How To Read And Do Proofs By Daniel Solow Pdf

[Mirtha Hinrichs]

DanielSolow is a professor of management for the Weatherhead School of Management at Case Western Reserve University. His research interests include developing and analyzing optimization models for studying complex adaptive systems, and basic research in deterministic optimization, including combinatorial optimization, linear and nonlinear programming. He has published over 20 papers on both topics.

The inclusion in practically every chapter of new material on how to read and understand proofs as they are typically presented in class lectures, textbooks, and other mathematical literature. The goal is to provide sufficient examples (and exercises) to give students the ability to learn mathematics on their own.

This book is designed to reduce the time and frustration involved in learning how to read, think about, understand, and "do" mathematical proofs and also to provide a description of other mathematical "thinking processes". In Part 1 of the book, the various techniques used in virtually all proofs, independent of subject matter, are identified and described. Students are taught not only how to use the techniques, but also when each technique is likely to be used, based on certain keywords that appear in the statement under consideration. Students are also taught how to understand a written proof by learning to identify the sequence of techniques that are used. In Part 2 of the book, other mathematical thinking process such as the following are described and illustrated with many examples:

This book is suitable as a text for an undergraduate transition-to-advanced-math course, as a supplement to any course involving proofs, or for self-guided reading (especially for Ph.D. students in math-related areas such as Statistics, Computer Science, Physics, Engineering, Finance, Economics, and Business). Students can view videos of my lectures for each of the first 15 chapters at:

A good textbook for learning proofs should have clear explanations and examples, provide practice problems with solutions, and cover a wide range of proof techniques. It should also have a logical progression of concepts and be easy to follow.

You should check the level of difficulty of the textbook and compare it to your own level of understanding. You can also look at the prerequisites listed and see if you have a solid foundation in those topics.

You can check the publication date of the textbook and see if it covers recent developments or advancements in the field of proofs. You can also read reviews or ask for recommendations from professors or peers who are familiar with the subject.

Synopsis: How to Read and Do Proofs (first edition) is a short introduction to the process of reading and writing mathematical proofs. It illustrates each concept through examples and exercises (with solutions). Recommended as a quick introduction or review to people who want to understand computer science white papers better or as a starting point to learning math/logic after some time away from it.

I find that lately I read a lot of white papers. While these are mostly about programming (usually Haskell, sometimes crypto), they often have proofs in them. Sometimes these proofs are reasonably well explained and I can follow them; often they are condensed, in the way that people who are writing for others in their field may condense their writing by using jargon or making assumptions that they know their audience will share.

There were times that I did not understand each step in the proof, particularly when they relied on knowing some fact about, oh, triangles or something. I do not remember most of those facts from my high school math classes, I am sorry to report. A couple of times I felt it was important to me to figure those steps out; usually I ignored those details. Few white papers I am likely to read these days require me to recall specific facts about equilateral triangles, so ignoring these details suited my purposes.

I study computer science at a university. My school offers several courses where various proofs are expected, but there is no course that introduces the fundamental concepts of proofs and how to write your own proofs. It's possible to "muddle through" some of these courses without really understanding proofs, but I want to spare myself the frustration and actually learn about them.

I want to start at the very basics of proofs. I'm only interested in proofs as they relate to computer science, since I will have to spend my spare time on this. However, if I really do need to learn about mathematical proofs in general first, then so be it.

The discussion and links here are also useful. In particular, the first edition of Bridge to Abstract Mathematics: Mathematical Proof and Structures, by Ronald P. Morash, is available for free download; I also taught from it a number of years ago and remember it as being quite decent.

Finally, some introductory discrete math texts devote significant space to introducing students to reading and writing proofs (and also contain mathematics that students in computer science ought to know). One such is Edward Scheinerman, Mathematics: A Discrete Introduction, which I recommend highly. Another is Susanna S. Epp, Discrete Mathematics with Applications.

