Book Description

In Figuring Space Gilles Châtelet seeks to capturethe problem of intuition of mobility in philosophy, mathematics andphysics. This he does by means of virtuality and intensive quantities(Oresme, Leibniz), wave-particle duality and perspectivediagrams, philosophy of nature and Argand's and Grassman's geometricdiscoveries and, finally, Faraday's, Maxwell's and Hamilton'selectrophilosophy. This tumultuous relationship between mathematics, physics andphilosophy is presented in terms of a comparison betweenintuitive practices and Discursive practices. Thefollowing concepts are treated in detail: The concept of virtuality;thought experiments; diagrams; special relativity; GermanNaturphilosophie and `Romantic' science. Readership: The book does not require any considerablemathematical background, but it does insist that the reader quit thecommon instrumental conception of language. It will interestprofessional philosophers, mathematicians, physicists, and evenyounger scientists eager to understand the `unreasonable effectivenessof mathematics'. ... Read more

Amazon.com

Francis Sullivan of the Institute for Defense Analysis said "Great algorithms are the poetry of computation"; David Berlinski calls the algorithm "the idea that rules the world." The Advent of the Algorithm is not so much a history of algorithms as a historical fantasia.Berlinski spins freely between semifictional accounts of historical figures, personal reminiscence, and mathematical proofs--without ever really defining an algorithm in so many words.

This is not the book for those who were maddened by Berlinski's A Tour of the Calculus; his style remains quirky, digressive, self-referential, and dense:

And then, by some inscrutable incandescent insight, Leibniz came to see that what is crucial in what he had written is the alternation between God and Nothingness. And for this, the numbers 0 and 1 suffice.

Twinkies and Diet Coke in hand, computer programmers can now be observed pausing thoughtfully at their consoles.

Berlinski's argument seems to be that algorithms--step-by-step procedures for getting answers--superceded logic, and will be superceded in turn by more biological, empirical, fuzzy methods. The structure of the book reflects this argument--sketches of people like Leibniz, Hilbert, Gödel, and Turing are interwoven with proofs and with characters of Berlinski's own invention. Berlinski's voice, closer to Hofstadter than to Knuth, remains unique. --Mary Ellen Curtin ... Read more

Reviews (35)

Does not quite compute, yet...
'The Advent of the Algorithm' contains useful data but one must wade through torrents of purple prose to find the nuggets. My impression is that Dr. Berlinski wanted us to feel the romance of the idea, but wasn't quite sure how to present that idea. The book seems to have been written in haste and by no means matches the excellence of his 'Tour of the Calculus.' Nevertheless, a novice, such as me, can benefit from Berlinski's discussion of the Turing and Post machines, Goedel's theorems, and Church's lamda calculus. The work of these logicians made precise the concept of 'algorithm,' says Berlinski. I think I know what he means, but he is a bit vague -- though admittedly he has the difficult task of trying to present a rarefied subject to a lay readership. We learn, for example, that if a problem is Turing computable, then an algorithm exists for its solution. So that means, I suppose, that, in this case, an algorithm is the set of instructions given that computes an answer on a Turing machine. To me, this does not quite get at the nitty gritty of what an algorithm is .... I realize that, in writing this book, Berlinski was balancing his poetic instinct against his mathematical streak. He doesn't quite succeed in the balancing act. Yet the discussion of the work of the four logicians makes the book useful. Perhaps Berlinski would have done a bit better had he, before writing his book, designed an algorithm to outline what he wanted to say. I recommend that he write another, more serious book focused specifically on the life and work of the logicians he cites.

The Advent of the Algorithm
David Berlinski has written two popular books on mathematics, the first entitled, A tour of the Calculus, and second, The Advent of the Algorithm. The theme of this duo of books is that mathematicians have produced two great ideas of 'the great scientific culture of the West.' Neither book requires the reader to have more than a high school level of mathematical knowledge. He does present proofs in appendices that a lay-person might find difficult or beyond their ability to follow; however, these are not required in order to understand the major ideas of the books.

The author's thesis, as stated at the beginning and end of both books, is that the analytical thought of Calculus has gone through it's cycle of growth and is now, for the most part, come to a stand-still, while the sort of mathematical logic embodied in the computer's use of the algorithm, has emerged as the succeeding great idea 'of the great scientific culture of the West.' Yet, the content of both books is not so much an argument in support of this thesis but a guided tour of the essential ideas of both mathematical methods.

Mr Berlinski is an emancipated professor of college mathematics and clearly knows his subject. He also is a sophisticated writer, presenting the reader with plenty of rhetorical devices in an attempt to make the terse matter of mathematical concepts easier to digest. These devices include imaginary reconstructions of plausible scenes and dialog he might have had with the great pioneering mathematicians, past professors and students. He also frequently meanders into metaphysical interpretations of the mathematical ideas, particularly between sections of the book bearing proofs. His choice of vocabulary can be challenging; I recommend having a pocket dictionary on hand.

Mr. Berlinski's second book, "The Advent of the Algorithm," describes the evolution of the second great mathematical idea: the algorithm. It begins with a portrait of the acknowledged founder of modern logic, Gottfried Leibnitz, who envisions a universal logic where all facts about the world can be organized and analyzed systematically and so avoid human corruption. Next, the reader encounters an attempt to achieve Leibnitz' vision within the field of mathematics. The first step towards the development of the algorithm are the attempts of 16th century mathematicians, such as Fourier and Leonhard Euler, to define irrational and transcendental numbers through the sums of numeric sequences. Advances in these sorts of studies led to critical ideas about how to define the limits of infinite sequences and the need for a logically cogent and encompassing numbering system. Logicians such as Peano, Georg Cantor and Gottlobe Frege set forth their answers in the form of well-constructed axioms that define limits, real numbers, and arithmetical systems. The dawn of the 20th century saw David Hilbert propose his famous 20 mathematical problems, in the hope of establishing all mathematics upon a few irreducible axioms and rules for working them. Just when the ponderous tomb, Prinicipia Mathematica of Bertrand Russell and Alfred Whitehead appeared to fulfill the holy quest, the Swiss, Kurt Godel published his famous Incompleteness Theorems that demonstrated the impossibility of such a system. The silver lining of Godel's work was his discovery of a remarkably succinct method for representing mathematical functions, by use of recursive algorithms. Something of the idea seemed to be stirring in the mid-twentieth century, for variations of these recursive algorithms were simultaneously discovered by Alonzo Church and Allan Turing. While Alonzo Church achieved his project through the design of a clever form of symbolic notation, Allan Turing conceived of a simple mechanical machine--a computer--that could carry out any computation that was logically possible. Curiously, while the Turing machine corroborated Godel's famous theorem, implying the impossibility of a set of axioms to account for all mathematical possibilities, he nevertheless felt certain that his machine represented the essential workings of human thought and that it was only a matter of time before a computer's output could be mistaken for the range and depth of human thought.

