The Concept of Mathematical Proof

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Proof is a funny word it can mean different things to different people. For example there is a difference in the concept proof between scientists and mathematicians.  The difference is actually quite subtle but without understanding it then you have little chance of understanding the work of any mathematician.

Classical mathematical proof is to start with a set of statements and axioms  which are true, or at least are self evidently true.  Then by using logic and deduction, you can move step by step to a conclusion.  As long as the underlying axioms are correct, and the logic is flawless then the conclusion should be undeniable.  The conclusion is termed the ’theorem’ by the mathematician.

A mathematical theorem is dependent on the process of logic and so should be true forever.  To the mathematician, their proofs are absolute there is no scale or level to this proof.  When you compare this to the scientists proof you see the underlying difference.  The scientist will put forward a hypothesis to explain a phenomenon  or occurence.  Then it is observed and compared with this hypothesis – the scientist will then gather evidence in favour of the hypothesis or against it.  Eventually evidence will way up either overwhelmingly in favour or against the hypothesis which will decide if it becomes part of scientific law.

The level of proof is not absolute in science, it is never approaching the level of mathematical absolute proof.  The scientists rely on perception and observation both of which can be subjective and are certainly fallible.

To quote Arthur C Clarke

“if an eminent scientist states that something is undoubtedly true, then it is likely to be proved false the next day.” 

Mathematical proof is absolute and devoid of any doubts, whereas scientific proof is frequently rewritten or proved to be false.  Take for instance the wonderful BBC program on Fermat’s Last Theorem and you’ll see some of the difficulties involved in attaining the level of mathematical proof.  It is often available on the BBC Iplayer if you check – if you live outside the UK – try this technique to watch Iplayer abroad.  It works by masking your true IP address and allowing you to use a false one in order to bypass the geotargeting restrictions.   You can even do this by simply using something called Smart DNS where you don’t even change your address –