Customer Reviews (2)

Read this book if you want to be a mathematician
I found this book to be a highly interesting biography of a good mathematician and great writer.I say good, because Paul Halmos denies the claim of being a great mathematician in the sense of Paul Cohen or Irving Kaplansky.Rather he says that he is a "professional" mathematician, and this book describes his professional life.But it does it in such a personal way that one cannot help but find it fascinating.I feel that this book not only influenced the way that I think about mathematics, but even the way that I think about life.

Wonderful look at mathematics, the times & the author
A Fantastic Book -- this 400+ page manuscript nicely mixes mathematical science with a historical view of the development (1930's through early 1980's) of mathematical research in the United States.This book is highlyreadable, extremely enjoyable and quite straightforward with details andopinions.One gets a first hand insight into how the author approached hisresearch, his career, and his life.Halmos has always been a brilliant andskillful writer but his contributions have mostly been in the technicalarena; this time he has provided a volume we can all enjoy.I found itdifficult to put this book down once I began its reading. ... Read more

Book Description
Useful both as a text for students and as a source of reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory which is most useful for its application in modern analysis. Topics studied include sets and classes, measures and outer measures, measurable functions, integration, general set functions, product spaces, transformations, probability, locally compact spaces, Haar measure and measure and topology in groups. The text is suitable for the beginning graduate student as well as the advanced undergraduate. ... Read more

Customer Reviews (4)

An excellent introduction to measure theory
There comes a time when the budding probabilist or statistician seeks a more comprehensive treatement of measure theory than is afforded in the first few sections of a graduate text in pstats. I chose Halmos's "Measure Theory" for this purpose for two primary reasons: i) Paul Halmos in my opinion is one of the best expository mathematics writers in history, and ii) years ago I paid $1 for the above-mentioned text (original Van Nostrand print) at a local thriftstore. My only perceived drawback was that likely some of his approach to measure theory may be outdated.

After reading the text (up to the chapter on probability) my opinion of Halmos as a writer and mathematician not only has been elevated, but the book delivered the thorough study of measure theory that I had hoped for. Indeed, the author does an excellent job in presenting measure theory in its entire generalitysemi-rings, rings, hereditary rings, algebras, sigma algebras and their extensions are all considered in detail, as well as measures on these set systems: finitely additive , sigma additive, inner measures, outer measures, sigma-finite measures, the completion of measures, regular measures). I especially enjoyed his presentation of Fubini's Theorem along with the concept of "section of a measurable set", which helped the theorem fall out effortlessly. I also found his presentation of different types of convergence (e.g. pointwise, uniform, almost uniform, in measure, in mean) very good and helped give me the bigger picture on modes of convergence. Theorem 22A is essentially a generalization of the Borel-Cantelli Lemma.

The book does have a few downsides. In particular pi and lambda-systems are not used and in some sense replaced by the older notion of semiring. Also, the author's definition of Lesbegue integrable seemed a bit more complicated than what is usually presented (e.g. for nonnegative measurable f to be integrable it requires a sequence fn of simple functions that is mean fundamental and converges in measure to f; compare this with the simpler definition of the integral of measurable f being the sup of Lesbegue integrals of simple functions g for which g <= f). But I consider these downsides minor and highly recommend this text to anyone who seeks a deeper understanding of measure theory.

My impression of measure theory has gone from seeing it asabstract mathematical machinery for simplifying analysis proofs, to a kind of mathematical philosophy that unifies the infinite with the discrete, and lays the proper foundations for inference, probabilistic reasoning, and learning; i.e. the foundations of cognitive intelligence.

Measure Theory -- where are the axioms??
Halmos was a lucid mathematical writer but his "Measure Theory" is dated 1950 and so antedates by two years Rohlin's definitive axiomatic discussion in Doklady Nauk USSR 1952.The attempted axiomatization by Halmos and Von Neumann in the the late 1940's failed and was withdrawn by its authors.Someone who is seriously interested to learn measure theory, with an approach based on the notion of measureable homomorphism, should visit Rohlin's publications and at least one other modern book.
WH Cobbs Narberth PA

A classic in the field
This book is an overview of measure theory that is somewhat dated in terms of the presentation, but could still be read profitably by someone interested in studying the subject with greater generality than more modern texts. Measure theory has abundant applications, and has even gained importance in recent years in such areas as financial engineering. Those interested in the applications of measure theory to financial engineering should choose another book however, since this one does not even mention the word martingale. After a review of elementary topology and set theory in chapter 1, the author begins to define the elementary notions of measure theory in chapter 2. His approach is more general than other texts, since he works over a ring instead of an algebra. Measures on intervals of real numbers is given as an example.Measures and outer measures are defined, and it is shown how a measure induces an outer measure and how an outer measure induces a measure.

