Mathematical Texts – Why There Are No Logic Symbols in Mathematical Texts

soft-question

Either in undergraduate or graduate textbooks on Mathematics (Real/Complex Analysis, General Topology, Differential Geometry, …), I never saw symbols $\Rightarrow$, $\iff$, $\forall$, $\exists$, etc. Instead, I just see their "read as" or explanation such as "if … then …", "if and only if", "for all", "there exists", etc. Before I started to study mathematics, I thought I would see a lot of logical symbols, but I didn't see any (even once!) even in the proofs. Why is that so? If they are for mathematical use and they take less space in texts, why are there none of them in mathematical textbooks?

Best Answer

Many mathematicians, and I want to be in that number, believe that

Let us fix any $\epsilon>0$. It follows from the assumptions that there exists a positive number $\delta$ with the property that $1/x<\epsilon$ whenever $x>\delta$

is more elegant than

$(\forall \epsilon>0)(\exists \delta>0)(\forall x)(x > \delta \Rightarrow 1/x<\epsilon)$.

Book authors often want to write good books, carefully written, elegant and pleasant to read. Book authors often think of themselves as artists, or professional writers: if, as others said in their answers, good English grants both style and scientific quality to a book, why not use it?

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