# Resource: Types of proofs

Methods/types of proofs exist for various purposes and scientific fields. Philosophical, psychological, medical, biological etc. Here we focus on mathematical and physical proofs. Mathematical proofs are in their nature accurate, and we are sure that their results are true, whereas physical proofs rather are methods for testing, checking, measuring and evaluating until we are sure beyond reasonable doubt. An accurate (and often mathematical) proof is called analytical, whereas a proof that is based on many experiments, measurements or much data is called numerical.

Some relevant terms regarding proofs are the following:[1,2]

• Theorem: Already-proven expression involved in a proof for something new.
• Lemma: “Side-expression” that is proven separately during a proof with the sole purpose of finishing that proof.
• Axioms: Assumptions that are not proven but defined or assumed true.
Types of proofs[4]
ProofDescriptionExample
Direct proofCombine something already known (axioms and theorems) to reach the resultIf $a$ and $b$ are even integers, then they are the double of two other numbers: $a+b=\underbrace{2c}_a+\underbrace{2d}_b=2(c+d)$. Their sum is also the double of another number, so it is also even
Proof by contradictionShow that if something is true, an absurd contradiction appears. So, it must be false.
Proof by inductionProve a base-case. Then prove an induction rule that says that the base-case counts for all other cases. (For an infinite row/series)
Proof by contrapositionIn order to know if a claim is true, prove its contrapositive (“inverted”) case true. If we want to prove that if $n$ is even, then $n^2$ is even, then we can instead say that if $n$ is odd then $n^2$ is odd. When proven true, then the contrapositive must also be true.
Proof by constructionFind one example that either proves (for an existence claim) or disproves (for a truth claim) something.Example: ‘I have an even number here, so even numbers do exist’. Counterexample: ‘I have an odd number here, so not all numbers are even’.
Proof by exhaustion / proof by casesProve each occurrence individually (for a non-infinite set)Claim: $n^2+n$ is always even, when $n$ is an integer. Prove this for even and then for odd $n$’s separately, covering all.
Geometric proofMake the result clear from accurate drawingDraw the limiting case of an obtuse triangle to see that it always will have two angles below $90^\circ$
Probabilistic proofShow that something must be true with 100 % probabilityThe cat must be inside the house, since it can’t get out
Plausibility proofShow that something most likely is true with very high probability The cat most likely is inside the house, since it could only get out through the chimney
Statistical proofCollect much data that shows a clear patternIt is a human feature to have 5 fingers, because 99.6 % of all babies (2996 out of 3000) are born with 5 fingers.[3]
Empirical proofPerform many measurements that all show the same resultThings will always fall down when dropped, because every time it has ever been tried, it happened.

Several of the proofs involve elements of falsification, such as the empirical proof and the proof by construction. We can call a proof combinatorial, when it includes more than one type (when more than one proof gives the same result, and we thus believe it to be true).

References:

1. What is the difference between lemma, axiom, definition, corollary, etc?’ (web page, answer to forum post), user 57Jimmy, Mathematics Stack Exchange, 2018, math.stackexchange.com/a/2716230/13230 (accessed Oct. 9th, 2019)
2. Mathematical Proof/Methods of Proof/Constructive Proof’ (encyclopedia), Wikibooks, 2017, en.wikibooks.org/wiki/Mathematical_Proof/Methods_of_Proof/Constructive_Proof (accessed Oct. 9th, 2019)
3. Genetic Overview of Syndactyly and Polydactyly’ (article), Humayun Ahmed, Hossein Akbari and others, Plastic and Reconstructive Surgery - Global Open, vol. 5, issue 11, 2017, insights.ovid.com/crossref?an=01720096-201711000-00008, DOI 10.1097/GOX.0000000000001549
4. Proofs and Mathematical Reasoning’ (article), Agata Stefanowicz, Joe Kyle and others, University of Birmingham, 2014, www.birmingham.ac.uk/Documents/college-eps/college/stem/Student-Summer-Education-Internships/Proof-and-Reasoning.pdf

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