The next chapter explores more carefully the relation between measures and outer measures. It is also shown in this chapter to what extent a measure on a ring can be extended to the generated sigma-ring. The all-important Lebesgue measure is developed here also, and the author exhibits an example of a non-measurable set.

In order to develop an integration theory, one must first characterize the collection of measurable functions, and the author does this in chapter 4. The convergence properties of measurable functions are carefully outlined by the author.

The theory of integration begins in chapter 5, wherein the author follows the standard construction of an integral by first defining integrals over simple functions. Then in chapter 6, signed measures are defined, and the Lebesgue bounded convergence theorem is proven and the Hahn and Jordan decompositions of these measures are discussed. The all-important Radon-Nikodym theorem, which gives an integral representation of an absolutely continuous sigma-finite signed measure, is proven in detail.

One can of course take the Cartesian product of two measurable spaces, and the author shows how to define measures on these products in chapter 7, including infinite products. The physicist reader may want to pay attention to the section on infinite dimensional product measures, as it does have applications to functional integration in quantum field theory (although somewhat weakly).

The author treats measurable transformations in chapter 8, but interestingly, the word "ergodic" is never mentioned. He also introduces briefly the L-p spaces, so very important in many areas of mathematics, and proves the Holder and Minkowski inequalities.

The next chapter is the most important in the book, for it covers the notion of probability on measure spaces. After an brief motivation in the first section of the chapter, probability spaces are defined, and Bayes' theorem is discussed as an exercise. Both the weak and strong law of large numbers is proven in detail.

Things get more abstract in chapter 10, which discusses measure theory on locally compact spaces. Borel and Baire sets on these kinds of spaces are defined, and the author gives detailed arguments on what must be changed when doing measure theory in this more general kind of space.

The book ends with a discussion of measure theory on topological groups via the Haar measure. This chapter also has connections to physics, such as in the Faddeev-Popov volume measure over gauge equivalent classes in quantum field theory. The author does a fine job of characterizing the important properties of the Haar measure.

an excellent book
If you want to stydy measure theory from scratch, I do recommend this book. This book is based on a ring, not an algebra, and is a little old-fashioned. So some people feel uncomfortable. But in particular,product spaces, the Fubini theorem and extension theorems are written veryclearly. I'm convinced this book will facilitate your learning in measuretheory and probability theory. ... Read more

Book Description
The theory is systematically developed by the axiomatic method that has, since von Neumann, dominated the general approach to linear functional analysis and that achieves here a high degree of lucidity and clarity. The presentation is never awkward or dry, as it sometimes is in other modern textbooks; it is as unconventional as one has come to expect from the author. The book contains about 350 well placed and instructive problems, which cover a considerable part of the subject. All in all, this is an excellent work, of equally high value for both student and teacher. Zentralblatt für Mathematik ... Read more

Customer Reviews (13)

Good Reading.
Halmos uses informal language and is not afraid to let his feelings show from time to time. There are a couple of exasperated remarks about the formalism that I find very funny. Because of this personal quality of the prose I feel affection for the work that I don't feel for many textbooks.

For me this book was just fine as the introductory text. But I was a physics student and must have had from that a working knowledge of vectors and transforms in orthogonal spaces, and I knew and liked the rigorous approach to math.

I still feel I would not like to be without the book.

A Classic for the mathematically-inclined.Good preparation for learning quantum mechanics.
This was one of the two textbooks (along with Rudin's Principles of Mathematical Analysis) that was used for the hot-shot freshman Math 218x course taught by Elias Stein at Princeton in the Spring of 1989.

It is a great book.It requires a bit of mathematical maturity, that is a love of mathematical proof and simplifying abstractions.This book abstractly defines vector spaces and linear transformations between them without immediately introducing coordinates.This approach is vastly superior to immediately extorting the reader to study the algebraic and arithmetic properties n-tuples of numbers (vectors) and matrices (n x n tables of numbers) which parameterize the underlying abstract vectors and linear transformations, respectively.

