Mathematical Writing Franco Vivaldi Pdf

[Zoraida]

As an applied mathematics student coming from a small university, I have not had an adequate course in writing/formulating proofs for problems in advanced calculus/real analysis (my university has an advanced calculus course, not a real analysis course). In the fall I will take the first of a two part course in advanced calculus. I believe we will use Fitzpatrick's book, Advanced Calculus.

So, in order to prepare for the rigor and proof writing that will be necessary in this course, is there a good book or pdf that will provide solutions to problems involving proofs? I've found plenty of books with tons of problems but finding solutions to check myself (or see if there is a more clever approach) has been difficult. I hope this isn't too broad of a question, maybe some of you coming from smaller universities will understand my dilemma.

Analysis With An Introduction to Proof, 5th Edition by Steven R. Lay. I thought this was a great book to learn how to write mathematical proofs. Shows you completely how to write proofs and the approach to take in how to start a proof.

I like the book Mathematical Writing by Franco Vivaldi, based on a course on writing offered to second year undergraduates at Queen Mary, University of London. It used to be offered online for free but seems to have been snapped up by Springer. Perhaps your library has a copy.

"A Logical approach to discrete math" teachs predicate and propositional calculus and even gives a "formal" epsilon-delta example, in chapter 9, in complete detail with reference to the quantifiers rules applied.

I learned to do math proofs in college. But recently I have begun studying more advanced math books and I've noticed some mathematicians frequently make assumptions that I don't. When I write proofs, I tend to state every detail possible. But mathematicians often omit details or repetition. I used to think this was annoying but having written more proofs I realize such omissions create concision and may aid in understanding as a result.

Remark 1: I have extensively studied general writing and verbal communication and very few of the principles used in explaining things (eg use of analogies or metaphors) work well for mathematics -- at least at higher levels. So math writing is clearly a distinct skill.

Remark 2:A very trivial, simple example of such concision is that when we refer to $\mathbbR^n$, we mean a set of numbers for $n=1$ but a set of ordered pairs for $n>1$. Even I wouldn't state this, but for younger learners, I'm sure that distinction isn't known initially. I certainly didn't and had to learn it.

Regarding the steps suppressed in mathematical writing, this is very sensitive to the audience one is addressing. Routine arguments that can be checked by a reader of appropriate background do not need to be elaborated on (provided you are absolutely sure they are correct). It is generally a good practice keep a draft with all the messy details, and abridge them in a final draft, in a way sensitive to one's intended audience.

I should elaborate a bit more on which details should be hidden. Generally, people read papers on topics they are interested in, either because they have nearby expertise or the topic has some bearing on their own work. Most of us immediately go looking for what is new in a paper, and appreciate finding it as soon as we can. If you don't know your audience and just write like an automaton, giving all details, it looks like you are a novice and have no idea what is routine and what is new. Worse yet, this sort of approach that avoids nuance makes your paper look more error-prone...because it looks like you are treating everything with the same amount of care. As a rule, one should omit or abridge those things that are sufficiently routine (after checking them to death), include those things that are delicate (i.e. potential pitfalls that should be treated with care...most experts are aware of these things in their areas...knowing these gives your paper credence with these experts) and last, but not least, highlight the new ideas so they are clearly visible. A paper that does these things looks more like a mathematician wrote it...indeed it would be very difficult for an amateur to effectively write such a paper convincingly!

Mathematical writing is very different from other kinds of writing, including other kinds of technical writing. Aside from low-level textbooks, the audience is generally assumed to be other competent mathematicians, with their expertise in the particular subfield depending on how technical the work is. It's awful form, and it will infuriate your readers, if you try to hold their hand and assume that they need helpful analogies and metaphors (which invariably just cloud the issue more) to deal with the concepts. Just present the data straightforwardly and without narration, giving enough details to be precise but not too much to be pedantic. (Note in particular that a math paper or book intended for other mathematicians is not a problem set to be graded; it's actually preferable to omit trivial details that a competent reader should be able to fill in themselves.) The line between the two is hard to define but easy to spot in practice. Think of it as something like a legal document: The audience is other lawyers, who are assumed to be competent enough to handle the material if it's presented directly, and who expect and want the terse style and precise vocabulary that presents the argument without beating them over the head with it.

I would also say in response to Remark 1 that analogies and metaphors remain valuable in mathematical writing, as they can aid understanding, but they must be used with care. For instance, if a metaphor contradicts an established mathematical definition, then the definition wins.

In general, writing mathematics is just like any other skill: The more you practice and the more you see others do it, the better you'll get. I find I take bits of pieces of styles from mathematicians that I read and admire most, thus creating my own unique style.

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The primary goal of this course is to communicate and discuss mathematics accurately and effectively. The student will improve in their ability to communicate and write clearly about mathematics. This includes understanding the background of the audience and modifying the narrative to fit the level of mathematical sophistication of the reader. Students will also gain a better understanding of how to structure and write a proof. The students will learn how to use LaTeX in order to typeset papers with mathematical content. LaTeX is the free software that mathematicians use to write mathematics. Using the mathematical constructs in LaTeX can help one to focus on how to arrange content logically and in a mathematically correct fashion.

In a writing-intensive course (W Course), writing should be integral to the learning goals and subject matter of the course. In the language of the General Education Guidelines at UConn, students should not write simply to be evaluated; they should learn how writing can ground, extend, deepen, and even enable their learning of course material. In addition to questions concerning strategies for developing ideas, clarity of organization, and effectiveness of expression and discipline specific stylistic norms, the W requirement should lead students to understand the relationship between their own thinking and writing in a way that will help them continue to develop throughout their lives and careers after graduation.

The purpose of a writing course in mathematics is to teach students how to communicate mathematics in a precise, concise, and clear manner. Throughout this course, your instructor will emphasize the best practices in writing mathematics, as it pertains to writing mathematical proofs in particular. The student will learn how to gauge the level of mathematical background of the audience (the reader) and learn how to modify a document to fit the mathematical level of the reader (e.g., how much background is necessary for each type of audience). The student of a W course will write drafts, revise drafts and resubmit. The reason why it is crucial to write a draft is so that the document can be peer-reviewed and critiqued by your instructor, so that a conversation can occur about what background may be necessary, and what level of detail is required when discussing a concept or a mathematical proof. In this course, there will be much emphasis on the structure of mathematical writing (examples, lemmas, theorems, corollaries, remarks, etc), and there will be discussions about the importance and educational power of properly chosen examples and diagrams, graphs, to illustrate a document and to illustrate a mathematical argument that may be otherwise hard to grasp by the reader.

Accordingly, your portfolio will not be considered complete unless you have made revisions addressing the points raised in the assessment of your initial submission and you will not pass the course without a complete portfolio that achieves a passing standard.

Five writing assignments (single-spaced and each at least 2, 2, 3, 4, and 4 pages long, respectively) will be assigned. Each assignment will be submitted, and then re-submitted once comments and feedback or insights by the instructor have been addressed. Each final draft is worth 20% of your grade. All assignments must be typeset using LaTeX. The writing portfolio will consist of the compilation of all the assignments completed throughout the semester.

(2 pages)In this assignment the student will learn some LaTeX basics, by elaborating on a handwritten text, and transforming it into a properly typeset LaTeX document. 2. Write a precise statement and proof of the quadratic formula.

(2 pages)In this assignment the student will create a LaTeX document that states a proper statement of the quadratic formula, and contains a complete proof of the formula, followed by worked out examples.3. Including graphics, diagrams, matrices, arrays, hyper-references, tables, and bibliography in your documents (tikz, Geogebra, etc)

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