In the final pages of this book, the author describes the onset of the biological sciences, with its discovery of genetic coding in DNA. The discovery is timely, for western science had just begun to experience man-made algorithms in working with computers. The author suggests that DNA is a sort of natural algorithm where enumerable variables are coordinated to produce living things. This brings us to the present state-of-the-art where biologists are attempting to make sense of the human gnome with the aide of the computer.

Not quite as good as A Tour of the Calculus
Purple prose aside, this book is not as illuminating as his Calculus book. But then again the subject itself is not as well developed as the Calculus, and the author's mastery of this area is not as sure. The fact is modern logic has never recovered from Russell's paradox and Godel's theorem is more often quoted then understood, or for that matter explained.

The author is also a bit chauvinistic in attributing the idea of algorithm to European root when the word algorithm itself came from the name of an Arabic mathematician who taught the world algebra, a fact never mentioned in the book. Also the assertion that calculus and algorithm are the only two great ideas in modern science is wildly exaggerated. Darwin's evolution theory may yet prove to be the greatest one of all.

All in all not as bad a book as many of the other reviews seem to imply. Definitely not a book for the impatient however...

Excellent ideas, buried under bad prose
Berlinski gathered the right set of ideas for this subject, and he explains them in thorough detail. In particular, I found the level of detail for Godel's, Church's, and Turing's work to be very informative, e.g. this was the first time I saw Godel's encoding scheme explained well enough for me to see just how it works.
However, the prose in this book was so bad that I just couldn't finish it. While it was good that he stayed away from a dry textbook style, the style he ended up with was better suited for a romance novel than a technical history. In addition to the prose style itself, the book is littered with fictional asides that are meant to illustrate subtle points but end up being a distraction. It seems that the author was trying to create both a good technical history and a work of art, but in the end he failed at both.

Better than reviews suggest; background knowlege useful
While the writing is from time to time over the top, the book is very enjoyable. What I appreciate most about the book is its treatment of the actual philosophy behind the development of the algorithm. This apparently is what many previous reviewers dislike. They might be used to books like Men of Mathematics or The Lady Tasting Tea - books that merely provide a brief description of the development of the topic accompanied by some life facts of the developers. Hence, this book is more of a philosophy book than a history book. And for those of us who would rather read a philosophy book than a history book, this book is a breath of fresh air. Reading of the book is helped, however, by a background in both logic and mathematics. Still, the book gets four stars from me as Berlinski would have done well to simplify his writing style. ... Read more

Amazon.com

As the leading American proponent and theorist of the software-design philosophy known as fuzzy logic, Bart Kosko, author of Fuzzy Future: From Society and Science to Heaven in a Chip, can be expected to have high hopes for the discipline. And it's not like it hasn't lived up to some of them already. Forsaking the binary either/or at the heart of digital computing, fuzzy logic's emphasis on the shades of gray between true and false makes it a valuable way to program microchips that guide factories, cars, household appliances, and other gadgetry that works with the physical world's nonbinary facts. It also makes for a pretty slick philosophical end run around the yes-or-no logic that has been the basis of Western thought for the last couple of millennia.

But here Kosko announces that fuzzy logic is ready to do more. Taxes, voting rights, abortion, warfare, genetic engineering, deep physics, computer-generated art, the quest for transcendent posthuman immortality--all of these and more, he tells us, may in the future be transformed by the powerful techniques of fuzzy thinking. The overall result: less government, ignorance, poverty, death; more power to the people. This of course is exciting news, and that may explain why Kosko sometimes seems less than interested in nailing down the details of what fuzz has to do with any of it. So if it's an education in fuzziness you want, look elsewhere--at Kosko's earlier, more introductory Fuzzy Thinking perhaps. But for a vivid snapshot of fuzzy thinking at its most ambitious, jump right on in. --Julian Dibbell ... Read more

Reviews (6)

the noncomputational barrier
God save us from yet another computer reductionist:

A provocative final chapter promotes the idea that digital networks will be able to hold our own (still-fuzzy) consciousnesses, putting an end to human death: "Biology is not destiny for the minds that will follow us.... Chips are destiny."

Fuzzy logic is still digital.
The human mind, and self-awareness, are noncomputational at their very core.
The prediction made by computer reductionists that we will eventually be able to download, or upload, our human self-aware minds into machines and computers is absolute nonsense.
The geeks who promote that idea have no proof, or evidence, whatsover, that mind uploading/downloading is feasible. It is all wishful thinking, persoanl bias, prejudice, and genuine fuzzy thinking on their part.
Most of these geeks are atheists, and the only method they can dream up for their minds continuing following their physical death, is to install their minds into machines.
The ultimate goal is immortality in the ultimate computer platform.
Sorry geeks; the non-computational barrier will prevent that form of surviving physical death.
The new paradigm bias is computaional reductionism. It is everywhere like a plague from hell. It is the present day predujice that sees everything in terms of computer logic. Life itself has been cheapened by this new prejudice. We are all numbers now, and sbsolutely nothing is sacred.
Let's hope that this new computer geek paradigm horror ends soon, before it reaches the stage of computational fascism.

Clarifies the Fuzzy Future
Kosko hits a home run here. Much better than his Fuzzy Thinking tome.

An Ambitious Attempt to Integrate Numerous Ideas
"The Fuzzy Future" is a wide-ranging work that attempts to integrate concepts from disciplines as diverse as physics, neurophysiology, and the social sciences.It's well-written, but not always easy to follow, due to the diverse subject matter. Definitely not "light reading"!

