Overview and comparison
Aristotelian (or Leibnizian) proofs and Platonic (or Cartesian) proofs are two disjoint notions. The former offers a reliable form of justification to philosophers and computer scientists, while the latter facilitates understanding for mathematicians.
| Aristotelian proof | Platonic proof |
|---|---|
| A deductively valid argument for a mathematical conclusion. That is, the proof can be verified by examining its form alone. | An argument consisting of clear and distinct ideas that facilitate understanding. That is, the argument not simply promise that a conclusion is true, but also why. |
- Are Platonic and Aristotelian ideas really in tension? Which should you ultimately believe (i.e., which should have epistemic priority)? For example, a formal proof is widely agreed to establish a conjecture, even if the proof is non-ideal (e.g., ugly) or useless, unable to be “surveyed” or “taken in.”
- Does an “ordinary” proof need to abbreviate a formal proof, or at least be sufficiently rational such that you know a formal proof exists? In general, it appears not!
Related: § Mathematical Logic and Formal Proofs, Meno’s paradox
Notes
- Original note, inspired by @2022viteri: A mathematical proof is a unidirectional, deductive argument from axioms to conclusion. A proof never explicitly advances an abductive argument in the opposite direction; rather, abduction in mathematical proofs is an active reasoning process used to evaluate and build confidence in the deductive argument.