Perhaps particularly in mathematics it is the case that if one meets a paradox, one should get concerned. I myself somehow got alarmed, felt some kind of awakening, when the students including myself were introduced the Russell’s paradox in the course of Eucledian spaces (Euklidiset avaruudet) in the year 1997.
I write now at very general level. If something is interpreted as a set somehow, but on the other hand it isn’t a set according to definition, it may be the case that definition may need to be verified or the philosophy should be developed further from the point of view of semantics. One must see ”that, that is”.
For example by understanding what the Russell’s paradox is at the level of logic, semantics and philosophy, one new concept is sufficient to see that there is no paradox!
One must deepen one’s view of the given problem and see beyond the known.
Generally speaking, if one meets a paradox, it may be meaningful to check the definitions, pay attention to the semantics; it may be the case that a new category of concept is needed, if the problem seems to be a paradox, but with deeper view of the problem it may be the case that it in fact isn’t!
As to Russell’s paradox, understanding it something else than a paradox may result in huge development in philosophy, perhaps even new programming languages can be created or new programming techniques – especially: What happens to the set theory?
Perhaps I’m just dreaming…
Zeno of Elea (ca. 490 – 420 BC) is known as Greek philosopher, who’s paradoxes have troubled also modern thinkers for a long time.
Zeno aimed at to show contradiction between plurality, motion and infinity with his paradoxes.
Motion is impossible, because whatever moves must reach the middle of its course before it reaches the end; but before it must have reached the quarter-mark, and so on, infinitely. Hence the motion can never even start.
So there are infinite number of sub-courses to go, before the end has been reached.
It was because of lacking concept of (mathematical) limit, these paradoxes saw the daylight. Arbitrary small changes couldn’t be explained. Though, there are views, that even modern calculus doesn’t explain completely Zeno’s paradoxes! So is the motion still an illusion? 🙂
Image courtesy of Stuart Miles at FreeDigitalPhotos.net
Isaac Newton founded differential and integral calculus in the 17th century.