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I am studying for a Master's degree of mathematics. There are some deduction in my thesis is written as the following form.

$A $

$\Rightarrow B $

$ \Rightarrow C $

$ \Rightarrow D $

But my adviser does not allow me write my thesis in such a form. He said there's nobody who writes a thesis like this, and requested that I use text rather than use mathematical symbol. But I found a book, "Abstract Algebra", written by Dipak Chatterjee that does.

enter image description here

I don't plan to take this book to my boss, because I want to graduate.

Here are some reasons from my classmate against me:

  1. There's nobody who writes a paper like this. But this is not reasonable for me. It doesn't mean this method is bad. In addition, If the professor adopts such form in his lecture, why I can't use such method in my thesis/dissertation?

  2. If you use text rather than mathematical symbols, then your article will be understandable for a non-professional reader. This is also not reasonable for me. Is there a non-professional reader who will read my thesis?

Would anyone give me some opinion?

Whatever support or against, thanks.

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    $\begingroup$ I have a hard time believing your adviser would oppose against writing with symbols as in the text book example. It makes the proof perfectly clear. If someone thinks otherwise, I would say that person is against the norm. On other instances, however, words might be more clear to most people. $\endgroup$
    – Git Gud
    Commented Jun 1, 2014 at 8:55
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    $\begingroup$ The advice of your advisor is entirely reasonable and proofs written in the style you advocate tend to be simply unreadable. Note that even the book you mention as your Exhibit A adopts this style in one or two occasions only (hence your "But I found a book ... that does" is not true, strictly speaking). $\endgroup$
    – Did
    Commented Jun 1, 2014 at 9:31
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    $\begingroup$ It's fine in the middle of a long computation that's just symbol manipulation, as in the example if you give. It's not a good style elsewhere, though, and it shouldn't be used in running text. For example, "If $f$ vanishes at some $x\in \mathbb{R}$, then there exists $x'\in \mathbb{Q}$ with $f(x) = 0$" shouldn't be replaced with, "then $\exists x'\in \mathbb{Q}$ with $f(x) = 0$," and "then $\exists x'\in \mathbb{Q}: f(x) = 0$" is worse. $\endgroup$
    – anomaly
    Commented Nov 19, 2014 at 23:18

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Personally, I totally agree with your advisor. I hate the use of symbols where words could be used to produce a better result. You can use symbols on the blackboard, but my opinion is that a thesis is like a book: it should be carefully written, and the final result should be aesthetically good. Many years ago, mathematics books were written with a particular care about the style. Take books like those written by Walter Rudin or Serge Lang: they even used words where symbols could have been perfectly legitimate. But you might be right: it is a matter of taste, above all.

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  • $\begingroup$ At a guess, many years ago, if the mathematician are used to write their paper prefer to use the implication symbol ``$\Rightarrow$'' rather than use words, then the books written by Rudin or Lang and the journal will be of the form today? $\endgroup$
    – bfhaha
    Commented Jun 1, 2014 at 9:01
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    $\begingroup$ It seems you move from an assumption that is not necessarily true: there are mathematicians that do not fill up their papers with symbols where words should be used. $\endgroup$
    – Siminore
    Commented Jun 1, 2014 at 10:20
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First, it is right that most mathematics is presented as written text and not as a series of logical symbols (the Pricipia Mathematica being one of the more famous examples - even the very formal books by Bourbaki are written mainly as text - basically, only introductionary books for undergraduates use formal manipulations).

I also agree that it is more appropriate to write a mathematical text rather than a series of logical symbols. Although it is correct that you should strive to be as precise and concise as possible, I think the reason to use text over symbols is that mathematical texts are much more about communicating ideas rather than facts. Having a proof in a formal and rigorous way is nice. But it is more difficult to extract the basic idea of a proof from a series of formal manipulations than from a written text. Hence, I would suggest that you try to write your thesis with the goal of explaining things as good as possible rather than "being as precise as possible". Although it may sound paradoxical, I suggest: Do not sacrifice clarity for precision.

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If you were writing your own book or if you were the only author of a paper you were submitting to a journal, I would say use whichever style you like but be aware that different journals/publication bodies may reject items which don't conform to their preferred style/template.

However, since you are a Master's student I would observe that what you write may be seen as a reflection on your advisor. Furthermore, your university may have some style requirements which bear on this issue.

If it makes you feel better I recall that I had put throughout my thesis "ploof" instead of "proof" and was forced by my advisor to use the latter. Not quite the same thing, but I was quite adamant.

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I agree with your thought. One of the objects in mathematics is to have things presented in a concise and precise manner. If the implication sign does the job nicely, continue using it (after you pass your thesis).

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Firstly, overuse of symbols does decrease readability, and not just to "non-professionals".

Secondly, consider these two distinct assertions:

  1. statement $D$
  2. statement $A$ is a sufficient condition for statement $D.$

Your suggested presentation

$A\\\Rightarrow B\\ \Rightarrow C \\ \Rightarrow D $

expresses option $(2),$ and notably does not actually conclude that statement D is true.

Your cited example

enter image description here

has the modus ponens structure \begin{align}&A \Rightarrow B \Rightarrow C \Rightarrow D\\\text{and}\:\:&(A\Rightarrow D)\Rightarrow P;\\\text{therefore}\:\:&P\end{align}

(the middle line is implicit), and genuinely utilises the implication $A{\Rightarrow}D$—rather than statement $D$ itself—to argue that a latter statement $P$ is true.

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