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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

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"I Want To Be A Mathematician" is an account of the author's life as a mathematician. It tells us what it is like to be a mathematician and to do mathematics.It will be read with interest and enjoyment by those in mathematics and by those who might want to know what mathematicians and mathematical careers are like.Paul Halmos is well-known for his research in ergodic theory, and measure theory.He is one of the most widely read mathematical expositors in theworld. ... Read more

Customer Reviews (1)

Thank you Mr. Halmos, for having wanted to be a mathematician...
I don't think my words of praise would do justice to this wonderful book. Halmos has strong opinions almost about everything and the way he talks about his examples are very wise. You don't need to be a would-be mathematician to enjoy the book. If you have ever wondered or invested some time in the world of mathematics, science and academia, Halmos provides you a very good account. If you are more than interested in math or maybe thinking about pursuing a Ph.D. this book will be much more valuable for you.

There are so many parts to be quoted from the book but I prefer to start a Wikiquote page for Halmos and pour sentences there. Halmos may not be one of the greats (according to his words) such as Euler, Gauss, Riemann, etc. but he is probably the greatest writer of such books.

All along the book I had a feeling: it was more like a frank and witty dialogue between me and the great mathematician (and lecturer) who had been there and done that. I kept on asking questions and Halmos kept on giving answers.

Thank you Mr. Halmos, for having wanted to be a mathematician, having been one of the best and having written such a nice book on what it was all about. ... Read more

Customer Reviews (18)

The Best First Step - But Only The First Step
Halmos' book was written forty years ago, so it is more than a bit out of date. However, it is still the best first step toward understanding set theory. Anyone who can add, subtract, multiply, and divide can pick this book up and start reading. Don't expect easy going: set theory isn't easy. You may need to read the book several times to fully understand it (fortunately it is so short this is not a problem - the entire book can be read in a couple hours).

After you have mastered this introduction you will be ready to move on to Suppes, Bernays, or even Quine.

And don't let the word "naive" fool you - this is an introduction to mathematical set theory - it just starts-out in the naive realm, but ends-up where it should, with the axiom of choice and other topics needed to model mathematics.

Absolutely essential for the beginner.

This book is ok but not that great
Pros:
1. This book is pretty concise and I appreciate the author's effort to only keep the essentials of naive set theory.
2. I like the presentation.There is a pretty smooth gradient in the increasing complexity of the topics.
3. This is a good introduction to more abstract mathematics.I used the first third of this book for a math lecture I gave at MIT.Of course, I had to do a fair amount of hand-holding to make sure everyone understood what was going on.

Cons:
1. There are not many problems following most of the chapters.At least having more worked examples would improve this book for autodidacts.
2. The writing style is pretty boring.One can easily lose the motivation to keep on reading.

Not worth the price
I already knew the main results of this book, but wanted to see how they related to each other and how the proofs went.However after finishing it I have to agree with the reader from Paolo Alto, this book is severely overrated.

Halmos is reputed to be a master expositor, but this book fails to meet basic levels of clarity and directness.First, the proofs consist mostly of English sentences rather than mathematical statements.In this respect it is similar to Spivak's Differential Geometry books, but even more verbose.I tend to have a hard time learning from books like these, but thankfully set theory is easier than differential geometry.Second, the book is written in a continuous narrative like a novel in that statements of theorems and their proofs are not separated or marked off from foregrounding passages, unusual for a math text.

Worst is Halmos's style.He is often willfully eccentric, always a sign of bad writing, and mixes a pedantic demand for notational purity with a shoot-from-the-hip approach to some of his most important results.I am surprised this book has won so many five star reviews, many seem to come from math enthusiasts rather than people engaged in serious study.In the end the material covered is adequate for such a book, and you will learn it, but only after spending more time than should be necessary.

Chapter one of Kolmogorov's real analysis book and Munkres' Intro to Topology both contain most of the set theory mathematicians typically use.If you want more, most of the books linked to this one will do a better job.

"Naive Set Theory" is A Masterpiece!
Naive Set Theory (Undergraduate Texts in Mathematics)

I discovered this book in 2004, 44 years after I had completed my Formal Education in Mathematics (ABD for Ph.D. JHU 1960) and 46 years after I'd completed a first year graduate course in Axiomatic Set Theory that was taught by "Ulian" and used Preprints of Suppes First Edition.

I absolutely adore Halmos' charming and irreverent style ("However important set theory may be now, when it began some scholars considered it a disease from which, it was to be hoped, mathematics would soon recover. For this reason many set-theoretic considerations were called pathological, and the word lives on in mathematical usage; it often refers to something the speaker does not like. The explicit definition of an ordered pair ( (a,b) = {{a}, {a, b}} ) is frequently relegated to "pathological" set theory.")

(I also studied his "Brand New" and long since "classic" "Abstract Vector Spaces" book as an undergraduate at MIT ca 1956.)

