1.1. From the Publisher.

These notes are based on a course of lectures given by Professor Nelson at Princeton during the spring term of 1966. The subject of Brownian motion has long been of interest in mathematical probability. In these lectures, Professor Nelson traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations. He continues through recent dynamical theories of Brownian motion, and concludes with a discussion of the relevance of these theories to quantum field theory and quantum statistical mechanics.

1.2. From the Preface.

These are the notes for the second term of a year course on stochastic processes. The audience was familiar with Markov processes, martingales, the detailed nature of the sample paths of the Wiener process, and measure theory on the space of sample paths.

After some historical and elementary material in \\1-8, we discuss the Ornstein-Uhlenbeck theory of Brownian motion in \\9-10, showing that the Einstein-Smoluchowski theory is in a rigorous and strong sense the limiting theory for infinite friction. The results of the long \11 are not used in the following, except for the concepts of mean forward velocity, mean backward velocity, and mean acceleration. The rest of the notes deal with probability theory in quantum mechanics and in the alternative stochastic theory due to Imre Fényes and others.

I wish to express my thanks to Lionel Rebhun for showing us the Brownian motion of a colloidal particle, to many members of the class for their lively and critical participation, to Elizabeth Epstein for a beautiful job of typing, and to the National Science Foundation for support during part of the time when these notes were written.

1.3. Author's Apology.

It is customary in Fine Hall to lecture on mathematics, and any major deviation from that custom requires a defence.

It is my intention in these lectures to focus on Brownian motion as a natural phenomenon. I will review the theories put forward to account for it by Einstein, Smoluchowski, Langevin, Ornstein, Uhlenbeck, and others. It will be my conjecture that a certain portion of current physical theory, while mathematically consistent, is physically wrong, and I will propose an alternative theory.

Clearly, the chances of this conjecture being correct are exceedingly small, and since the contention is not a mathematical one, what is the justification for spending time on it? The presence of some physicists in the audience is irrelevant. Physicists lost interest in the phenomenon of Brownian motion about thirty or forty years ago. If a modern physicist is interested in Brownian motion, it is because the mathematical theory of Brownian motion has proved useful as a tool in the study of some models of quantum field theory and in quantum statistical mechanics. I believe that this approach has exciting possibilities, but I will not deal with it in this course (though some of the mathematical techniques which will be developed are relevant to these problems).

The only legitimate justification is a mathematical one. Now "applied mathematics" contributes nothing to mathematics. On the other hand, the sciences and technology do make vital contributions to mathematics. The ideas in analysis which had their origin in physics are so numerous and so central that analysis would be unrecognisable without them.

A few years ago topology was in the doldrums, and then it was revitalised by the introduction of differential structures. A significant role in this process is being played by the qualitative theory of ordinary differential equations, a subject having its roots in science and technology. There was opposition on the part of some topologists to this process, due to the loss of generality and the impurity of methods.

It seems to me that the theory of stochastic processes is in the doldrums today. It is in the doldrums for the same reason, and the remedy is the same. We need to introduce differential structures and accept the corresponding loss of generality and impurity of methods. I hope that a study of dynamical theories of Brownian motion can help in this process.

Professor Rebhun has very kindly prepared a demonstration of Brownian motion in Moffet Laboratory. This is a live telecast from a microscope. It consists of carmine particles in acetone, which has lower viscosity than water. The smaller particles have a diameter of about two microns (a micron is one thousandth of a millimetre). Notice that they are more active than the larger particles. The other sample consists of carmine particles in water they are considerably less active. According to theory, nearby particles are supposed to move independently of each other, and this appears to be the case.

Perhaps the most striking aspect of actual Brownian motion is the apparent tendency of the particles to dance about without going anywhere. Does this accord with theory, and how can it be formulated?

One nineteenth century worker in the field wrote that although the terms "titubation" and "pedesis" were in use, he preferred "Brownian movements" since everyone at once knew what was meant. (I looked up these words [1]. Titubation is defined as the "act of titubating; specif., a peculiar staggering gait observed in cerebellar and other nervous disturbances." The definition of pedesis reads, in its entirety, "Brownian movement.") Unfortunately, this is no longer true, and semantical confusion can result. I shall use "Brownian motion" to mean the natural phenomenon. The common mathematical model of it will be called (with ample historical justification) the "Wiener process."

