Reviews (4)

Oops
"Philosophical and Mathematical Correspondence" was published 13 years after Heijenoort's.

Just a comment.
In response to Jay Miller's question below there is a book titled "Philosophical and Mathematical Correspondence. Gottlob Frege" that has 21 letters between Russell and Frege over a period of 10 years beginning with Russell's observation of his famous paradox in 1902. This wonderful collection of correspondence was published 20 years before "From Frege to Godel" and includes letters from many of the same mathematicians and logicians.

Essential reference in the history of logic and computing
The second part of my review title may shock some, but the excellent collection of papers that Van Heijenoort has edited (and in many cases translated!) is also an excellent reference in the history of computing. Everyone appreciates that mathematical logic gave rise to computer science; the papers in this collection from Hilbert, Herbrand, Gödel, and others will show why.

If your interest is instead the history of logic, all the classics in the range specified by the work's title are here, complete with their own ideosyncratic notation. van Heijenoort's wonderful introductions to each piece will interelate the works, provide references to other literature and situate everything in a wonderful intellectual climate.

Be warned, however, that the foundational papers in this still growing field continue for another 15 years or so; these are reprinted in Davis' (alas, out of print) anthology _The Undecidable_.

Isbn: 052129648X
Sales Rank: 33883
Subjects:  1. General    2. Mathematics    3. Methodology    4. Philosophy    5. Research    6. Philosophy / General    7. Philosophy of mathematics    8. Social & political philosophy   

Isbn: 0198505248
Sales Rank: 1283157
Subjects:  1. Intuitionistic mathematics    2. Logic    3. Mathematical And Symbolic Logic    4. Mathematics    5. Philosophy Of Mathematics    6. Science/Mathematics    7. Mathematical logic   

Isbn: 019824035X
Sales Rank: 456111
Subjects:  1. Logic    2. Philosophy    3. Philosophy of mathematics    4. Philosophy of science   

Isbn: 0198239750
Sales Rank: 955508
Subjects:  1. Logic    2. Mathematics    3. Science/Mathematics    4. Philosophy of mathematics   

Reviews (2)

The Differentiation of Reality through language
This is the Calculus we all ought to be studying in school. Not since Whorff have we come this close to the heart of Languages.

Some remarks on Remarks...
If you are interested in the philosophy of mathematics, this is the book for you! No special knowledge of mathematics or philosophy is needed to read and understand this book. Some prior experience with philosophy wouldbe helpful, though,just to get you used to discussing philosophy. Somereading of other philosophy books by Wittgenstein, especially PhilosophicalInvestigations or On Certainty would be helpful, to get you used toWittgenstein's unusual style of writing. But if you haven't read anyphilosophy before reading Remarks on the Foundations of Mathematics, don'tworry, it isn't necessary.

Wittgenstein seems to think that mathematicsis a language as German, or French, or English are languages. Mathematicsis a human language, he says.He discusses the way we learn mathematics andalso what constitutes proofs of mathematical theorems.

This is anextremely interesting book to read, but it is not something you can sitdown and read in one sitting. It will take a few sittings to get through itall. For myself, this was because I had to stop frequently and think of theimplications of what Wittgenstein was saying. For example, he says that themathematiciam creates essences. I had to re-evaluate my conception ofmathematics after reading this statement.

Reviews (10)

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.

Isbn: 0195080300
Sales Rank: 555417
Subjects:  1. Foundations Of Mathematics    2. Logic    3. Logic, Symbolic and mathematic    4. Logic, Symbolic and mathematical    5. Mathematical And Symbolic Logic    6. Philosophy    7. Mathematical logic    8. Philosophy of mathematics    9. Western philosophy, from c 1900 -