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Could it be true that Whitehead and Russell's Principia Mathematica is the most influential book written in the 20th century?Ask any mathematician or philosopher--or anyone who understands the impact these fields have had on modern thinking--and you'll get a short answer: yes.Their goal, to set mathematics on a firm logical foundation, was revolutionary, and their tools and rigor continue to influence modern professionals. Using Peano's symbolic logic, they formalized axioms and produced theorems (including the famous "1 + 1 = 2") in orderings, continuous functions, and other areas of mathematics.

Although the Principia is far from comprehensive, Whitehead and Russell's method and program captivate their readers. The audacity to hope to formalize all of mathematics logically was inspirational and helped to give great boosts to math and logical philosophy.Though Gödel proved in 1931 that any such program is doomed to incompleteness, the tools found in and developed from the three volumes helped build the atomic bomb and the Internet. It may not be summer vacation reading (for most), but Principia Mathematica will reward the dedicated student with a deeper understanding of how we got here.--Rob Lightner ... Read more

Reviews (13)

A monument of mathematical logic
This book is the ultimate attempt to derive all of mathematics from logic while avoiding paradoxes of the sort that Russell himself sprang on Frege--and in passing, it gives in rigorous symbolic form Russell's "theory of descriptions."

Just as Bach took the baroque style of music about as far as it could go, Russell and Whitehead took this attempt to put mathematics on a firm logical footing about as far as it could go (and Goedel's incompleteness theorem killed off the hopes that mathematicians such as Hilbert had for the goal). Nevertheless, like any really good problem, it turned up worthwhile byproducts.

Alas, my exposure to the full three-volume set is confined to time spent at a university library; I could only afford the paperback volume of the first fifty-six chapters. I hope to eventually buy a copy of this classic work in its entirety.

Mostly of historical interest
The notation of PM is hard to read by anyone who learned logic post 1960, say. The typesetting is archaic. Hundreds of theorems are proved, but it is not clear where
they all lead. Russell and Whitehead are guilty of a number of major philosophical confusions, such as use and mention, between meta- and object language, and their confused notion of "propositional function." Their choice of axioms can be much improved upon. The PM theory of types and orders is a complicated horror; Chwistek, Ramsey, and others later showed that it could be radically simplified. R & W think they can substitute the intensional for the extensional, and ultimately define sets and relations in logical terms. PM does not have a clue about model theory or metatheory. There is no hint of proofs of consistency, completeness, categoricity, and Loewenheim-Skolem. In this sense, the fathers of modern logic are Skolem, Goedel, Tarski, and Church. And Goedel did indeed prove that there must exist mathematical truths that cannot be proved true using the axioms of PM, or any other finite set of axioms.

But this is still one of the greatest works of mathematics and philosophy of all time. The long prose introduction is a philosophical masterpiece. The collaboration between Russell and Whitehead may be the greatest scientific collaboration in British history. Whitehead, who was trained as a mathematician, went on to become one of the shrewder philosophers of the 20th century, and supervised Quine's PhD thesis. PM's treatment of the algebra of relations (a brilliant generalisation of Boolean algebra that
has not received the study it deserves) is perhaps the most thorough ever.

Mathematical logic is indeed the abstract structure that underlies the digital electronics revolution. And PM is still perhaps the greatest work of math logic ever penned.

A spoiler!
The denouement in which we discover that the Vicar was murdered by the Butler, in the Conservatory, with a Candlestick was weak. But the sex scenes, on pages 183 - 879 were the most sensitive yet erotic that I have ever read (except for page 1334 of the "Catalogue of Insects, Arachnids and Marsupials vol XXIV").

Top work, Whitehead and Russell! I eagerly await volume 4.

