Logic and probability in quantum mechanics

Library of Congress Cataloging in Publication Data Main entry under title: Logic and probability in quantum mechanics.. On the other hand, to many philosophers the problems that are dis

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, Boston University

DONALD DAVIDSON, Rockefeller University and Princeton University

GABRIEL NUCHELMANS, University of Leyden

WESLEY C SALMON, University of Arizona

VOLUME 78

LOGIC AND PROBABILITY IN

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,

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

During the academic years 1972-1973 and 1973-1974, an intensive inar 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

sem-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 so-licited explicitly from individuals who are actively working in the foun-dations of quantum mechanics

Seventeen of the twenty-one articles appeared in Volume 29 of these Four additional articles and a bibliography on -the history and

Syn-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

ROLAND FRAisSE / Essai sur la logique de l'indeterminisme et la

GAR Y M HARDEGREE / The Conditional in Quantum Logic 55

D J FOULIS and c H RANDALL / Empirical Logic and Quantum

RICHARD J GREECHIE / Some Results from the Combinatorial

PART II / PROBABILITY

ZOL T AN DOMOTOR / The Probability Structure of

BAS C V A N FRAASSEN / The Einstein-Podolsky-Rosen Paradox 283

PA TRICK SUPPES and MARIO ZANOTTI/Stochastic

ROBERT W LATZER / Errors in the No Hidden Variable Proof of

DAVID J ROSS / Operator-Observable Correspondence 365 JEFFREY BUB / Randomness and Locality in Quantum Mechanics 397 WILLIAM DEMOPOULOS / Fundamental Statistical Theories 421 ALFRED LANDE / Why the World Is a Quantum World 433

PA TRICK SUPPES and MARIO ZANOTTI / On the Determinism of Hidden Variable Theories with Strict Correlation and Condi-tional Statistical Independence of Observables 445

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

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 nation On the one hand, the case is overwhelming that quantum me-chanics is the most important scientific theory of the twentieth century

exami-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 mental 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

funda-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

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

knowl-in that it is knowl-in an essential sense a nonmechanistic theory Here anistic 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 ar-bitrary nonrecursive measures of distances, masses, and forces In the classical case, however, the usual rational approximations have a recur-sive 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

nonmech-The second article by Fraisse raises general issues about the logic of indeterminism and the extent to which the fundamental results of quan-tum 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 chanics 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

me-The epistemological status of the laws of classical logic has been sieged 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 dis-tinguishing feature of the attack that has been launched from a quantum mechanical base is that it is an attack that rests upon an empirical sci-entific theory of an advanced and complicated nature That a challenge

be-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 sophical 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-

philo-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 es-sential 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 me-chanics 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 ar-ticle 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-sentation of quantum logics, in particular, orthomodular, partially dered sets, by means of structured families of Boolean algebras He brings

or-to his analysis of these structures methods that have been used in the study of manifolds by geometers In the third article, Terrence Fine pro-poses 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 ability in the discrete model that he would use for formulating the fun-damental principles of quantum mechanics Bastin's frontal attack on continuity assumptions and use of a continuum in quantum mechanics

prob-is perhaps the most salient feature of hprob-is approach to foundations He replaces the continuum by a discrete model that is hierarchical in cliar-acter 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 rea-son 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

quan-tum 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 chanics He examines critically the use of such experiments by von Neumann, Mackey, Piron, and Jauch

me-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

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 de-fined, 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 at-tempted to show that quantum mechanics is not complete in the sense that additional variables are required for the theory to have the appro-priate features of causality and locality The par~dox arises from mea-surements 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 measure-ment on the other particle is taken to violate our ordinary ideas of cau-sality 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 ad-vocating 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 con-sistently realized or satisfied by any hidden-variable theory He argues that his theory of statistical variables provides just the right sort of com-pleteness 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 contro-versy Van Fraassen focuses almost entirely on the Einstein-Podolsky-

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 chanics is stochastically incomplete We mean by this that, when time-dependent phenomena are examined, the predictions of the theory give only mean probability distributions as a function of time and do not de-termine a unique stochastic process governing the motion of particles

To illustrate how a stochastic approach may be applied in quantum chanics, 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

me-In the fourth article, Latzer examines in detail the well-known variable' proof of Kochen and Specker and finds several serious difficul-ties with their conceptual formulation and mathematical development

'hidden-of the problem 'hidden-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 ele-mentary 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 pleteness

com-The four articles that are included in the present volume and that were not included in the original issue of Synthese deal essentially with prob-lems 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 ics, 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

mechan-on the foundatimechan-ons of quantum mechanics The final article, which is the second article by Zanotti and me, is concerned to show that any hidden-variable theory with strict correlation and conditional statistical inde-