I plan to waste your time by considering the history of nineteenth century work on Brownian motion in unnecessary detail. We will pick up a few facts worth remembering when the mathematical theories are discussed later, but only a few. Studying the development of a topic in science can be instructive. One realises what an essentially comic activity scientific investigation is (good as well as bad).

1.4. Review by: Henry P McKean Jr.
Mathematical Reviews MR0214150 (35 #5001).

The theory of Brownian motion seeks to describe the incessant spontaneous motion of small particles immersed in a fluid. The phenomenon is familiar to any microscopist. Chapters 1-4 review the historical development of the theory, associated with the names of Einstein, Smoluchowski, Langevin, Perrin, and Ornstein-Uhlenbeck. This part is excellent. The mathematical complements are less successful and cannot be recommended to beginners owing to an uneven mixture of the elementary and the sophisticated, though aficionados will find a number of points to interest them.

1.5. Review by: J L Doob.
The Annals of Mathematical Statistics 39 (20 (1968), 686.

This charming little book, notes for a course on stochastic processes, discusses Brownian motion 'as a natural phenomenon'. The audience had already en-countered measure theoretic probability. The natural phenomenon turns out to involve sophisticated mathematics but the stress is frequently on physical significance. The author thinks that stochastic process theory is now in the doldrums, that it can be rescued by the introduction of differential structures, and that a study of dynamical theories of Brownian motion can help in the rescue.

The book starts with an entertaining and enlightening discussion of the early history of Brownian motion analysis, including quotations from authors as far apart as Brown and George Eliot. The first analysis of Browian motion from a modern physical point of view was by Einstein in 1905 and the theory was first put into rigorous mathematics about 20 years later by Wiener. In 1908 Langevin introduced stochastic differential equations into the analysis to get a truly dynamical approach. The Langevin equation was modified by Ornstein and Uhlenbeck in 1930 to derive a new mathematical process which is a second approximation to the physical one.

The author derives the original Brownian motion (Wiener) and the Ornstein-Uhlenbeck processes, allowing external forces, by sophisticated semigroup arguments, leading to the usual differential operators. There is a careful discussion of the kinematics and dynamics of the processes considered, including heavy doses of stochastic integrals and Ito's stochastic differential equations. The concluding chapters discuss the role of probability in quantum theory and 'Brownian motion in the aether' (the latter from a point of view going back to Fényes (1952)) giving on the one hand a quantum mechanical interpretation and on the other a stochastic mechanical interpretation of the Schrödinger equation. The relations between these two interpretations are not yet fully understood.

The unusual combination of sophisticated but intuitive mathematical and physical reasoning makes this book instructive and interesting to readers with the most diverse interests.

1.6. Preface to the Second Edition (2001).

On 2 July 2001, I received an email from Jun Suzuki, a recent graduate in theoretical physics from the University of Tokyo. It contained a request to reprint "Dynamical Theories of Brownian Motion", which was first published by Princeton University Press in 1967 and was now out of print. Then came the extraordinary statement: "In our seminar, we found misprints in the book and I typed the book as a TeX file with modifications." One does not receive such messages often in one's lifetime.

So, it is thanks to Mr Suzuki that this edition appears. I modified his file, taking the opportunity to correct my youthful English and make minor changes in notation. But there are no substantive changes from the first edition.

My hearty thanks also go to Princeton University Press for permission to post this volume on the Web. Together with all mathematics books in the Annals Studies and Mathematical Notes series, it will also be republished in book form by the Press.

Fine Hall
25 August 2001

2.1. From the Publisher.

These notes are based on a course of lectures given by Professor Nelson at Princeton during the spring term of 1966. The subject of Brownian motion has long been of interest in mathematical probability. In these lectures, Professor Nelson traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations. He continues through recent dynamical theories of Brownian motion, and concludes with a discussion of the relevance of these theories to quantum field theory and quantum statistical mechanics.

2.2. From the Preface.

These are the lecture notes for the first part of a one-term course on differential geometry given at Princeton in the spring of 1967. They are an expository account of the formal algebraic aspects of tensor analysis using both modern and classical notations.