Principia
I decided to write a review, because, when reading the existing ones,- I realized their incorrectness. Leaving out the "Customer from Christchurch New Zealand", the rest shows an evident shallowness of mind. The reader "La-la land" utilizes an enormous mass of epithets discrediting Russell and Whitehead, which could be valuable in a form, but instead,- he shows a stupid prejudice that must have learned in his Mathematical-logic "polytechnic" course. I will only refute his last thought( which is the base of his "thesis"), because the others refute themselves. He presents Russell as a "Fruitless Mathematician", and even more stupid, compares him with Hilbert, saying: " at least he proved himself worthy.....". Throughout all Mathematics history we have individuals with enormous logic-constructive aptitudes, who although creating fundamentals results, were unable to understand their significance. Two perfect examples are Newton and Leibniz, both creators of the "infinitesimal calculus". One went on to construct the modern mechanistic view of physics in his "Principia". The other, with a much more profound understanding of logic, a superficial "monadic-substantial" and teleological ontology. Newtonian physics was a major episode in modern science, and Leibniz "subject-predicate" logic is the first glance at mathematical-logic.But their incorrect understanding of the infinitesimal calculus made them see, in it, the proof of an omnipotent god: they both conceived a universe with its first cause as god, and the human aptitude is, within it, merely an "algorithmic" one, which could never fully calculate god's creation. Hilbert, also providing fundamental results in constructive knowledge, went on to expose a somewhat "Hegelian" conception of mathematics, giving an almost silly definition of numbers. Both of this errors cause enormous damage, which I don't have space to describe now. Russell's "Principia Mathematica", although written with the wrong "motivation"( that is: to reduce the whole of mathematics into axiomatic form, finding the "universal method"), achieved unquestionable logic-mathematical results: The most valuable and original, the "theory of descriptions". in an abridged explanation, these theory comprehends the next: "algorithmic" function in logic and mathematics. when you say, " this is black", the theory of descriptions shows that you are only saying something about "this", which is a subject-variable(x), and black is an element-predicate, calculable within the conjunct "this". The theory permits mathematical-logic understand algorithmic functions, and is, also, what makes possible via your computer processor to read codified information. The result is more than a "fruit". it gives you the possibility of grasping that, like any other mathematical fruit, men is able of creating it,- and of reading it(calculate it). these means: Mathematical creations are only valuable as a source of human power, not as mystic ontological formulae,- that stupid motivation in all pseudo "Mathematicians".
In terms of actuality, the axiomatic system, the method, has been perfected, simplified, and transcended. If I had to recommend some books on the matter, I would say Tarski's: "Introduction to logic and to the methodology of the deductive sciences", Patrick Suppes:"axiomatic set theory", continued by the reading of the: "Gödel proofs" by Raymond Smullyan, some other text dealing whith "boolean algebra" such as: "logic as algebra" by Halmos. This would give any self-educated person, the basic models he needs to comprehend math-logic, the "method" with which he can possibly contribute to this "powerful trend of modern thought" as described by tarski. Remember that Russell and whitehead say in the introduction that they not claim having the most perfect axiomatic reduction, only that the one presented was enough to reduce mathematics into that form, which was, until godel, true, or at least "thought possible"(completely). Is important to undersatnd that "principia mathematica" made "possible" the incompleteness proofs of Godel: his original paper was named "on formally undecidable propositions of principia mathematica and related systems"(see dover edition), and although he uses mostly the axioms of peano in his system, if someone as Russel had not attempted successfully such axiomatic construction of math, godel would have never found or seen the incompleteness of arithmetic's. Something similar could be said of the later notions of completeness of first order logic, metamathematics, etc. The few works (few only in number) independent from principia may be the ones of: 1) the polish masters: Lukasiewicz, Lesniewski, and the last king Tarski. 2) the forgotten Richard Martin's and Rudolf Carnap's logic-syntaxic-semantic conception of math-logic. The rest walked, continued walking the path of principia. Individual example: Quine. ...

If you don't know know this book then you don't need it
Let me try to give a balanced review.

First this is a monumental work and one of the most influential works of the 20th century. I am not giving it five stars: this book earned them. With that said I don't think is the most influential book of the 20th century because such a book doesn't exist. In my opinion that kind of debate is totally misleading.

However the five stars do not suggest that you should buy this book. With the exception of libraries and scholars specializing in Russell or related subjects, I can't see anybody else spending [this amount] on a copy of this work. That is unless they like to collect books. For a math or philosophy student the paperback copy to *56 is all you need.

Unless you are a mathematician, a logician or a philosopher with a strong background in logic and philosophy of mathematics and aware of the issues surrounding the problems in the foundations of mathematics at the beginning of the 20th century then you are not going to benefit from STUDYING this book. The emphasis in studying is important because this book needs to be studied not just read like some reviewers may suggest.

If you are not an expert in this area and you want to learn about the subject then you may want to start with Bertrand Russell's "Introduction to Mathematical Philosophy". It summarizes the major points of this work for the layman and is Russell at its best (he won a Nobel prize mostly due to this book). Read it with a critical mind and then you can continue reading Quine, Putnam, Brower, Heyting and the rest. You can get a good bibliography from Benacerraf and Putnam's "Philosophy of Mathematics".

Finally if you are a mathematician, a logician or a philosopher you already know about this book and you don't need this review. Moreover you know you can borrow a copy from the university library for study...that is unless you like to collect books. ... Read more

Book Description

This book is the first to present a new area of mathematical research that combines topology, geometry, and logic. Shmuel Weinberger seeks to explain and illustrate the implications of the general principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity. He provides applications to the problem of closed geodesics, the theory of submanifolds, and the structure of the moduli space of isometry classes of Riemannian metrics with curvature bounds on a given manifold. Ultimately, geometric complexity of a moduli space forces functions defined on that space to have many critical points, and new results about the existence of extrema or equilibria follow.

The main sort of algorithmic problem that arises is recognition: is the presented object equivalent to some standard one? If it is difficult to determine whether the problem is solvable, then the original object has doppelgangers-that is, other objects that are extremely difficult to distinguish from it.

Many new questions emerge about the algorithmic nature of known geometric theorems, about "dichotomy problems," and about the metric entropy of moduli space. Weinberger studies them using tools from group theory, computability, differential geometry, and topology, all of which he explains before use. Since several examples are worked out, the overarching principles are set in a clear relief that goes beyond the details of any one problem.

