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>> Russell realized this, and recognized that Gödel's results invalidated his project. > One particular project. A mathematical proof applies to all cases, no
by lutusp 11y ago
>> Russell realized this, and recognized that Gödel's results invalidated his project.
> One particular project.
A mathematical proof applies to all cases, not just one. Gödel's result addressed Russell's project, but it applies universally, as do all valid mathematical results. Many mathematicians regard Gödel's result as the most important mathematical finding of the 20th century.
> That doesn't mean we've stopped doing math.
Not the topic.
> A human can look at Coq and say "Yes, this is correct" or "No, this is wrong" just fine, without running afoul of the halting problem, Gödel's incompleteness theorem, or anything else.
This is false. If the Halting problem is insoluble, it is also insoluble to a human, and perhaps more so, given our tendency to ignore strictly logical reasoning. It places firm limits on what we can claim to have proven. Surely you aren't asserting that human thought transcends the limits of logic and mathematics?
> Of course they've proven things just fine if they're able to convince other human mathematicians that they've proven things.
Yes, with the caution that the system they're using (and any such system) has well-established limitations as to what can be proven. You may not be aware that some problems have been located that validate Gödel's result by being ... how shall I put this ... provably unprovable. The Turing Halting problem is only one of them, there are others.
- c22 11y ago> Surely you aren't asserting that human thought transcends the limits of logic and mathematics? Given that both logic and mathematics are creations of human thought, is this really so untenable?
- geofft 11y ago> A mathematical proof applies to all cases, not just one. Gödel's result addressed Russell's project, but it applies universally, as do all valid mathematical results. What? I have no idea what you mean by "applies universally" and "all valid mathematical results." A straightforward reading of that sentence implies that Pythagoras' Theorem is not a "valid mathematical result" because it applies only to right triangles. That can't be what you're saying, is it? I am acknowledging that Gödel's second theorem states that any mathematical system that attempts to prove itself is doomed to failure. I am also stating that Russell's project involved a mathematical system that attempts to prove itself. Therefore, Gödel's second theorem applies to Russell's project, but it does not necessarily say anything else about other projects. > Many mathematicians regard Gödel's result as the most important mathematical finding of the 20th century. So what? This is argumentum ad populum. How important the mathematical finding is is irrelevant to whether it matters to the discussion at hand. > This is false. If the Halting problem is insoluble, it is also insoluble to a human OK, so do you believe that modern mathematics is invalid because Russell's project failed?