Superb discussion of Aristole
591621
Well written, balancing general discussion, review of critical discussion, and detailed examination of specific passages.  Offers convincing interpretation of problematic texts, especially De Anima and The Metaphysics. A standard text that encourages examination of the original texts.
1448658
Howard Eves presents this five-star story of mathematics as two intertwined threads: one describes the growing content of mathematics and the other the changing nature of mathematics. In exploring these two elements, Eves has created a great book for the layman. I find myself returning to his book again and again.
My few semesters of calculus, differential equations, and other applied math failed to formally introduce me to abstract algebras, non-Euclidian geometries, projective geometry, symbolic logic, and mathematical philosophy. I generally considered algebra and geometry to be singular nouns.  Howard Eves corrected my grammar.
"Foundations and Fundamental Concepts" is not a traditional history of mathematics, but an investigation of the philosophical context in which new developments emerged. Eves paints a clear picture of the critical ideas and turning points in mathematics and he does so without requiring substantial mathematics by the reader.  Calculus is not required.
The first two chapters, titled "Mathematics Before Euclid" and "Euclid's Elements", consider the origin of mathematics and the remarkable development of the Greek axiomatic method that dominated mathematics for nearly 2000 years.
In chapter three Eves introduces non-Euclidian geometry. Mathematics is transformed from an empirical method focused on describing our real, three-dimensional world to a creative endeavor that manufactures new, abstract geometries.
This discussion of geometries, as opposed to geometry, continues in chapter four. The key topics include Hilbert's highly influential work that placed Euclidian geometry on a firm (but more abstract) postulational basis, Poincaire's model and the consistency of Lobachevskian geometry, the principle of duality in projective geometry, and Decartes development of analytic geometry. For the non-initiated these topics may seem daunting, but Eves' approach is clear and quite fascinating.
Chapter five, which might have been titled "The Liberation of Algebra", may at first be a bit overwhelming to those unaware of algebraic structures like groups, rings, and fields.  But take solace as even mathematicians in the early nineteenth century still considered algera to be little more than symbolized arithmetic. As Eves says, non-Euclidian geometry released the "invisible shackles of Euclidian geometry".  Likewise, abstract algebra created a parallel revolution. (Again, don't be intimidated by the terminology.  Eves is quite good.)
The remaining four chapters look at the axiomatic foundation of modern mathematics, the real number system, set theory, and finally mathematical logic and philosophy.  Eves concludes with the surprising discovery of contradictions within Cantor's set theory as well as Hilbert's unsuccessful effort to define procedures to avoid inconsistencies or contradictions within an axiomatic system.
Eves mentions Godel's fundamental contribution to mathematical logic, but stops short of delving into Godel's Proof. For additional reading I highly recommend "Godel's Proof" by Ernest Nagel and James R. Newman.
I also highly recommend Richard Courant's and Herbert Robbins' classic, "What is Mathematics?", a more detailed examination of the development of fundamental ideas and methods underlying mathematics. I would suggest that most readers, particularly non-math majors, first read Eves and later tackle Courant and Robbins.
I have read "Foundations and Fundamentals of Mathematics" at least twice.  I gave my son a copy for Christmas. He says that the book is great and he even claims to be reading it as he walks across his campus between classes. The price is great. It belongs in your book collection.
997366
I return to this book again and again just out of sheer pleasure. The depth of scholarship of the authors shines through on every page and the choice of historical material is fascinating. Topics like compactness and uniform convergence  can here be seen to have arisen out of genuine necessity-they are not (as would seem from other books)mere names in a standard syllabus. If you have any mathematica interest at all, take this book on holiday and sleep with it under the pillow to extract more from it by osmotic pressure overnight.
3128577
Charles Sanders Pierce urged mathematicians to present their results with demonstrations that showed how they actually got there, rather than the usual way, which is to tidy up a nicely polished pyramidal proof that leads enexorably, pontifically from axioms to Q.E.D., and then to throw away the scaffolding and sweep away all traces of the parallel paths, false starts, and tangents of the actual process. This suggests a difference between mathematical reasoning as practiced and mathematical reasoning as mathematicians feel it ought to be. We may also distinguish mathematical reasoning from mathematical knowledge, as for example the toolbox of rules and formulae that every high school student of algebra is asked to master.

Nickerson is a research psychologist inquiring into how reasoning is done in the diverse fields of mathematics as an outsider, as he presumes us, his readers, to be. This book is very readable, and also very thorough. One must be prepared to commit mental resources to taking in a lot, and one very much needs the readability lest the scope, variety, and depth of coverage appear to be too daunting. Nickerson's coverage is impressive, and his intent is very ambitious.