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

LOGIC

G KREISEL

A NOTION OF MECHANISTIC THEORY

I INTRODUCTION The notion in question is suggested by the words 'mechanism' or 'ma-chine' 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 chanistic theories, will be useful for such physical theories as classical

nonme-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) exten-sions of the sphere of theory consists in this,: constants are calculated theoretically which were previously obtained by 'empirical' usually'ap- proximate 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 restrie<ts the class of meaningful theories, loses much of its (restrictive) force The principle requires that theoretical relations corresponding to functions, mapping data to (other) observ-

ables, 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

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 re-cursive) 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 func-tions is sufficiently flexible to satisfy the following condition, for any of the familiar styles of approximation (by general Cauchy sequences, de-cimal or binary expansions, etc.): If everything in sight is restricted to

be recursive and if a real number e is recursive for one style of tion, e is also recursive for the other styles For example, if, has a re-cursive approximation p,./qn with a recursive modulus of convergence,

approxima-that is, a recursive v such approxima-that, for all m and n,

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-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 e: 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

nega-tive, 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 ogy for computational purposes arise with Yes-No questions (or, gen-erally, functions with values in a discrete space); cf III.2(b) and IV.2 below

topol-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

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 mech-anistic?

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 set-

up, 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 matics - and surely correctly analyzed by Turing - this- just means that the physicist's theory is mechanistic (at least as far as the context covered

mathe-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 sev-enteenth century, for example, in continuum mechanics (so to speak, the opposite extreme to discrete digital computers), turned out to be mech-anistic: in this century methods of approxima~on were found to deter-mine solutions recursively from the parameters (data) Or, when it was shown that there are no such solutions for certain ranges of the param-eters, 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 Math- ematical developments arising from the Hilbert problems, at De Kalb,

May 1974, and that several problems proposed by Ar'nold (one of the

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 e which are satisfied by some e, but in general

by no recursive e However, if e is isolated it is automatically

recursive

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 con-

tinuous; that is, it is supplied with a recursive modulus of continuity Inspection of the proof of Specker (1959) shows that the function I con-structed there satisfies the conditions above, but I does not attain its maximum at any recursive e (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 eo, this can be recursively computed by trial and error If eo is isolated there is a rational interval [a, b] where a < b

and eo 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 eo cannot

lie By continuing this procedure one locates eo

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 e of the plane is afixed point of f By Brouwer's fixed-point theorem there is such a e -As Brouwer knew, though he stated the result in different terms, e does not depend con-

A NOTION OF MECHANISTIC THEORY 9 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)<s & v= -lor f(s)=s & v=O or

f(s»s & v= + 1

(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 math-ematical structures, such as those of the natural numbers with the suc-cessor 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-

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 ories, 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 coun-terexamples are very sharp We do not merely ask whether existing the-ories 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

the-Returning then to the axiomatic theories of the mathematical tures 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

struc-(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 pleteness theorem supported the proposals in the case of logic, Tarski's elimination of quantifiers did the same in the case of (the first-order the-ory for) the field of real numbers, and Godel's incompleteness theorems refuted the proposals in the case of number theory

com-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?

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 ordi-nary 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 sup-port 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 appli- cations which accumulated in the course of nature (here: in number-the-oretic 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, formaliza-tion 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 re-search! 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 sidering systems E with the property that if E I- 3mR (ii, m) (where

con-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 m for 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

of nonrecursive objects, such as sequences of natural numbers, or, equivalently, examples of recursively unsolvable problems, came from logic; the decision problems for first-order arithmetic and for first-order predicate logic were the easiest examples to find As time went on, prob-lems with a more mathematical look were considered, and the matter

of recursiveness became more interesting for mathematics (even though some of the interesting examples were 'easily' reduced to the logical ones; that is, without any new mathematical construction after some imagina-tion was applied to spotting an interesting case) The mathematically most powerful result in this area is due to Matyasevic who found di-ophantine equations in two parameters, which define an enumeration

of all recursively enumerable sets (This enumeration is of course not recursive.) However, diophantine equations seem far removed from con-temporary mathematical physics, and so one tries to spot consequences with a more physical look - We shall first describe some of those con-sequences with a physical or at least geometrical look and then discuss the advantages of using Matyasevic's result, compared with earlier un-solvability results

1 (a) One example6 comes from the geometry of continuous curves (defined by continuous functions on, say, [0,1]) Such a curve attains

its maximum value, and there is a left-most point at which this value is attained Several examples of such curves determined by recursive data

are known, for example, in Specker (1959); they are recursively ous curves which do not attain their maximum at any recursive point

continu-A fortiori, the left-most point is not recursive

The proof in Specker (1959) uses functions defined by primitive cursive schemata By use of Matyasevic's results the same idea of proof allows one to find examples of such functions defined by more restricted means (more familiar from ordinary mathematical practice)

re-The principal obstacle to using this example for our topic (of showing that existing theories are not mechanistic) is not connected with the dif-ficulty of measuring the position of the left-most point (by means of the first method of measurement that comes to mind) For contrary to simple-minded positivistic accounts it is a principal task of theory to determine appropriate means of measurement, and, in particular, appropriate (statis-tical) evaluation of probable errors In the present context, of the geometry

of curves, the principal issue is this:

A NOTION OF MECHANISTIC THEORY 13

By which data will the curve be given?