If I taught a serious linear algebra course using this book then there are a few deficiencies I would try to correct:

1. The polar decomposition is covered but the singular value decomposition (for linear transformations between different inner product spaces) is omitted.This is a pretty big gap in terms of applications, although it's easy to get the singular value decomposition if you have the polar decomposition.

2. The identification of an reflexive vector space with its double-dual was a stumbling block for me when I took the course in 1989.There was no mathematical definition of "identify", and so I was confused.Perhaps a good way to remedy this is to give a problem with the example of the Banach space L^p (perhaps just on a finite set of just two elements), and show how L^p is dual to L^p'.

3. The section on tensor products should be improved and expanded, especially in light of the new field of quantum information theory.

4. It would be nice to have a problem (or take-home final) where the reader proves the spectral theorem using minimal polynomials without recourse to determinants, and introduces the functional calculus just using polynomials.It is disturbing to see how many physics grad students are so hung up thinking of eigenvalues only as roots of the characteristic polynomial that they can't understand properties of the spectrum of a self-adjoint transformation A by considering polynmomials of A.

Except for property (3) above, this is a good book for students who are interested in taking a quantum mechanics or quantum computing course in the future.Halmos wrote this while Von Neumann was his advisor at Princeton, and von Neumann put quantum mechanics proper on a rigourous mathematical footing.

If you read this book and like it, then in the future you might want the following graduate-level textbooks:

Bhatia's book "Matrix Analysis".

Reed and Simon's "Methods of Mathematical Physics", especially volume 1 on functional analysis.(This is the infinite-dimensional version of Halmos's book.)

Halmos's "A Hilbert Space Problem Book"

You'll certainly need to learn some analysis before tackling the last two books, though!

A classic for advanced students
This book is very concise and covers everything you learned in undergraduate Linear Algebra course and much more. Pre-reqs I would say have to be a solid prep in Linear Algebra and knowing how to do proofs (a course or two in Real Analysis wouldn't be a bad idea). The book contains a handful of examples and exercises but most of the exercises are proofs. Recommended for seniors and graduate students only.

succinct and elegant descriptions
This book is correctly regarded by many mathematicians as a definitive introductory text on its subject. Be aware that it is not well suited as YOUR first text. If you have never dealt with finite dimensional vector spaces and are, typically, an undergrad, then a more standard and longer text will be an easier read.

But, once you have some knowledge of vector spaces, it pays to read Halmos. He provides a succinct exposition that explains the main ideas in an engaging and understandable style. Standard texts, directed towards a broader maths audience, often require more verbosity.

The only drawback with Halmos' book is the relative paucity of problems. However, if you have another text, that should typically provide you with the necessary problem sets.

overrated, but still good
This book is a good reference, because it contains a lot more material than is contained in most courses, but I don't think I'd want to use it for an intro to linear algebra. It's got stuff that other books don't have, like Hilbert spaces & some analysis stuff in an appendix, tensor products, multilinear forms... It's good as a reference or supplement, but not as a main text, IMO. For an intro, I liked Axler's Linear Algebra Done Right or the Hoffman/Kunze book. ... Read more

Book Description
Can one learn linear algebra solely by solving problems? Paul Halmos thinks so, and you will too once you read this book. The Linear Algebra Problem Book is an ideal text for a course in linear algebra. It takes the student step by step from the basic axioms of a field through the notion of vector spaces, on to advanced concepts such as inner product spaces and normality. All of this occurs by way ofa series of 164 problems, each with hints and, at the back of the book, full solutions. This book is a marvelous example of how to teach and learn mathematics by 'doing' mathematics. It will work well for classes taught in small groups and can also be used for self-study. After working their way through the book, students will understand not only the theorems of linear algebra, but also some of the questions which were asked which enabled the theorems to be discovered in the first place. They will gain confidence in their problem solving abilities and be better prepared to understand more advanced courses. As the author explains, 'I don't think I understand a subject until I know the questions ... I wrote this book to organize those questions, problems, in my own mind.' Try this book with your students and they too will be able to organize and understand the questions of linear algebra. ... Read more

Customer Reviews (5)

Not as good for second year graduate students
It is a good book for beginners but not as great for graduate students.

