(March 1, 2018 at 2:50 pm)SteveII Wrote:(March 1, 2018 at 1:42 pm)polymath257 Wrote: Nope that is NOT what the symbolism means. It is not just a convenience. Those are actual, exact, equalities.
I assure you, you don't want the technical definition. You wouldn't be able to handle the logic of the definition.
What you have then is a limit of mathematics to account for the difference. It would mean the two are mathematically equivalent. This does not mean that the two are logically equivalent.
When we are talking about spanning infinite series of things (as in Zeno's paradoxes) we are decidedly NOT talking about mathematics.
And don't tell me next that mathematics and logic are the same thing. https://en.wikipedia.org/wiki/Logic#Types
Do you see mathematical logic listed there by any chance? In fact, Aristotelian logic (syllogistic logic) is clearly described as having little more than historic value and being obsolete.
Logic is a subset of mathematics. You see, mathematics is the study of formal systems and logic consists of a few such systems. Propositional logic, predicate logic, even modal logic are *formal* systems and so are a part of math.
The two expressions have no meaning outside of mathematics. So the mathematical equivalence is all there is. The issue doens't even arise in propositional logic since propositional logic isn't strong enough to even talk about whole numbers, let along real numbers.
And, yes, if we are talking about an infinite series of things, we are most certainly talking about mathematics.