I think that mathematicians as well should benefit from reflecting on how they reason, perhaps agreeing at some times and then when they disagree it may be finding themselves on occasion brought to an unfamiliar but possibly fruitful self-reflection. Also, mathematicians naturally specialize, and a survey of the field can be like looking up from myopic concentration at a desk and gazing out the window to scan a distant and opening horizon.

I haven't yet read enough to comment on the success of Nickerson's project, but what I have read has led me to recommend it a few times, and I now recommend it to you.
4533399
I am not aware of too many texts directed specifically at the finite volume and Riemann solver approaches, and so finding any book discussing details of these methods is hard to come by, and finding a very good one is just rare. But this book presents an excellent review of these kinds of numerical schemes for hyperbolic problems. I used this text (both the first and third edition) extensively throughout graduate school. I kept the library copies well beyond what I was allowed to. I will always keep it within my arm's reach. It is an advanced text; I had to start with LeVeque's book (<a data-hook="product-link-linked" class="a-link-normal" href="/Finite-Volume-Methods-for-Hyperbolic-Problems-Cambridge-Texts-in-Applied-Mathematics/dp/0521009243/ref=cm_cr_arp_d_rvw_txt?ie=UTF8">Finite Volume Methods for Hyperbolic Problems (Cambridge Texts in Applied Mathematics)</a>) to understand a lot of the basic concepts (LeVeque's book is also an excellent text on these topics). The approaches and discussions in Toro's text are formulated in a very mathematically rigorous language, which can be daunting to read, but I found that this language helped significantly in my transition from knowing basic concepts to specifying the details precisely in a rigorous fashion. I have not experimented at all with the Numerica libraries that the book discusses, but I found most parts of the text readable enough to write my own code from.
1085366
This book is a massive piece of work edited by Anthony Kenney, Jan Pinborg and the late Norman Kretzmann. Moreover, Eleonore Stump is an associate editor of this text as well. The book itself is over 1000 pages. It is a history of late medieval philosophy from about 1100 to 1600. The book is essentially a series of essays that deal with the various philosophical trends, ideas, issues, etc. that were prevalent within the aforementioned dates. The book deals with Aristotelian logic, logic in the middle ages (semantic theory), logic in the high middle ages, metaphysics and epistemology, natural philosophy, philosophy of mind and action, ethics, politics, and scholasticism. Thus, the reader of this text will gain a greater understanding about Abelard and old logic to free will and free choice to God's knowledge of future contingents, and much more. The Islamic philosophers and their influences and ideas are covered as well as the Jesus society philosophers of the late reformation period (i.e. Molina, Suarez, etc.). The text itself is not for the beginning philosophy student, thus the contributors expect their readers to have some back ground knowledge in the issues at hand. This book is meant for the serious student of this period. Therefore, if you love to study the philosophers of the Medieval period, then you will love the detail that this book provides. I highly recommend it.
491333
Considering how opaque and difficult to read the typical technical discussion is in philosophy of mathematics, this book deserves six stars.  It is so extremely easy to read that it can be read profitably in a couple of hours.  I learned a number of useful facts about Gdel's work.  I also found the discussion to be somewhat insightful, although there were some points of difficulty.  I think that even a very young child could read this book and derive some useful insights.  I was very pleased with this book.  I strongly recommend it for its clarity and insight into some of the most important philosophical work of the twentieth century.  The book is clear, short, and easy to read, while still exposing the reader to some of the depth of Gdel's work.
527486
Some historical figures are known to all cultured persons.  Leo
Tolstoy, Johannes Brahms, Vincent Van Gogh -- these are names of
which most of us would be embarrassed to be ignorant.  On the
other hand, few of us would think to add the name of Jules Henri
Poincare to this list; and yet, if we did, it would tower over
all the rest.  For Poincare was a mathematician of the very
highest rank, an astonishingly fertile creative genius whose
prescient insight and technical mastery utterly transformed the
art to which he devoted his life.  Among his predecessors in the
three-thousand-year history of mathematics, only Archimedes,
Newton, Euler, and Gauss can be said to have equalled or
exceeded him.  By any standard, Poincare was one of the truly
great.
If for no other reason than this, THE VALUE OF SCIENCE is well
worth reading.  It is, after all, quite rare to find a book that
collects the thoughts of one of the very few genuinely profound
intellects in human history.  But what makes THE VALUE OF
SCIENCE truly wonderful is that it is not merely worth reading:
it is, in fact, a joy to read.  For in addition to his uncanny
mathematical gifts, Poincare had the knack of expressing