Great book!
I agree with the following reviewer in that Halmos' books are always entertaining and inviting! Fun to read, concise yet clear! The book contains most of the elementary yet important problems in an undergrad course with solutions in the back. It's very helpful working through all these problem because it will tremendous enhance your understanding of the subject. Also, if you want a hardcore problem oriented approach to linear algebra, check out Proskuryakov's Problems in Linear Algebra. Some of the problems in the book are Putnam like. Virtually any type of Putnam taste problems in Linear Algebra can be found in Proskuryakov.But this one, contrast to Halmos', is the least entertaining--that's why it is called HARDCORE PROBLEM APPROACH! You would be a great linear algebra problem solving machine working through both books!

Start out with one of the best
Halmos is one of the great mathematical expositors of the 20th Century, and his book "Finite Dimensional Vector Spaces" stands as the definitive introduction to the subject for budding mathematicians. This book, "Linear Algebra Problem Book", is perhaps best described as an engaging and semi-informal invitation and complement to that original work, which grew out of lectures given by the legendary John von Neumann. In contrast to typical treatments of linear algebra, "Finite Dimensional Vector Spaces" is abstract (introduces determinants through alternating forms), rigorous, concise, and demands a certain level of mathematical maturity. This book, "Linear Algebra" is exactly the opposite. Starting from very little assumed background, it all but gives away the store, written in plain language, anticipating students' questions and misconceptions, and leading them to a deeper understanding of mathematics through the Socratic method. This is not a problem book in the Schaum's outline sense; there is no drill or rote calculations. Every question is carefully chosen to illustrate a point or expose a potential misunderstanding in the student's knowledge or to exercise the student's intuition and ability to make connections. The answers are given as detailed explanations, integral to the exposition, which go far beyond merely answering the questions posed, raising deeper implications and questions. This is an excellent book for beginning students of higher mathematics, and a very user friendly guide to Halmos' classic text.

Clear, interesting, easy to follow.
I'm no mathematician but, I own a number of linear algebra and abstract algebra books and this book makes them all pale in comparison. I really enjoyed working through it. Maybe a bit pedantic for more advanced typesbut, if you want to get a handle on the concepts this is the book.

Clear, interesting, easy to follow.
I'm no mathematician but, I own a number of linear algebra and abstract algebra books and this book makes them all pale in comparison. I really enjoyed working through it. Maybe a bit pedantic for more advanced typesbut, if you want to get a handle on the concepts this is the book. ... Read more

Book Description
From the Reviews:"...He (the author) uses the language and notation of ordinary informal mathematics to state the basic set-theoretic facts which a beginning student of advanced mathematics needs to know. ...Because of the informal method of presentation, the book is eminently suited for use as a textbook or for self-study. The reader should derive from this volume a maximum of understanding of the theorems of set theory and of their basic importance in the study of mathematics." Philosophy and Phenomenological Research ... Read more

Customer Reviews (11)

Naive? just the publisher's "come on"first class work
Quite a good introduction to set theory, ideal for the person too busy
to read all the latest texts produced by academia. Also for the person
scared by too much mathematical symbolism, makes them realise thatwhile
symbolism is an aid in math it is not of course absolutely necessary.
In physics we see the current downgrading of quantum physics, so we
we should see a reduction in "math trash" as a consequence.
So irrespective of the background, all who read this book should be
rewarded with a greater grasp of math and how it is developed.

Not quite perfect
There is no escape from Set Theory in mathematics, and by extension, in physics. I finally realized that and went to the basics and bought this book and I am glad I did. Every little piece of knowledge I have in mathematics now appear to me in a brighter light.

The book starts from scratch in that it assumes no prior knowledge in mathematics at all. It does, however, assume knowledge of basic pure logic. Set Theory is developed through the introduction of the axioms, one by one, where the axioms are taken as universal truths which cannot be derived (from previously introduced axioms).

This development goes through various theorems valid for all sets, like De Morgans laws, the formation of new sets from old ones, like the power set and cartesian products, relations a other more specialized constructs, like functions.

Special sets are developed, e.g. the natural numbers. It is an amazing experience the first time one realizes that all sets one need (that I know of) in mathematics can be constructed from the emtpy set. Even more amazing is the fact that most of the symbols used in mathematics are actually sets.