Kosko's Predictions for the Future of Technology
Kosko predicts the future within the framework of a paradigm shift from binary thinking to fuzzy logic.There is an extensive index to allow for easy reference and about 100 pages of footnotes that keep the technicaljargon out of the primary text.The story flows like a science fictionnovel in which the author is constantly surprising the reader with newinsights into the way things may be.A great book that leaves you feelingenlightened and just plain smarter.

Tour de Fuzz
Worth the wait!

Gray rules ... Read more

Book Description

For many philosophers, modern philosophy begins in 1879 with the publication of Gottlob Frege's Begriffsschrift, in which Frege presents the first truly modern logic in his symbolic language, Begriffsschrift, or concept-script. Danielle Macbeth's book, the first full-length study of this language, offers a highly original new reading of Frege's logic based directly on Frege's own two-dimensional notation and his various writings about logic.

Setting out to explain the nature of Frege's logical notation, Macbeth brings clarity not only to Frege's symbolism and its motivation, but also to many other topics central to his philosophy. She develops a uniquely compelling account of Frege's Sinn/Bedeutung distinction, a distinction central to an adequate logical language; and she articulates a novel understanding of concepts, both of what they are and of how their contents are expressed in properly logical language. In her reading, Frege's Begriffsschrift emerges as a powerful and deeply illuminating alternative to the quantificational logic it would later inspire.

The most enlightening examination to date of the developments of Frege's thinking about his logic, this book introduces a new kind of logical language, one that promises surprising insight into a range of issues in metaphysics and epistemology, as well as in the philosophy of logic.

Reviews (1)

Insufficient historical perspective
Two important books have been published in the last fifteen years, both of which have implications for any discussion of Frege: A. Garciadiego, BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC PARADOXES (1992), and I. Grattan-Guinness, THE SEARCH FOR MATHEMATICAL ROOTS (2000).

Macbeth's book refers to neither of these works.The result is that she marginalizes her own book.Most glaringly, she grants some logical content to Russell's paradox, and so messes up a discussion of the famous interaction of Russell and Frege.This is one historical episode you don't want to get wrong, but she does.She hasn't read Garciadiego, so she doesn't realize that Russell's "paradox" is a meaningless formulation, devoid of logical content.

Because her work is not grounded in math history at all, she also fails to put Frege's ideas in context.Whatever the different ideas of Dedekind, Russell, Frege or Cantor, they all come out of a very ancient mathematical school of "natural" mathematics.This approach--which never manages to form a logical part of any mathematical argument--seeks to "avoid" or "solve" paradoxes.Macbeth has missed the bus, in that she isn't aware that the most important work going on now in philosophy and mathematics is the reexamination of these paradoxes.Garciadiego famously began this work, drawing up a "list" in his book of nonexistent paradoxes, Russell's, Richard's, Burali-Forti's, Cantor's.Among other things, it has destroyed Godel's theorem, which relies--for its distinction between truth and provability--on Richard's paradox having at least some logical content.Richard's paradox has no logical content whatsoever.

This movement has proved so disturbing that it has brought substantive work to a halt, in both philosophy and mathematics, while we see where we are (if we're anywhere!).

And math and philosophy are not the only disciplines which have been brought to a halt."Natural" mathematics lies at the heart of Sraffa, Kimura, chemistry (in ways which are just beginning to be revealed) and even Einstein, who bought into the "natural" math polemics of a book for which he had high regard, Poincare's SCIENCE AND HYPOTHESIS.Where is it in the relativity of simultaneity?Here:

Up to now our considerations have been referred to a particular body of reference, which we have styled a 'railway embankment.' We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated....People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises: Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A -> B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M' be the mid-point of the distance A -> B on the travelling train. Just when the flashes (as judged from the embankment) of lightning occur, this point M' naturally coincides with the point M, but it moves towards the right in the diagram with the velocity v of the train.

This translation is accurate (the French and Italian are not). Einstein really does say "fallt zwar...zusammen." That is, he says that one point "naturally" coincides with another. The "naturally" reveals the intuitionist expression of the concept, for it reflects the belief that the formulations of geometry do not express facts.

Obviously, the logical problem with it is that, regardless of what Einstein may "feel" about mathematical expressions, nowhere in Einstein's writings--either in the 1905 papers or after--is any meaning assigned to 'naturally.' The failure to do so, destroys the idea, and it is easy to see why. If we retain the concept without meaning there is no logical basis on which to proceed beyond it. If we eliminate it, we wind up with a contradiction: the two assumed coordinate systems collapse into one. What is more, when we place this train experiment next to the various other thought experiments, we see that they are simply translations of the same problem into other terms, just as the false 'paradoxes' turn out to be subject to the same problem Richard indicated (reference to an infinite domain which destroys the meaning). In special relativity, natural coincidence can only be defined by infinitely many words. So the distinction collapses.

Macbeth knows none of this.Typical scholar so deeply into her subject that she loses track of her object. ... Read more

Reviews (2)

Excellent introduction to logic
I'm not a big logic person, but this book helped me get a firm grasp on the logical structure of language. It helped me do very well in a class I was struggling through. The author's style is very clear and accessible- it hardly feels cold and mechanical like other logic books! Also, I found the web page nifty and very helpful too: ruccs.rutgers.edu/~logic/MeaningArgument.html Check out the logic in action animations, they really help!

the best intro-level logic book i've read
first things first: ernest lepore is the author of the book, NOT the editor.

secondly, read the editorial review. i don't have much to add to it, but in the spirit of pitching in my two cents...

i have had the pleasure of taking a logic course with dr. lepore, and his book is a perfect reflection of his lecturing style - concise, explicit, and fun. while the other introductory books i've read have attacked logic from a decidedly sterile and proof-oriented perspective, lepore's method aligns itself with a subject already familiar to us: language. viewing logic through the framework of language renders a seemingly difficult topic easier to comprehend. it is, if you will excuse the pun, the logical way to approach the subject.

i highly recommend this book to any introductory-level student of logic. ... Read more

