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This digital document is an article from Encyclopedia of Philosophy, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 702 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.Explores major marketing and advertising campaigns from 1999-2006. Entries profile recent print, radio, television, billboard and Internet campaigns. Each essay discusses the historical context of the campaign, the target market, the competition, marketing strategy, and the outcome. ... Read more

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This digital document is an article from Encyclopedia of Philosophy, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 7044 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.Explores major marketing and advertising campaigns from 1999-2006. Entries profile recent print, radio, television, billboard and Internet campaigns. Each essay discusses the historical context of the campaign, the target market, the competition, marketing strategy, and the outcome. ... Read more

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This digital document is an article from Encyclopedia of Philosophy, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 1163 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.Explores major marketing and advertising campaigns from 1999-2006. Entries profile recent print, radio, television, billboard and Internet campaigns. Each essay discusses the historical context of the campaign, the target market, the competition, marketing strategy, and the outcome. ... Read more

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Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician. ... Read more

Customer Reviews (3)

Will be using this book for a looong time.
Loved the book, It's definately going to be a go-to book for my other advanced classes. It is wrote in an extraordinarily user-friendly way and is actually somewhat fun to read.

A systematic and gentle approach to explaining the main ideas of mathematics
To be successful in mathematics, your mind must perform operations that are unlike the operations needed to do most other things. You must be able to hold abstract ideas, sometimes several at a time, as well as see the relationships between multiple concepts. Furthermore, those abstract ideas are built on other abstract ideas; for example, most of mathematics is built on the fundamental abstract idea of the use of a variable.
However, being different and at times being hard does not mean that the ideas of mathematics are incomprehensible. Humans excel at understanding abstract ideas, a strong argument can be made that such a skill is the very definition of human intelligence. In this book, Houston has created a primer on the fundamental abstract ideas of mathematics; the primary emphasis is on demonstrating the many principles and tactics used in proofs. The material is explained in ways that are comprehensible, which will be a great help for people who seem to hit the wall regarding what to do when confronted with the creation of a proof.
Many students are capable of leaping the comprehension hurdle; yet hit a wall when it comes time to apply the concepts in order to generate a proof. In this book, Houston takes a systematic and gentle approach to explaining the ideas of mathematics and how tactics of reasoning can be combined with those ideas to generate what would be considered a convincing proof.

Published in Journal of Recreational Mathematics, reprinted with permission

Great Introduction to Mathematical Method
I am reading a great book that is in author's words intended for undergraduate students of mathematics, but that in my opinion offers much more to a motivated reader.

"How to Think Like Mathematician" by Dr. Kevin Houston is a very engaging, readable and pragmatic text on mathematical "technique". It is a non-pompous, well written, valuable, easy to follow and understand valuable set of lessons and tips on understanding and adopting mathematical method, language, theorems, proofs and techniques.

It is an introductory text, so for more in depth treatment of the subjects such as proofs and number theory you may need to look further into books such as Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition) orThe Princeton Companion to Mathematics but if you are looking to understand the mathematical method and how to be able to read and write "serious" math this is an ideal book.

I specially have to point out Dr. Houston's writing style - it is engaging, humorous, but substantial, pedagogical and never trivial. ... Read more

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Written especially for scientists, engineers and mathematicians, this book has been extensively updated and revised to conform to 1998 ANSI/ISO C++ Standard and include all the recent developments in C++. Softcover. ... Read more

Customer Reviews (10)

Great Book
This book is brilliant for anybody that has an interest in the implementation of numerical methods in C++. I bought this book at the start of my PhD. I knew virtually nothing about C++ and within a week of getting this book I was fluent in pointer aritmetic, dynamic memory allocation and classes.

This book is great because it is geared towards people who just want to start programming numerical applications and are not too bothered with the finer points of computer programming. With the aid of this book I coded a 3d finite element solver, amongst other things. It also has plenty of examples and exercises to get you going.

This book is by no means perfect. It falls down in a couple of areas namely graphics and file I/O. It has nothing on graphics, which are essential for numerical applications. The file I/O that is discussed is great for an introduction. It introduces streams and how to import and export simple sets of data. But apart from these two applications the book is fine.

I would recommend it highly to anybody that wants to start programming in C++ with emphasis on numerics.

