Hi all,
I would like to know how to generate mathematical proofs and theory. Is there any material that I can study? ..What do you think?
There are a number of publications relevant to this. A classic one is Polya's "How to prove it". There are other suggestions at https://www.quora.com/What-are-some-good-books-on-proofs.
There are a number of publications relevant to this. A classic one is Polya's "How to prove it". There are other suggestions at https://www.quora.com/What-are-some-good-books-on-proofs.
Thank you so much for your reply. The problem is that I am an engineer and as you both know we practice math but we do not go into how to make proof so I am always feel like I miss something in math which I do not know and I feel like I am not good in math though I practice it always. I will search for your books suggestion and I hope that it can fill the gape. One more thing, about Wulf's point of view, I am self learner so it is hard to find feed back from teachers. Do you think that Arthur Engel’s Problem-Solving book is the best start for me?
Dear Peter Breuer,
Thank you for detailed answer. Saying that "math is engineering in the same way that computer programming is engineering" is totally new for me. I never think of math and engineering in that way. I always thought that math is a tool that we use in our study. Maybe that is because of the way of teaching. On the other hand, some engineering departments need more math than the others. I am mechanical engineer so I feel that control or computer engineer practice more math then me.
I maybe wrong though.
Dear Wulf Rehder,
I started to read "How to read and do proofs" by Danial Solow. I think the book will show me the methodology of doing proofs. After that, I may read Arthur Engel book. What you said about Engineering mathematics is true. In the college, it was never said to us where we can use equations. The teacher used to teach us Engineering mathematics like abstract mathematics. As a result of that, we loose both. I think that is part of my problem.
I believe Polya's famous book is "How to Solve It." It introduced the idea of heuristic approaches to problem solving. "How to Prove It: A Structured Approach" by Daniel J. Velleman appears to be highly-rated and it might be valuable for non-mathematicians.
A question that needs to be answered has to do with identifying the domain in which mathematics is being applied and how proofs are employed in that area. It is useful to start with proofs that are already offered in the domain of interest, learning what the mathematical basis is, and understanding how the proofs are constructed.
I would add to Peter Breuer's observation that another purpose for a proof is demonstration to others that a particular condition is satisfied. This is important because it is communities of expertise that review and refine offered proofs, clearing any defects. Also, one should be careful about using mathematical proof in the context of concrete systems.
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." -- Albert Einstein, 1921.
I think you meant that I should search for mathematical proofs in my domain which is engineering , is that right?
I chose "How to read and do proofs" by Danial Solow because I need to understand the construction and methodology of proofs before I go into the proofs itself. I may read other books when I finish the book mentioned previously. What is said by Enistein is seemed to be confusing. "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Is it mean that Mathematics can not be certain about reality? and if it certain then it is not real, not practical? interesting
To Ayad Khudhair Al-Nadawi
1. Yes, proofs in applications to engineering matters.
2. The full text of Einstein's commentary is in "Geometry and Experience," a lecture delivered in 1921. You can find links in my detailed consideration of this idea at http://orcmid.com/blog/2010/02/abstraction-einstein-on-theoryreality.asp
3. A recent book, "Plato and the Nerd" makes useful distinction between theories (i.e., as models, possibly of two kinds) and engineering in order to relate computing and human activity. Here's one appraisal and source https://www.facebook.com/MiserProject/posts/1997521747129065
4. I've been doing some work in computation theory. An informal presentation of the use of a mathematical theory and its connection to a concrete computational interpretation is being presented at https://orcmid.wordpress.com/2018/07/07/miser-project-hark-is-that-a-theory-i-see-before-me/ although the distinction between the theory and contingent reality will unfold as part of the subsequent posts. I am using first-order logic with equality to express the theory and to conduct proofs/deductions with the theory.
To Dennis Hamilton
Thank you so much for your responses. Be sure I will check these links. I am happy for having these details.
Ayad Khudhair Al-Nadawi might find the following blog article to be relevant, since it differentiates between proof and problem-solving: https://anthonybonato.com/2018/10/17/problem-solving-vs-proving/
@ Dennis Hamilton
Very interesting article. I see now as an engineer, they taught us to be problem-solvers and not generate proofs. That may seem sense from engineering point of view since we use theorem but never generate them.
Thanks
@ David A Miller
Thank you for your suggestion. Definitely , you are right and I will try the online courses.
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