Book Description

The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in mathematical logic. This edition includes an extensive appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains. The text contains numerous exercises and an appendix furnishes answers to many of them.Introduction to Mathematical Logic includes:opropositional logicofirst-order logicofirst-order number theory and the incompleteness and undecidability theorems of Gödel, Rosser, Church, and Tarskioaxiomatic set theoryotheory of computabilityThe study of mathematical logic, axiomatic set theory, and computability theory provides an understanding of the fundamental assumptions and proof techniques that form basis of mathematics. Logic and computability theory have also become indispensable tools in theoretical computer science, including artificial intelligence. Introduction to Mathematical Logic covers these topics in a clear, reader-friendly style that will be valued by anyone working in computer science as well as lecturers and researchers in mathematics, philosophy, and related fields. ... Read more

Reviews (5)

A classic of math logic, sans philosophy
This book is a bit of an elegy to a dying world: the math logic of the 20th century.
It does not cover any nonclassical or philosophical logic, directions heavily researched in recent decades. Algebraic logic is slighted, even though Mendelson was an authority on Boolean algebra. Nor does he do justice to the model theoretic perspective, although the treatment of the Tarski semantics for first order logic in chpt. 2 is a bit of a classic. The treatment of recursion in chpts. 3 and 5 are thorough. The set theory of chpt. 4 is a bit unconventional (NBG rather than ZF) but is well exposited. My overall complaint is the crabbed notation, altho he's come a long way since the first edition. The book also cries out for a more graceful English style and page layout. Here Machover (1996) stands out.
Mendelson's bibliography is wonderfully long and rich. Finally, this text contains perhaps the gentlest extant introduction to second order logic.

A Classic Textbook Now In Its Fourth Edition
Nearly forty years after it was published (1964), Elliot Mendelson's Introduction To Mathematical Logic still remains the best textbook on the principal topics of this subject. Although the book does not presuppose any background in the subject or in any particular branch of mathematics, the reader should have some degree of "mathematical sophistication."

The first chapter starts with truth tables and ends with a completeness proof of a given formal system for propositional logic and an independence proof of the axioms of this system. Chapter Two is the study of quantification theory. Topics include quantificational completeness, Hilbert's Second Epsilon-Theorem, various topics from model theory, such as compactness and Lowenheim-Skolem Theorems, theorems on submodels and ultrafilters and non-standard analysis. The new fourth edition adds a very nice section on interpretations of quantification theory that allow the empty domain. Chapter Three presents an axiom system for number theory, recursive functions and proves (among other theorems) the famous Godel Incompleteness theorems, Tarski's indefinability of Truth Theorem and Church's Undecidability Theorem. Chapter Four is devoted to elementary set theory. Topics include an axiom system for set theory, ordinal and cardinal numbers, the axiom of choice and regularity, and alternative axiom systems of set theory. The new fourth edition includes an axiom system with urelements, something rarely presented, and an interesting note on the historical application of such a system in the construction of the first independence proof of the axiom of choice. The fifth chapter is the study of computability. The chapter begins with the notion of an algorithm and Turing Machines and builds up to the Kleene-Mostowski Hierarchy. The new fourth edition concludes with an excellent appendix on second-order logic.

I have used Mendelson's book to teach a one-semester course to advanced undergraduate and graduate students with great success. Such a course is centered on the first three chapters, omitting from Chapter Two anything beyond quantificational completeness. If time permits, I recommend either the rest of Chapter Two, the beginning of Chapter Five, or the appendix on second-order logic. Set theory, the content of Chapter Three, is usually offered as a separate course.

My first reference in logic
In my work as a math teacher, researcher, author and journal editor, I often encounter problems with a logical component. When that need arises, my first choice of reference is always this book. It is the most concise and readable introductory text I have ever encountered and it is a rare occasion when I fail to find the background material needed to solve the problem. It is also an excellent source of problems and I have pulled the ideas for many test questions from it over the years. Those problems have appeared on tests in both mathematics and computer science.
The topics are fairly standard, starting with the propositional calculus and covering quantification theory, formal number theory, axiomatic set theory and computability. What differentiates this book is the clarity of the description, making it ideal for an introductory course for undergraduates. Solutions to some of the problems are given in an appendix.
Logic is a fundamental component of mathematics and all mathematicians need some exposure to it. It is also a critical part of computer programming, as many sections of programs can be directly deconstructed into logical statements. Workers in both areas will find this book of enormous value.

Outstanding Organization and Clear Style
I was sufficiently fortunate to have taken Professor Emeritus Mendelson's famous logic course at Queens College, the City University of New York, just two semesters before his retirement. I was, and continue to be, astonished by Dr. Mendelson's precise yet easy style, and the beautifully efficient organization of the subjects. Everything from the expository prose to the system of notational conventions has been carefully thought through so as to make the book both very substantive and very readable. In my opinion, it's the best introduction to serious mathematical logic currently on the market, and thanks to the genius of its author, it is likely to remain so for a long time. The buyer will not be disappointed.

Great Book on Logic and Meta-theory
This book has come in handy. Although it is a bit difficult, it is, relative to other books on mathematical logic, very accessible. (The learning of logic always takes dedicated time!). The introduction has a summary of certain set-theoretic notions, etc., and the book covers First-Order propositional calculus and Quantifier-predicate calculus, as well as Second-Order logic, and a good deal of Meta-theory to show completeness and soundness, etc. I used this book as a side-reference studies and it helped significantly. ... Read more

Book Description

In 1824 a young Norwegian named Niels Henrik Abel proved conclusively that algebraic equations of the fifth order are not solvable in radicals. In this book Peter Pesic shows what an important event this was in the history of thought. He also presents it as a remarkable human story. Abel was twenty-one when he self-published his proof, and he died five years later, poor and depressed, just before the proof started to receive wide acclaim. Abel's attempts to reach out to the mathematical elite of the day had been spurned, and he was unable to find a position that would allow him to work in peace and marry his fianc

But Pesic's story begins long before Abel and continues to the present day, for Abel's proof changed how we think about mathematics and its relation to the "real" world. Starting with the Greeks, who invented the idea of mathematical proof, Pesic shows how mathematics found its sources in the real world (the shapes of things, the accounting needs of merchants) and then reached beyond those sources toward something more universal. The Pythagoreans' attempts to deal with irrational numbers foreshadowed the slow emergence of abstract mathematics. Pesic focuses on the contested development of algebra--which even Newton resisted--and the gradual acceptance of the usefulness and perhaps even beauty of abstractions that seem to invoke realities with dimensions outside human experience. Pesic tells this story as a history of ideas, with mathematical details incorporated in boxes. The book also includes a new annotated translation of Abel's original proof. ... Read more