If the only way we know of describing the curve is by the usual mations (made explicit in the uniform convergence topology) the left- most position for a maximal value is simply not determined approximately

approxi-by approximate data Specifically, the approximations consist of finitely

many overlapping rectangles between the abscissas (a;, a;t), and the

ordinates (b;, b;) where a;+l <a;, and b;;+l <b;:

pic-a point on the curve inside the nth rectpic-angle is interpreted to mean that the part of the t:urve between the abscissas (a;, a;) lies wholly within the rectangle ('Theoretical' is taken in the wide sense here, including global visual impressions of the look of the curve.)

Clearly, theory could go much farther, in sQme particular situation

in which continuous curves arise The class of theoretically possible curves may be so much restricted that a few rough measurements are interpreted

to mean that the curve in question is precisely the one defined in Specker

(1959) In this case we have a quite new (valid) method of 'measuring' the left-most position at which the curve attains its maximum: the calculation from the data It would simply be a better method than the familiar 'direct' method (which will of course still be required to be

consistent with the sophisticated method)

A hackneyed, but perhaps useful, illustration ofthe situation considered can be found within pure mathematics too Suppose the function f is

regular in, say, the circle C of the plane, If(z)I>N- 1 for some N on C

and J is bounded, say, by M, in some concentric circle C' containing C

in its interior In particular, suppose we are given a definition of f, say,

by power series or Dirichlet series (which permits the calculation of J

on C to any desired degree of accuracy) To calculate

1 f f'(z)

2ni J(z) dz

c

we need very little on the basis of the data above! Approximate values

of J(zn) for zneC for suitably many Zn will give us enough to approximate the values of f' on C and to calculate the integral to accuracy < t This

is enough to get a precise value since, by complex Junction theory, the value must be among 0, 1,2, (This is a better method of calculation than, say, getting approximate values of f' from the definition and cal-

culating the integral numerically to accuracy r 100000 Put differently,

it is better to operate directly on the definitions of f, given by

(approxima-tions to a suitable finite number oQ the coefficients of the power series

or Dirichlet series expansions, than to operate on (approximations ,to) the numerical values of f This is the illustration referred to in the Ped- agogic remark in II(c) concerning 'specially adapted' representations.) (b) Other examples of nonrecursive objects with a physical look are

to be found in the papers by Scarpellini (1963) and Richardson (1968) They consider classes 3's and 3'R resp of quite elementary 7 functions

of a real variable such that

each element of 3's and 3'R is given by finitely many rational numbers, and the following decision problems are recursively unsolvable:

do we find a situation involving Je3'su3'R with data permitting us to

fix J? Otherwise we have a violation of Hadamard's principle since the two properties of J considered above are certainly not continuous

if the data are subject to the topology of uniform convergence

2 Besides simply recognizing that some questions with a physical look

A NOTION OF MECHANISTIC THEORY 15 violate Hadamard's principle (a fact illustrated in IV.I), it is necessary

to say a word about suitable reformulations of questions, to make them continuous in the data - but possibly leaving open whether the reformula-

tion has a recursively continuous solution Here, analogues to the

distinc-tions in I(c) will be essential to avoid trivialities (connected with the fact that a Yes-No question which is continuous on the usual topology

for the rea1s has always the same answer since the data space is ed) The well-known three-body problem seems to provide a good illustra-

connect-tion, even apart from its glamor

We consider three point masses ml' m2, m3 at time t=O, at points '1>

'2' '3' respectively (in three dimensions), with velocity vectors Vb V2, V3

moving subject to the inverse square law The scales are so chosen that the force of attraction is mptJ!(rijf The problem is to determine the

positions and velocities of the three point masses at time t 1•

Clearly, (given the standard theory of differential equations) there is

no problem if we are given the information that there is no collision for

times ~tl In fact there will then be neighborhoods U I in phase space (that is, masses near mb positions and velocities near 'I and Vi at time

t = 0) such that there is no collision before time h for initial conditions

in Ui So the principal question is to decide whether or not there is a collision for t~tl This question has a Yes-No answer, which clearly cannot be continuous in the data (for the ordinary metric on masses, points, and velocities)

To reformulate the problem in accordance with Hadamard's principle

we take as data

and

intervals U I in phase space with rational end points (instead

of mb 'I' Vi lying in Ui)

an interval T with rational end points t', ttl (instead of t 1,

either the 'decision' is 0; each liie U j (i = 1,2,3) and for every

initial position in V there is no collision at times ~ t,

or the 'decision' is 1; the distances between lii and U j are

< n -1 and for some initial position in V there is a collision

before time t

The reformulation satisfies Hadamard's criterion (simply because the rational data are taken to be known or 'given' precisely) But for fixed

nand Uj, a 'small' change in T may, for example, flip the decision from

o to 1 and place lii outside U j instead of inside U j •

I do not know if the reformulated collision problem has a recursive

solution (The results, e.g., by Moser, 1973, do not seem to settle the matter.) If it does not, classical mechanics of point masses is not likely to

be mechanistic in our sense This notion of 'mechanistic' has led (one)

in a, I believe, completely straightforward way to (re)formulate the collision problem as above So inasmuch as the formulation has intrinsic appeal as judged by the light of nature, the notion has the kind of uses envisaged at the end of the Introduction

