Introduction to Mathematical Thinking

This is a great book for filling the gap that we face in making a transition from grade school to more advanced, college-level math. It is very short, and so can be read and studied as a supplement for other math-related studies. It is basically about the important shift in perspective from using math as a calculational tool to understanding and communicating math to others. This transition mainly requires some knowledge and skill at proofs, and a basic appreciation of why this "grammar" (which is not an area that normally impedes us much in understanding or communicating in a spoken language like English) becomes so important in math. Professor Devlin addresses this ably, and evinces in this short book, his immense insight into teaching math. He accomplishes a lot in this short book, and I think it is extremely important to learn the types of basic concepts and methods he discusses at the earliest possible age at which a student can appreciate this. Although a short work, his book contains numerous exercises, and it would be wise for the beginning student to at least attempt many. Proofs, and an appreciation of the value and significance of "pure" mathematics are extremely important in progress in many fields of science, math and engineering. Because he is very precisely addressing one of the most serious problems in making a transition to advanced math, and in a very lucid, intelligent way, I can strongly recommend this book. Many people want to use math only intuitively and/or as a tool, but to progress in math, we must absolutely confront the extreme importance of rigorous mathematics, and understanding math adequately. Of course, there is a vast difference between the basic skills, of, for example, proving relatively simple theorems, and coming up with proofs, which amounts to a skill that is not uncommonly associated with the highest levels of art, ingenuity, insight, and, not infrequently, genius. But, climbing that hill is one far beyond the scope of what Prof. Devlin is trying to communicate in his book. While a really great book, it is probably best to accompany studying it with some assistance from a tutor or a teacher.Here is a bit of perspective on exploration and adventure in math, and the world Professor Devlin is trying to help us, as far as opening our minds: As a teacher, I have been working at my own study in math. I finally got a hand calculation I was working on for what I am studying with Lagrangians (with respect to calculus of variations and the theory of PDEs) to work out. I have to say that there are the moments of fun, when one does something really nice or gets to use some really nice software (like the stuff that the number theory enthusiasts use to detect Mersenne primes for calibrating supercomputers and for cryptography), but hand calculations are not always so much fun. Furthermore, in order to learn physics and math, and have an understanding of what the abstractions mean, you have to make some contact with calculations, with experiments, and do some hard work. It's not just a matter of reading about cool theory. We all like the adventure and the great exploration that math and physics and engineering represent (and really, that is at the heart of what our species is about and what makes us humans a little different from other organisms), but there is always the price of hard work that has to be paid. We can't always let others pay that price, or expect AI or computers to make it easy, if we want to understand and progress. You can spare yourself a lot of time and effort if you are extremely clever, or a genius like Feynman or Villani, that's for sure. But I think the adventure of physics and math is for all of us, not just the extremely clever or the geniuses.

The first few chapters where the author defines the linguistic forms axioms, theorems etc. take and the discussions of connectors and quantifies are decent. The explanation of implication is over complicated due to avoidance of truth tables and the standard definition (even though the author already introduced both truth tables and negation) in the attempt to build intuition. There are some good parts around symbolic manipulation, but the proof sections are lacking in explanation of the mental models and oscillate between too simple and “figure it out without any answers/hints though you are learning this”. In fact there is a lot of “figure it out yourself” without actually teaching you a new way of thinking. Not the best work of this author for sure. I would recommend reading through the first few chapters and ignoring all the sections on proofs. One can then pick up the supremely awesome “Proofs: A long-form mathematics textbook” by Jay Cummings and really do the work while savoring the fun of the subject.

This goes out as a "thank you" to Professor Devlin, but should prove informative to people who can relate to my situation,All my life, I absolutely hated math. I mean the deep pains in my cortex, blood boiling derision of the concepts kind of hatred. I remember being 8 years old and absolutely loathing the thought of studying times tables. Then, variables were added - great, numbers and letters now! No one took the time to explain why we were studying such seemingly needless information. I understand now that I am quite the pragmatist, needing a defined practical application or goal for almost anything I do. Hence the reason I hated mindless calculations so much.My disdain for the field grew to a point in my teenage years that I could no longer even look at the subject without my well-established biases taking over, driving me away. It affected my SAT score, with a perfect score in verbal comprehension being blemished by an embarrassingly subpar result in math. My educational experience was inhibited for almost two decades, simply because I saw no use for math past counting how many apples I wanted to buy at the grocery. After law school, I became interested in finance. From fairly light interactions with financial valuation methods grew a shameful realization that my hatred for math had caught up with me, finally affecting my daily life. I was a quantitative infant in the worst way, entirely handicapped in a vast and important arena.Then I found Coursera, and through its curriculum I came across this course. Though I have not finished it, as it has just started, the book has opened my eyes. Math is a language, just like the ones we speak and master. Its algorithms are logical thought, its concepts proven by deductive and inductive reasoning. All you need is a good handle on core concepts and, if you are like myself, a tangible application (even if its somewhat metaphorical or anodyne), and you will be able to appreciate what Professor Devlin has done with this course.If you are willing and able to learn past the age of 20, hated math your entire existence but are finding useful applications for it as you age, then this book and the corresponding course are entirely for you.Thank you, Professor Devlin, for doing what no teacher was able to for decades - finally get through to a smart kid who just lacked the appropriate impetus to apply himself.

This is an excellent and inexpensive companion to Prof. Keith Devlin's Coursera course.Here, "mathematical thinking" refers to the kind of formal logic that one might find early-on in an undergraduate mathematics course, but don't be put-off by that as it doesn't mean you need much of a background in high school mathematics; it's basic, logical, deductive reasoning expressed formally in mathematical shorthand. It's a valuable practical tool for everyday thinking and this ought to be taught in our primary schools as a foundation for many other things - and not just maths and sciences.In this respect, Devlin's book is a concise and inexpensive alternative to Burger & Starbird's "The Heart of Mathematics: An Invitation to Effective Thinking" (another book associated with an excellent on-line course). While Burger & Starbird is lavish, it's also very expensive.While it seems that school mathematics is increasingly about learning how to "turn a handle" to do things in a hurry and without much real insight, Devlin places particular emphasis on taking things slowly in order to think about and understand the concepts. The exercises are an essential part of the book and although there are no solutions, you'll mostly know when you have the right answer if you've understood what's going on - and if you don't, then go back and think about it; as Devlin says repeatedly: don't rush!. In addition, Devlin's Coursera on-line course is very well worthwhile (and free) and discusses many of the problems.

I have the print copy but decided to get the EBook for my Kindle Fire. Firstly this is an excellent introduction to logic and mathematical proof. However, being a 'mathematical' text there are a lot of Greek characters - some of which do not reproduce on the Kindle Fire - leaving a series of empty squares. Strangely enough this is not reproduced on my Kindle for Ipad app. Other than that flaw it woukld get 5 stars.

Excellent alongside Coursera on-line course by Keith Devlin himself.

A good and accessible introduction to mathematical reasoning, with exercises to test your comprehension. It covers logic and mathematical proof, and does not require much previous mathematical knowledge, so it is accessible to lay readers as well as people studying maths at school or university.

Keith Devlin makes mathematics interesting. This is something none of my maths teachers achieved when I was at school. If you think you are not very good a maths don't worry this book will awake a hidden interest in the subject. I read it because I took the Coursera course on the Introduction to Mathematical Thinking. So if you would like to think like a mathematician or, as in my case aspire to, then this book is essential reading.