Reviews (1)

Unsolvable yet quite graspable
To me, Abel's Proof successfully bridges the difficult gap that separates math books from fun books. Being one who appreciates the history and development of ideas and who is not afraid of a few equations, my needs as a reader were tastefully satisfied. If you, like me, find yourself enticed by some of the more subtle problems in math and science, while at the same time, have not the recourse to explore each one to their fullest, this book will be a welcome guide. Pesic uses Niels Abel's proof (1824) regarding the general insolvability by radicals of fifth degree equations as the central trunk of a robust tree whose branches contain delightful episodes of mathematical examples, human dramas, twists of fate, and historical parades. As much a biography as anything else, I could feel the personalities of the mathematicians evinced through their contributions to the question of solvability. From the near misses of Ruffini and Gauss to the final QEDs of Abel and Galois, one sees the human elements of struggle, triumph, anger, and success, set thoughtfully alongside the mathematical details. Carefully arranged mathematical sidebars allow this book to be read with as much technical intent as one chooses to bring; the math is there for the taking (little goes beyond a basic familiarity with algebra). In short, this book offers a delightful way to see some intriguing math and the characters who made it happen. ... Read more

Book Description

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. ... Read more

Reviews (10)

The fundamental work on what mathematics really does
Definitions, examples, theorems, proofs -- they all seem so inevitable. But how did they come to be that way? What is the role of counterexamples? Why are some definitions so peculiar? What good are proofs?

In this brilliant and deep -- yet easy to read -- book, Lakatos shows how mathematicians explore concepts; how their ideas can develop over time; and how misleading the "textbook" presentation of math really is.

Fascinating for anyone who has seen mathematical proofs (even high-school Euclidean geometry) and essential for anyone studying mathematics at any level.

(I wrote this review in 1996, before Amazon kept track of reviewers' names... some additional notes:)

If you'd like to read more discussion about Lakatos and the intellectual context of P&R, you'll be interested in Brendan Larvor's "Lakatos: An Introduction".

The fundamental work on what mathematics really does.
Definitions, examples, theorems, proofs -- they all seem so inevitable. But how did they come to be that way? What is the role of counterexamples? Why are some definitions so peculiar? What good are proofs?

In this brilliant and deep -- yet easy to read -- book, Lakatos shows how mathematicians explore concepts; how their ideas can develop over time; and how misleading the "textbook" presentation of math really is.

Fascinating for anyone who has seen mathematical proofs (even high-school Euclidean geometry) and essential for anyone studying mathematics at any level.

nice reading for the general public
Very nice book if you are in high school or in college and would like to see how mathematics evolves. It makes a very pleasant reading although the mathematical ideas behind are not trivial.

It discusses polyhedra in 3 (or more) dimensions and Euler's formula that describes their numbers of vertices, edges, faces, e.t.c. The challenge is to determine what specific kinds of polyhedra satisfy the formula and conversely, how one could generalize the formula so as to describe more (if not all) polyhedra. Lots of historical references illustrate the fact that the discussion is not naive and that reflects the actual history of the subject.

One can realize through this book that math people are not Gods and do not produce theories out of nowhere, but they experiment with their objects like any other scientist, and then try to summarize in an elegant/rigorous way.

a study in mathematical thought
I want to add a few words to the brief comment by the reader in Monroe (who gave this book one star). I tend to agree that "Proofs and Refutations" isn't a primer in mathematical proof-writing; it's certainly not a textbook for beginning mathematicians wanting to know how to practice their craft.

However, for those readers (including beginning mathematicians) who are interested in the broader picture, who are interested in the nature of mathematical proof, then Lakatos is essential reading. The examples chosen are vivid, and there is a rich sense of historical context. The dramatised setting (with Teacher and students Alpha, Beta, Gamma, etc) is handled skilfully. Now and then, a foolish-seeming comment from one of the students has a footnote tagged to it; more often than not, that student is standing in for Euler, Cauchy, Poincare or some other great mathematician from a past era, closely paraphrasing actual remarks made by them. That in some ways is the most important lesson I learned from this book; "obvious" now doesn't mean obvious then, even to the greatest intellects of the time.

Although "Proofs and Refuatations" is an easy book to begin reading, it is not an easy book per se. I have returned to it repeatedly over the last ten years, and I always learn something new. The text matures with the reader.

Excellent Critical Reasoning Framework
As a lay reader of mathematics, I am prone to read for more for analogy and thought methods instead of, for example, the real implications of variations on Eulers Formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges.

Displaying solid content with artful execution, this book interested me in both the math of the thing and the acompanying thought processes.

Content: This book has near-poetic density and elegance in arguing a non-linear approach to mathematical development and, for me, to just plain thinking. Our tendency (as born worshippers of linearity and causality) is to discover a brick for the building then immediately look for the next to stack on top. Lakatos contends that PERHAPS you have discovered a brick worthy of the building, now let's see what truly objective tests we will put to this brick and before giving it a final stamp of approval. It seems obvious to say "always question", but the exercise in this book will take you through the process and show you what you may take for granted in this simple concept. For example, do you observe HOW you question? See his discussion throughout on global vs. local counterexamples, just as a start.

Execution of the text: This is the beautiful part. Mr. Lakatos has written this book as theater: characters with definite identities, plot, drama. The narrative flows in the voices of students and a professor who proves to be a sound moderator, intervening at timely points, i.e. those where questions may be crystallized or thoughts prodded to that point. This is where learning takes place, in a heated, moderated debate over Euler's formula. What was most interesting to me about this method was that it lent itself easily to isolating a particular thread of discussion. I literally chose certain characters to research from beginning to end in order to follow the evolution or confirmation of their thinking.

You emerge with a good framework that makes this book excellent reference material for problem-solving.