I gave the course primarily to myself. One difficulty in learning differential geometry (as well as the source of its great beauty) is the interplay of algebra, geometry, and analysis. In the first part of the course, I presented the algebraic aspects of the study of the most familiar kinds of structure on a differentiable manifold, and in the second part of the course (not covered by these notes), I discussed some of the geometric and analytic techniques

These notes may be useful to other beginners in conjunction with a book on differential geometry, such as that of Helgason, Nomizu, De Rham, Sternberg, or Lichnerowicz. These books, together with the beautiful survey by S S Chern of the topics of current interest in differential geometry (Bull. Am. Math. Soc. 72 (1966). 167-219) were the main sources for the course

The principal object of interest in tensor analysis is the module of C∞ contravariant vector fields on a C∞ manifold over the algebra of C∞ real functions on the manifold, the module being equipped with the additional structure of the Lie product. The fact that this module is "totally reflexive" (i.e., that multilinear functions on it and its dual can be identified with elements of tensor product modules) follows - for a finite-dimensional second-countable C∞ Hausdorff manifold - by the theorem that such a manifold has a covering by finitely many coordinate neighbourhoods.

I wish to thank the members of the class, particularly Barry Simon, for many improvements ...

2.3. From Princeton University.
https://dof.princeton.edu/people/edward-nelson

Ed served for a number of years as director of graduate studies, reflecting his deep commitment to graduate education. He polished four of his graduate courses into books published in the Mathematical Notes series by Princeton University Press. The books are truly marvellous but cannot capture the verve and originality of Ed's teaching. One of the books, Tensor Analysis, based on a course he gave on differential geometry, includes a theorem that says - literally - that it is possible to park in any parking space just slightly longer than your car if you use enough iterations of the parallel parking manoeuvre. To do this, one would like to be able to "slide" sideways into the parking place. Ed modelled the configuration space of the car as a certain four-dimensional manifold and studied the interactions of four vector fields on this manifold, showing how "steer" and "drive" could be used to produce "wriggle" and then "slide." It is enormously fun reading, for mathematicians anyhow, but it had a serious purpose as an introduction to the subject of "holonomy." As delightful as the version in the book is, the version in class was illustrated with a toy car Ed purchased at the Woolworth's, then on Nassau Street, for this purpose. He got a "sitting ovation" at the end of his lecture!

2.4. Review by: G A Reid.
The Mathematical Gazette 54 (387) (1970), 99-100.

These notes may be well described as comprising the algebra of differential geometry. They give a complete and well-written summary of precisely this, without, however, going into the geometrical definitions in detail at all. Indeed, it is suggested, most appropriately, that these notes may be read in conjunction with a book on differential geometry proper, such as Helgason, de Rham, or Sternberg.

After the fundamentals of tensor algebra and exterior algebra are set up and the central notion of Lie module is introduced, there follows a treatment of covariant differentiation, with affine connections being introduced algebraically following Koszul. There is then a chapter on holonomy; and in the final three chapters the algebraic aspects of Riemannian structures, symplectic structures, complex structures, and Kähler structures are discussed

There are many helpful and illustrative remarks throughout the text, and an instructive lesson on how to park, or unpark, a car, showing that provided the space exceeds the length of the car this can always be done, any inability in practice being rather a matter of more human ineptitude.

This book is written for people learning differential geometry, which is effectively to say research students and beyond. For them I would recommend it.

3.1. From the Publisher.

Kinematical problems of both classical and quantum mechanics are considered in these lecture notes ranging from differential calculus to the application of one of Chernoff's theorems.

3.2. From the Preface.

These are the lecture notes for the first term of. course on differential equations, given in Fine Hall the autumn of 1968.

3.3. Review by: Clark Robinson.
Mathematical Reviews MR0282379 (43 #8091).

These notes develop the local theories of flows on finite dimensions and one-parameter groups of unbounded operators in a Hilbert space. They include a review of calculus on Banach spaces, existence of solutions of differential equations, Sternberg's linearisation, sums and Lie products of vector fields, spectral theorem and Stone's theorem for unbounded operators, commutative multiplicity theory, Friedrichs' extension theorem, and sums and Lie products of self-adjoint operators. A more extensive treatment of flows on finite dimensions can be found in many differential equations books, and unbounded operators are treated in many books, such as that of F Riesz and B Sz-Nagy [Leçons d'analyse fonctionnelle (1965)]. However, these notes put the two subjects together and try to emphasise their similarities. For example, the two sections on sums and Lie products are very parallel. Also the flow on finite dimensions is treated more like the infinite-dimensional case than it usually is. Another feature of these notes is that several new proofs of theorems are given. These proofs are usually very good, such as the one for the converse of Taylor's theorem. However, the one for Sternberg's linearisation contains an error.

