quantum mechanIcs.
His examination of a set of inconsistent axioms that underlie many elementary discussions of quantum mechanics brings into concrete focus
the peculiar problems of operator-observable correspondence that exist
in quantum mechanics and that are often central to discussions of completeness.
The four articles that are included in the present volume and that were
not included in the original issue of Synthese deal essentially with problems relevant to Part III. The article by Bub is directly concerned with
randomness and locality in quantum' mechanics, especially in relation
to hidden-variable theories. Demopoulos examines the sense in which
quantum mechanics can be regarded as a fundamental statistical theory.
He examines Bub's earlier account of completeness of quantum mechanics, which itself assumes knowledge of the earlier work of Kochen and
Specker. The Kochen and Specker work, of course, is examined in great
detail in an earlier article by Latzer in this volume. The next article, by
Lande, summarizes in somewhat different form his well-known views
on the foundations of quantum mechanics. The final article, which is the
second article by Zanotti and me, is concerned to show that any hiddenvariable theory with strict correlation and conditional statistical inde-

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INTRODUCTION

xv


pendence of observables must be deterministic. The central point is that
conditional statistical independence of observables seems to be too
strong a condition to impose on properly stochastic hidden-variable
theories.
The volume closes with an extensive bibliography prepared by Nilson
on the history and philosophy of quantum mechanics.
P A TRICK SUPPES

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PART I

LOGIC

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G. KREISEL

A NOTION OF MECHANISTIC THEORY

I. INTRODUCTION

The notion in question is suggested by the words 'mechanism' or 'machine'. Unlike the usual meaning of 'mechanistic', that is, deterministic
in contrast to probabilistic, the notion here considered distinguishes
among deterministic (and among probabilistic) theories.
The general idea is this. We consider theories, by which we mean such
things as classical or quantum mechanics, and ask if every sequence of

natural numbers or every real number which is well defined (observable)
according to the theory must be recursive or, more generally, recursive
in the data (which, according to the theory, determine the observations
considered). Equivalently, we may ask whether any such sequence of
numbers, etc., can also be generated by an ideal computing or Turing
machine if the data are used as input. (This formulation explains our
terminology 'mechanistic'.) The question is certainly not empty because
most objects considered in a (physical) theory are not computers in the
sense defined by Turing; in fact, so-called analogue computers are not
Turing machines; at best their behavior may be simulated by Turing
machines. They will be, according to theory, if the particular theory of
the behavior of the analogue computers considered happens to be
mechanistic in the sense described above.
.
The stress on the proviso 'according to theory' in the preceding
paragraph is intended as a warning: We are here primarily interested
in a distinction between classes of theories, not classes of phenomena.
The reader should not allow himself to be confused at this stage by
doubts about the validity of a theory with regard to the phenomena
for which it is intended. Naturally such doubts imply doubts about the
relevance (to those phenomena) of any results about the mechanistic
character of the theory. It remains to be seen whether the notion of
mechanistic theory, that is, the division into mechanistic and nonmechanistic theories, will be useful for such physical theories as classical
P. Suppes (ed.). Logic and Probability in Quantum Mechanics. 3-18. All Rights Reserved
Copyright © 1976 by D. Reidel Publishing Company. Dordrecht-Holland

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4


G. KREISEL

or quantum mechanics. It would be a bit odd if it were not since the
distinction is useful in such mathematical theories as geometry and
topology which, after all, are also theoretical idealizations.
Over the last decade, I have published scattered remarks concerning
the question whether existing (physical) theories are mechanistic, with
special emphasis on specific familiar problems which are reasonable
candidates for counterexamples. The most recent publication is Kreisel
(1972) where back references are to be found in the third paragraph on
p. 321. 1 The results are not conclusive, except for showing convincingly
that the subject lends itself to precise formulations. For what it is worth:
I still have the impression that some (of the unsolved) problems of
current physical theories have nonmechanistic solutions.
The purpose of the present note is to discuss a quite specific aspect
of the extension of theoretical knowledge, which is liable to introduce
nonmechanistic elements in a perhaps not altogether trivial way. One of
the most striking features of the whole business of (fundamental) extensions of the sphere of theory consists in this,: constants are calculated
theoretically which were previously obtained by 'empirical' usually'approximate measurement. In particular - and this case seems most relevant
for our purposes - according to theory, some quantities may have to be
integral (multiples); in this case quite rough measurements are sufficient
to fix a precise 'quantized' value. In this situation, a famous principle,
due to Hadamard, which restriemuch of its (restrictive) force. The principle requires that theoretical
relations corresponding to functions, mapping data to (other) observables, must be continuous in the data; this condition becomes empty if
the theoretically permissible values for the data are 'quantized' (discrete)
since then every function is continuous.
Superficially the kind of quantization mentioned would seem to be
most relevant in connection with the quantum theory. But I do not know

enough about the subject to judge with confidence the significance of
(mathematical) examples. So I have used examples from geometry and
celestial mechanics to illustrate the state of affairs described in the last
paragraph. This is done in Section IV. The examples use the Background
Information in Section II, but not the discussion of Section III which is
intended to remove some sources of malaise (concerning the notion of
mechanistic, or rather of nonmechanistic theories) which I have found