... Read more

Book Description

This book provides a way to understand a momentous development in human intellectual history: the phenomenon of deductive argument in classical Greek mathematics. The argument rests on a close description of the practices of Greek mathematics, principally the use of lettered diagrams and the regulated, formulaic use of language. ... Read more

Book Description

"This book provides an in-depth treatment of classical nonmonotonic systems, in particular Default Logic. But its salient feature is a description of exciting recent results on Inference Relations, Belief Revision and their relations." -- Daniel Lehmann, Professor of Computer Science, Hebrew University

Nonmonotonic reasoning provides formal methods that enable intelligent systems to operate adequately when faced with incomplete or changing information. In particular, it provides rigorous mechanisms for taking back conclusions that, in the presence of new information, turn out to be wrong and for deriving new, alternative conclusions instead. Nonmonotonic reasoning methods provide rigor similar to that of classical reasoning; they form a base for validation and verification and therefore increase confidence in intelligent systems that work with incomplete and changing information. Following a brief introduction to the concepts of predicate logic that are needed in the subsequent chapters, this book presents an in depth treatment of default logic. Other subjects covered include the major approaches of autoepistemic logic and circumscription, belief revision and its relationship to nonmonotonic inference, and briefly, the stable and well-founded semantics of logic programs. ... Read more

Reviews (1)

The book provides a clear & detailed insight to the subject
The book covers all aspects of default reasoning, nonmonotonic reasoning , autoepistemic logic and circumscription in sufficient details. A little background of first order predicate logic is adequate to read this rich and valuable text.I offer it to our graduate program on nonmonotonic reasoning at ETCE department, Jadavpur University, Calcutta. My research scholars also like the book very much especially for its clear but elegant presentation. ... Read more

Book Description

Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic tchniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic. One device for making this book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry--Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992--1999. ... Read more

Book Description

This book contains an introduction to symbolic logic and a thorough discussion of mechanical theorem proving and its applications. The book consists of three major parts. Chapters 2 and 3 constitute an introduction to symbolic logic. Chapters 4-9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis. ... Read more

Reviews (2)

very readable introduction to automated theorem proving
Best introductory book on automated theorem proving available. Although it was written in the early 70's, it is written in a very clear, but mathematically precise, manner. It does not drown a reader with an abundance of symbols and definitions. It is a clear and well written exposition on automated theorem proving based on resolution. Unlike some recent text books, it does NOT use sequentzen logic systems. It covers all aspects of resolution-based theorem proving: different forms of resolution, deletion strategies, unification.

One of the Best in theorem proving...
If you are interested in Artificial Intelligence or you are one of those crazy guys who likes the Computer theory area this is a good book for you, mechanical theorem proving is an important subject into the AI area, all you want to know is Mathematical Logic, first order logic and predicate calculus. Very good book but very hard too (specially if you have to make some Mechanical theorem provers by yourself as a school project :) ) ... Read more

Book Description

Book Description

Cellular automata, dynamic systems in which space and time are discrete, are yielding interesting applications in both the physical and natural sciences. The thirty four contributions in this book cover many aspects of contemporary studies on cellular automata and include reviews, research reports, and guides to recent literature and available software. Chapters cover mathematical analysis, the structure of the space of cellular automata, learning rules with specified properties: cellular automata in biology, physics, chemistry, and computation theory; and generalizations of cellular automata in neural nets, Boolean nets, and coupled map lattices.

Current work on cellular automata may be viewed as revolving around two central and closely related problems: the forward problem and the inverse problem. The forward problem concerns the description of properties of given cellular automata. Properties considered include reversibility, invariants, criticality, fractal dimension, and computational power. The role of cellular automata in computation theory is seen as a particularly exciting venue for exploring parallel computers as theoretical and practical tools in mathematical physics.

The inverse problem, an area of study gaining prominence particularly in the natural sciences, involves designing rules that possess specified properties or perform specified task. A long-term goal is to develop a set of techniques that can find a rule or set of rules that can reproduce quantitative observations of a physical system. Studies of the inverse problem take up the organization and structure of the set of automata, in particular the parameterization of the space of cellular automata. Optimization and learning techniques, like the genetic algorithm and adaptive stochastic cellular automata are applied to find cellular automaton rules that model such physical phenomena as crystal growth or perform such adaptive-learning tasks as balancing an inverted pole.

Howard Gutowitz is Collaborateur in the Service de Physique du Solide et Rsonance Magnetique, Commissariat a I'Energie Atomique, Saclay, France. ... Read more

Reviews (1)

comment by the editor of the book
This book is 10 years old, but in this
slow-moving field where results are hard won,
most of it is still worthwhile. Just the
fact that you are contemplating buying this
book puts you in a very rare class IV of
individuals. Confirm your status and buy
a copy! 2000 were printed, looks like a few
still left. get 'em while they're hot. ... Read more

Book Description

"The best introduction to logic you will find."—Martin Gardner

Penetrating and practical, Logic Made Easy is filled with anecdotal histories detailing the often muddy relationship between language and logic. Complete with puzzles you can try yourself and questions you can use to raise your test scores, Logic Made Easy invites readers to identify and ultimately remedy logical slips in everyday life. Even experienced logicians will be surprised by Deborah Bennett's ability to identify the illogical in everything from maddening street signs to tax forms that make April the cruelest month. Designed with dozens of visual examples, the book guides readers through those hair-raising times when logic is at odds with common sense. Logic Made Easy is indeed one of those rare books that will actually make you a more logical human being. ... Read more

Reviews (1)

Learn logic in a friendly way
I really enjoyed this book. I've taken Critical Thinking, Philosophy, and Discrete Math at University but this book clarified ideas to me that I found difficult in formal classes. The book is a friendly, popular version of formal logic. The author presents history, nuances, and examples of logic that we easily misunderstand. Please realize the book is a readable subject of logic and not meant to replace expensive course textbooks. Afterall, look at the difference in price. If you truly love logic as I do, you'll find the book interesting and informative. If you read this book, then you will learn more about logic than you already know. ... Read more

Book Description

Reviews (9)

Great stuff.
First, this isn't one of Smullyan's popular puzzle books- its a serious mathematics text. Second, don't use this as your first exposure to first-order logic (note the title doesnt say "Introduction to ...")- although logically self-contained, it requires some experience to appreciate what a neat little book this is.