4.1. From the Publisher.

Stochastic mechanics is a description of quantum phenomena in classical probabilistic terms. This work contains a detailed account of the kinematics of diffusion processes, including diffusions on curved manifolds which are necessary for the treatment of spin in stochastic mechanics. The dynamical equations of the theory are derived from a variational principle, and interference, the asymptotics of free motion, bound states, statistics, and spin are described in classical terms.

In addition to developing the formal mathematical aspects of the theory, the book contains discussion of possible physical causes of quantum fluctuations in terms of an interaction with a background field. The author gives a critical analysis of stochastic mechanics as a candidate for a realistic theory of physical processes, discussing measurement, local causality in the sense of Bell, and the failure of the theory in its present form to satisfy locality.

4.2. From the Preface.

These are the revised lecture notes of a course given in June, 1983 for the Troisième cycle de la physique en Suisse romande. This course was given at a time when my thinking about stochastic mechanics was in a state of flux. There are many loose ends; This is not a treatise. I publish it in the attempt to express a viewpoint about the nature of quantum fluctuations, and in the hope that others will be encouraged to work on the many mathematical problems that are left unresolved here.

4.3. Contents.

Chapter I. KINEMATICS OF DIFFUSION
1. Differentiable Manifolds
2. Affine Connections
3. Measures on Path Space
4. Martingales
5. Diffusion
6. Markovian Diffusion
7. Continuity of Paths
8. Stochastic Integrals
9. Stochastic Action
10. Stochastic Parallel Translation
11. Existence of Diffusions

Chapter II. DYNAMICS OF CONSERVATIVE DIFFUSION
12. Newtonian Dynamics
13. Lagrangian Dynamics
14. Stochastic Quantization
15. Nodes

Chapter III. STOCHASTIC MECHANICS
16. Gaussian Processes
17. Interference
18. Momentum
19. Bound States
20. Statistics
21. Spin

Chapter IV. PHYSICS OR FORMALISM?
22. Measurements
23. Locality
24. Fields

A LIST OF OPEN PROBLEMS

4.4. Review by: Eric A Carlen.
Mathematical Reviews MR0783254 (86f:81039).

This stimulating book is a concise survey of stochastic mechanics, a subject whose development was largely initiated by the author's earlier book [Dynamical theories of Brownian motion (1967)]. The present book is full of original approaches to problems old and new in both mathematics and physics.
...
Stochastic mechanics provides in this way a description of quantum phenomena in terms of classical probability theory. Instead of indeterminism due to "collapse of the wave function", one has indeterminism due to random perturbations by the "quantum fluctuations". This raises a number of questions both mathematical (Are there actually diffusions with transport fields as singular as those prescribed by (a formula given above)?) and physical (What is the nature of the quantum fluctuations; that is, what is fluctuating and why?). The four chapters of this book are devoted to such questions.
...
At the end of the book is a list of 16 open problems suggested by stochastic mechanics. Progress has since been made on some of these, and indeed on many of the questions discussed in the last two chapters. The subject is currently experiencing too rapid a development to allow for a definitive treatise. (It was not the author's intention to produce such; see the preface.) Nonetheless, the slight datedness of the last two chapters will be well mitigated by the appearance of the author's lectures at the Ascona conference on stochastic behaviour in classical and quantum systems, to be published in the Springer Lecture Notes series.

4.5. Review by: John L Challifour.
Science, New Series 229 (4714) (1985), 645-646.

For myself, this volume of lecture notes presents the foreground between stochastic mechanics and quantum mechanics with clarity, economy, and elegance - virtues that I have come to expect from Nelson's writing. The reader will need to have a background in mathematics and physics at the level of first-year graduate courses and some knowledge of tensor calculus on Riemannian manifolds. Otherwise the book is self-contained. There can be little doubt that the mathematics of stochastic mechanics is interesting and worthy of and that as an area of ​​physics stochastic mechanics might provide a context for the study pursuit of classical phenomena in random backgrounds, such as electromagnetic phenomena that result from random sources. Whether one views stochastic mechanics as a possibly correct description of quantum fluctuations, without a flat contradiction between the two, is liable to depend upon whether one views quantum mechanics as a correct description of nature. A conservative view holds quantum mechanics to be only an approximation at the atomic scale of a more fundamental theory valid at subnuclear scales perhaps quantum field theory or quantum statistical mechanics. These differing views need not be contradictory, but a significant amount of work will be needed to make a convincing argument for stochastic mechanics as a fundamental physical theory. In this monograph the endeavour is in skilful hands.