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A NOTION OF MECHANISTIC THEORY

5

in the literature and in conversations with logicians. The (pedagogically)
most important part of Section III concerns our experience with theories
of (a) familiar mathematical objects as conceived in so-called classical
mathematics and of (b) less familiar abstract constructions considered
in so-called intuitionistic mathematics. There is a good deal of literature
on the question whether those theories are mechanistic: we point out a
number of systematic errors which slowed up progress.
II. BACKGROUND INFORMATION

We need a few facts about recursive functions of natural numbers,
recursive real numbers, and recursive functions of (not necessarily recursive) real numbers. (a) Most modern texts on mathematical logic
contain enough information fo.r our purposes about recursive functions
of natural numbers. A reader who has not met the notion should replace
'recursive' by - what he imagines to be - systematically computable. (b)
Recursive real numbers will be approximated by recursive sequences of

rational numbers p,./qn or equivalently of pairs (Pn, qn) of natural numbers.
The principal point to remember here is that the class of recursive functions is sufficiently flexible to satisfy the following condition, for any of
the familiar styles of approximation (by general Cauchy sequences, decimal or binary expansions, etc.): If everything in sight is restricted to
be recursive and if a real number is recursive for one style of approximation, is also recursive for the other styles. For example, if, has a recursive approximation p,./qn with a recursive modulus of convergence,
that is, a recursive v such that, for all m and n, .

e

e

n>v(m)-Ip,./qn- Pv(m/qv(m)1 (where m- 1 replaces the usual e) then, also has a recursive binary
expansion. This part of the subject is also easily accessible.
(c) A little care is needed in connection with recursive functions of
real numbers, for two reasons. First of all, the equivalences mentioned in
(b) are simply discontinuous: there is no ii such that for all , and all
its Cauchy expansions (equipped with a modulus of convergence) the
nth binary digit of, is determined by the first ii terms of the expansions;
neither if we decide to use the expansion of p2- Q with a tail of O's nor
if we use the different, but 'equivalent' expansion with a tail of l's. Sec-

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6

G. KREISEL

ond, some functional equations can be solved by continuous functions
(defined on the representations) which do not preserve equivalence, but

cannot be solved by functions which do. The standard example, familiar
from the theory of separating roots of polynomials, is provided by

(*)

x 3 -3x=c.

For each real number c there is an x which satisfies (*), and for each
recursive c there is a recursive x (in fact, for recursive c, all x which
satisfy (*) are recursive). But there can be no c I-+X which is continuous
for the topology usually associated with the real numbers. For when
c=O, (*) has three solutions (-.J3, 0, .J3); if Icl is large and c is negative, there is only one solution, say e-(c) and e-(c)-+ -.J3 as c-+O,
while for the corresponding e+, e+(c)-+ +.J3 as c-+O. However, it is
clear how to define e. if c is given by binary expansions, say eb, making
use of the fact that if c=p/2', then c has two binary expansions, for
which eb may take different values; such a eb will still be continuous
In the binary approximations. 2 - Other defects of the usual topology for computational purposes arise with Yes-No questions (or, generally, functions with values in a discrete space); cf. III.2(b) and IV.2
below.
Pedagogic remark. The reader is recommended to look carefully at
the example above and to make up some other examples for himself
(e.g., concerning the inadequacy of binary expansions for the addition
of real numbers). He should not allow himself to get paranoid about
(needing) 'general definitions' or even general 'criteria' to decide which
representations are to be used in different situations. He should take
it on trust that a few representations, that is, styles of approximations,
will be sufficient in most applications (and that in any case the most
interesting problems will no doubt require representations specially
adapted to the particular problem, as illustrated by the example at the
end of IV. 1(a) below). Where the literature referred to below does not
state explicit conditions on the style of approximation for which the

work is valid, the reader should give a moment's thought to the matter.
He should take it on trust that, ·in most cases, an author clever enough
to give an interesting construction on approximations will also have
been clever enough not to have relied on some peculiar (or, as one says,
accidental) feature of the approximation.

e:

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A NOTION OF MECHANISTIC THEORY

7

III. EMPIRICAL EVIDENCE AND SOURCES OF SYSTEMATIC ERROR

We give examples to avoid possibly premature generalities, and informal
comments to put the examples in perspective. (The reader should here
think of some substantial theory he happens to know, such as classical
or quantum mechanics.) Our main question is this:
What evidence do we have, from our experience with the
theory in question, for supposing that the theory is mechanistic?