I have found a few very minor "flaws" in his presentation of "Naive Set Theory", including a "too clever" and redundant way of introducing the Null Set first by way of a "special" Axiom asserting that "There exists a set." and later as part of the "Axiom of Infinity".

My only concern with this book is related to the fact that he derives the Natural Numbers, and proves the Peano Postulates associated with them in such an "elementary way" (without first putting the reader through the rather difficult theory of Ordinal Numbers as Suppes does and more importantly as "The Supreme Master" Paul Cohen does in his out-of-print Monograph "Set Theory and the Continuum Hypothesis" (in which Cohen provides the solution of Hilbert's "First Problem" concerning the Continuum Hypothesis, that got him (Paul Cohen) the Field Medal), that I am fearful that Halmos' treatment of the Natural's is "too good (i.e. elementary) to be correct" and there may a be flaw or gap in Halmos' development of the Natural Numbers and proof of the Peano Postulates that I have been unable to detect.

If any reader of this review can detect such a flaw or gap in the treatment of the Natural numbers or any other part of the book, or confirm that there are no serious flaws or gaps in the book - I would appreciate such confirmation especially if it comes from someone who is familiar with Suppes' "Axiomatic Set Theory" and especially with Cohen's "Continuum Hypothesis" monograph, which is out of print, but also available on AMAZON!

what a wonderful read
Paul Halmos is regarded as one of the few mathematicians who can write well and engage his audience.He does not disappoint here.It is wonderful overview for us, naive persons, of the main logical and set theory axioms that are used today as a basis of everyday mathematics. The ideas of statements, relations and axioms gave me what I needed to understand a bit better my main interest....diff. geometry....and it's extensive set notation.Countable sets and powers of infinity helped with the ideas of analysis and it's sets of infinite series and sequences.So if logic and set theory exist only as union and intersection operations from your elementary school days, and is what got you you're engineering degree, then this is the book for you.It will give you a good idea of how set theory proofs are conceived and written.It gives you a taste of just how deep the rabbit hole goes. ... Read more

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All the Mathematical Reviews entries having operator theory (MR classification number 47) as a primary or secondary classification between 1980 and 1986 appear in these volumes. ... Read more

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Les achats comprennent une adhésion à l'essai gratuite au club de livres de l'éditeur, dans lequel vous pouvez choisir parmi plus d'un million d'ouvrages, sans frais. Le livre consiste d'articles Wikipedia sur : John Von Neumann, János Bolyai, Paul Erdős, Abraham Wald, Béla Bollobás, András Sárközy, Cornelius Lanczos, Paul Halmos, George Pólya, Raoul Bott, Miklós Laczkovich, László Lovász, Marcel Grossmann, Frigyes Riesz, Lipót Fejér, Rózsa Péter, Peter Lax, Alfréd Rényi, Julius König, Endre Szemerédi, Farkas Bolyai, Dénes Kőnig, Éva Tardos, Sándor Csörgő, Alfréd Haar, Mario Szegedy. Non illustré. Mises à jour gratuites en ligne. Extrait : John von Neumann (né Neumann János, 1903-1957), mathématicien et physicien américain d'origine hongroise, a apporté d'importantes contributions tant en mécanique quantique, qu'en analyse fonctionnelle, en théorie des ensembles, en informatique, en sciences économiques ainsi que dans beaucoup d'autres domaines des mathématiques et de la physique. Il a de plus participé aux programmes militaires américains. Benjamin d'une fratrie de trois, il s'appelle tout d'abord Neumann János Lajos (les Hongrois placent les noms de famille en tête) à Budapest en Autriche-Hongrie. Il est le fils de Neumann Miksa (Max Neumann), un avocat-banquier, et de Kann Margit (Marguerite Kann). Il ne prête guère attention à ses origines juives, sinon pour son répertoire de blagues. János est un enfant prodige : à six ans, il converse avec son père en grec ancien et peut mentalement faire la division d'un nombre à huit chiffres. Une anecdote rapporte qu'à huit ans, il a déjà lu les quarante-quatre volumes de l'histoire universelle de la bibliothèque familiale et qu'il les a entièrement mémorisés : doté d'une mémoire eidétique, il sera capable de citer de mémoire des pages entières de livres lus des années auparavant. Il entre au lycée luthérien de Budapest (Budapesti Evangélikus Gimnázium) qui était germanophone en 1911. E...http://booksllc.net/?l=fr ... Read more

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Originally published in 1957.

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This is a volume in honour of the 75th Birthday of Paul R.Halmos. The majority of the volume consists of two kinds of articles - broad surveys of specific areas of mathematics and historical reminiscences about mathematical life during the past 50 years. Additionally, it contains approximately 40 halftone photographs of mathematicians taken over the last 50 years. This monograph on general topics, math education, analysis and the history of mathematics is intended for mathematicians, and students of mathematics. ... 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

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A clear, readable introductory treatment of Hilbert Space. The multiplicity theory of continuous spectra is treated, for the first time in English, in full generality. ... Read more