Remark Whatever the mathematical merits of the formulation (of the

collision problem) given above may be, further work is needed to see

if it can be used to establish the nonmechanistic character of classical celestial mechanics Specifically it is necessary to describe (an ensemble of) experiments and their statistical analysis for which the most probable outcome of the experiments is determined by the solution to our prob-lem In other words, if our problem has no recursive solution the most probable outcome of the experiments should be nonrecursive too The difficulty consists in the fact that our formulation is asymmetric It

distinguishes between (a) the actual absence of a collision (if the initial data are suitably sharpened to lie in V) and (b) the possibility of a colli-sion (if the initial data are fuzzy to degree n- 1); 'possibility' in the sense that a collision cannot be excluded on the basis of those data It is by

no means clear (to me) what principles of statistical analysis are applicable here, especially since the treatment of the data above is quite unrealis-tic, not at all related to any particular distribution of errors of measure-ment.8

Stariford University

A NOTION OF MECHANISTIC THEORY 17

NOTES

1 Proofs ofthe results at the end of the review of Kreisel (1972) appeared in Mints (1974)

2 In fancy language, continuous in the real number generators (for c) or, again, continuous

for the totally disconnected Baire space topology on binary sequences The reader may like to write down a system of neighborhoods for binary sequences which corresponds

to the usual topology on iii (but evidently the two distinct sequences 0·1 and 1·0 cannot

S Trivially, if one states Newton's laws in terms of the usual notions of space and time, the intention is that they hold for objects and distances of arbitrary real measures - not

merely as some kind of shorthand concerning (rational) approximations But we discover

that, in most applications of Newtonian mechanics, the consequences of the theory cerning approximations have a recursive, that is, mechanistic character Put differently, there exists an 'autonomous' mechanistic theory for those approximations, albeit not the intended theory (Here the logician naturally thinks of the parallel provided by the intended abstract notion of logical validity and the mechanical enumeration of the set of formulas

con-in the language of first-order predicate logic which are, abstractly, valid.)

6 Already mentioned in Section m, and an apparent exception to m.2 where, roughly speaking, uniqueness implies recursiveness (when the data are recursive) The reader in-

terested in the matter may wish to see whether uniqueness in a wider 'sample' space implies recursiveness

7 For a precise description, see the papers loc cit The functions are included among rational, trigonometric, exponential functions, and possibly their inverses (The papers were written before Matyasevic's work, and the results can be sharpened, roughly speak- ing, by suppressing closure of 8' under exponentiation to the base 2.) - It should perhaps

be mentioned that Richardson (1969) provides, as a foil to Richardson (1968), some cidability results for the integral case which have been siitlplified in Ehrenfeucht (1973)

de-8 The difficulty may be related, as P Suppes suggested in-conversation, to the weakness

of mathematical formulations in stability theory which also, usually, neglect the statistical

aspects of errors of measurement - It should perhaps be mentioned that there are open problems in the quantum theory, nonrecursive solutions of which would be easier to interpret (even if perhaps more difficult mathematically) Suppose we find a SchrOdinger equation of a - presumably large - molecule such that the (dimensionless) ratio ) 2/) 1 of its second to its first eigenValue is not recursive (in the data) Then there is no difficulty

in finding a corresponding experimental setup to show that the quantum theory is mechanistic in the sense of this note

non-REFERENCES Ehrenfeucht, A., 'Polynomial Functions with Exponentiation Are Well Ordered', Algebra universalis 3 (1973), 261-263

Gandy, R 0., 'The Concept of Computability', in R Harre (ed.), Scientific Thought

1900-1960, Clarendon Press, Oxford, 1969

Kreisel, G., 'Which Number Theoretic Problems Can Be Solved in Recursive Progressions

on nt-Paths Through 01', Journal of Symbolic Logic 37 (1972), 311-334 (reviewed in Zentralblatt 255 (1973), 28-29)

Mints, G E., 'On E-theorems', Zapiski 40 (1974), 101-118

Moser, J., 'Stable and Random Motions in Dynamical Systems, with Special Emphasis

on Celestial Mechanics', Annals of Mathematics Studies, Vol 77, Princeton University

Richardson, D., 'Solution of the Identity Problem for Integral Exponential Functions',

Zeitschrift fUr mathematische Logik und Grundlagen 15 (1%9), 333-340

Scarpellini, B., 'Zwei unentscheidbare Probleme der Analysis', Zeitschrift fUr matische Logik und Grundlagen 9 (1%3), 265-289

mathe-Specker, E P., 'Der Satz vom Maximum in der rekursiven Analysis', in A Heyting (ed.),