5.1. From the Publisher.

This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed.

5.2. From the Acknowledgements.

This work was begun in December, 1979, at the Institute for Advanced Study. The next year I lectured on some of this material at Princeton University, and benefited from the comments of David Anderson, Jay Hook, Patrick Smith, Hale Trotter, Brian White, and Mitsuru Yasuhara. I also had helpful conversations with Laurence Kirby and Simon Kochen, and especially with Sam Buss, who found many errors and made many suggestions for improvements. The work was partially supported by the National Science Foundation. I am grateful to these people and institutions.

5.3. The impredicativity of induction.

The induction principle is this: if a property holds for 0, and if whenever it holds for a number $n$ it also holds for $n + 1$, then the property holds for all numbers. For example, let $\theta(n)$ be the property that there exists a number $m$ such that $2m = n.(n+1)$. Then $\theta(0)$ (let $m = 0$). Suppose $2.m = n.(n+1)$. Then $2.(m+n+1) = (n+1).((n+1)+1)$, and thus if $\theta(n)$ then $\theta(n + 1)$. The induction principle allows us to conclude $\theta(n)$ for all numbers $n$. As a second example, let $\pi(n)$ be the property that there exists a non-zero number $m$ that is divisible by all numbers from 1 to $n$. Then $\pi(0)$ (let $m = 1$). Suppose $m$ is a non-zero number that is divisible by all numbers from 1 to $n$. Then $m.(n + 1)$ is a non-zero number that is divisible by all numbers from 1 to $n + 1$, and thus if $\pi(n)$ then $\pi(n + 1)$. The induction principle would allow us to conclude $\pi(n)$ for all numbers $n$.

The reason for mistrusting the induction principle is that it involves an impredicative concept of number. It is not correct to argue that induction only involves the numbers from 0 to n; the property of n being established may be a formula with bound variables that are thought of as ranging over all numbers. That is, the induction principle assumes that the natural number system is given. A number is conceived to be an object satisfying every inductive formula; for a particular inductive formula, therefore, the bound variables are conceived to range over objects satisfying every inductive formula, including the one in question.

In the first example, at least one can say in advance how big is the number $m$ whose existence is asserted by $\theta(n)$: it is no bigger than $n.(n+1)$. This induction is bounded, and one can hope that a predicative treatment of numbers can be constructed that yields the result $\theta(n)$. In the second example, the number m whose existence is asserted by $\pi(n)$ cannot be bounded in terms of the data of the problem.

It appears to be universally taken for granted by mathematicians, whatever their views on foundational questions may be, that the impredicativity inherent in the induction principle is harmless that there is a concept of number given in advance of all mathematical constructions, that discourse within the domain of numbers is meaningful. But numbers are symbolic constructions; a construction does not exist until it is made; when something new is made, it is something new and not a selection from a pre-existing collection. There is no map of the world because the world is coming into being.

5.4. Review by: Pavel Pudlák.
The Journal of Symbolic Logic 53 (3) (1988), 987-989.

Since the appearance of set theory, every mathematical theory has been subjected to the test of whether it can be modelled in set theory. In this context, set theory is almost always identified with Zermelo-Fraenkel set theory (ZF) or its extensions. A possible explanation of what modelling means is based on the concept of relative interpretation. Usually we choose intuitively some set of properties of the concept studied and we want to show that the concept is meaningful by constructing it from sets. The properties can be thought of as axioms and the construction as an interpretation. Though it is not always quite so (as we have many second-order theories), we may ask why we ever bother with the interpretations in set theory, if we already have some intuition of the concept in question. The reason is that we are worried about the consistency of our theory and we want to prove its relative consistency with respect to ZF. The theory ZF is considered to be "semantically true" and hence consistent or "sufficiently well tested by the mathematical practice against a possible contradiction." Nominalists cannot accept semantic reasons and the practical reasons are not satisfactory for them either. Therefore, the activity described above seems meaningless to them: we reduce the consistency of one theory to the consistency of another that is even more likely to be contradictory. What can a nominalist do once we have the second incompleteness theorem?