himself beautifully in writing.  Even in translation, his prose
has an admirable lucidity and grace, and his aphoristic style
often makes him highly quotable.  When he speaks of mathematical
creation, as he does in a celebrated essay of that name, he
offers general readers a fascinating glimpse into the depths
of his own extraordinary mind at the peak of its imaginative
frenzy; his (scrupulously non-technical) account of one of his
greatest mathematical discoveries supplies an unforgettable
intellectual thrill, a sort of electric shock for the soul.  For
anyone interested in the psychology of creation, this is simply
irresistible stuff.
A final brief caveat: although Poincare was clearly the greatest
mathematician of his time, he was not the greatest physicist.  A
curious wrong-headedness kept him from beating Albert Einstein
to the creation of special relativity, and general relativity
eventually proved Poincare wrong in some of his opinions on the
relation between physics and geometry.  So when Poincare speaks
of physics, bear in mind that some of his positions no longer
seem really tenable.
But this is no reason not to read every word of THE VALUE OF
SCIENCE.  It's a marvelous book, and its author was a marvelous
man.  Get to know them both.  You'll be glad you did.
4357893
Having previously read Chesterton's The Dumb Ox in a large battered tome of Chesterton's collected works, I much appreciated reading it again in this slim, sturdy, attractive, modestly-priced edition, one of the first offerings of Angelico Press. Chesterton wrote a similar small study on St. Francis of Assisi ten years before the publication of St. Thomas Aquinas. Taken together, Chesterton's short books on the two most remarkable Italian friars in history give an unparalleled insight into the high Middle Ages. For those who might be intimidated by a book on the world's premier Catholic philosopher and theologian, there is no reason to fear. As I found to my great relief (being a bit philosophically challenged myself) St. Thomas Aquinas is not a technical treatise but rather a highly accessible overview of Thomistic philosophy and its historical significance. Above all, it is an animated and perceptive reflection on the stolid Schoolman, that "huge, heavy bull of a man" whose efforts to reconcile Aristotle and Christianity resulted in his definitive masterpiece, the Summa Theologica,  As Chesterton explains throughout the book with his characteristically colorful metaphors and illustrations, Thomism was a system of "colossal common sense," "a working and even a fighting system; full of common sense and constructive confidence, and therefore normally full of hope and promise."  It is fitting that Chesterton, "the Apostle of Common Sense," as Dale Ahlquist called him, celebrated for his eminently practical, rational conclusions, should have given St. Thomas Aquinas such a spirited and affectionate introduction to the modern age. The reader suspects early on that Chesterton who, like Aquinas, was a man of considerable girth and intellectual aptitude, had a great affinity for the Dumb Ox of Sicily. Chesterton's short work on Aquinas was criticized by some for its lack of philosophical analysis and detail, but his intuitive understanding of the saint, his keen commentary on the well-known and not so well-known events in Aquinas' life and his clearly demonstrated fondness for the Angelic Doctor are the most valuable aspects of the book. Being far more inclined towards poetry than philosophy, I was delighted with Chesterton's whimsical details and masterful reflections. St. Thomas Aquinas wrote the Church's most beautiful Latin poetry, reknowned for its elegance, potency and profundity, and perhaps this was the link between the medieaval Italian scholar and the modern English man of letters----that Chesterton, an accomplished poet himself, could recognize a kindred soul in the inscrutable churchman with the "Napoleonic head and a dark bulk of a body" and labored in his book to paint a fully dimensional and sensitive portrait of the noble Thomas of Aquino.
4895710
An exhaustive if not exhausting explication, summary, and analysis of Aristotelian philosophy. The main parts of this book are:  I "The Philosopher"; II Knowledge and Science; III Physics and Metaphysics; IV Ethics and Politics; V The Reception. All in 18 chapters with multiple sections.
After writing what is arguably the best book on Kant, Hoeffe sets his sights on Aristotle. Because of his familiarity with Kant we get quite a bit of comparison between Aristotle and Kant. Kant is challenging and Hoeffe's work on him is great in clarifying things. With Aristotle, on the other hand, the traditional picture taught at universities and written about is not of a difficult philosopher, just one who can be cumbersome to read. Of course that picture of a simple, clear, and direct Aristotle is only possible when one ignores all the nuances in Aristotle. Hoeffe, rather than clearing things up simplistically, aims to present us the entire Aristotle- all of it. And that is by no means the easy Aristotle with which we are familiar. What we get is a picture of an Aristotle who is not as systematic, holistic, clear as we expect. We are introduced to every single two-three-four-and-fivevold distinction he has ever made, often at two or three levels, in every