Running out of Algebraic Symbols I'm currently writing my thesis in computer science. My work contains a large amount of equations to do with linear algebra, data transformations and calculus. On receiving feedback on my first draft from my supervisor, he has told me that I am reusing symbols all over the place. So I am going back over things, but I'm finding that I'm running out of both English and Greek characters and I'm having to constantly search back through the text to see if the symbol I want to use has already been defined. For example:

Take two vectors X and Y with lengths m and n respectively.


All of these symbols have previously been used. So my question is, once a symbol has been defined once, can it not be reused for the whole thesis, and how then should I deal with this problem. 

Thanks.
Running out of symbols is a fairly common scenario, and it is usually caused by assigning symbols too early without due consideration to the amount of work yet to come. As such, the main thing to do is to
Restructure
Think about the entire set of symbols you're using.

For which ones is the specific symbol in use crucial for the clarity of the text? There will usually be a core of symbols that you really don't want to reassign, but this is probably smaller than you think. Earmark those symbols for these concepts, and disregard other occurrences (even if they happen earlier in the text).

Keep this core small. Use variations on the symbols (upper/lower case, primes, subscripts) to fit as many variations of the concepts into the key symbols.

Serialize. If you have some sequence of objects which you initially called a, b, c, d, ..., use an appropriate subscript or parenthesis notation to collapse them into a single symbol. Over-expansive naming of sequences of this sort is probably the number-one reason for running out of symbols in the first place.


The other thing you should do is
Encapsulate
It is perfectly OK to have the same symbol appear twice, with different meanings, in the same extended piece of work, as long as there is no confusion between them. This is particularly easy to do if the symbols appear, for example, inside the proof of a theorem. Thus, if at some point you use the symbol x in a context like

Proof. Let x be ... ... ... . ▮

then it is probably OK to re-use the same symbol in a similar encapsulated environment, as long as (i) you clearly define the symbol on its first appearance in each environment, and (ii) you don't (excessively) reference the symbol once you're outside of the encapsulated environments.
Similarly, it is usually OK to reuse symbols in different chapters as long as the topic of the two chapters is sufficiently different compared to the similarity of the symbols and the amount of weight you put into the choice of symbol. This is a very common practice: look at any linear algebra textbook and you'll see the symbol v used a number of times, to refer to vectors that are not necessarily compatible. The techniques they use to avoid confusion are forms of encapsulation.
From the phrasing of your question, it seems that this is partly the problem with your supervisor. One way to go about this is to step up the encapsulation, either by being more clear with the text, by introducing formal environments, or even using graphical means like boxes around examples to make it clear that the symbol choice is local to that environment.
If all else fails
Some amount of creativity is OK
This includes things like multi-letter symbols for important context, though you should be careful with how you typeset them (usually in upright script). However, you should avoid using letters from non-standard alphabets, as they will only further the confusion: few of your readers will know how to pronounce them (which then extends into how they internally vocalize them while reading, and makes it harder for your readers to discuss the paper with each other), and it makes it harder for others to adapt and extend your notation.