There is a natural answer to this question: we can only minimise the risk of inconsistency by considering theories as weak as possible. Recent research into weak fragments of Peano arithmetic has revealed two relevant facts. First, arithmetic with induction restricted to bounded formulas, so-called bounded arithmetic, is suitable for developing an interesting part of number theory and finite combinatorics. Second, bounded arithmetic is interpretable in Robinson's arithmetic $Q$ (which has a very simple set of axioms without any induction at all). Thus, we can show that a nontrivial part of mathematics is consistent relative to a theory that can hardly be weakened any more. This is the starting point for Edward Nelson in Predicative arithmetic.

Nelson's proposal is to develop mathematics into theories that are interpretable in $Q$. He has another reason for doing this, and he stresses it even more than the fact that $Q$ is very weak. It is a particular form of interpretation that is used to show that bounded arithmetic and certain extensions of it are interpretable in $Q$. These interpretations are constructed simply by restricting the universe of numbers, i.e., we take some definable initial segments of integers (with the same arithmetic operations). He calls such constructions predicative, in contrast to the impredicativity of accepting the full scheme of induction. This explains the title of the book

In his criticism of classical mathematics based on belief in the semantical correctness of ZF, Nelson goes further than I have sketched above. One can show that bound arithmetic with exponentiation (i.e., with an axiom saying that exponentiation is a total function) is not interpretable in Q. Nelson considers this fact to be an indication that such a theory might be inconsistent, hence Peano arithmetic, and so forth, would be inconsistent too. (He writes that he has really tried to prove this; see p. 177). In case this is not so, he expects that there will still be other good reasons for working in theories such as predicative arithmetic; he stresses especially the problems of feasibility of computations on computers.
...
Considering the book as a whole, I think the book deserves attention from those who are interested in the foundations of mathematics. The book outlines an approach to the consistency problem that is in a very fair sense: it does not try to solve it by referring to more obscure facts or using tacit assumptions.

5.5. Review by: A J Wilkie.
Bulletin of the American Mathematical Society 22 (2) 1990), 326-331.

This book presents a formalist account of the foundations of arithmetic and "to one who takes a formalist view of mathematics", Nelson reminds us in his penultimate chapter, "the subject matter of mathematics is the expressions themselves together with the rules for manipulating them - nothing more." This view is expressed even more forcefully in the final sentence of the book: "I hope that mathematics shorn of semantical content will prove useful as we explore new terrain." Now these views are not, of course, new or even particularly extreme but the reader who has reached this point in the book will have realised just how much of conventional mathematical reasoning, and even reasoning usually accepted as totally finitary, Nelson regards as containing unjustifiable semantic elements. Let me, therefore, now turn to the beginning of the book and present some examples of arguments that Nelson finds problematic.

Many logicians would argue that finitary mathematical statements are adequately captured by formulas of the predicate calculus in the language containing a constant symbol for zero (0) and function symbols for the successor, addition and multiplication of natural numbers ($S$, + , ., respectively), and that finitary arguments are adequately modelled by formal proofs (using classical logic) in the system of first-order Peano Arithmetic (PA). (Statements and arguments about finite objects other than numbers can be coded into this system, but our concern here is with the natural numbers themselves.)
...
Now by Gödel's second incompleteness theorem the consistency of PA cannot be proved within PA (and hence, if the comments above are correct it has no finitary proof at all) but this consistency hardly seems a controversial issue. After all, if we regard a natural number as being something that we eventually reach in constructing the sequence 0, SO, SSO, SSSO, ... then surely this description carries with it the fact that the induction axioms are simply true. Certainly there seems to be no appeal here to any non-formal notions such as a completed infinite set (a view reinforced perhaps by the fact that Peano Arithmetic is equivalent (or rather, bi-interpretable with) the system obtained from Zermelo-Frankael set theory by replacing the axiom of infinity by its negation). Nelson disagrees. He argues that since we have specified a certain predicative construction of the natural number sequence (and it does seem impossible to formulate a finitary justification of the principle of induction without using some notion of 'constructing') then the only instances of the induction scheme for which the justification above is valid are the corresponding predicative ones.
...
Nelson has done a good and careful job at presenting the huge number of formal proofs necessary for the development of his theory. Even so, I would not recommend anyone to read this book unless he or she had already acquired some intuition on weak sub-systems of Peano Arithmetic. Suitable references for this are provided by Nelson and indeed most of his technical material has received more conventional treatments elsewhere. However, I think that the book is a valuable addition to the literature both for mathematical logicians because of the systematic and exhaustive account of interpretation in weak systems of arithmetic, and for philosophers of mathematics for the way in which conclusions compatible with strict finitism are deduced from assumptions based purely on a formalist viewpoint