work he has written and because Aristotle wrote on the same subject at different times, we get an Aristotle who is not at all free from inconsistencies and incoherences. This is not easy reading as some sections are quite abstruse. Some of it can be blamed on the translation. German is not easy to translate but unfortunately it's obvious the translator chose to translate literally rather than to emphasize readability. If this were the work of original philosophy, then a literal translation would make sense, but in translating a secondary work, there is no need to sacrifice readability. There are also some curious choices in diction which, while literal, make little sense. One can't but admire the effort and work Hoeffe has done in presenting such a comprehensive summary of Aristotle. The main virtue of this text is that Hoeffe objectively presents us Artistotle works and how influential he was. This is not about how witty the author is, or how great his personality is, or how original his interpretation. For the most part he stays out of things, unless he sees some errors in traditional interpretations. In those cases Hoeffe will present his case with plenty of support from sources. When applicable, Hoeffe brings in other elements to compare, for example, the chapter on politics discusses modern democracy, republics, nations to compare with Aristotelian notions. The book itself, like most SUNY books is gorgeous, printed on excellent smooth paper, using some gorgouse types, although the main text looks somewhat dense and compressed. An indispensable companion to Aristotelian texts, recommended for all serious students of Aristotle and Greek philosophy.
3532939
I do want to point out that this is a very good book.  I did give it five stars after all.  It is a great book for getting a good view of several scientists and their contributions to the sciences and engineering from the late renaissance period to modern times.
The topic our kind author, John Gribbin, is tackling is very large and no one book can hope to cover the entire topic.  Even, as is the case here, in a book of almost 700 pages. Gribbin has chosen to focus on the interplay among the scientists, mainly when he wants to segway from one scientist to another, and the social implications of their discoveries... including much on the political realties of the time.  Especially in the case of Galileo Galilei, where the political issues can be as important as his scientific discoveries.
I would say that the great weakness to this approach is that he focus's a lot of those who invented things, and less on those who developed new ways of thinking about the world.  He claims to be doing the later, and does do a good job of it at times, but he appears to ignore the implications of a quote from Galileo he likes to use a lot: "science is written in the language of mathematics".  Gribbin almost totally ignores the contributions the people he covers made to mathematics, and pure mathematicians have trouble even getting a mention from him.  For example, in discussing Newton he could have discussed Gottfried Leibnitz a little, but instead just mentions that Newton and him argued about who discovered Calculus as leaves it at that.
The second great weakness of this book is there is no quick introduction to the best of the Greek and Roman philosophers who did a lot of science.  Aristotle was thought highly of for a lot of good reasons by people in the middle-ages, and he was right about a lot of things.  It is easy, especially now-a-days, to think that the ancient Greeks sure missed the boat on a lot things. Aristotle gets mentioned, now and again, is passing, but I haven't even run across a mention of Archimedes (when discussing Newton and Calculus it would have been nice to point out that Archimedes came close to discovering it almost 2000 years before.  Some think the only reason he missed was the faulting numeral system he had to work with), Pythagorus, or Euclid.
Hindsight is 20/20, and the Greeks and other ancients got most of the things correct, especially when you consider that they didn't have many of the tools that the later true scientists had access too.
Then there are the two great oversights among great scientists that I think would have site well into the mold he was using: Pasteur and Goddard.  Not minor folks in the history of science.
Because he couldn't cover everything, I overlook these problems and still give it a high rating.  People should know that their is a lot of ground to cover in science and the history of science and seek out more information on the topic.  Especially since an informed populace is the requirement and basis for a democracy.  People need to understand how science and true scientists work in order to have informed opinions about many of the issues facing society today.
Still a fine work and worthy of anybody library.
3348544
I really enjoyed this book and am now starting it again to ponder it deeper.  I am amazed that someone from the 1600s could be so current in his thinking.  It will take a second (and third) reading to really process all of his ideas.  Kreeft's commentary on Pascal is extremely well thought out (and he throws in a few points of humor too!).
150471
This book is a must have for anyone interested in the debate with the new atheists, or anyone interested in the value of classical philosophy.

