A formal proof system defines two things with complete precision: a fixed set of starting statements, called axioms, accepted without further justification, and a fixed set of mechanical inference rules that specify exactly how new statements can be derived from statements already established. Because both the axioms and the inference rules are defined purely in terms of symbol patterns, syntax, rather than what those symbols are actually supposed to mean, a proof's validity can be checked mechanically, symbol by symbol, without any genuine understanding of the statements' actual meaning being required at all.
A proof is really just a sequence of statements each justified by an explicit rule
Within a formal proof system, a proof is a finite sequence of statements where each individual statement is either one of the system's starting axioms or follows from earlier statements in the sequence by one of the system's explicitly defined inference rules. Checking whether a proposed proof is actually valid means checking, line by line, that each statement really does follow correctly according to the rules, a task that requires only comparing symbol patterns against the rulebook, not evaluating whether the statements are actually true in any deeper sense.
This purely mechanical checkability is exactly what makes formal proofs so reliable
Because checking a formal proof's validity comes down to purely mechanical pattern-matching against a fixed rulebook, the process doesn't depend on a checker's own interpretive judgement, intuition, or even genuine comprehension of what the proof's symbols represent, which is exactly why a computer, with no understanding of meaning whatsoever, can verify a formal proof's correctness just as reliably as a trained mathematician. This mechanical reliability is precisely what makes formal proof systems the gold standard for absolute logical certainty, since a formally valid proof's correctness doesn't rest on anyone's subjective judgement at all.
What we're still unsure about
That formal proof systems allow purely mechanical, syntax-based verification of a proof's validity is well established, rigorously developed mathematical logic confirmed since the early twentieth century. What's more genuinely a matter of ongoing philosophical interest is exactly what relationship this purely mechanical, syntactic notion of proof actually bears to genuine mathematical understanding or truth, since a system can mechanically verify a proof's validity without that verification process itself capturing anything about why the underlying mathematical claim is actually true or meaningful, and philosophers of mathematics continue to debate exactly how syntactic proof and semantic truth relate to each other, rather than that relationship being fully settled.
This sits inside Formal Proof Systems, one of seven topics in Logic, one of five domains in Philosophy, one of seventeen subjects the app can quiz you on.