A shoddy piece of workmanship
It is missing the first 16 pages, and I bought it new. Needless to say, I am very angry.

A Pleasure for Numerical Applications!
This book is fantastic for learning C++ using numerical-analytic examples. It is not exactly an easy introduction to the C++ language, which is admittedly vast and profound. Especially someone with little or no programming background would be rather disappointed with it.

I recommend coming to it after doing a first course in C++ itself using something like Deitel & Deitel (C++ How to Program) or Ivor Horton's Beginning C++. Once that preparation is in place, this book will not only provide a very useful revision of the basics (first learnt elsewhere), but also prove exceedingly successful in teaching numerical applications. C++ is so vast that it is impossible to digest many of its aspects from a single source only. Therefore, the "introductory" tone of the author is incredibly useful (for a second pass, I must stress).

Somewhat equal weight is given to both the procedural and object-oriented aspects of the language with coverage of legacy C concepts such as function pointers as well as state of the art OO concepts like encapsulation, inheritance, overloading. Moreover, there are dedicated chapters to such quintessentially C++ issues as namespaces, exception handling and STL. Last but not least, the treatment is fully ANSI/ISO compliant, so the code should run on most modern compilers.

A broad menu of numerical examples are illustrated ranging from simple arithmetic to root-finding, interpolation and matrix-algebra based computations. Most exercises are do-able and quite illuminating.

Very badly organized and written
I think those who love this book are probably familiar with C++ and are impressed by the coverage of topics. I admit that the book has excellent scope, but there are many other books about which the same can be said. What makes this book very poor is that the author doesn't seem to understand the notion that one must explain something in the order in which it is used. For example, many examples in the book use the standard library and employ the 'using' namespace operator, but the author glosses over this until much later in the book. In fact, even though it is used throughout, stream IO is only substantially covered in the last chapter. Odd choice!

I'm not sure why this book is entitled '... for Scientists, Engineers and Mathematicians' unless the publisher was just trolling for readers. Except for a slight nod in the choice of examples, there is NO significant effort dedicated in this book towards scientific computing. It's basically just a C++ primer, and a rather mediocre one at that. In fact, the author only spends two pages on the standard library complex data type! Finally, there is no mention of optimization techniques to make numerical code run faster, a subject of specific importantance for any book that purports to be dedicated to scientific computing.

Excellent book
I have the second edition of this book, and have found it to be the best book of its type on the market (and I have bought several...). Though it may not suit undergraduates being exposed to C++ for the very first time, for research scientists and engineers actually wishing to use C++ in practice it provides an excellent grounding, and reasonably comprehensive coverage of all parts of the language. Also, unlike some books with similar titles it doesn't just present C type code with a chapter on classes (objects) tacked on as an afterthought. If I have a slight criticism, it is that perhaps there should be some in-depth examples of scientific/engineering problems programmed in an object-oriented way (i.e. as in Barton & Nackman's useful but now rather dated C++ textbook). ... Read more

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Customer Reviews (2)

Eye-opening
This is an important work and makes clear yet more aspects of the impact of the Holocaust on the course of modern history.It may be a bit hard to follow for people to whom the names of prominent 20th Century mathematicians are not well-known, but it shows how the Nazis changed the course of scientific history by scattering and murdering so many in one important discipline.

Interesting but pedantic
When the author begins by saying "We need to define what a mathematician is", you know you are in for a slog. The book consists of case studies, which are quite varied. No clear gestalt emerges from the individual situations or even if one is possible. ... Read more

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These autobiographical memoirs of Neal Koblitz, coinventor of one of the two most popular forms of encryption and digital signature, cover many topics besides his own personal career in mathematics and cryptography - travels to the Soviet Union, Latin America, Vietnam and elsewhere, political activism, and academic controversies relating to math education, the C. P. Snow two-culture problem, and mistreatment of women in academia.

The stories speak for themselves and reflect the experiences of a student and later a scientist caught up in the tumultuous events of his generation.

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This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. The book tries to bridge the existing gap between the different languages used by mathematicians and physicists. For students of mathematics it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which is beyond the usual curriculum in physics. It is the author's goal to present the state of the art of realizing Einstein's dream of a unified theory for the four fundamental forces in the universe (gravitational, electromagnetic, strong, and weak interaction).