Ed Fesser has written a book that can be taken as a great overview of Greek philosophy and the insights of Aquinas and the Scholastics. He shows the incoherence of the "materialist-mechanistic" view which seeks to banish the existence of non-material entities such as mind and soul. He writes in a down to earth manner, always defining his terms and giving examples that are accesible to laymen interested in the subject. He brilliantly defends Aristotelian thought, showing that it is relevant and true even in the present day. He shows how the "new atheists" have misrepresented Aquinas and set up strawmen to attack.

If you want to be able to defend religion and traditional morality, the arguments that Feser presents are a Godsend.  As he makes clear, there is no appeal to "faith", his entire argument is based on reason and rational inferences. I cannot praise this book too much.  Read it and you will see for yourself.
2654156
REVIEW OF WHEN ATHENS MET JERUSALEM authored by John Mark Reynolds

I appreciated reading this book.

I found it very readable and clear.  His writing is insightful in this exploration of Greek philosophical thinking (Athens) and Judeo-Christian convictions (Jerusalem.)

His introduction points to the debt our Western civilization owes to Greek thinking being informed by Judeo-Christian convictions (Athens meeting Jerusalem.)  Insightfully in the introduction, he points to our present cultural temptation to jettison knowledge or even an interest in either Athens or Jerusalem.

In the main work, Dr. Reynolds explores delightfully the history of Greek thinking from Homer to Plato.  He really shines in his discussion of Socrates and Plato.  Though I was a science major in college, I did have to read some Plato.  I never read the Phaedo, Republic or Timaeus, because I had the impression they were dense, difficult and time-consuming.  But Dr. Reynolds opens these works up wonderfully in an engaging, bright way.  I think it is the clearest, most insightful treatment of Plato that I have read.  I wish I had had something like this when I was going through college.

The book then looks some at Aristotle and later Greek philosophers around the time of Christ, including a short section on the apostle Paul.  In his Postscript, Dr. Reynolds reviews the historical progression of Greek thinking with its limitations and stagnation.  He points to the deep solutions that connect with the coming of Christ.

Dr. Reynolds is smart, brilliant and knowledgeable.  It was a worthwhile pleasure reading this work.  I would think this book would be of value to teachers, ministers, students, philosophy buffs, as well as any wishing to expand their horizons by "going forward to the past" to make more sense of the present.

I hope he writes a second book that will explore even further how Christianity has deeply influenced Western Civilization and indeed, our present world.

I appreciate the author's pensive, perceptive and at times almost poetic work.  For me this is one of those books that will stay in my study where I will refer to it further to savor and digest its insights.
827633
Just as Ayn Rand corrected, completed, and extended the Aristotelian project into a new, brilliant, organic - and true - philosophic whole; so too retired professor of mathematics, Robert Knapp - applying Ayn Rand's theory of knowledge - has corrected, completed, and extended the Euclidian postulates into a new, brilliant, organic - and true - mathematic whole.

Dr. Knapp has synthesized his decades of research in modern academic mathematical theory together with Ayn Rand's theory of knowledge into a stunning work in which he has created the framework for a new philosophy of mathematics.

Chapter I, "Euclid's Method" lays the groundwork for a reinterpretation of Euclid's method that is solidly grounded in reality.  After more than 2,000 years, Euclid's mathematical treatise has finally received the proper objective foundation necessary for modern mathematical theory to be set aright.  This single chapter deepened my understanding of both the foundations of geometry and AR's theory of concepts.

While I do teach math to children, I found that having a mathematical background was not necessary to follow Dr. Knapp's arguments.  Moreover, no prior knowledge of Ayn Rand's theory of knowledge was required either.  Dr. Knapp's writing is crystal clear as he shuttles back and forth between abstract points and concrete examples.  I found the book's illustrations extremely helpful in grasping the material.

My favorite chapter was "Geometry and Human Cognition" for its insightful application of AR's idea of "measurement omission" to mathematical diagrams and the resulting cognitive rewards.  Surprisingly, I found my mind wandering to AR's theory of stylization and art while considering Dr. Knapp's views.  There is much here for the serious thinker.

If you are interested in the philosophy of science in general, or the philosophy of mathematics specifically - this is the book that all future mathematical theorists will have to take into account.  As for students of AR's philosophy, here we have an application of her view of knowledge to heretofore unexplored territory - a territory rich with untapped wealth.

Dr. Knapp's book can but only be the necessary beginning of a philosophy of mathematics for the 21st century - and a brilliant beginning it is.
3681345
Charles Trinkhaus is a master of understanding and interpreting the thought and philosophy of the Renaissance. In this brief, though extremely insightful and intelligent work, Trinkhaus provides us with a glimpse into the true significance of Petrarch as the archetype of Italian humanism, poetry as philosophical expression, and the then burgeoning understanding of the significance of humanity and the individual. An excellent read for all students of the Renaissance in general and Petrarch in particular.
560407
this is a nice edition with lots of discussion, including discussion by the author, prefacing the music at the beginning. mikrokosmos is a great way to learn piano and be exposed simultaneously to a modal oriented harmonic language