The development goes through ordinal numbers and their arithmetic, and end with a brief introduction to cardinal numbers. Along the way one gets some insight into the precise meaning of infinite numbers and it's a thrill to discover that it's clear that one infinite number can be very much larger than another. In the same context it's also a little amusing to see that one can't push things too far even when one is in the realm of uncountably infinite numbers (quote "...there is no set that big...").

This book clearly deserves five stars, there is no doubt about that. I agree with what most other positive reviews say, but I would like to point out a few shortcomings:

The book could have been clearer; there are in my oppinion sometimes too many scentences and too few equations. In the same way I believe that there are too many words in the equations that are there. Longer statements with the ubiquitous "If and only if" and "for some" and the like become tiresome and even bring linguistic intricasies into the picture. They can and should be replaced by symbols.

Negative numbers aren't even mentioned. Rational numbers, and of course, the real numbers, aren't mentioned. This is in line with the rest of the book. Halmos even warns the sensitive reader at one point that he might be shocked because the number (e.g. set) 2 is to be used.

The axiom of choice is introduced through the cartesian product, the elements of wich are special functions. This is confusing on a first reading because functions are introduced (before that) as subsets of cartesian products.

The Classic Introduction to Set Theory
This is still the indispensable introduction to the subject for the student of mathematics, although specialists in logic and set theory will want to dig deeper into the subject. It's style is conversational, yet rigorous and can be either lightly browsed or studied more deeply. Although somewhat dated, it should still be a valuable resource in every mathematician's education.

Excellent writing! Great naive book
This is one of the most popular book on introdutory set theory that has been around for decades. Famous book! I was surprised to come across a problem book that is to company this Naive guy which is by Sigler. Very good readind indeed!!

Non-intimidating introduction to set theory
I find set theory to be the most intimidating subject in math.It seems so removed, but underpins every assumption I make in mathematics.Many other set theory books are dense and not very clear, but Halmos clearly expounds set theory.

Set theory, as is most mathematics, is hard, so be prepared to think.This book has only 102 pages in it and has just about everything I ever needed to know about set theory for me to feel confident using this theory to understand and prove things in other branches of mathematics.

Halmos's Naive Set Theory is the type of book I look for most, when I'm interested in a topic outside my specialization, but would like to know it better to apply it to my research.It's a clear, concise introduction to set theory, getting to the meat of it, without all the little asides and interesting things that distracts from learning the core of the subject.

This book should be on the bookshelf of every serious (and amateur) mathematician. ... Read more

Product Description
This classic book is based on lectures given by the author at the University of Chicago in 1956. The topics covered include, in particular, recurrence, the ergodic theorems, and a general discussion of ergodicity and mixing properties. There is also a general discussion of the relation between conjugacy and equivalence. With minimal prerequisites of some analysis and measure theory, this work can be used for a one-semester course in ergodic theory or for self-study.ReadershipGraduate students and research mathematicians interested in number theory.Table of ContentsIntroduction Examples Recurrence Mean convergence Pointwise convergence Comments on the ergodic theorem Ergodicity Consequences of ergodicity Mixing Measure algebras Discrete spectrum Automorphisms of compact groups Generalized proper values Weak topology Weak approximation Uniform topology Uniform approximation Category Invariant measures Invariant measures: the solution Invariant measures: the problem Generalized ergodic theorems Unsolved problems References ... Read more

Product Description
The book is a complete collection of Paul Halmos's articles written on the subject of algebraic logic (the theory of Boolean functions). Altogether, there are ten articles, which were published between 1954-1959 in eight different journals spanning four countries. The articles appear in an order that allows the reader unfamiliar with the subject to read them without many prerequisites. In particular, the first article in the book is an accessible introduction to algebraic logic. ... Read more

Book Description
This book contains four essays on expository writing of books and papers at the research level and at the level of graduate texts. The authors were the four members of the AMS Committee on Expository Writing ... Read more

Customer Reviews (3)

A must read for mathematicians and scientists
This short booklet contains four essays about how to write mathematics papers and books. The essays by Steenrod and Halmos, two prominent figures of 20th century mathematics, stand out for their common sense, depth and lucidity. They bring forth essential strategic issues, such as the need to maintain a clear separation between the formal and informal parts of mathematical papers, as well as useful tactical issues such aschoosing notation. In my opinion, the essays transcend the field of mathematics, and the principles that they delineate are applicable to all areas of scientific writing.

multi-author approach works
The various authors of "How to Write Mathematics" set out to describe their own personal writing styles, beginning with an admission that there is a lack of consensus on this topic. Indeed, the book closes with a couple of pages by the last author critiquing the advice of the first. On including detailed proofs in your publications, for example, you will find one author prescribing and another proscribing the practice.