5.6. Review by: Constantine Dimitracopoulos.
Mathematical Reviews MR0869999 (88c:03061).

The author's motivation for developing arithmetic predicatively, i.e. without the induction principle, is his mistrust of this principle; as he explains in Chapter 1, this principle assumes that the natural number system is given, which is unacceptable to a nominalist. The base theory for the author's considerations is R M Robinson's theory $Q$. The two main theories studied are denoted by $Q_{2}$and $Q_{4}$. Loosely speaking, $Q_{2}$ is the theory obtained from $Q$ by adding the symbol ≤, its defining axiom and an induction scheme for manifestly bounded formulas; $Q_{4}$ is obtained from $Q_{2}$ by adding the binary function symbol # ($x\#y$ can be thought of as $2^{\log x-\log y}$) and some axioms to ensure that it has nice properties, and by extending the induction scheme to all manifestly bounded formulas of the new language. These theories are subtheories of predicative arithmetic, since they pass the author's test for predicativity (they are locally interpretable in $Q$); this is shown by exploiting a relativisation scheme due to R Solovay.

After an extensive study of $Q_2$ and $Q_4$, we get an idea of the limitations of predicative arithmetic in Chapter 18. Using the Hilbert-Ackermann consistency theorem, the author shows that neither Solovay's nor any other relativisation scheme can incorporate exponentiation - see also the before-mentioned work of Paris and the reviewer; there is an impassable barrier, the author says, that separates the predicative from the impredicative.

At this point, one naturally wonders: Which results of mathematical logic can be obtained within predicative arithmetic? In particular, can the consistency theorem and Gödel's second theorem be established predicatively? To answer these questions, the author embarks upon the arithmetisation of syntax, following closely J. R. Shoenfield's presentation [Mathematical logic (1967)]. After a lot of work, he succeeds in proving predicatively the arithmetisation of the consistency theorem in Chapter 30. Since this theorem can be used to prove the consistency of $Q$, it follows that Gödel's second theorem cannot be proved without the use of impredicative methods.

In the last chapter but one, it is shown that one cannot prove in predicative arithmetic that exponentiation is total. Clearly, this is unsettling for mathematicians, who are convinced that exponentiation is total; this conviction, argues the author, is based on the semantic view of mathematics, which is unacceptable to a nominalist. So the nominalist is led to the question: Which formula is it more profitable to adjoin to predicative arithmetic, the formula expressing that exponentiation is total or its negation?

In the final chapter, the author proposes a modified Hilbert program, which is "to build up, parallel to classical mathematics, a demonstrably consistent elementary mathematics such that most results in the core of classical mathematics have an elementary analogue with the same scientific content, and such that the equivalence of the classical to the elementary result is easily provable by classical means". He also proposes a candidate for such an elementary mathematics.

On the whole, this is a very interesting book that deserves to be read for presenting defects of semantical reasoning in an age of increased interest in computational complexity problems.

6.1. From the Publisher.

Using only the very elementary framework of finite probability spaces, this book treats a number of topics in the modern theory of stochastic processes. This is made possible by using a small amount of Abraham Robinson's nonstandard analysis and not attempting to convert the results into conventional form.

6.2. From the Preface.

More than any other branch of mathematics, probability theory has developed in conjunction with its applications. This was true in the beginning, when Pascal and Fermat tackled the problem of finding a fair way to divide the stakes in a game of chance, and it continues to be true today, when the most exciting work in probability theory is being done by physicists working on statistical mechanics.

The foundations of probability theory were laid just over fifty years ago, by Kolmogorov. I am sure that many other probabilists teaching a beginning graduate course have also had the feeling that these measure-theoretic foundations serve more to salve our mathematical consciences than to provide an incisive tool for the scientist who wishes to apply probability theory

This work is an attempt to lay new foundations for probability theory, using a tiny bit of nonstandard analysis. The mathematical background required is slightly more than that taught in high school, and it is my hope that it will make deep results from the modern theory of stochastic processes readily available to anyone who can add, multiply, and reason.