From the reviews: "… Quantum field theory is one of the great intellectual edifices in the history of human thought. … This volume differs from other books on quantum field theory in its greater emphasis on the interaction of physics with mathematics. … an impressive work of scholarship." (SIAM Review 2008)

Customer Reviews (2)

Pretty good
If is not too unfair to say that quantum field theory is a kind of bag of tricks, but considering its monumental experimental success it is apparent that these tricks have worked. From a mathematical standpoint they are very suspect, and the professional mathematician who studies the formalism of quantum field theory will be aghast at its non-rigor, especially in the area of renormalization. That is not to say that no attempts have been made to put quantum field theory on a rigorous mathematical foundation. This has occupied the time of many researchers, but as of yet, such a foundation has not been found.

In spite of this, and justifiably so, quantum field theory goes on, and people taking up its study are faced at times with making choices between mathematical rigor and physical relevance. The author of this book, the first of a planned series of 4 volumes, realizes this and has attempted to give the reader a first glance of quantum field theory that emphasizes both the physics and the mathematical relationships that are part of its narrative. In general the author does a fairly good job, even though at times the details of certain subjects are left out, while others are developed to excess.

From selective chapters, this reviewer found that the following discussions stand out:

A Glance at Topology:
This chapter does not contain a whole lot of insights or material that cannot be found in other books. The author it seems wants to whet the reader's appetite but clearly wants to put off detailed discussion of the mathematics until Volume 3 or 4 of this series. There are a few places in the chapter though that deserves some comment:

- The concept of transversality is introduced for curves when discussing intersection theory but the author does not use the tangent space. Instead, he alludes to the `generic' property of transversality, namely that it can be obtained by sufficiently small perturbations which are `stable". Stability is a measure-theoretic concept and so the author explains this briefly by Sard's theorem.

- The first Chern class of the tangent bundle of the sphere is stated but not motivated. Readers will have to search intensely for this motivation in the original literature, and will find that it is relatively scant, as is the entire literature on characteristic classes in this regard. There are many books and monographs on the subject, but most, if not all, are purely formal and do not motivate the concepts in a way that they can be truly appreciated. If the author is able to pull this off in the later volumes, it would be a major advance in the education of quantum field theory.

- There is an interesting discussion of the Hopf fibration of the 3-dimensional sphere, but emphasizing the physics rather than the mathematics. The author uses the Hopf fibration to characterize the quantum states of a non-relativistic electron, but again defers the proof of the important results to Volume IV. Essential to his discussion is his claim that representing electron quantum states in this fashion is an indication of "nonlinearity."

Many-Particle Systems in Mathematics and Physics
- The author cannot resist relating the partition function of statistical physics with number theory and the Riemann zeta function. His discussion is fitting given the current work on proving the Riemann hypothesis using ideas from quantum field theory.
- Modular forms are briefly discussed here, which again is fitting because of the work of a few mathematical physicists who are attempting to show connections between the Langlands program in algebraic geometry and number theory and the notion of duality in gauge theories and string theory.
- The discussion on the Casimir effect is interesting, mainly because the author explains how physicists view the notion of convergence in a series. Physicists are actually more receptive to using divergent series, with the renormalization program in quantum field theory being the best example of this. Several summation methods for divergent series have been invented and used by physicists. The author gives a very detailed, superb discussion on the analytic number theory behind the Casimir effect.

Rigorous Finite-Dimensional Magic Formulas of Quantum Field Theory
- The title of this chapter is interesting, given the infinite-dimensional nature of quantum field theory. But apparently the author wants to build the reader's confidence and intuition using ordinary linear algebra of finite-dimensional Hilbert spaces (with a bit of Dirac calculus and discrete path integrals thrown in).
- One of most useful relations for calculations in quantum field theory, especially gauge theory, namely that DetA = Exp(Tr(Ln(A))) is discussed here in the context of Lie groups.
- The Dyson series, the bread-and-butter of perturbation calculations in quantum field theory is discussed and viewed as a consequence of the principle of superposition.
- The treatment of the functional calculus is rigorous, but this may mislead the newcomer to quantum field theory as being extendable to the full infinite-dimensional quantum field theory.