I am an electrical engineer and trying to make a transition into machine learning. I read in multiple articles that I have to learn data structures and algorithms, before this I have to learn about mathematical proofs. I started studying it on my own using the material available on MIT's OCW, while I did grasp the concepts of induction and well ordering etc..

I've been struggling with the exercises for a very long time and it's really frustrating. I can easily deal with any type of proofs that I saw before (e.g. once I saw the proof of a recurrence question I became pretty good at proving them). My problems start when I face an unusual question. I feel like I am memorizing the proofs rather than learn how to prove.

Is there any way (or any resources) that can improve my proving skills in a way that whenever I see an unusual question (like the checkers tiles and chess tiles type of questions) I don't have to stare at them for 2 hours before giving up?

You can't learn "how to prove". "Proving" is not a mechanical process, but rather a creative one where you have to invent a new technique to solve a given problem. A professional mathematician could spend their entire life attempting to prove a given statement and never succeed.

I can easily deal with any type of proofs that i saw before ( eg. once i saw the proof of a recurrence question i became pretty good at prooving them). My problems start when i face an unusual question.

That is normal. Any mathematics "proofs" course isn't designed to teach you how to take an arbitrary problem you've never seen before and be able to solve it (since nobody, not even the best mathematics professors can do that). Rather, your learning goals are

If you are working on a new, unknown problem, it is normal that you might not be able to solve it. However, knowing and having memorized other proof techniques may help you. Often proofs involve combining a new idea with existing known proof techniques. The more, and the more varied the proofs you already know are, the better your chance of being able to solve the given problem.

You are on the right track. You should simply keep studying proof techniques. The exercises you are doing are good. Don't worry if you get stuck. As you get more experienced and your "toolbox" of techniques grows, you will be able to solve exercises that are less "alike" the previous ones you have seen.

As other authors have mentioned, partly because proofs are inherently hard, but also partly because of the cold fact that proofs are not written for the purpose of teaching, even in most textbooks. Rather, most proofs are written out of a kind of obligation, as a sort of run-away argument; not presenting proofs at all is considered unacceptable, but writing them in exhausting details would burn the author out as well as endanger the reader getting lost in the woods. Hence, most proofs are succinct on purpose, leaving a lot of dots solely for the reader to connect themselves. While some people find this a helpful exercise, many readers like you and me find it making mathematics unnecessarily challenging. This is also why classroom pedagogy in a university setting is indispensable for professional mathematic learning as the tools of dialogue can fill in the blank of textbook proofs.

I can certainly recommend the book of G. Polya's, How to Solve It. It is a standard classic, not to be missed. There is a newer book How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow that may be more accessible.

In any event doing proofs is entirely Unnatural for humans. It is a discipline that requires careful thought we do not normally use. We are used to making many assumptions to get through our days and our lives. If we had to justify the first of them we could not get out of bed. A mathematical proof strips away the assumptions and lives on only what you can show clearly and unambiguously.

I had the similar trouble with problems over trigonometric identities. Trying to get from the start to the finish is easy when there is a known, learned method. Identities may require multiple steps in unknown directions without much sense of direction. Proofs are a bit easier since the logical methods are fairly limited and known (if you read the books). Keep at it.

As for resources, you might like G. Polya's book How to Solve It. Looks like the Wikipedia article gives a nice and somewhat detailed overview. Basically, the book will offer you a strategy or methods for dealing with mathematical statements and their proofs.

My guess is you'll also want to learn about algorithms' space and time complexity, as quantified in big O notation. Time complexity, in particular, hints at why proofs are hard. If I promised you there is a proof of at most length $n$ of a given statement, how would you find it? In theory, you could go through all proofs of length $\le n$ until you find one, which would take exponential time, say $O(ne^cn)$ (I've included a factor of $n$ for reading time). That's far too inefficient for our purposes, unless $n$ is very small. There might be a much better algorithm, but no-one's found a particularly efficient general one. That's why proving things remains a "creative" exercise, by which we mean "we don't know in pseudocode terms how such thinking works".

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