One last, but important note. This book will have you praising the lowly footnote. I would buy it for that alone. You will read along with the discussion, then get off and examine a footnote, and then pick the dialogue back up not having lost a step. On the contrary, Mr. Lakatos deepens your context with on-point explanations and math history. ... Read more

Book Description

Carefully focused on reading and writing proofs, this introduction to the analysis of functions of a single real variable helps readers in the transition from computationally oriented to abstract mathematics. It features clear expositions and examples, helpful practice problems, many drawings that illustrate key ideas, and hints/answers for selected problems.Logic and Proof. Sets and Functions. The Real Numbers. Sequences. Limits and Continuity. Differentiation. Integration. Infinite Series. Sequences and Series of Functions.For anyone interested in Real Analysis or Advanced Calculus. ... Read more

Reviews (2)

Definitely a good first text
I bought this book because I have been looking for a Introductory analysis text that isn't too advanced, but yet doesn't gloss over the essential stuff, and I found it in Lay's book. For the self-studier, this book is excellent! I have several books on analysis: Shilov, Kolomogorov, Rosenlicht, Ross,etc... For the beginner, this book is superior to all of them. A plethora of examples. Also, a good range of problems:from straight forward problems requiring only the use of a definition to more advanced problems requiring a little thought. If you already have had some Analysis, then this book is probably not for you. But, if you are a student who wants to learn Analysis on your own, then this book would be hard to beat. After this book, one should be able to tackle "Papa Rudin". For according to Rudin, all that is needed to study his "Real and Complex Analysis" tome, is the first seven chapters of his "Principles of Mathematical Analysis". This book covers all that Rudin covers with the exception of Riemann-Stieltjes integration. On the whole, this is a great start! If proof-based math is new to you, then you will appreciate the first chapter on proofs. Would have given five stars, but I would have liked to seen Riemann-Stieltjes integration. That's really only nit picking, though.

This book was surprisingly good
I didn't think this book was going to be very good, but the author has "proved" me wrong ;-) This book starts out so basic that in my class (which was the first analysis course in our math department) we actually skipped the first 1/3 or so of the book. The first 9 or 10 sections consist of stuff like basic set theory, logic, definition of a function, etc. I would think that even the most elementary Analysis books would completely leave this out and expect that the reader is already familiar with this. So if you need it, this book will be a good resource for you.

Then the book goes into a very nice introduction to topology. Basic concepts like open/closed sets, accumulation points, compact sets, etc. Topology can be a little intimidating simply because it's _so_ abstract, but this book makes the basic concepts very easy to understand, and prepares one for a more advanced course in topology. Alot of (good) Elementary Analysis books leave topology out, but I'm glad this book contained it. It is a very interesting subject.

All the material in the book is explained probably about as easily as the concepts CAN be explained. If you still have trouble with it, you might consider a different major. Not to say that this book transforms a very difficult subject into a pathetically easy piece of cake because that's impossible, but the material is presented probably as easily as it can be in order to maintain precision and detail (which is the whole point of Analysis).

The book is definitely not running short in the examples or end-of-section problems department, so that is another plus. The problems at the end of each section range in difficulty from problems that almost exactly match an example worked in detail in the section, to fairly challenging problems. With enough time though the average student could probably do every problem at the end of every section.

I'd recommend this book for self study as well as a supplement to any introductory analysis course. If you have already have exposure to rigorous proof of calculus theorems, then this book will probably be too basic for you.

The reason this book got 4 stars instead of 5 is because of its utterly ridiculous price. Just as good is Elementary Analysis: The Theory of Calculus, ISBN: 038790459X, except that it doesn't include the section on Topology ... ... Read more

Book Description

This is the first textbook on formal concept analysis. It gives a systematic presentation of the mathematical foundations and their relations to applications in computer science, especially in data analysis and knowledge processing. Above all, it presents graphical methods for representing conceptual systems that have proved themselves in communicating knowledge. Theory and graphical representation are thus closely coupled together. The mathematical foundations are treated thoroughly and illuminated by means of numerous examples. Since computers are being used ever more widely for knowledge processing, formal methods for conceptual analysis are gaining in importance. This book makes the basic theory for such methods accessible in a compact form. ... Read more

Book Description

Gödels Incompleteness Theorems are among the most significant results in the foundation of mathematics. These results have a positive consequence: any system of axioms for mathematics that we recognize as correct can be properly extended by adding as a new axiom a formal statement expressing that the original system is consistent. This suggests that our mathematical knowledge is inexhaustible, an essentially philosophical topic to which this book is devoted.

Basic material in predicate logic, set theory and recursion theory is presented, leading to a proof of incompleteness theorems. The inexhaustibility of mathematical knowledge is treated based on the concept of transfinite progressions of theories as conceived by Turing and Feferman.

All concepts and results necessary to understand the arguments are introduced as needed, making the presentation self-contained and thorough. ... Read more

Book Description

The study of random graphs was begun by Paul Erdös and Alfred Renyi in the 1960s and now has a comprehensive literature. A compelling element has been the threshold function, a short range in which events rapidly move from almost certainly false to almost certainly true. This book now joins the study of random graphs (and other random discrete objects) with mathematical logic. The possible threshold phenomena are studied for all statements expressible in a given language. Often, there is a zero-one law that every statement holds with probability near zero or near one. The methodologies involve probability, discrete structures and logic, with an emphasis on discrete structures. The book will be of interest to graduate students and researchers in discrete mathematics. ... Read more

Book Description

"First-Order Logic and Automated Theorem Proving" is a treatment of classical logic that presents fundamental concepts and results in a rigorous mathematical style. It also considers applications to automated theorem proving, to the point of providing usable programs (in Prolog). This material will serve both as a first text in formal logic and as an introduction to automation issues, and is intended for those interested in computer science and mathematics at the beginning graduate level.

The book begins with propositional logic, then treats first-order logic, and finally, first-order logic with equality. In each case the initial presentation is semantic: Boolean valuations for propositional logic, models for first-order logic, and normal models when equality is added. This defines the intended subjects independently of a particular choice of proof mechanism. Then many kinds of proof procedures are introduced: tableau, resolution, natural deduction, Gentzen sequent and axiom systems. Completeness issues are centered in a Model Existence Theorem, which permits the coverage of a variety of proof procedures without repetition of detail. In addition, results such as Compactness, Interpolation, and the Beth Definability theorem are easily established. Implementations of tableau theorem provers are given in Prolog, and resolution is left as a project for the student.

