How many words (i.e. not "math" symbols") should I use in my proofs? ${}{}$

Solution 1:

Your plan is exactly backwards.

All proofs should be readable as English prose, i.e. sentences arranged into paragraphs. Symbols may be used as needed, but they need to be human-readable. If you've defined enough symbols, you can write parts of the proof entirely in symbols, provided that they can be parsed back into English.

For example, $$\forall\ x\in\mathbb{R}, \exists\ y\in \mathbb{Z}: x\ge y$$ reads as "For all real numbers $x$, there is an integer $y$, such that $x$ is greater than or equal to $y$."

Solution 2:

Don't worry, be happy. It is okay. There are symbols that can be used, but using more of them doesn't really make the logic better. $$\begin{array}{|l}\lVert C\rVert=k\\\lVert\,\bigcup_{B\in C} B\,\rVert=k+1\\\hline~\begin{array}{|l}\forall B\in C~(\lVert B\rVert\leq 1)\\\hline\lVert\,\bigcup_{B\in C}B\,\rVert \leq k\\\bot\end{array}\\\neg\forall B\in C~(\lVert B\rVert\leq 1)\\\exists B\in C~(\lVert B\rVert>1)\end{array}$$ "Given that we have a collection of boxes and the total number of items in the boxes is one more than the number of boxes, then since if each box had no more than one item inside, there could be no more items than the number of boxes, contrary to what was given, therefore there must be some box(es) in the collection with more than one item inside."

Solution 3:

A lot of this also comes down to writing style. These sorts of proofs aren't just a computational exercise; they're meant to express an idea, to be read and understood, just like any prose writing in any philosophy book. Different writers will want to emphasize different parts, and the same writer may want to express the same proof differently for different audiences. Different branches of mathematics will use different symbols to different degrees.

One key difference between mathematical writing and other writing is that we've incorporated lots of symbols as a way of eliminating ambiguity and removing redundancy, because as mathematicians we are very concerned with being unambiguous, since so much of our work depends on precise definitions. But the symbols are not inherent to mathematical writing, they developed over millennia.

So the symbols aren't necessarily instead of than words, they're in addition. Symbols are only another way to express what we could already express through words and diagrams. How then should you know whether to use more symbols or fewer? The same way you would pick up any writing style: by reading lots of different examples (e.g., textbooks and papers) by different authors, by writing a lot, and by finding trusted writers (published mathematicians) to read and critique your work.