It's not a general mathematical logic text- there is no model theory (beyond basic Skolem-Lowenheim), incompleteness, recursion theory, or set theory. It covers tableaux (this alone is worth the price of the book), Hilbert-style axiomatic systems (briefly), sequent systems, Gentzen's Hauptsatz and Extended Hauptsatz, Craig's and Beth's theorems, and more. But the heart of the book is completeness theorems, their proofs, and closely related material such as compactness and Herbrand-like theorems. Smullyan shows there are two main approaches to completeness (analytic vs. synthetic), breaks each into stages, provides nice abstracted formulations, and usually gives several different proofs of each result. The centerpiece is his "Fundamental Theorem of Quantification Theory", a theorem associating a truth-table tautology with every valid first-order sentence (check out the amazingly slick proof of completeness for the the Hilbert-style system that this provides). Similar constructions such as magic sets are also discussed. All this forms a much more extensive and illuminating look at completeness proofs than I've seen elsewhere.

The first-order logic used in the book has no equality and no function signs. There are few exercises, most of them simple. Smullyan writes clearly and with an appropriate amount of rigor (but its not as polished as his later books). Makes a great supplement to more general-purpose introductory mathematical logic books. If you haven't seen the tableau method yet buy this book immediately. Experienced readers will appreciate the sophisticated coverage of completeness proofs.

An Oddity But a Good-ity. Wait, that's terrible.
The reviewer from Illinois gave a very good characterization of Smullyan's style here:
"Smullyan has divorced logic from its roots: logics are simply recursively-defined sets of sentences and mappings, and that is that. No discussions, ala WvO Quine, on the history or linguistic difficulties of a concept, just definition and proof."
Readers familiar with Smullyan's enormous talent for popular exposition may be expecting the same herein: not so. This is very much for people who have attained what medical professionals call "mathematical maturity" (which is about as difficult to attain as zen, yet perhaps amounts to little more than the ability to read VCR instruction manuals). For example, the very first section is a wiz-bang treatment of trees (not the usual graph-theoretic ones), defined in the abstract/axiomatic fashion.
Of course, people who spend perhaps way too much of their time steeped in math are attracted to treatments of just this sort.
A structural characterization in terms of sets and mappings is much more meaningful, interesting, and aesthetically pleasing to those with these unusual inclinations (compulsions?) than a characterization framed significantly by historical motivation (please understand that I'm speaking roughly here). This is why I gave a positive review. A star was witheld for the selfish reason that I'm not sure I'll find much use for such an odd treatment of model theory, the topic for which I was seeking a more mainstream treatment when I purchased this. Regrets are nonetheless few: time spent reading Smullyan is never a waste.

Wonderful--Why Can't I Assign It 4.5 Stars?
This is a great book, and served as my introduction to tableaus. I think it strikes a good balance between the conciseness of a math text and the verbosity often found in philosophy texts; it's also very reasonably priced. My only complaint is one that I noticed in a previous review: some exercises are too difficult and there are no solutions (the former wouldn't be a problem if the latter weren't the case). Also, this book isn't for the total rookie-some prior knowledge is assumed. My choice for introductory material would be Copi's Symbolic Logic or even his Introduction to Logic (with Cohen) for those with no or limited background in mathematics.

a classic
I mainly bought this book because of the influence it has had on numerous modern-day logic texts. If you are unfamiliar with the tableaux method for structural proofs, then you will gain alot from reading this, as it provides a different perspective from the more popular Hilbert-system approach. Tableaux systems, of course, have been made popular because they are easy to program with a computer. Please see Gallier's "Logic for Computer Scientists" for more on this matter.

Great as a Reference, Probably Not for True Beginners
This is an excellent reference! It has more material covered in just 155 pages than most other works address in twice as much space. I refer to it very often.

However, I doubt it would be appropriate for someone that has not previously been introduced to the material. If a truly introductory text is required, I would look elsewhere. ... Read more

Book Description

Model theory investigates the relationships between mathematical structures ('models') on the one hand and formal languages (in which statements about these structures can be formulated) on the other. Example structures are: the natural numbers with the usual arithmetical operations, the structures familiar from algebra, ordered sets, etc. The emphasis is on first-order languages, the model theory of which is best known. An example result is Löwenheim's theorem (the oldest in the field): a first-order sentence true of some uncountable structure must hold in some countable structure as well. Second-order languages and several of their fragments are dealt with as well. As the title indicates, this book introduces the reader to what is basic in model theory. A special feature is its use of the Ehrenfeucht game by which the reader is familiarised with the world of models. ... Read more

Book Description

Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Lévi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence of unconscious mental processes in mathematical invention and other forms of creativity. Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field, remains an important tool for exploring the increasingly complex problem of mental life.

The roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. His discussion of this process comprises a wide range of topics, including the use of mental images or symbols, visualized or auditory words, "meaningless" words, logic, and intuition. Among the important documents collected is a letter from Albert Einstein analyzing his own mechanism of thought.