Construetwity in Mathematics, North-Holland, Amsterdam, 1959

L'hypothese de la ramification modifie radicalement notre tion de l'univers, notre logique physique; cependant, depuis dix-sept ans, eUe n'a ete reprise et discutee que par de rares auteurs Citons De Witt (1968, 1970, et 1971); et citons Graham (1971) eette conspiration

concep-du silence autour de l'idee de 1a ramification, parait due en premier lieu

au caractere peu rentable du sujet, sur Ie plan de l'arrivisme universitaire

En effet 1a notoriete du physicien theoricien, ou plus modestement son avancement dans la hierarchie universitaire, sont lies it 1a possibilite d'une verification experimentale rapide de ses idees Or une telle verification est it premiere vue impossible pour la ramification: comment prouver experimentalement que notre present se ramifie en plusieurs futurs, puis-que chacun de ces futurs evoluera pour son propre compte, sans agir sur les autres, bref se conduira comme s'il etait l'unique futur? Apres reftexion, nonS verrons, dans retude qui suit, que 1a situation n'est pas

aussi desesperee qu'il semble it premiere vue I,.a ramification conduit it une conception nouvelle de revolution et de l'effacement de l'onde de probabilite 11 en resulte d'abord que cette onde respecte les imperatifs relativistes, et notamment ne se developpe ou ne s'efface qu'it des vitesses inferieures ou egales it celIe de la lumiere: il est permis de lui accorder une existence physique, au meme titre qu'it ronde electromagnetique par exemple De plus l'interaction entre ronde porteuse d'un corpuscule et, par exemple, Ie miroir sur lequel elle se refiechit, ne se calcule pas de la meme fa~on, avec la ramification, que dans une theorie traditionnelle avec unicite du futur: voir paragraphes III et IV ci-dessous 11 y a la des verifications experimentales possibles dans un avenir raisonnable, des que l'on aura les moyens de multiplier les experiences aux memes condi-

tions initiales, concernant un petit nombre de corpuscules, et de trier par ordinateur celles qui auront donne un resultat improbable, tel que

la localisation d'une dizaine de photons envoyes vers l'ecran, sur un meme bord dudit ecran

Et puis, it n'est pas exclu d'esperer pour un avenir lointain, la couverte de mini-interactions entre les futurs issus d'un meme present: pour prendre un exemple 'concret' malgre son allure 'science-fiction', it n'est pas exclu que ron obtienne un jour des photographies de branches d'univers paralleles, done des photographies du possible auquel nous avons echappe Par exemple une photo des ruines de Paris dans les branches d'univers ou la guerre thermo-nucleaire a detruit l'humanite

de-a lde-a suite de lde-a crise internde-ationde-ale de 1962, notre survie etde-ant due de-a lde-a localisation favorable d'un photon dans une cellule nerveuse d'un di-rigeant: ce qui n'a nullement empache Ie cataclysme dans des branches d'univers defavorisees par une mauvaise localisation du meme photon

On se rendra compte, a la lecture de ce qui suit, que l'idee de la tion n'exige que peu de preparation technique, roais plutot un minimum

ramifica-de curiosite philosophique et ramifica-de souplesse d'esprit Je crois utile ramifica-de poser a rna fa~n de logicien, pourvu seulement de minirudiments de physique quantique, parce qu'il y a deja beaucoup a preciser du simple point de vue qualitatif Notamment la ramification acheve de faire dis-paraitre la notion de corpuscule, dans la mesure ou elle pouvait encore rappeler Ie 'point materiel' de la mecanique classique, ou meme relativiste Que Ie photon unique porte par une onde plane s'impacte en un point

l'ex-a n'entrl'ex-aine l'ex-absolument pl'ex-as qu'il l'ex-ait eu dl'ex-ans un pl'ex-asse immedil'ex-at une

trajectoire aboutissant en a: il n'a eu aucune trajectoire, ou si ron veut, toutes les lignes d'univers orthogonales aux surfaces d'onde sont des trajectoires De meme il n'y a pas lieu de se demander par quel trou est passe Ie photon, puisque ledit photon est une onde, meme au cours

de ses localisations: onde tres localisee a l'instant de remission, puis de plus en plus diffuse et passant par tous les trous qui se presentent, enfin onde effacee progressivement a partir de l'atome qui absorbe Ie photon;

au benefice d'une 'trajectoire d'arrachement' electronique, qui est encore une onde, et au co-detriment d'une onde orbitale electronique qui est egalement effacee Finalement, les seules traces qui subsistent de l'aspect corpusculaire, sont: Ie 'nombre de corpuscules portes par une onde', qui est un entier abstrait; la localisation a partir de laquelle une onde est

ESSAI SUR LA LOGIQUE DE L'INDETERMINISME 21 creee ou effacee, et la transmission de l'energie-impulsion, de l'onde ef-facee a l'onde creee

Signalons qu'en consequence de l'existence objective des ondes de probabilite, la normalisation d'une onde, qui ramene a 1, ou plus gene-ralement au nombre d'occupation, Ie flux du quadrivecteur densite de presence a travers un espace, la normalisation n'est plus une liberte of-ferte a chaque instant Du fait de son existence objective, l'onde est normee une fois pour toutes aussitot apres la localisation qui lui donne naissance Apres quoi sa valeur est fixee non a un facteur pres, mais absolument, en depit d'absorptions partielles par des ecrans, de reflec-tions, refractions, diffractions ou autres avatars; jusqu'a son effacement par une nouvelle localisation

