1

When we are in the context of pure mathematics, quantifiers are everywhere. When we are in the context of mathematical modelling, quantifiers usually disappear. For instance, in statistics we often encounter sentences such as "Let $X$ be a normally distributed random variable with mean $\mu$ and variance $\sigma^{2}$" and the reader tacitly understands $\mu$ as a real number and $\sigma^{2}$ as a real number $>0$. But this style seems logically unsatisfactory. To make it logically accurate we may add "for some real $\mu$ and some real $\sigma^{2} > 0$", which is, however, cumbersome.

Do you feel uncomfortable with this style in the context of mathematical modelling? What opinions you would suggest so that one can look after both conciseness and logical precision?

Yes
  • 20,910

1 Answers1

2

Nobody uses quantifiers if the context makes it clear what the letters stand for. For instance, in the context of real analysis saying "if $x<y+\varepsilon $ for all $\varepsilon >0$, then $x\le y$" is perfectly precise and clear. In the context of set theory, saying "for all $S$, $|\mathcal P (S)|=2^{|S|}$" is perfectly precise and clear. And so on and so on. The aim of logic is not to become a trap of mindless formality. Proofs are communicated at a degree of precision that is sufficient to accurately convey whatever you want to convey but is loose enough so as not to be so constrictive as to annoy everybody.

In short, the distinction you allude to between pure maths and mathematical modeling quite simply does not exist.

Ittay Weiss
  • 81,796
  • Well, I think I catched the point you made. Nevertheless, the examples you gave are different from the example I gave; for the examples of yours admit no free variables, while in my example $\mu$ and $\sigma^{2}$ become free variables :) – Yes Sep 02 '14 at 11:50
  • and what do you call the $x$, $y$, $\varepsilon$, and $S$ in what I wrote above? – Ittay Weiss Sep 02 '14 at 21:51