Rigorous Finite-Dimensional Perturbation Theory
- This chapter is very interesting in that it views the issues in the renormalization of quantum field theory as a problem in bifurcation theory. In fact the author makes the somewhat radical statement that this is similar to the problems in chaotic classical dynamics where the motions of asteroids for example are complicated because of the resonances that can occur. "The complexity of phenomena in quantum field theory is caused by resonances," he states. Readers more knowledgeable about quantum field theory, such as this reviewer may find this somewhat novel and will no doubt want to study the chapter in more detail than the others to see just how good this analogy is.In this regard, it must be remembered that this discussion takes place in finite dimensions, and so the analogy to bifurcation on second glance may not be too surprising.

Another Road to Reality
I'm in love with QFT. I noticed this planned monumental 6-volume work some time ago but I bought this book from Amazon UK after reading Quantum Field Theory Demystified and looking for more thorough ab initio treatment of QFT. Upon its arrival I immersed myself into it and in my opinion the first volume is like The Road to Reality book but more mathematically oriented with proofs, numerous examples, historical notes, generous citations and references.

Thanks,
Dmitry Vostokov
Founder of Literate Scientist Blog ... Read more

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Customer Reviews (56)

Cost overrun
This is irrelevant to the content of this book, as I have not read it.The reason I haven't is the cost.Something is wrong when the hardcover book costs $9.89, and the electronically delivered Kindle edition is $9.99.I know that the publisher sets the electronic book delivery price, and Amazon sets the price of the hardcover, but I just can't help thinking that something's wrong here, when a no cost, electronically delivered book costs more than a hard copy of the same book, with its underlying price to the publisher consists of the cost of paper, ink, binding, shipping and handling, postage, and the labor required to produce and deliver the hard book.Unless I find it in my local library, I won't be reading it.From reading the other reviews, I don't think that I'll be missing much.

Ordinarily, it upsets me to read complaints about the price in a space reserved for comments on the content of the book, but something about this pricing structure (one dime difference) just rubbed me the wrong way.Sorry for the diatribe.

Superficial and Sarcastic
Paulos gives us the latest entry in the New Atheist series, the purpose of which is to show how religious belief is foolish, absurd, and irresponsible.Just to give you a sampling of Paulos' approach to religion: it is "silly blather," "abracadabra", "inane", "seductive nonsense," "fatuous," "bogus", "childlike," a "verbal magic show," "obstinate blindness," and "infected" with "linguistic disease."Paulos does not even try to hide his disdain for religious believers, but proceeds to show how (in his view), every argument for religion (ontological, design, cosmological, etc.) is entirely false and ridiculous.As with so many of the New Atheists, he seems to assume that all religious believers are equivalent to bible-thumping fundamentalist literalists; it seems never to occur to him that there is a wide range of religious belief.Also, as with most New Atheists, he seems to have gotten his view of religion entirely from the popular media, and not have read any actual theology or serious religious philosophy.For him, religion means Creationism, faith healing, and televangelists.Apparently for him religion is simply so stupid that one need not know anything about it in order to mock and ridicule it. In fact, though there is to be sure a lot of nonsense out there in the religious world, there is also some very sophisticated religious thinking.And the arguments for religion (e.g. the ontological or design argument) are by no means mere verbal tricks; they are thought-provoking and worth taking quite seriously.But Paulos thinks he can dismiss religion entirely in 150 pages (without even having to do any research on the subject either).In short, if you want to engage in mockery and ridicule of religious believers, this is the book for you.If you want a serious thoughtful engagement with the question of religion, then this book is a waste of time.

A Mathematician Cashes in on the New Atheism
I enjoyed several of Paulos' previous books, especially Innumeracy but I was more than a little disappointed with this one. It is a very short book, with little or no original thought. I imagine the author purchasing several of the recent books by the New Atheists, reading them over the weekend, then spending the next few days jotting down a few quick thoughts and calling it a book.

