First,<p>&gt; some sliver<p>Second, I for the most part agree with you in the ways that are relevant to the point you&#x27;re trying to make. However, the knowledge&#x2F;tool distinction is a vague (in that they may fall on a spectrum) and ambiguous (both words tend to be moving targets), which is why I chose the term instrument instead. Mathematics is a tool only in the sense that it is knowledge that is applied towards some benefit, e.g., efficiency. Also,<p>&gt; Compare Knuth: &quot;Beware of bugs in the above code; I have only proved it correct, not tried it.&quot;<p>This is again &quot;proof&quot; in the day-to-day mathematical sense of a convincing argument, not in the formal sense for writing down an expression per line such that each line can be logically deduced from some preceding lines through the the use of axioms. A type checker is literally a search procedure through the space of derivation trees to find a formal proof that this term has this type.