Reviews (3)

The Psychology of Invention in the Mathematical Field
Not only is this book fascinating, it's the only one of it's kind. The book has also proved very useful to me in life. As a graduate student I used Poincaré's implicit `advice' (described in the book) in the following way. In electrodynamics we had a long problem sheet to hand in every two weeks. I started by writing down answers to all problems that I knew. Then, I thought about the next-easiest problem each day walking twice to and from the University (about 1 1/2 hours altogether). When the answer came I wrote it down and iterated the process. Before the end of two weeks most of the problems (from Jackson) had been solved. Poincari's advice is very good about giving the unconscious a chance to work. Phooey and double phooey on the silly, uncreative skinner-box types and other behaviorists who don't recognize the unconscious as the source of creativity!

The Psychology of Math
The Mathematician's Mind is a study on how research mathematicians go about the business of advancing their field.Jacques Hadamard, a prominent mathematician, wrote this psychology text over 50 years ago, after having done his best work 50 years prior. Although in some ways dated, both in content and in writing style, the book provides an interesting examination of the role of the conscious and subconscious in solving a problem, particularly the process of incubation and (seemingly) sudden inspiration. He brings up the roles intuition and logic play in the way various mathematicians go about their business.Hadamard also examines the influence of aesthetics in not just choosing a problem, but in solving it. He studies the choice of research direction, with the interesting comment that Hadamard himself avoided areas of research where there was already a great deal of activity.

The book is short enough that if the subject interests you, it is worth your time.

The text is also published under the title "The Psychology of Invention in the Mathematical Field."

A study of the mental workings of some great mathematicians
This is a short study of how creative thought works.Hadamard, a world-class mathematician best known for his proof of the prime number theorem in 1896, wrote this in the 40's, basing it on correspondence with many of the great living mathematicians of his time.The actual questions he posed are preserved in an appendix.

Most of his respondents were mathematicians (and he limited his correspondence to the best minds in the field), but he did get information from several other fields, and cites data about physicists (a letter from Einstein forms another appendix), chemists, physiologists, metaphysicians, and so on.What he is trying to examine is a slippery subject, perhaps best explained by a quote.Here is a discussion of Sidgwick, an economist: "His reasonings on economic questions were almost always accompanied by images, and the images were often curiously arbitrary and sometimes almost undecipherably symbolic.For example, it took him a long time to discover that an odd symbolic image which accompanied the word 'value' was a faint, partial image of a man putting something on a scale."

Hadamard gives his own mental images that accompany his following through the steps of Euclid's famous proof of the infinitude of primes.I won't reproduce that here for space reasons, but the contrast with Sidgwick's--and with other reports of mental activity--is fascinating.Many other examples are given, from Mozart to Polya to Galton to Poincare. Hadamard makes it clear that language and thought are not the same thing, contrary to a commonly expressed view among linguists.He cites Max Muller's comments equating thought and language, and acknowledges that for Muller it may be so, but convincingly demonstrates, by quoting numerous other mathematicians, that it is not true for everyone.The further conclusion, that the process of creative thought, while following similar patterns in similar discipline, can vary dramatically, is as far as Hadamard can go with the data he has.

One other note: this book was originally titled "The Psychology of Invention in the Mathematical Field" and is available under that title from Amazon, published by Dover Books.It's not immediately clear from the Amazon page that this is so.The Dover edition is substantially cheaper.

A fascinating and informative book. ... Read more

Book Description

This is the first single-volume edition and translation of Frege's philosophical writings to include all of his seminal papers as well as substantial selections from all three of his major works.It is intended to provide the essential primary texts for students of logic, philosophical logic, philosophy of language, and philosophy of mathematics.It contains in particular Frege's four papers "Function and Concept", "On Concept and Object", "On Sense and Reference", and "Thought", and new translations of key parts of the Begriffsschrift, The Foundations of Arithmetic, and the Basic Laws of Arithmetic. The editor's substantial introduction provides the reader with an overview of the significance and development of Frege's philosophy, while the footnotes, appendices and glossary facilitate understanding of some of the more difficult elements of Frege's thought. ... Read more

Reviews (3)

Comment on The Frege Reader:
What a great book this is! The Frege Reader is not for everybody, that's for sure. But when/if you get into the "right space" - then please read this book.

I can't remember when I first heard the name "Frege". But I do know how my reading and study began that eventually brought me to stumble across this mathematician, logician, and philosopher. You see I'm a software developer, more specifically a database guy. I have read much of Chris Date and Hugh Darwen's work. They say that programming languages and databases are considered to be "formal systems", that is to say, a formal system of logic. Date and Darwin go on to say that what we are really doing when we call the database to create an answer set is "instantiating the predicate". So, I started on a path to learn what a "predicate" is. It did not take long before the names: Russell, Whitehead, Wittgenstein, and finally, Frege came up.

There are many fine authors who have written about Frege's logic and philosophy. But, until you read his words (and his words are really, really good!) you really don't get a sense for what this man was really trying to say. This book is not just talking about numbers. This book is about everything we can talk about. Using Frege's "perfect language" we learn to distinguish between "objects", and what we say about those "objects".

So, I learned from this book that when I "instantiate my predicate" I am (in Frege's words) finding the content of the concept, saturating the concept, finding its meaning, its "Bedeudung", returning thoughts to my user.

In his book, LOGIC, LOGIC, and LOGIC, George Boolos quotes one of his professors. The professor said that the way to seduce good students to philosophy is to teach them Russell's and Frege's concept of number. Programmers and DBAs can also be "seduced" by reading Frege. So, if you want to be "seduced" to philosophy, then read The Frege Reader.

