How I Know What Is True
Proof Versus Evidence
- Established idea
- Formal theory
- Working interpretation
A mathematical proof, a worked example, a simulation and a personal story each support a different strength of claim. I want to celebrate a discovery without overstating what is actually known.
How much certainty does this particular result deserve?
Why it attracts me
I love the moment of discovery, and I have seen how easily that excitement inflates a claim. Being exact about the kind of support behind a result is how I try to keep both the joy and the honesty.
The idea
Think of a ladder. A personal story shows that something happened once. A simulation shows what a model does. A check of many cases shows that a pattern holds where it was checked. A proof shows that a statement must hold, given its assumptions. Each rung supports a stronger claim. None is worthless, but each should be described as what it is.
An example
The Agentic Solvers side quest is careful about this line. It certifies that every tree (a branching network with no loops) with up to 23 points has a particular kind of labeling, with each case checked by a separately written program. But the page says plainly that "no structural proof for all orders is claimed." It is a strong, checked result for a limited range of sizes, still awaiting expert review, and not a proof about trees of every size. The related Agentic Maths project shows the other route: there, a limited set of computer-checked cases forms the base of a full proof, itself checked by computer, that covers every size from 17 upward. It too awaits expert review.
Where it connects
Unsolved Problems in Mathematics is where this matters most, because finding a pattern is not proving a theorem. Formal verification in Proofs a Computer Can Check makes the line visible by checking proofs step by step. And Knowing the Limits of What I Know is the attitude that this node turns into practice.
Questions I'm still exploring
- How can someone without expert training judge what kind of support a claim has?
- When is strong evidence without proof good enough to act on?
- Which of my own beliefs rest mostly on stories rather than tests?
Sources and further reading
- George Pólya, Mathematics and Plausible Reasoning (Princeton University Press, 1954) — Separates the plausible reasoning that guides discovery from the reasoning of proof.
Working interpretation: drafted from my notes and interests for review. It is not a direct quotation, and I may still change it.