Archive for September, 2012

The Concept of Mathematical Proof

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 – http://www.theninjaproxy.org/ninja/change-ip-address-region-free-smart-dns/

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Math Subjects Taught For Students At School

We are trained to fix statistical issues in a variety of different ways. The pad and document methods are the most popular way we show kids to fix applied mathematics issues. Learners are also requested to understand how to use psychological statistical to quickly fix mathematical issues without the use of document and pad. Finally, students are requested to implement the commercial calculator and Solceller to finish the majority of their calculations issues. The commercial calculator is the method that has triggered a lot of conversation. This question is always asked: At what age should we use this tool to show our kids to fix mathematical problems?

Some people believe that the Solceller enables the kids to focus more on the mathematical knowing and ideas instead of a bit of your efforts on training calculations abilities. This machine can help kids finish complicated statistical functions. The instructor can take a longer interval training mathematical principles, thus more arithmetic can be trained each category interval. Instructors are always under pressure to show a certain variety of principles each category interval, and if students have to invest lots of your efforts finishing easy calculations, how can they show new concepts?

Furthermore, some students become disappointed because they don’t have enough chance to finish their mathematical issues. This is in part due to the fact that they don’t have a good knowing of statistical calculations abilities. This may cause students to be troublesome or less focused in college. Some of these students have been permitted to implement the Solceller calculator in college because the instructor doesn’t want to invest longer training to students on the basic abilities that they skipped in their earlier attribute. So, the calculator is used to allow the instructor to continue training the class. Research has also said that kids can use the calculator at any quality stage, provided that they are utilized properly. The analysis says that the calculator should be used as a supplement to learning and not as an alternative. In addition, the instructor must be knowledgeable as to how to utilize the calculator in the educational setting training. Research also declares that most teachers are not trained in the use or neglect of hand calculators in educational setting training.

This information is important, but let us not ignore, if students start using a calculator to fix easy mathematical calculations at a young age, they will becoming reliant on them in the future? When will they understand their time tables? In Asia, where kid’s statistical ratings are very great, kids are prohibited to use hand calculators until they reach the junior great university stage. Even then, students use the calculator occasionally. I suppose they use them at the secondary university stage in the higher-level sessions, like calculus. Students who become reliant upon using hand calculators are losing their psychological calculations abilities. This can hurt them in the lengthy run when more psychological calculations may be needed. The old saying, “if you don’t use it, you lose it”, is true. Learners who quit utilizing their mind to do mathematical calculations will ignore easy statistical projects, thus mathematics will become tougher for them.

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