Aaron Cotnoir has suggested that people think that paradoxes are meaningless. I think they are lucky that they hadn’t suggested that to me unless they wanted to see me freak out.
It was my good fortune to have my first real exposure to the work of Frege, Russell and Wittgenstein be from Thomas Ricketts. I can’t remember verbatim what he said, but this is close:
No one knows how long it took Frege to understand what Russell had written in his letter (Russell’s Paradox), be it a few seconds, a minute, ten minutes or a few hours. But we do know that at that moment his entire universe collapsed.
Only out of gross ignorance of history can anyone believe that paradoxes are meaningless. Frege’s project up until Russell came along and spoiled everything was, at least in part, to give a firm foundation for mathematics based solely upon logic. With just a few laws coupled with his newfound quantification he was able to provide a seemingly consistent theory and then also provide sophisticated philosophy of language to bolster his views.
There was probably a moment when Frege allowed himself to dare to think he’d solved one of the greatest mysteries of the universe. Not only had he legitimately and demonstrably changed mathematics forever, but the ramifications of his theory were obviously far-reaching into philosophy and science. Then Russell sent him that letter that struck at the very axioms of his theory. It was a jugular shot and I can’t see Frege feeling other than like all the blood had been drained from his body. Everything he had worked for was put in jeopardy.
So if anyone believes that paradoxes are meaningless, I suggest to go read some history. Paradoxes can destroy. Any theory that comes along and says paradoxes are meaningless, is garbage.