Basics of Proofs (2017) [pdf]
When I was first introduced to mathematical proofs, I was perplexed by how fuzzy and intuition-based the notion of proof was. A &quot;convincing argument&quot;, really?! There is no knowing, as a novice, how detailed those &quot;arguments&quot; have to be, what parts you can simply assert without further justification. Usually the teachers themselves can&#x27;t explain what the criteria for an &quot;obvious&quot; and &quot;not obvious&quot; step is, they just know it intuitively from experience. Writing proofs, then, is a lot like having to learn to ride a bike: Instructions are mostly unavailable or useless.<p>I later learned that there is indeed a precise way of learning proofs which doesn&#x27;t rely on intuitions of what counts as a rigours inference step: Formal logic together with a natural deduction proof system. Natural deduction is a formal proof system which resembles actual (&quot;natural&quot;) proofs in mathematics, unlike other proof systems.<p>In such a proof system, inference rules, like modus tollens or universal instantiation, are strictly defined. Only the given inference rules (and those which are provable from the given rules) may be used. Coming up with such proofs still requires creativity, there is no algorithm. But there is no ambiguity in what counts as a valid or invalid proof or inference step.<p>Of course, this is far too tedious for actual mathematical proofs, since every little step needs to be done explicitly, e.g. even applications of modus ponens (rule: &quot;A, if A then B, therefore B&quot;). Moreover, mathematicians rarely prove anything from axioms, they start from other statements which are considered more trivial.  But I think it would be helpful for many people to first learn logic and deduction &quot;the precise way&quot;, and <i>then</i> do actual mathematical proof where you can jump over more obvious parts.<p>But that&#x27;s not how it is teached in mathematics or computer science. Students are thrown into the cold water, and only receive tips&amp;tricks, but no rigorous introduction. Ironically, the one subject which often teaches formal logic as an early introductory class for undergraduates isn&#x27;t mathematics, computer science, or physics, it&#x27;s philosophy.