II me reste a souhaiter que les affirmations et arguments qui suivent interessent quelques physiciens, et les incitent aux discussions, refuta-tions ou precisions quantitatives

I LE PARADOXE D'EINSTEIN ET LARAMIFICA TION DE

L'ESPACE-TEMPS Dans tout ce qui suit, nous nous pla~ons dans Ie cadre de la relativite restreinte Considerons une onde porteuse d'un unique photon, tombant sur un ecran, que nous supposons etre une plaque photographique, apres d'eventuelles interferences ou diffractions Nous recoltons un impact et un seul, du fait que Ie photon se localise, avec toute son energie,

en un point de l'ecran Plus precisement, une reaction en chaine est declenchee a partir de cette localisation; elle aboutit a la formation d'une petite tache sur l'ecran-plaque photo; cette tache etant perceptible par notre oeil, ou au microscope: c'est l'impact La localisation, donc la position de l'impact, est imprevisible, etant entendu qu'elle se produit sur la figure de diffraction, et avec la probabilite calculee par la meca-nique ondulatoire Autrement dit, si nous recommen~ons l'experience

un grand nombre de fois, l'ensemble des impacts dessinera la figure de diffraction, d'autant plus fidelement que Ie nombre de ces impacts sera grand

Nous pourrions d'abord croire que Ie caractere imprevisible de la localisation, est dft a notre ignorance; qu'en realite, l'impact se produit

au point spatio-temporel Ie mieux predispose, compte tenu de la

repar-tition de probabilite de presence du photon dans son onde porteuse Ainsi un vieux drap se dechire en son point de moindre resistance, compte tenu de la repartition des forces de traction Mais une telle conception suppose que les diiferents points spatio-temporels de l'ecran, dont certains peuvent etre simultanes, pour un referentiel donne, et tres eloignes les uns des autres, puissent en quelque sorte se concerter in-stantanement, pour determiner l'unique point de localisation Or d'apres

la relativite restreinte, aucun signal, aucune information ne peut se propager plus vite que la lumiere, ou pillS exactement, a une vitesse supl:rieure a la vitesse limite (nollS dirons 'vitesse limite' plutot que 'vitesse de la lumiere', non seulement pour abreger, mais pour preserver l'hypothese de Louis de Broglie, d'apres laquelle les photons pourraient avoir une masse au re~ '"'s non nulle, donc des vitesses legerement infe-rieures a la vitesse limite, leur energie, ou couleur, etant fonction de leur vitesse par les formules usuelles de la relativite restreinte~ Le paradoxe ainsi expose est celui d'Einstein, presente au Conseil Solvay en 1927 L'hypothese de la ramification est u~e solution logique de ce paradoxe, consistant a dire que la localisation du corpuscule se produit partout lOU cela est possible d'apres la forme de l'onde porteuse et celIe de l'ecran

II existe donc, pour notre unique corpuscule, un tres grand nombre N

de localisations, donc d'impacts Mais l'observateur, ainsi que son riel et plus generalement, tout Ie contenu physique de la portion d'espace-

mate-temps situee dans Ie futur des impacts, se trouve demultiplie en N

exem-plaires par un ramifieur, analogue a une onde de choc se propageant a

la vitesse limite; et naturellement chacun des exemplaires de vateur ne per~it qu'un seul impact Le nombre N est une constante

l'obser-dependant vraisemblablement de la seule nature du corpuscule (photon, electron, meson, etc.); nollS l'appellerons Ie nombre d'Everett, un peu par analogie avec Ie celebre nombre d'Avogadro

Precisons comme suit notre hypothese Supposons l'ecran constitue par un nombre fini d'elements tridimensionnels, ou elements d'hyper-plans En eifet chaque element de l'ecran peut avoir deux dimensions spatiales et une temporelle, comme pour un ecran au sens usuel, qui est

un element de plan avec une certaine duree; soit encore trois dimensions spatiales: par exemple si notre corpuscule, dont la presence est poten-tielle dans une onde porteuse, est oblige de se localiser a un instant im-pose par l'expl:rimentateur, en raison de la detente dans une chambre de