There are so many better books on this subject. Try instead any of the following:

The End of Faith: Religion, Terror, and the Future of Reason by Sam Harris

The God Delusion by Richard Dawkins

God Is Not Great: How Religion Poisons Everything by Christopher Hitchens

The New Atheism: Taking a Stand for Science and Reason by Victor J. Stenger

Letting Go of God by Julia Sweeney

Well done, but to what end?
Nothing between the covers of this slim book hasn't been said before.So why write it?JP argues cogently, and Irreligion is entertaining enough.But who is its intended readership?Those who incline to JP's views have no need for a book like this -- and can find far more interesting treatments of the subject matter elsewhere.Those who incline to different views will likely a) not read this book, or b) reject the both the reasoning and the conclusions it sets forth.The objective, in writing books like this, is, it seems to me, to sway on-the-fence religious adherents to a secular humanist world view, or at least to coax them towards a deeper understanding and respect for it.All well and good, if anyone were actually on the fence.I doubt, however, that even one heart or mind is swayed by books of this sort.In the end, though they make for interesting/entertaining reading, such books do little more than preach to the converts who read them.They may be targeted to the 'out'-crowd (very wishful thinking), but they're invariably read by the 'in'-crowd.As such, they serve little discernible purpose.

Worth about $3.50
Impressions:

1. This book was easy to read, *at the beginning*, but as it went on it took too many digressions and seemed like babbling. In fact, the exact point at which the book started to go downhill was the "Argument From Subjectivity" chapter. Mercifully, it was short enough to keep it from turning into total torture.

2. The greatest use of the book was to summarize some of the different ideas that philosophers have used to try to prove the existence of God. (Reading Philosophy texts is more than a notion, and most people would have perished of boredom before even making it through 3 of the texts whose essence Paulos has summarized.)

3. Paulos seems to believe that there exists some type of "universal morality" at some very basic level, and that that includes such things as "basic honesty." He has obviously never visited China. Or most of Africa, for that matter.

4. The devotion of a mini-chapter to each school of thought was a neat and clarifying idea. The book can be read out of order, and certain ideas that need re-reading are much more easily found.

5. It seems that Paulos started out trying to write something that was not overly-philosophical.....and he succeeded until about halfway through. He could have toned down the $5 words. We all know that you can use a thesaurus! (Salubrious? Hirsute? As if none of us had ever heard of "healthy" or "hairy.")

6. I wonder why he focused so obssessively on the church (this review is coming from another atheist). If this book was simply about poor reasoning and religious fervor, then it could have been extended to include Muslims or even environmentalists. If it was about the inaccuracy of historical witnesses/ texts being the basis for some beliefs, then he could have taken on any Chinese source (most of which are of doubtful accuracy). But on the other hand, he might then have had even less to put into this already relatively short book.

This book would have been worth about $3.50 to me if I had to purchase it all over again. I bought it for about $7 (including shippping) and feel that I may have spent a bit too much. ... Read more

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Numerous scientists have taken part in the war effort during World War I, but few gave it the passionate energy of the prominent Italian mathematician Volterra. As a convinced supporter of the cause of Britain and France, he struggled vigorously to carry Italy into the war in May 1915 and then developed a frenetic activity to support the war effort, going himself to the front, even though he was 55. This activity found an adequate echo with his French colleagues Borel, Hadamard and Picard. The huge correspondence they exchanged during the war, gives an extraordinary view of these activities, and raises numerous fundamental questions about the role of a scientist, and particularly a mathematician during WW I. It also offers a vivid documentation about the intellectual life of  the time ; Volterra’s and Borel’s circles in particular were extremely wide and the range of their interests was not limited to their field of specialization. The book proposes the complete transcription of the aforementioned correspondence, annotated with numerous footnotes to give details on the contents. It also offers a general historical introduction to the context of the letters and several complements on themes related to the academic exchanges between France and Italy during the war.

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This is the second volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. This book seeks to bridge the existing gap between the different languages used by mathematicians and physicists.

For students of mathematics it is shown that detailed knowledge of the physical background helps to discover interesting interrelationships between quite diverse mathematical topics. For students of physics fairly advanced mathematics, beyond that included in the usual curriculum in physics, is presented.

The present volume concerns a detailed study of the mathematical and physical aspects of the quantum theory of light.

Customer Reviews (2)

Grand plan, poorly executed.
My background in math and physics is not good enough to do a good review of this book. It looks at QED from a big picture point of view which I like. He probably does a good job of it, I'm not in a position to say. I hope someone will come along soon who can do a better job of reviewing the good parts of the book.

