&gt; Russell realized this, and recognized that Gödel&#x27;s results invalidated his project.<p>One particular project. The specific project of formalizing mathematics within itself is unachievable. That doesn&#x27;t mean we&#x27;ve stopped doing math.<p>And this is in fact the approach modern mathematics has taken. The fact that one particular book (the proof of mathematics&#x27; own consistency) can&#x27;t be found on the shelves doesn&#x27;t mean there isn&#x27;t value in keeping the rest of the books well-organized.<p>&gt; Yes, true, but for the reason that we have no such confidence (and can have no such confidence). We cannot make that assumption, because the Turing Halting problem and its theoretical basis prevents it.<p>No! Humans aren&#x27;t developed in either Coq or ML. A human can look at Coq and say &quot;Yes, this is correct&quot; or &quot;No, this is wrong&quot; just fine, without running afoul of the halting problem, Gödel&#x27;s incompleteness theorem, or anything else. The fact that you can have no <i>Coq-proven</i> confidence in Coq&#x27;s correctness does not mean that you can have no confidence of Coq&#x27;s correctness.<p>(Of course, this brings up whether a human should be confident in their own judgments, but that&#x27;s a completely unavoidable problem and rapidly moving to the realm of philosophy. If you are arguing that a human should not believe anything they can convince themselves of, well, okay, but I&#x27;m not sure how you plan to live life.)<p>This is like claiming that, because ZF isn&#x27;t provably correct in ZF (which was what Russell, Whitehead, Gödel, etc. were concerned about -- not software engineering), mathematicians have no business claiming they proved anything. Of course they&#x27;ve proven things just fine if they&#x27;re able to convince other human mathematicians that they&#x27;ve proven things.