Stephen A. Wilson
sawilson3@att.com

Nice collection of an important philosopher
This is a nice selection of excerpts and full essays written by Frege. The book is a pleasure to read, however, not only becaues of the selections and the fine introductory section, but because Frege is such a clear writer and thinker himself. I particularly enjoyed Frege's Begriffshrift - you can see modern quantificational logic being born.

All you need and more
The Frege Reader is an excellent collection of Frege's works. The texts are edited carefully and the editor has supplied extremely helpful footnotes throughout. The introduction and appendices are clear resources that the reader will consult often as she works through the text.

The excerpts from many of Frege's letters are a great addition as these shed light on the development of his project. This work will remain for years the standard first place to turn for Frege. ... Read more

Reviews (2)

Very good but be aware of omissions
This book is indeed much shorter than Principia, mainly because it is derived for lecture notes for a 1 semester PhD course. It is also a lot clearer than PM. But the notation is largely the same, which makes for hard reading if your are under 50. Quine's proof format doesn't take up much space, but has always eluded me. This book contains the best treatment of truth functional and quantificational logic prior to natural deduction and truth trees.

I like the set theory of this book, but I warn you that it is very nonstandard. Even ardent lovers of Quine's NF theory hate
the ML theory of this book.

The weakness of this book is its treatment of metatheory:
consistency, completeness, decidability, categoricity. The treatment of Godel's incompleteness is detailed and highly original (altho' it owes more to Tarski than to Godel). But it is very difficult, and Smullyan (1991) is much better.
Quine also had no clue re model theory or recursion.

I respect the historical remarks a lot. Just one big omission: Quine, like nearly everyone of his generation, missed that
math logic as we know and love it does not descend from Frege, but from an 1885 article by C S Peirce.

In Depth Look at Logic
Try this book when you know a bit about the basics of logic. The descriptions are much more lucid than those in Principia, even if the ideas are less earthshattering for there time. Quine, as he always does, gives a masterful, detailed look at logic. If you are a fan of logic and the foundations of math, this book is not to be missed. ... Read more

Book Description

This book, written by one of philosophy's preeminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics.This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. Hintikka's new logic is highly original and will prove appealing to logicians, philosophers of mathematics, and mathematicians concerned with the foundations of the discipline. ... Read more

Reviews (7)

The importance is evident even for an amateur
I'm no matematician, no philosopher, just found the book in a bookstore and got curious about the presentation of a 'new logic' (the little I know was from philosophy class in high school). As an amateur you are faced with formulae and notation that you have to figure out on your own, which makes it harder to read it, but I did manage to figure it out with the help of the Equation Editor in the Office Suit. Just a single page explaining the notation would have been enough to make the book self-contained, even for an amateur. The language is simple enough even for a non-native English speaker (it is typically easier reading books written by non-native speakers, since their language is less complicated).
Regarding the subject, it is very, very interesting, because of the profound implications this will have--I'm sure of that--on the development of sicence, once we get a new generation of scientists who are not entrenched. It was like reading something from the future. It's been years since I read it, but I still have it in fresh memory, and the information-independent logic is the only logic I would consider using, if it came to that. When working with incomplete information, I realize that it is essential to be aware of the lack of information. And what better way to introduce the tools for handling incomplete information, than on the basic, logical level?
I submit, that in a few decades, the ideas of this book will pervade software in our computers.

A pleasure to read
This book re-examines first-order logic as it has been applied to the foundations of mathematics. But it is much more than that. If you are interested in the human decisions behind why logics were built as they were, if you want to understand the impact of Godel's Incompleteness Theorem, if you are interested in understanding philosophy of inferential systems in general, then you will find this book quite profound and useful.

Requirements for reading this book are roughly: a general knowledge of syllogistic and first-order predicate logic, an idea of what Godel's theorem is about and the concept of godel-numbering, some philosophy (e.g., ontology vs. epistemology), but mostly a keen interest at learning about logic and it's foibles and potentials.

Chapter 1 begins with the Hilbert program, and the attempt at axiomatization in general. Chapter 5 clears up alot of confusion about the Godel Incompleteness theorem and what it really means. He delineates between descriptive, semantic, deductive and Hilbertian completeness notions, and describes their inter-relatedness and Godel's theorem's role. These chapters alone are useful for gaining deeper understanding of the problems that arise in syntactic axiomatic deductive systems.

Chapter 7 is on the Liar Paradox, and he offers a unique solution to that based not upon Austinian notions, but rather based upon Hintikka's IF ("independence-friendly") first-order logic which avoids resorting to infinities or relying on any semantic re-interpretation (Hintikka uses a simple formal statement "~T[d]" where d is the godel-number of that statement, as the basis of the discussion).

He then goes on to discuss the presumed role of axiomatic set theory and chips away at it's pretense as a secure foundational approach.

But this merely scratches the surface. The book is primarily about the human decisions that were made, the reasoning behind them and why/where they failed. This is part of what makes it so readable and engaging. For Hintikka, logic and math seem to be very human activities, and there is no attempt to sanitize logic as being something pure or absolute.

As an explication of human decision-making in logic, I think this book has important insights buried within and consequences for the inferential world of logic and mathematics, as well as reasoning in general. It will take several readings to grasp it's profound implications.

'IF logic' itself (chapter 3) is a ridiculously simple and brilliant enhancement to first-order predicate logic, produced merely by lifting the mandatory left-to-right scoping restrictions Frege had placed on quantifiers in the syntax. And he extends (no pun intended) that notion by similarly lifting restrictions on mandatory scoping across operators as well. What arises looks very much like ordinary predicate logic, but the scoping independence opens up new vitality to the logic that makes it's applicability broader, as well as philosophically more interesting.