In this new edition, the author has added material on AE calculus, Herbrand's Theorem, Gentzen's Theorem, and related topics. ... Read more

Book Description

In this volume, Lambek and Scott reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory.In Part I, they show that typed lambda-calculi, a formulation of higher-order logic, and cartesian closed categories, are essentially the same.Part II demonstrates that another formulation of higher-order logic, (intuitionistic) type theories, is closely related to topos theory.Part III is devoted to recursive functions.Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained.Detailed historical references are provided throughout, and each section concludeds with a set of exercises. ... Read more

Reviews (1)

This is definitely Higher!
I was looking for a book for my girlfriend this Christmas and stumbled upon this one. At first I thought it would be too light but was I ever mistaken!! This book is so high that it would make Jack Kerouac dizzy. It begins with a treatment of basic category theory and ccc's and then goes on to present toposes and intuitionistic type theory. The authors take care to annotate their turnstile with the set of free variables (Hah! I bet you thought I had no idea what this book was about!) so that they can deal with empty types in a reasonable way. The treatment of presheaf models is very lucid and the discussion of internal languages and lambda-calculi is excellent. In fact many papers of Koymans are just exercises from this book worked out. The book is slightly out of date, no treatment of linear logic or symmetric monoidal-closed categories. Overall this book is highly recommended for the beginner and expert alike. ... Read more

Book Description

Here is an introduction to modern logic that differs from others by treating logic from an algebraic perspective. What this means is that notions and results from logic become much easier to understand when seen from a familiar standpoint of algebra. The presentation, written in the engaging and provocative style that is the hallmark of Paul Halmos, from whose course the book is taken, is aimed at a broad audience, students, teachers and amateurs in mathematics, philosophy, computer science, linguistics and engineering; they all have to get to grips with logic at some stage. All that is needed to understand the book is some basic acquaintance with algebra. ... Read more

Reviews (3)

A superb introduction to the glories of Boolean algebra
This book reviews some ideas Halmos worked on in the 1950s: the algebraization of predicate logic. The result was polyadic algebra, which has been unfairly neglected since. Tarski, Henkin, and their Berkeley students worked on a rival research program that culminated in the better known cylindric algebras. The treatment remains at the undergrad level, because Halmos stops short short of polyadic predicates. Halmos's "Algebraic Logic," which AMS keeps in print and is a fine read, contains all of Halmos's professional writings on polyadic algebra.

While Halmos does not cover all of first order logic, he does an excellent job of introducing the reader to the great power and depth of Boolean algebra, revealed by Marshall Stone and Tarski in the 1930s, and other Poles in the 1950s. By this I mean Boolean algebra coupled with the notions of filters, ideals, generators, and quotient algebras. The metatheory of the propositional calculus has a very elegant Boolean representation.
For that matter, the completeness of first order logic has a nice polyadic algebra translation.

Lattice theory is an extremely powerful generalization of Boolean algebra that has not attracted the attention it deserves. If Halmos had written a text on lattice theory, that situation would in all likelihood have ended. Halmos and Givant include an all-too-brief tantalizing chapter on lattices.

If this book has a drawback, it is the relative unsophistication of its first 40 odd pages, an introduction to logic. This is especially disappointing given that Givant is a logician, and an excellent one at that, being a student of Tarski's.

The books main asset is Halmos's lively prose style, unparalleled in modern mathematics. Math PhD students should study this book closely as a superb example of how to exposit mathematics.

Interesting view on logic
In his "automathography" Halmos described his views on logic which he had in the 1960's. He felt that logic, as usually stated, was very un-profound, unrigorous, combinatorial amusement. He felt additionally that logic could be put on a firm algebraic footing through the theory of Boolean rings. At that time he interpreted many things in logic in terms of Boolean rings. This book is, in some sense, the child of these labors. Halmos created this book in his usual easy to read style, and when he said that few prerequisites were assumed, he meant it. I found the (short) book very interesting, but I also found the introductory pages seemed to drag. Perhaps this is because I already know something about logic, but the rest of the book was interesting and self contained. This book was lighter than most logic books I've seen. These books were mainly written by philosophers in some capacity or other, and they never stopped their thick prose.

A Builder of a Solid Foundation in Mathematics
It can be strongly argued that logic is the most ancient of all the mathematical sub-disciplines. When mathematics as we know it was being created so many years ago, it was necessary for the concepts of rigid analytical reasoning to be developed. Of the three earliest areas, geometry was born out of the necessity of accurately measuring land plots and large buildings and number theory was required for sophisticated counting techniques. Logic, the third area, had no "practical" godfather, other than being the foundation for rigorous reasoning in the other two. In the intervening years, so many additional areas of mathematics have been developed, with logic and logical reasoning continuing to be the fundamental building block of them all. Therefore, every mathematician should have some exposure to logic, with the simple history lesson automatically being included. This short, but excellent book fills that niche.
The title accurately sets the theme for the entire book. Algebra is nothing more than a precise notation in combination with a rigorous set of rules of behavior. When logic is approached in that way, it becomes much easier to understand and apply. This is especially necessary in the modern world where computing is so ubiquitous. Many areas of mathematics are incorporated into the computer science major, but none is more widely used than logic. Written at a level that can be comprehended by anyone in either a computer science or mathematics major, it can be used as a textbook in any course targeted at these audiences.
The topics covered are standard although the algebraic approach makes it unique. One simple chapter subheading, 'Language As An Algebra', succinctly describes the theme. Propositional calculus, Boolean algebra, lattices and predicate calculus are the main areas examined. While the treatment is short, it is thorough, providing all necessary details for a sound foundation in the subject. While the word "readable" is sometimes overused in describing books, it can be used here without hesitation.
Sometimes neglected as an area of study in their curricula, logic is an essential part of all mathematics and computer training, whether formal or informal. The authors use a relatively small number of pages to present an extensive amount of knowledge in an easily understandable way. I strongly recommend this book.

Published in Smarandache Notions Journal reprinted with permission. ... Read more

Book Description

Now in its fourth edition, this book has become a classic because of its accessibility to students without a mathematical background, and because it covers not only the staple topics of an intermediate logic course such as Godel's Incompleteness Theorems, but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem.John Burgess has enhanced the book by adding a selection of problems at the end of each chapter. ... Read more

Reviews (16)

Great textbook, poor text
I can hardly imagine a better introduction to the topics covered than this book. It discusses virtually everything the intermediate logic student could want: diagonalization, Turing machines, undeciability, indefinability, incompleteness, forcing, and on and on. Although the first few chapters are a bit awkward, the style is generally crystal clear and the examples and metaphors vivid. It's far and away the best read of any text on logic I've yet encountered.

