Social Research Glossary   A B   Citation reference: Harvey, L., 2012-18, Social Research Glossary, Quality Research International, http://www.qualityresearchinternational.com/socialresearch/ This is a dynamic glossary and the author would welcome any e-mail suggestions for additions or amendments. Page updated 24 January, 2018 , © Lee Harvey 2012–2018. A fast-paced novel of conjecture and surprises

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Proof

core definition

Proof refers to the evidence, or the act of assembling evidence, or the presentation of argued evidence, in order to establish a fact, the veracity of an event or logical proposition, or the cause of a phenomenon.

explanatory context

Proof is tied to the notion of objective facts, independent of the observational context, that can be verified by the senses or by recourse to established bodies of knowledge or extant theorems.

Proof is applied to singular phenomena in everday activity (as for example in proving someones involvement in an event in a court of law). It is also used to establish scientific generalisations (as in using controlled experiments to prove scientific laws). It is also used in the sense of providing a justification for a mathematical assertion as in a mathematical proof.

In the case of scienfiic generalisationsense the objectivism associated with proof is allied with assumptions about generalisable cause and effect.

It is argued by falsificationists that inductive proof from observation is impossible because not all possible observations can be made, and they prefer to establish disproof of working theories.

Less positivistly inclined scientists argue that neither proof nor disproof is possible given the theory-laden nature of observation.

analytical review

Hunt (2000) discussing mathematical proof states:

What precisely is a proof? The answer seems obvious: starting from some axioms, a proof is a series of logical deductions, reaching the desired conclusion. Every step in a proof can be checked for correctness by examining it to ensure that it is logically sound, and you can tell that you've proved a theorem once and for all by making sure that every step is correct.

This might sound simple enough, but one problem is that humans (and even computers) are fallible: what if the person checking a proof for correctness makes a mistake and thinks that a step which is logically incorrect is in fact correct? Obviously somebody else will need to check that the person doing the checking didn't make any mistakes; and somebody will need to check that person, and so on. Eventually you run out of people who could check the proof: and in theory they could all have made a mistake!

The question "When is a proof not a proof?" is one which philosophers have debated for centuries. The problem is compounded by the fact that some proofs are so specialised that very few people in the world can even understand them; and some proofs are so long that very few people in the world have time to read them! In this article we shall explore some famous examples.

Fermat's Last Theorem, a problem which has been around since 1637 when Pierre de Fermat wrote it into the margin of one of his books, was finally proved in 1993 by Andrew Wiles of Princeton University, USA, while he was visiting the Isaac Newton Institute for Mathematical Sciences in Cambridge. The proof is extremely intricate, and relies on the reader having studied the most modern and highly advanced number theory. It's also quite long, taking up over 100 pages in print. But only a handful of people in the entire world can understand the proof. To the rest of us, it's utterly incomprehensible, and yet we are quite happy to announce that "Fermat's Last Theorem has been proved". We have to believe the experts who tell us it has been, because we can't tell for ourselves.

In no other field of science would this be good enough. If a physicist told us that light rays are bent by gravity, as Einstein did, then we would insist on experiments to back up the theory. If some biologists told us that all living creatures contain DNA in their cells, as Watson and Crick did in 1953, we wouldn't believe them until lots of other biologists after looking into the idea agreed with them and did experiments to back it up. And if a modern biologist were to tell us that it were definitely possible to clone people, we won't really believe them until we saw solid evidence in the form of a cloned human being. Mathematics occupies a special place, where we believe anyone who claims to have proved a theorem on the say-so of just a few people - that is, until we hear otherwise.

In fact, the first proof of Fermat's Theorem which Wiles produced had a flaw in it. The mathematicians who checked his proof for errors with a fine-toothed comb found a gap in the proof, and insisted that it had to be corrected. It took Wiles another year to find a way of circumventing the error, so it wasn't until 1994 that a full complete and correct proof was published. Or was it? Nobody knows whether there might be another error lurking in his proof: the whole proof might be completely wrong. But for the moment we are happy to say that "it has been proved"....

associated issues

related areas

evidence

generalisation

theory-laden nature of observation

Researching the Real World Section 1.4

Sources

Hunt, R., 2001, 'The origins of proof IV: The philosophy of proof' available at http://plus.maths.org/content/origins-proof-iv-philosophy-proof, accessed 28 March 2013, still available 24 December 2016.

copyright Lee Harvey 2012–2018

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