Proofs and Refutations: The Logic of Mathematical Discovery [Paperback]

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.

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 Lots of historical references illustrate the fact that the discussion is not naive and that reflects the actual history of the subject.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.

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.