The essays were written independently and this shows in the overlap and the contradictions. Whereas I have been highly critical of multi-author books in the past, the approach seems to work in this collection. Recalling my own efforts in writing a book on a topic within another field, I found myself agreeing or disagreeing with the authors on the various points. I was relieved to note that as early as at least three decades ago, authors before me shared identical conflicts with editors and copyreaders.

Based on my limited experience in writing, I had developed a number of half-formed rules of thumb. Given the recommendations of writers more experienced than I as so boldly presented, I am likely to be more aggressive in applying these rules. I would recommend this book to those authors with at least a few writing efforts under their belts. Given the specific nature of the examples in the essays, however, it is likely that only mathematicians will fully appreciate this book.

The paper by Halmos is worth the price of the book
Paul Halmos gives wonderful advice on how to write in mathematics.Get it. ... Read more

Customer Reviews (1)

Deserves 10 stars
This book should have been titled "A Hilbert Space Idea/Problem Book" as it not only challenges the reader to work out interesting problems in operator theory and the geometry of Hilbert space, but also motivates the essential ideas behind these fields. It is definitely a book that, even though out-of-print, will be referred to by many newcomers to operator theory and quantum physics. The insight one gains by the reading of this book is unequaled in any other books in existence on operator theory. It is becoming more rare as mathematics advances, to find books that attempt to explain the intuition behind the abstractions that are manifested in any area of mathematics. The problems in the book deal with both concrete examples and general theorems, and the reader should attempt to try and solve them without looking at the hints. The solutions found by the reader can then be compared with the author's, and some interesting differences will occur.

There are so many interesting discussions in this book that to list them all would probably entail listing everything in the book. The reader will find excellent discussions of the origin of normal operators on infinite dimensional Hilbert spaces as analogs to matrices on finite dimensional spaces; why the weak topology in infinite dimensions is not metrizable; the non-emptiness of the spectrum and why the spectral radius can be computed even though the spectrum cannot; the impossibility of isolated singular operators; the non-continuity of the spectrum: the existence of an operator with a large spectrum and the existence of operators with small spectra in every neighborhood of the large spectrum. The author then goes on to show that the spectrum is an upper semicontinuous function, thus preventing the existence of small spectra arbitrarily close to large spectra. This is an excellent discussion on the meaning and intuition behind semicontinuity; the result that every normal operator is unitarily equivalent to a multiplication and its equivalance to the spectral theorem. The author goes on to explain how one gives up the sigma-finiteness of the measure when doing this, and the origin of functional calculus; the difference between infinite and finite dimensions when attempting a polar decomposition for operators and its connection with partial isometries; the origin of compact operators and their connection with integral equations. The author shows how even the identity operator is not an integral operator on the space of square-integrable functions with Lebesgue measure.

In discussing the spectral theorem in chapter 13 the author statesmost profoundly: "In some contexts some authors choose to avoid a proof that uses the spectral theorem even if the alternative is longer and more involved. This sort of ritual circumlocution is common to many parts of mathematics; it is the fate of many big theorems to be more honored in evasion than in use. The reason is not just mathematical mischievousness. Often a long but 'elementary' proof gives more insight, and leads to more fruitful generalizations, than a short proof whose brevity is made possible by a powerful but overly specialized tool." In these few sentences the author has characterized the problem with current methods of teaching advanced mathematics. Too often the formalism masks the true meaning and intuitive motivation behind the mathematics. And even though mathematics is being applied to many different areas at an unprecedented rate, pure mathematics seems to be trapped in a local minimum, and I beleive this is due to the reluctance of authors to explain in detail the essentials of their ideas. This book is a perfect example of how mathematics can be taught that requires much thought and creativity on the part of students, without spoon-feeding them and thus encouraging a passive attitude to the learning of mathematics. I salute the author in his achievements in research and in teaching...one can only hope that his approach will be followed in all future works of mathematics. ... Read more