As a mathematician, I was concerned about the books' emphasis on logic rather than mathematics (the text is aimed at philosophy students, too). But the introduction to foundations flows so easily and naturally that I could never complain. Anyone interested in the topic, regardless of their background, could hardly do better (or cheaper) for an introduction.

P.S. - I wanted to give this five stars, but, as other reviewers have pointed out, there are simply too many typos. C'mon, get an editor.

Very lucid explanations
This book is regarded as a 'classic' and rightly so. It assumes a minimal background, some familiarity with the propositional calculus. Even this can be dispensed with, if the reader is sufficiently motivated, as there is a well-written review of the first-order logic that one typically learns in an introductory formal logic course.

The book is highly readable. Each chapter begins with a short paragraph outlining the topics in the chapter, how they relate to each other, and how they connect with the topics in later and earlier chapters. These intros by themselves are valuable. The explanations though are what stand out. The authors are somehow able to take the reader's hand and guide him/her leisurely along with plentiful examples, but without getting bogged down in excessive prose. And they are somehow able to cover a substantive amount of material in a short space without seeming rushed or making the text too dense. It's nothing short of miraculous.

What made the book especially appealing to me is that it starts right out with Turing Machines. As a topologist who recently got interested in computational topology, I needed a book that would quickly impart a good, intuitive grasp of the basic notions of computability. I have more "mathematical maturity" than is needed to read an introductory book on computability, so I feel confident in saying that most of the standard texts on computability revel in excessive detail, like defining Turing Machines as a 6-tuple -- something that serves no purpose other than pedantry. This book is different. I particularly liked how the authors stress the intuitive notions underlying the definitions. For example, they lay special emphasis on the Church-Turing thesis, always asking the reader to consider how arguments can be simplified if it were true.

One should note that the emphasis of this book is more towards logic. While it starts with issues of computability, it moves into issues of provability, consistency, etc. The book covers the standards such as Goedel's famous incompleteness theorems in addition to some less standard topics at the end of the book. A small set of instructive exercises follows each chapter.

Not much to add, but
it should be noted that this book is not intended for the auto-didact. Like other good logic texts-Jeffrey's Formal Logic or Pollock's Technical Method's (out of print, but available in PDF on his website)-there is very little commentary in the brief chapters, so it is useful if you are already very familiar with the material or if you have a very worthy guide. An advantage of the short chapters is that material is broken down in finer increments; a disadvantage is that material is presented with spare guidance at times. I was also disappointed by the sparsity of examples. Like many logic and math students, I learn better from examining a few examples than I do from either lectures or text: give me three examples of something and I'll usually have it down. I would have liked to see more examples in this text. The exercises are ample and creative, which I appreciate, but often go so far beyond the text it's mind boggling. They often require extensive extrapolations from the text sometimes even proving theorems or lemmas not in the text just for use in the exercise. I should say that I'm a philosopher and not a mathematician (I suspect the other reviewers are primarily mathematicians), so my estimation of the difficulty will differ. I aced Symbolic Logic, Modal Logic, Deviant Logic, and Advanced Symbolic Logic and still had difficulty with some of this material, even though I had a prior acquaintance with Godel's proof. Note that the first reviewer, who thought it was a breeze, described himself this way "As a topologist who recently got interested in computational topology..." Good for him, but if you are not a professional mathematician this book will probably be quite challenging at times, even if you are otherwise good at mathematical logic. Note also that the second five-star review refers to the older edition-it has not necessarily improved with age. I firmly agree with the reviewer from Brooklyn that the proofs could have had more forecasting and with the reviewer from Raleigh that a solution set, say to the odds, would have been very useful, especially for the auto-didact, from whose perspective I am writing.

almost great
Except for the scores of typos. Previous reviewers have observed this already; one has added that Burgess maintains an errata file on his website at Princeton. In fact he has two (for 1st and 2nd printings). But note that the errata file, at least for the 1st edition, is far from complete. I've noticed at least a dozen (potentially very confusing) typos that he has not yet catalogued. It's very frustrating to have to check the errata file (over 40 pages!) everytime one gets confused.

Two more points (1) the proof of compactness could have been better organized, and thereby made less tedious. (2) In general, there could stand to be more meta-level discussion about what's going on in the book. I find it's mostly trees, very little forest. (I'm not asking for _Godel, Escher, Bach_ here; I mean: where is this proof headed? Where did these satisfacton properties come from? etc)

On the positive side, the book is comprehensive, with very little handwaving, and the chapters are usually short and sweet. I prefer this text to Mendelson's. Enderton's is not bad.

Nice Revision
This 4th edition text requires much diligence and patience, even given the author's clarity and excellent presentation. The problem sets are very useful to readers/scholars of most levels. Some answers (not just hints) would be helpful, though.

The most damning feature of the book is its typos and errors. One of the authors (John Burgess) has an errata sheet online, but it is hardly reasonable for thousands of individuals to review these corrections and make them in thousands of texts when a competent editorial staff could have done the work. ... Read more

Book Description

This book introduces several topics related to linear model theory: multivariate linear models, discriminant analysis, principal components, factor analysis, time series in both the frequency and time domains, and spatial data analysis. The second edition adds new material on nonparametric regression, response surface maximization, and longitudinal models. The book provides a unified approach to these disparate subject and serves as a self-contained companion volume to the author's Plane Answers to Complex Questions: The Theory of Linear Models. Ronald Christensen is Professor of Statistics at the University of New Mexico. He is well known for his work on the theory and application of linear models having linear structure. He is the author of numerous technical articles and several books and he is a Fellow of the American Statistical Association and the Institute of Mathematical Statistics.Also Available: Christensen, Ronald. Plane Answers to Complex Questions: The Theory of Linear Models, Second Edition (1996). New York: Springer-Verlag New York, Inc. Christensen, Ronald. Log-Linear Models and Logistic Regression, Second Edition (1997). New York: Springer-Verlag New York, Inc. ... Read more