My concern is that at the page level, it has a lot of problems. There are a large number of typos, sometimes in strategic places. I suppose that can't be helped. However, there is at least one definition that is simply incorrect. Fortunately, that becomes obvious right away. Although Schwartz spaces and tempered distributions are used in several places, they are not defined at all. Instead it refers you to the definition in volume I which I don't have. On the other hand, the Heaviside function is defined five times. Some of the definitions of mathematical objects are ambiguous, obviously not written by a mathematician for a mathematician. I presume that Professor Zeidler's native language is German, and while his English is quite good, it is not quite good enough for this book. He practially says as much himself. An intersting, but unimportant side-note: in one place I found the word 'und', where 'and' was meant. The notation is inconsistent from chapter to chapter, sometimes changing within a page and in at least one case changing in mid-sentence. It seems that whereever there is a similarity of ideas in two places, there lacks a corresponding similarity of presentation.

Some things I like about the book. Not every proof is given in the book. Instead, a citation is given to where the proof can be found in another book. This allows the author to cover more ground and the reader to choose whether or not to find the proof. I haven't been looking up these proofs myself. I hope that the diligent reader doesn't need an entire library to complete the proofs. A large range of mathematics is presented. Even if you don't learn any physics, you will learn a lot of math by reading this book. What is more, you will have physical motivation for the mathematics. I think this helps to understand the meaning behind the abstraction.

A Bridge Over Troubled Water
A Bridge Over Troubled Water


Professor Eberhard Zeidler has set upon himself the monumental task of uniting mathematics and physics, thereby building a bridge between two "enemy" ommunities.This is as difficult an undertaking as Shakespeare's attempt to unite two lovers, and thereby create a bridge between their feuding families.The reason is that Zeidler's task, like Shakespeare's, is beset with certain inherent difficulties and contradictions.

Mathematics, and mathematical education, start with a contradiction.By its very technical nature, mathematics is divided into various compartments and sub-compartments.Yet, to my knowledge, no other field is more united and inseparable than mathematics.Even physics can be divided into somewhat independent fields, but mathematics cannot.This makes math unique among all branches of human knowledge.

Soft fields like literature and history are very loosely bound.You can start with any author, like Shakespeare, Sartre, or Steinbeck, or any historical period, like British, Indian, or America history, and from there you can continue in any direction you want like a free bird.You can't do that in maths.You have to begin at a very specific point, 1,2,3,... and follow a very tightly bound order - just try to teach calculus to a guy who doesn't know trigonometry.In other words, soft fields like literature are like beautiful cities - you can visit their attractions in any order you want.But mathematics is like a maze or labyrinth - you have only one entry point, and once inside and in panic, you are forced to chase every nook and corner to reach for the exit before you go mad.This is the agony and the ecstasy of mathematics.

Now enters physics onthis "murky and stormy" stage.Again there are inherent differences in the approach of mathematicians and physicists to their respective fields.The most important thing in mathematics is a proof.In fact, I could not justify a lot of "weird" ideas in mathematics till I read their proofs.Often it is the necessity of proving things that introduces a lot of "weird" and counter-intuitive notions in math (at least for me).

Recently proofs in mathematics have become longer and longer.Many of Harish-Chandra's proofs in semi simple Lie groups and algebras are more that 100 pages long.Wiles proof of Fermat theorem is more than 200 pages long.And the Classification Theorem of finite groups requires more than 5000 pages of proof.As a physicists I have not even tried to read those proofs - I am incapable of that.But in my opinion a logical question arises:

Have these proofs to be so long, and if so, why?Do they reflect the strength or the weakness of the human mind?

I personally believe this is a fundamental question, and its very answer - whatever it may be - will clarify further the nature of mathematics, and even more , it would "throw some light" on the relationship between math and physics.

Now to physics.

Physics is all about Nature.And mathematics is just a language used to explain Nature.The relation between mathematics and physics is that of poetry and the language used to express that poetry.There is no inherent relation.This means that math and physics are not tightly bound, just as poetry is not bound to any specific language.Shakespeare could have used any language to express the same beautiful thoughts.And if it ever turns out that extraterrestrials have a better language to explain Nature I for one would rush to master it and throw our terrestrial mathematics into the dust-bin.This hints at a mismatch between math and physics.