ESSAI SUR LA LOGIQUE DE L'INDETERMINISME 23 Wilson ou une chambre a buIles; l'ecran est alors constitue par Ie volume inteneur a la chambre, a l'instant de la detente L'ecran peut d'ailleurs comprendre Ies deux sortes d'elements: par exemple Ie corpuscuIe, dont l'onde porteuse evolue a l'interieur d'une chambre, peut se loca1iser sur l'un des murs, pendant une premiere partie de l'experience; apres un delai qui lui est accorde, s'il ne s'est pas localise sur un mur, une detente provoquee par l'experimentateur l'oblige a se localiser a l'inteneur Notre ecran est donc une hypersurface qui separe l'espace-temps en deux regions, Ie passe et Ie futur: toute ligne d'univers, prolongee inde-finiment vers Ie passe et vers Ie futur, et ayant en chaque point une vitesse strictement inferieure a la vitesse limite, est partagee par l'ecran

en un intervalle initial allant de l'infini passe jusqu'a un point section avec l'ecran; un intervalle final allant a l'infini futur, et un inter-valle median pouvant traverser plusieurs fois l'ecran (dans ses regions bispatiales-temporelles) ou avoir avec lui une infinite de points communs Notons a, b, les elements d'hyperplans qui constituent l'ecran, et

d'inter-soit N Ie nombre d'Everett Que l'onde porteuse d'inter-soit scalaire et obeisse

a l'equation de Klein-Gordon, ou qu'elle ait quatre composantes et obeisse a l'equation de Dirac, il existe de toutes fa~ons un quadrivecteur densite de presence, defini en tous points de l'espace-temps et de flux conservatif, dans toute la region anterieure a la traversee de l'ecran La probabilite de presence, ou de localisation, du corpuscule sur l'element

a de l'ecran, est Ie flux, a travers cet element a, du quadrivecteur, une fois l'onde normalisee, c'est-a-dire lorsque Ie flux total a travers un

esp~ce est raniene a la valeur 1 Nous nous placerons dans un cas ou Ie flux est toujours positif, l'hypersurface ecran ~tant orientee dans Ie sens

du passe au futuro Cela est automatiquementrealise avec l'equation de Dirac, qui donne a travers tout element trispatial un flux positif, donc

une probabilite de presence positive Multiplions par N la probabilite

concernant l'element a: nous obtenons un nombre reel positif dont la

partie entiere (Plus grand entier inferieur ou egal) sera notee n ; la somme des n pour tous les elements a de l'ecran, etant au plus egale a N

Notre hypothese consiste a dire qu'en chaque point u de temps, posterieur a l'un au moins des elements de l'ecran, l'espace-temps, avec son contenu physique, est ramifie en N branches Notant a, b,

l'espace-ceux des elements de l'ecran qui sont anterieurs a u les N branches

en u se repartissent en n branches a-impactees, c'est-a-dire dans

les-queUes on observe la localisation du corpuscule en a, avec toutes les

consequences qu'elle entraine: echange d'energie entre Ie corpuscule et l'element a, reaction en chaine aboutissant a la formation d'un impact photographique, etc Toujours en u, nous avons nb branches b-impactees, evidemment distinctes des a-impactees La somme na + nb + relative

a tous les elements d'ecran anterieurs a u, est inferieure aN; il reste en u

des branches au nombre de N - (na + nb + ) que nous appelons vierges,

parce qu'on n'y observe aucune localisation du corpuscule, les vateurs situes en u et dans ces branches sont dans l'attente de l'impact

obser-Le rapport nJN du nombre de branches a-impactees au nombre total

de branches, materialise la probabilite, calculee par la mecanique latoire, pour que l'impact ait lieu en a; de meme pour chaque element

ondu-b, de l'ecran Ce rapport n'est serieusement inferieur a cette lite, ou n'est nul, que si Ia probabilite de presence sur l'element est tres

probabi-faible, de l'ordre de liN Ainsi la probabilite retrouve sa definition

ele-mentaire, elle est Ie rapport du nombre des branches favorables au bre total des branches Mais nos branches d'Qnivers ne sont pas seule-ment des 'cas', des 'eventualites', dont une seule se realiserait Tou'tes sont egalement reelles; l'observateur appartient au contenu physique

nom-de l'espace-temps, il est a ce titre demultiplie en N exemplaires dont na

observent l'impact en a, et nb l'observent en b, etc.; chacun de ces plaires etant abusivement tente de croire qu'il est Ie seul reel, du fait que les autres exemplaires sont a jamais separes de lui

exem-En un meme point u, Ie contenu physique de l'espace-temps differe

d'une branche a l'autre Etant donne u et une branche de u, ce contenu

est entierement defini par Ia position de l'impact ou par l'absence pact Par exemple la valeur du champ electromagnetique en u et dans une branche vierge, est obtenu par Ia simple utilisation des equations de

d'im-ce champ et des conditions anterieures Alors que, dans une branche a-impactee du point u, it faut en plus tenir compte des consequences de

l'impact en a: l'ecran ayant pu etre programme a l'avance, pour qu' une localisation en a declenche l'allumage d'une lampe, et parconsequent une modification du champ electromagnetique dans Ie futur de a, et en

particulier au point u

De la description precedente, it resulte que l'impact en a efface l'onde porteuse dans tout Ie Jutur de a, mais seulement dans les branches a- impactees de ce futur: la probabilite de presence du corpuscule, dans ces