1. Large parts of the theory are bound to be mechanistic - in fact,
all those which, naively, we regard as worked out! If the theoretical
physicist has obtained, say, a differential equation for some physical setup, the traditional mathematician will regard the theory as worked out
only if he has a systematic method for computing approximations to the
solution of the equation. But in the sense of 'method' used in mathematics - and surely correctly analyzed by Turing - this- just means that
the physicist's theory is mechanistic (at least as far as the context covered

by the differential equation is concerned~ The reverse procedure, to use
the physical setup as an analogue computer which, according to theory,
computes approximations to the solution, is not a mathematical method.
Two obvious points stand out On the one hand, (partial) differential
equations which were set up in the eighteenth or even back in the seventeenth century, for example, in continuum mechanics (so to speak, the
opposite extreme to discrete digital computers), turned out to be mechanistic: in this century methods of approxima~on were found to determine solutions recursively from the parameters (data). Or, when it was
shown that there are no such solutions for certain ranges of the parameters, discontinuities were discovered of the kind violating Hadamard's
principle m~ntioned in the introduction. On the other hand, the subject
contains a lot of unsolved problems (and we know that counterexamples
to the mechanistic character of the theory must be sought among such
problems). Here it should perhaps be mentioned that the American
Mathematical Society recently solicited a list of open problems from
distinguished mathematicians in connection with its Symposium Mathematical developments arising from the Hilbert problems, at De Kalb,
May 1974, and that several problems proposed by Ar'nold (one of the

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G. KREISEL

8

leading workers in mechanics) concerned the possibility of algorithmic,
that is, what we call 'mechanistic', solutions to questions in stability
theory, a particularly recalcitrant subject. 3
I have no good idea for evaluating this situation, especially in view
of the experience discussed in 111.3.
2. There is a whole class of theoretical results which either provides
evidence for supposing that familiar theories are mechanistic or at least
excludes a whole lot of candidates for counterexamples. The results have

the following character.
We do have examples of (familiar) properties, say p(e), of
real numbers which are satisfied by some but in general
by no recursive However, if is isolated it is automatically
recursive.

e
e.

e,

e

The following two cases give the flavor of such results.
(a) Let P, be the property: the continuous function I attains its
maximum in, say, [0, 1]. Suppose that the function is recursively continuous; that is, it is supplied with a recursive modulus of continuity.
Inspection of the proof of Specker (1959) shows that the function I constructed there satisfies the conditions above, but I does not attain its
maximum at any recursive (We shall make further uses of this fact
in Section IV.)
But clearly, under the conditions oni stated above, if the set g:p,(e)}
has an isolated point, say
this can be recursively computed by trial
and error. If is isolated there is a rational interval [a, b] where a < b
and
is the only element of {e:P,(e)}n[a,b]. We trisect [a,b] and
compute approximations to the maxim.um value of I in each of the three
parts: this requires only a modulus of continuity. After a finite number
of steps, the approximations to those values will be close enough to
determine at least one of the three parts of [a, b] in which
cannot

lie. By continuing this procedure one locates
We shall return to physical applications of this and related examples
in IV.l.
(b) Let P, be the property (of the continuous mapping I of say the
unit circle into itself): the point of the plane is afixed point of f. By
Brouwer's fixed-point theorem there is such a
As Brouwer knew,
though he stated the result in different terms, does not depend con-

e.

eo

eo

eo,

eo

eo.

e

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e. e


9

A NOTION OF MECHANISTIC THEORY

tinuously on f with regard to the uniform convergence topology. An
even sharper counterexample was constructed in Orevkov (1964), for a
particular recursively continuous f which has an additional property
(needed for bisection arguments and the like): f is supplied or, as one
sometimes says, 'equipped', with a recursive f* where the arguments of
f* are pairs, of rational intervals I and rational points r, and the values
of f* are pairs, of rational points s and v e { - 1, 0, + I} such that, if
f*(1, r)=(s, v)

seI;

f(s)f(s»s & v= + 1

f(s)=s & v=O

or

(where < is the relevant partial ordering of points in the plane). - N.B.
In general, f* cannot be extended continuously (on the usual topology
for Ihl) to all real values of r since then the discrete-valued component
v would have to be constant. We could extend f* continuously (for the
Baire space topology) to binary sequences in the sense ofII(c),just before
the Pedagogic remark.
Orevkov's particular f has no recursive fixed point. 4
However, the isolated fixed points of any recursively continuous J,
supplied with a recursive f* as above, must be recursive. This will be
clear to anyone who knows a standard proof of Brouwer's theorem which