One of the differences between physics and mathematics is that physics is more intuitive, and long mathematical proofs won't help if the intuition is not there.In fact, physics is a reality-check on math - it controls maths runaway flights into pure logic.I guess this is what Einstein and Feynman always said.And it is exactly here that a lot of bridges between math and physics have collapsed.Probably the mismatch lies in the brains of mathematicians and physicists.So, paradoxically again, in order to build true bridges between math and physics, some psychological study of the minds of mathematicians and physicists is needed.This gives some indication of the difficulty of building these bridges.It is not a bilateral but a multilateral task - a sort of multidisciplinary study.Seeing that Prof. Zeidler needs 6000 pages to deal with this bilateral task alone, it is no wonder that people dare not plunge into the murky waters of multilateralism - bilateralism is dark enough a territory to mess with.

The reason I am excited about "Zeidler Program" is that almost every time I read something from the 2 volumes of his series, I end up seeing some connection between physics and math which I did not see before, and this
sets me "thinking" ( at least in my limited ways).This cannot be said of many other books where all you can do is stare at the "most general formalism expressed in extremely terse notation".All you can carry away from such "masterpieces" is pseudo-knowledge consisting of some memorized terms and this is the most dangerous form of knowledge.

Yet the reason I am notpassing a technical judgement on the first 2 volumes of his program is that I want to wait till volume 3 (Gauge theory) and volume 4 (Quantum Mathematics) appear on the stage.They will be the making or breaking points of "Zeidler Program".Only they will demonstrate what kind of bridges he has built between poor, old, and senile Electromagnetism and his new, flashy, and rich cousin called Principal Fiber Bundle, and between a man called Witten and a man called Witten again, that is, between Witten the physicist and Witten the mathematician.It is over such troubled waters that many a bridges have been washed away in the past.So I wish Prof. Zeidler all luck.

To be continued after the appearance of volumes 3 and 4......

Rattan Mann
Oslo, Norway
... Read more

Product Description

The book starts with an elementary introduction to formal languages appealing to the intuition of working mathematicians and unencumbered by philosophical or normative prejudices such as those of constructivism or intuitionism. It proceeds to the Proof Theory and presents several highlights of Mathematical Logic of 20th century: Gödel's and Tarski's Theorems, Cohen's Theorem on the independence of Continuum Hypothesis. Unusual for books on logic is a section dedicated to quantum logic.

Then the exposition moves to the Computability Theory, based on the notion of recursive functions and stressing number{theoretic connections. A complete proof of Davis{Putnam{Robinson{Matiyasevich theorem is given, as well as a proof of Higman's theorem on recursive groups. Kolmogorov complexity is treated.

The third Part of the book establishes essential equivalence of proof theory and computation theory and gives applications such as Gödel's theorem on the length of proofs. The new Chapter IX, written for the second edition, treats, among other things, categorical approach to the theory of computation, quantum computation, and P/NP problem. The new Chapter X, written for the second edition by Boris Zilber, contains basic results of Model Theory and its applications to mainstream mathematics. This theory found deep applications in algebraic and Diophantine geometry.

Yuri Ivanovich Manin is Professor Emeritus at Max-Planck-Institute for Mathematics in Bonn, Germany, Board of Trustees Professor at the Northwestern University, Evanston, USA, and Principal Researcher at the Steklov Institute of Mathematics, Moscow, Russia. Boris Zilber, Professor of Mathematics at the University of Oxford, has been added to the second edition.

Product Description

This book presents biosensor development and modeling from both a chemical and a mathematical point of view. It contains unique modeling methods for catalytical (amperometric, potentiometer and optical) biosensors. It examines processes that occur in the sensors' layers and at their interface, and it provides analytical and numerical methods to solve enzymatic kinetic and diffusion equations. The action of single enzyme as well as polyenzyme biosensors is studied, and the modeling of biosensors that contain perforated membranes and multipart mass transport profiles is critically investigated. Furthermore, it is fully described how signals can be biochemically amplified, how cascades of enzymatic substrate conversion are triggered, and how signals are processed via a chemometric approach and artificial neuronal networks. The results of digital modeling are compared with both proximal analytical solutions and experimental data.