ESSAI SUR LA LOGIQUE DE L'INDETERMINISME 25 branches, etant un sur l'element a et zero ailleurs Ainsi, dans l'hypothese

de la ramification, l'effacement de l'onde porteuse respecte les imperatifs relativistes, en se propageant a la vitesse limite Par contre, dans l'hypo-these traditionnelle de l'unicite du futur, l'onde porteuse devrait etre aneantie partout et instantanement lorsque l'impact apparait; cela de-vant etre vrai pour n'importe quel observateur, donc pour tout referen-tiel, on voit que l'onde porteuse devrait etre aneantie non seulement dans Ie futur de l'impact, mais dans tout son ailleurs: cela releverait de l'intervention divine ou de la magie, puisque l'effacement se ferait Ie long du cone passe du point d'impact, donc a la surface d'une sphere diminuant a la vitesse limite et aboutissant a l'impact

Revenons a la ramification; dans les branches vierges, l'onde porteuse continue son evolution, conformement a la mecanique ondulatoire Cette onde porteuse est une realite physique, puisqu'elle provoque, dans ces branches vierges, l'apparition d'un impact, avant de s'effacer dans

Ie futur de cet impact Ainsi, avec la ramification, la question de la rea lite physique de l'onde porteuse, si controversee par les tbeoriciens, est tranchee ajfirmativement; a tout Ie moins, les principes relativistes ne s'opposent plus a une reponse affirmative

Nous avons, jusqu'ici, considere l'espace-temps point par point, en decrivant la repartition en branches en un point quelconque de l'espace-temps Le lecteur peut demander comment ces branches se 'raccordent' les unes aux autres lorsqu'on voyage, -ou, ce qui est relativistement equivalent, lorsqu'on attend Repondons que Ie seul chemin physique-ment parcourable par un observateur-voyageur, est Ie parcours d'une ligne temporelle d'univers, a vitesse stricteme~t inferieure a la vitesse limite Soit donc u un point de l'espace-temps, v un point posterieur a u,

et joignons u a v par une ligne temporelle Notons a, a',""" les elements

de l'ecran anterieurs a u; notons b, b', les elements de l'ecran anterieurs

a v mais situes dans Ie futur de u ou dans l'ailleurs de u Alors disons que

l'ensemble des na branches a-impactees du point u, conduit a l'ensemble des na branches a-impactees du point v; ou que Ie deuxieme ensemble

provient du premier De meme pour chaque element d'ecran a',."" rieur a u Notons n<u la somme des entiers-na, n a"," •• relatifs aux elements d'ecran anterieurs au point u Disons que l'ensemble des N -n<u bran-ches vierges en u, conduit a l'ensemble des nb branches b-impactees du point v, et a l'ensemble des nb' branches b'-impactees du point v,"""' et

ante-enfin Ii l'ensemble des N -n<v=N - (na+na,+··· +nb+nb,+ ) ches vierges du point v On voit que l'ensemble des branches vierges du point u conduit Ii un ensemble du meme nombre de branches du point v,

bran-reparti en plusieurs sous-ensembles correspondant aux divers impacts

b, b', qui peuvent etre per~llS au cours du voyage, ou correspondant

Ii l'absence d'impact

Du point de vue d'un observateur-voyageur situe en u sur une branche vierge, et decidant d'aller en v, la probabilite, calculee par la mecanique ondulatoire, de percevoir l'impact en b au cours du voyage pro jete, est materialisee par Ie rapport nb/(N -n<u) du nombre des branches favo-rabIes (branches du point v avec impact en b), au nombre total des branches vierges du point u; ou, ce qui revient au meme, au nombre total des branches du point v qui proviennent des branches vierges du

point u La probabilite de ne percevoir aucun impact au cours du voyage projete, est materialisee par Ie rapport (N -n<v)/(N -n<u) du nombre des branches favorables (branches vierges du point v), au nombre total des branches vierges du point u, ou nombre total des branches du point

v provenant des branches vierges en u L'apparition du denominat<mr

N - n<u au lieu de N, est connue SOllS Ie nom de reduction du paquet de probabilites

Le caractere fondamental de l'indeterminisme, est garanti par Ie fait que les diverses branches vierges en u sont indiscernables physiquement:

cela n'a aucune signification physique de dire que l'on se trouve sur une

branche, vierge en u et conduisant Ii une branche b-impactee du point v,

ou conduisant Ii une branche vierge en v NOllS pouvons dire que les

N - n<u observateurs-voyageurs venant des branches vierges du point

u et allant vers v, franchissent la frontiere du cone futur de b, au cours de leur voyage: autrement dit, ils sont traverses par la sphere spatiale issue

de b et croissant Ii la vitesse limite Pendant ce franchissement, tout se passe comme si les N - n<u observateurs etaient extermines et imme-diatement remplaces par un nombre egal d'observateurs ayant Ie meme stock de souvenirs; parmi ces nouveaux observateurs, nb per~oivent

l'impact situe en b, et les autres ne per~oivent aucun impact Une telle mort subite suivie d'une resurrection collective, sans que cela ait un sens de dire qui revit en qui, equivaut physiquement Ii la continuite de

la vie, assaisonnee de l'alea imprevisible de l'impact en b, ou de son

absence