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This is commonly believed, but Gödel didn't identify a logical error at the heart of the whole enterprise, he proved astonishing theorems revealing limitations
by steppi 2mo ago
This is commonly believed, but Gödel didn't identify a logical error at the heart of the whole enterprise, he proved astonishing theorems revealing limitations of any sufficiently powerful formal system. One can kind of think of the Principia as a science experiment to find the extent to which known mathematics could be proven from foundational axioms that could be thought of as "laws of logic". To make their system work, Russell and Whitehead themselves had to add extralogical axioms, such as their Axiom of Reducibility [0] and the Axiom of Infinity, giving empirical evidence (but not a proof) that "laws of logic" alone were not enough. They were also aware of limitations in their own system, such as the inability to define the cardinal $\aleph_\omega$ [1].
Like the article says, what they did was ahead-of-its-time, and a monumental influence on all subsequent work on formal systems, including Gödel's work, regardless of whether Russell and Whitehead achieved their initial aims.
[0] https://en.wikipedia.org/wiki/Axiom_of_reducibility https://en.wikipedia.org/wiki/Axiom_of_reducibility
[1] https://www.gutenberg.org/files/78255/78255-h/78255-h.htm#Page_190 https://www.gutenberg.org/files/78255/78255-h/78255-h.htm#Pa...
- gumby 2mo agoThe flaw was the belief that it was possible (though they were not sure that they two could pull it off). Gödel showed that it was not an impossibly difficult task but an actually impossible task.
- steppi 2mo agoA failed hypothesis is not the same thing as a logical error. Would they have been more free of flaws if they believed it was impossible and thus didn't try? We'd all be the poorer for it
- jibal 2mo agoIt cannot be overstated how flawed your own statements here are. You shouldn't be talking about logical error regarding a work of logic when you don't even know enough logic to know what a logical error is. And there was no "flaw" in attempting to resolve foundational paradoxes in set theory, primarily to address Frege's error and deal with Russell's paradox, or in attempting to prove that all mathematics can be derived from pure logic. Seeking to demonstrate a hypothesis that is later shown to be erroneous is not "flawed", else the entire knowledge-seeking enterprise is "flawed". And switching from "logical error" to the very vague notion of a "flaw" is goalpost moving that looks a lot like bad faith. I won't comment further.