How to write a proof? Use English or mathematical notations?

Question

I am now considering writing a mathematics paper. One thing that bothers me a lot is the following question

How far should I use notations instead of English sentence?

On one hand, it is easy for me to make mistakes in writing notations because it needs a lot of accurate details. Other people who use different notations may find it hard to understand my notations.

On the other hand, notations are less ambiguous that English prose. It ensures the accuracy of my statement in some way. If a use notation in a clear manner, it shows that my thought is clear.

So how can I keep a balance?

Answer 1

The combination of writing and notation depend on your personal style. Regardless, you should attempt to write everything in a form that is as clear as possible. You should certainly use a combination of words and notation. You must use complete sentences even if you use mostly mathematical notation. Why not try to write a paper and ask specific questions about difficulties you encounter?

Answer 2

Well, like everybody else said, there is no general answer. But if you really want a generalization I want to share this. From my experiences I've learnt that in doubt a beginner should stick to formal notations. Like you said, it ensures the accuracy of your statement.

Some of the easiest mathematical statements are very hard to formulate in regular sentences. You would have to doublecheck this kind of prose: is it clear and unambiguous? is it consistent? And is it really readable? Notations are. And if you are afraid that people who use different notations may find it hard to understand your notations, you just have to make clear what your notations mean. Nothing is easier than this. And readers who are used to notations will be able to follow.

Your one Pro Argument for english sentence is this: 'it is easy for me to make mistakes in writing notations because it needs a lot of accurate Details'.

I have to tell you that you will make more mistakes without notations. Especially if you are not aware that this will require at least the same amout of 'accurate Details'.

(If you are a beginner, do some exercising: Translate some notations into prose. I bet you will learn more from that than from any possible answer in here.)