IF logic, in particular, is more amenable to being about imperfect information, and information independence (hence "independence-friendly logic"). Hintikka's version of truth-definition is about a verification game (as in game theory), not a Tarskian retreat to a metalevel of formalism. Throughout, there are these kinds of comments and concepts on relating logic back to the world.

IF logic is an intriguing example of how a subtle change in rules of syntax can have large consequences, and Hintikka is definitely pushing for it as -the- preferable first-order logic (actually, family of logics) over standard predicate logic. (And for game theoretical semantics and model theory as his preferred meta framework.) However, Hintikka's salesmanship aside, the insights in the book are not dependant on IF as being -the- alternative, but as a demonstration of those insights.

As a non-mathematician/non-logician, I had braced myself for a slog through a dry, tough read (particularly since there are nearly two decades of rust accumulated on my predicate logic skills) despite the positive reviews I had read on Amazon, but was pleasantly surprised at the lively writing style and also the modicum of formulae, with no tedious proofs to sweat over. Even the final chapter on "Epistemology of Mathematical Objects" is quite readable. And with some chapter headings like "Who's Afraid of Alfred Tarski?" and "Axiomatic Set Theory: Fraenkelstein's Monster?" you know the author enjoys his subject matter. :)

BORING !
I read this book for preparation to enter City College of San Francisco's Math 840-Elementary Algebra Class. It took me like 3 months to get through this and it didn't help me out at all. On top of that it was so boring I was afraid I might die of boredom at times. This book is of no help at all for understanding the principles of mathematics. If this guy was my teacher, I'd give him a lousy evaluation.

Throwing down the gauntlet!
I am a working mathematician with a PhD in Differential Geometry who has recently become interested in the foundations of Mathematics. I came to this book circuitously via a pdf by Jaakko called "A revolution in Logic". Jaakko is on the offensive in this book, and he certainly can be polemic, to say the least. Having said that, the book is a tour de force: as a book on first order logic and IF, on the philosophy of mathematics, and the nature of mathematics and its objects and structures---it has gotten me thinking on many different levels. I would guess English is not his first language (due to easily corrected spelling, grammar, Etc), but despite this the writing style is quite smooth, with some interesting turns of phrases. I get the feeling this is one of the most brilliant books that I will ever have the pleasure of reading AGAIN and AGAIN! My 2cents David Thompson Assistant Professor of Mathematics and Computer Science Olivet College, Michigan

A philosophical and logical milestone
I read this book last year and I found it both entertaining and interesting, but only recently I was struck by how profound it is. I find Mr. Hintikka's insight into the dual use of logic (description vs. inference) and the conflict it caused in the development of logics (expressivity vs. axiomatic completeness) fascinating. He decisively resolves this conflict in favour of expressivity by introducing a brand of logic which is amazingly expressive yet non-axiomatizable. So, instead of proving propositions, we validate them through calculations in the semantic model.

I don't think this book will have the impact it should, only because the philosophical-logical establishment is already entrenched in certain ways of thinking that it can not abandon. And I think Mr. Hintikka is painfully aware of this. His tone is polemical, almost vitriolic at times, and it has a certain voice-of-reason-crying-in-the-wilderness streak to it. While entertaining, the style detracts from the importance of the book.

I consider "The Principles of Mathematics Revisited" one of the most important books on logic ever. Its impact will not be immediate but it should eventually be momentous. ... Read more

Book Description

Book Description

[see attached for complete text]{\it Calcolo Geometrico}, G. Peano's first publication inmathematical logic, is a model of expository writing, with asignificant impact on 20th century mathematics. Kannenberg's lucid andcrisp translation, {\it Geometric Calculus}, will appeal to historiansof mathematics, researchers, graduate students, and general readersinterested in the foundations of mathematics and the development of aformal logical language. In Chapter IX, with the innocent-sounding title"Transformations of a linear system," one finds the crown jewel of thebook: Peano's axiom system for a vector space, the first-everpresentation of a set of such axioms. The very wording of the axioms(which Peano calls "definitions") has a remarkably modern ring, almostlike a modern introduction to linear algebra. Peano also presents thebasic calculus of set operation, introducing the notation for'intersection,''union,' and 'element of,' many years before it wasaccepted. Despite its uniqueness, {\it Calcolo Geometrico} has beenstrangely neglected by historians of mathematics, and even by scholarsof Peano. The book has never been reprinted in its entirety, and onlytwo chapters have ever been translated into English. In part, thisneglect has been due to Peano's organization of the work. That is, thesection on mathematical logic bears almost no relation to the rest ofthe book, and the material there was superseded only a year after itspublication by Peano's second book.Since all but this first sectionwas generally thought to be expository rather than original work, itwas regarded lightly, if noticed at all, and ultimately all butforgotten. Only in very recent years have the book's unique meritsbegun to be recognized. ... Read more

Book Description

Hardbound. ... Read more

Book Description

The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to English-speaking students. This book is an elaborate version of the course on 'Conjecture and Proof'. It gives miniature introductions to various areas of mathematics by presenting some interesting and important, but easily accessible results and methods. The text contains complete proofs of deep results such as the transcendence of e, the Banach-Tarski paradox and the existence of Borel sets of arbitrary (finite) class. One of the purposes is to demonstrate how far one can get from the first principles in just a couple of steps. Prerequisites are kept to a minimum, and any introductory calculus course provides the necessary background for understanding the book. Exercises are included for the benefit of students. However, this book should prove fascinating for any mathematically literate reader. ... Read more