involves the calculation of the so-called Brouwer degree of a point (if
f(x)#=x the degree of x with regard to f can be calculated from our
data in a recursive manner, for curves sufficiently close to x).
The significance of (b), for our problem, depends on the fact that, first
of all, many differential equations occurring in physical theories are
solved by means of (generalizations of) Brouwer's fixed-point theorem
and, second, stability of the solution tends to require that the solution
be isolated (in the relevant spaces).
3. Finally, we shall try to see what can be learned from experience with
the best-known examples of, demonstrably, nonmechanistic theories,
namely, the axiomatic theories of Frege and Dedekind of specific mathematical structures, such as those of the natural numbers with the successor relation or the ordering of the real numbers. The theories consist
of familiar so-called second-order axioms, known as 'Peano's axioms'
for arithmetic and 'Dedekind's axioms' for the continuum. The theories
are nonmechanistic because, on the one hand, they determine the struc-

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10

G. KREISEL

tures uniquely and, on the other hand, some of the familiar predicates
of natural numbers are, demonstrably, not recursive.
Remark for a reader who feels bothered at this point by orthodox
'doubts' about the objectivity of mathematics or about the existence of
mathematical objects. Whatever the merits of those doubts (in other
contexts), they do not discredit our proposed use of (experience with)
the axiomatic theories mentioned. We want to use them for orientation,
in connection with analyzing the mechanistic character of existing theories, say in physics. Whether we like it or not, existing theories use

freely concepts from the arithmetic of natural and real numbers which
have nonmechanistic theories. Furthermore, our requirements on counterexamples are very sharp. We do not merely ask whether existing theories are intended 5 to be nonmechanistic (an intention which one may
like or dislike; actually they are, at least in the sense that they are not
intended to be only about digital computers). We want to know whether,
according to existing theories, there are observable sequences which are
not recursive in the data, in other words, sequences which simply do
not possess any mechanistic theory at all.
Returning then to the axiomatic theories of the mathematical structures mentioned before the Remark, we have a situation which is, at least
superficially, quite similar to that described in IILl. The huge bulk of
the mathematical problems that were regarded as solved, had formal,
that is, mechanically computable, solutions. There were formal systems
(replacing Frege's and Dedekind's second-order axioms) which were
proposed as mechanical means of proving all theorems, in elementary
(also called: first-order) logic, the field of real numbers, number theory.
A great deal of 'evidence' for these proposals was said to be provided,
for example, by PM.
We all know that, within a couple of years around 1930, Godel's completeness theorem supported the proposals in the case of logic, Tarski's
elimination of quantifiers did the same in the case of (the first-order theory for) the field of real numbers, and Godel's incompleteness theorems
refuted the proposals in the case of number theory.
Even using hindsight, it is not/at all clear (to me) how the 'empirical
evidence' available in the twenties could properly be used to prepare
us for the results. In particular, in connection with number theorY,would
it have been more reasonable for Godel to have tried first to prove
completeness?

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A NOTION OF MECHANISTIC THEORY

11

Finally - and this is perhaps most directly relevant to the role of
'empirical' evidence - even today we do not have any theorem in ordinary number-theoretic practices which cannot be proved in PM. And
though this empirical fact does not contradict incompleteness (since there
are many open problems in ordinary number theory), it may seem to support the following hypothesis (which is of course also refuted by Godel's
argument): if a number-theoretic proposition can be proved at all, it
can be proved in PM. In short, the nonmechanistic nature ofthe axiomatic
theory of natural numbers was discovered, not by sifting existing applications which accumulated in the course of nature (here: in number-theoretic practice) but by looking for unusual or neglected applications (here:
to metamathematical questions); applications specifically chosen for
their relevance to questions of mechanization or, equivalently, formalization. This - it seems to me - is the principal lesson to be learned from
our experience with axiomatic theories of mathematical object; for use
with our present problem concerning the mechanistic character of (other)
scientific theories.
Perhaps it is worth adding (at least for the reader familiar with the
subject of constructive mathematics) that an apparently systematic error
was introduced even in those metamathematical studies which made the
mechanistic character of constructive theories a principal subject of research! Specifically, the question whether there is a proposition
'Vn3mR(n, m) which is constructively provable, but 'VnR(n,f(n)) is not
provable if f is recursive. The systematic error which precluded the
possibility of firiding such an example was this: people insisted on considering systems E with the property that .if E I- 3mR (ii, m) (where
ii=O, 1, ... ) then for some m, R(ii, m) is provable in E itself. The error
lies in this. Of course, if E I- 3mR (ii, m) and E is constructively sound,
there is some mfor which R(ii, m) can be proved constructively in some
En' e.g., in Eu {R(ii, m)}. But it would be a petitio principii to assume
that En depends recursively on n; cf. p. 328 of Kreisel (1972), where this
kind of error is analyzed.
IV. PRINCIPAL EXAMPLES

The examples concern mathematical definitions of nonrecursive objects

and the requirements imposed by Hadamard's principle. The first examples

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