What makes this possible is the decision to leave the results in nonstandard form. Nonstandard analysts have a new way of thinking about mathematics, and if it is not translated back into conventional terms, then it is seen to be remarkably elementary.

Mathematicians are quite rightly conservative and suspicious of new ideas. They will ask whether the results developed here are as powerful as the conventional results, and whether it is worth their time to learn nonstandard methods. These questions are addressed in an appendix, which assumes a much greater level of mathematical knowledge than does the main text. But I want to emphasise that the main text stands on its own.

6.3. Contents.

Preface
Acknowledgments
1. Random variables
2. Algebras of random variables
3. Stochastic processes
4. External concepts
5. Infinitesimals
6. External analogues of internal notions
7. Properties that hold almost everywhere
8. $L^{1}$ random variables
9. The decomposition of a stochastic process
10. The total variation of a process
11. Convergence of martingales
12. Fluctuations of martingales
13. Discontinuities of martingales
14. The Lindeberg condition
15. The maximum of a martingale
16. The law of large numbers
17. Nearly equivalent stochastic processes
18. The de Moivre-Laplace-Lindeberg-Feller-Wiener-Lévy-Doob-Erdös-Kac-Donsker-Prokhorov theorem

6.4. Review by: Tom L Lindström.
Mathematical Reviews MR0906454 (88k:60001).

The purpose of this elegant little book is to give an introduction to the theory of continuous-time stochastic processes with as little technical machinery as possible. In only eighty pages, the author takes the reader from scratch (the prerequisites are hardly more than high-school algebra) through random variables, Brownian motion, martingales and the law of large numbers to a very general version of the central limit theorem, which, in addition to the sufficiency and necessity of Lindeberg's condition, also includes the Lévy-Doob characterisation of Brownian motion and various invariance principles. He achieves it by replacing the conventional approach based on measure theory by one based on rudimentary nonstandard analysis. The advantage of this method is that all the main arguments can be carried out in a (formally) finite setting, and thus many of the technical complications of the conventional approach disappear. In an appendix (which requires more mathematical sophistication), the author shows how his nonstandard results can be reinterpreted in a measure-theoretic context.

Who should read this book? Although it formally presupposes little more than high-school algebra, the book is tersely and succinctly written, and requires some mathematical maturity on the part of the reader. But for everybody from the bright mathematics major to the curious professional mathematician who wants to find out about either the fundamental ideas of the theory of stochastic processes or the potential of nonstandard analysis, this is an excellent place to start. And even if neither of these two arguments appeal to you, the book may still be worth a try; the author is both one of the masters of his field and a great expositor, and it is a pleasure simply to follow the flow of his arguments. For the professional probabilist or nonstandard analyst the book has extra benefits. To the probabilist it offers a research tool of great potential; experiences over the last fifteen years have shown that nonstandard methods are often of great use in formulating and proving probabilistic results. To the nonstandard analyst it offers an interesting alternative to the way hyperfinite probability theory is usually presented; the author does not make use of the Loeb measure.

7.1. From Internal Set Theory.

Ordinarily in mathematics, when one introduces a new concept, one defines it. For example, if this were a book on "blobs," I would begin with a definition of this new predicate: x is a blob in case x is a topological space such that no uncountable subset is Hausdorff. Then we would be all set to study blobs. Fortunately, this is not a book about blobs, and I want to do something different. I want to begin by introducing a new predicate "standard" to ordinary mathematics without defining it.

The reason for not defining "standard" is that it plays a syntactic, rather than a semantic, role in the theory. It is similar to the use of "fixed" in informal mathematical discourse. One does not define this notion, nor consider the set of all fixed natural numbers. The statement "there is a natural number bigger than any fixed natural number" does not appear paradoxical. The predicate "standard" will be used in much the same way, so that we shall assert "there is a natural number bigger than any standard natural number." But the predicate "standard" - unlike "fixed" - will be part of the formal language of our theory, and this will allow us to take the further step of saying, "call such a natural number, one that is bigger than any standard natural number, unlimited."

We shall introduce axioms for handling this new predicate "standard" in a consistent way. In doing so, we do not enlarge the world of mathematical objects in any way; we merely construct a richer language to discuss the same objects as before. In this way, we construct a theory extending ordinary mathematics, called Internal Set Theory that axiomatises a portion of Abraham Robinson's nonstandard analysis. In this construction, nothing in ordinary mathematics is changed.