Reviews (1)

The basics of proof techniques covered in sufficient depth
As the title indicates, the legendary Paul Erdos was involved in the creation of this book. In 1983, Erdos and other Hungarian mathematicians started the Budapest Semester in Mathematics (BSM), a program for American and Canadian undergraduate students. One of the courses in this program involves creative problem solving, which was the motivation for the material in this book. As is the case with books on problem solving, no particular area of mathematics is examined. The emphasis is on proof techniques, which are largely independent of the mathematical topic.
Of course, the quality of any book of this type is largely dependent on the choice of problems that are described, and in this case the chosen topics are excellent. The book is split into two main sections, which are further split into the following subsections:

I) Proofs of Impossibility, Proofs of Nonexistence.
1) Proofs of Irrationality.
2) The Elements of the Theory of Geometric Constructions.
3) Constructible Regular Polygons.
4) Some Basic Facts About Linear Spaces and Fields.
5) Algebraic and Transcendental Numbers.
6) Cauchy's Functional Equation.
7) Geometric Decompositions.

II) Constructions, Proofs of Existence.
8) The Pigeonhole Principle.
9) Liouville Numbers.
10) Countable and Uncountable Sets.
11) Isometries of R^n.
12) The Problem of Invariant Measures.
13) The Banach-Tarski Paradox.
14) Open and Closed Sets in R. The Cantor Set.
15) The Peano Curve.
16) Borel Sets.
17) The Diagonal Method.

While each of these topics is introduced, that does not mean that the coverage is superficial. The book is advertised as having more than elementary coverage, and I concur with that assessment. Detailed proofs of the main ideas are included with exercises at the end of each section. Hints for the solution of many of the problems are included in an appendix.
This is an excellent short introduction to many of the proof techniques that are the staple of working mathematicians. I strongly recommend it as a primary or secondary text for any course where the goal is to teach basic proof techniques to advanced undergraduates. ... Read more

Book Description

This is the best text on complexity theory I have seen, andcould easilybecome the standard text on the subject...This is thefirst modern text on the theory of computing. ---William Ward Jr, Ph.D,University of South Alabama

Taking a practical approach, thismodern introduction to the theory of computation focuses on the studyof problem solving through computation in the presence of realisticresource constraints. The Theory of Computation explores questions andmethods that characterize theoretical computer science while relatingall developments to practical issues in computing. The book establishesclear limits to computation, relates these limits to resource usage,and explores possible avenues of compromise through approximation andrandomization. The book also provides an overview of current areas ofresearch in theoretical computer science that are likely to have asignificant impact on the practice of computing within the next fewyears.

Highlights Motivates theoretical developments by connectingthem to practical issues. Introduces every result and proof with aninformal overview to build intuition. Introduces models through finiteautomata, then builds to universal models, including recursion theory.Emphasizes complexity theory, beginning with a detailed discussion ofresource use in computation. Includes large numbers of examples andillustrates abstract ideas through diagrams Gives informalpresentations of difficult recent results with profound implicationsfor computing.

The writing style is very literate and careful. Thisis a well-written book on theoretical computer science, which is veryrefreshing. Clear motivations, and lucid reflections on theimplications of what the author proves abound. ---James A. Foster,Ph.D., University of Idaho ... Read more

Reviews (3)

If you love formalisms...
This is the assigned text for a graduate class in Foundations of Computation that I'm currently taking.I have thus far struggled through the first five chapters, and feel confident about making the following statement: Unless you *already* have a strong intuitive grasp of the concepts that the book covers AND are very comfortable with mathematical formalisms, you will find this to be a very unrewarding book (as I have).

I found it absolutely necessary to supplement my reading with the Sipser book (Introduction to the Theory of Computation), which delivers the essential concepts much more cleanly and powerfully.The problem with the Moret book is that the formalism adds almost nothing, while making the concept so much more difficult to extract.For example, the book certainly defines the mathematical notions of recursive and recursively enumerable sets, but the treatment will (I think) only resonate with those already broadly familiar with Turing decidability and recognizability.Otherwise, the discussion seems terribly abstract and unimportant.

I'm not being complete fair, because Moret doesn't leave things completely at the abstract level, but what I'm saying is that when you open this book, you have to already know what you're looking for.So if you already are familiar with issues of decidability and are ready for a more formal development (which may have no practical value whatsoever), you may get something from the book.If you open the book to see what theory of computation is all about, you will probably regret it.

If you're getting started, get the Sipser book.

-- Big Muggle

Beware the bad binding
I don't think I can include myself in the book's target audience, so Iwill withhold a review of the content.

However, I want to warnpotential buyers that the construction of the book may be very poor.Ofthe 15 or so used copies of the book in the Rutgers University bookstore,almost all had broken bindings.This despite no evidence whatsoever of anyabuse (no bent corners, scarred covers, ripped pages, etc.).The bindingon the used copy that I purchased also broke after about two weeks ofextremely gentle use.

If you think (as I do) that an $80 book ought tohave as robust a construction as technology allows, you may want to avoidthis edition.

Unsuccessful
Always I can find something worse than typor in the book. Though the author want to introduce some nice topic on the subject, too many errors and confusion make it not enjoyable. ... Read more