Download e-book for iPad: A Mathematical Bridge: An Intuitive Journey in Higher by Stephen Hewson

By Stephen Hewson

Even supposing better arithmetic is gorgeous, ordinary and interconnected, to the uninitiated it might probably think like an arbitrary mass of disconnected technical definitions, symbols, theorems and strategies. An highbrow gulf should be crossed earlier than a real, deep appreciation of arithmetic can advance. This publication bridges this mathematical hole. It makes a speciality of the method of discovery up to the content material, top the reader to a transparent, intuitive knowing of the way and why arithmetic exists within the method it does. The narrative doesn't evolve alongside conventional topic strains: every one subject develops from its easiest, intuitive start line; complexity develops obviously through questions and extensions. all through, the booklet comprises degrees of rationalization, dialogue and fervour not often visible in conventional textbooks. the alternative of fabric is in a similar way wealthy, starting from quantity thought and the character of mathematical idea to quantum mechanics and the background of arithmetic. It rounds off with a range of thought-provoking and stimulating workouts for the reader.

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22 A Mathematical Bridge combinations of true, false, AND, OR, NOT and implication applied to the axioms in the development of proofs. This, in a nutshell, is the process of mathematics. 4 Proof Proving a result in mathematics can be quite similar to the process of a criminal trial. The mathematician wishes to prove a certain result to be true; the prosecution lawyer wishes to prove that the defendant is guilty of a certain crime. In both cases, evidence for the result is collected and a case for the result is then logically constructed.

At the start of the new millennium mathematics has become a very deep and complex affair, with global cooperation between thousands of mathematicians being used to probe the boundaries of mathematics. In most topic areas it takes years of study to reach the point at which a research contribution is possible, and then typically only in a very narrow area of expertise. Many proofs of results use theorems of other mathematicians which, for practical reasons of time, need to be taken on trust. Some proofs are literally thousands of pages long and incredibly complex.

Whilst this final complete clarification is essential to be mathematically certain of our results, simply looking at the final presentation fails to give an appreciation of the creative process of actually doing the mathematics and the struggle and toil which went in to creating the proof, which can take days, weeks, months or even years to complete. Without an appreciation of the process, only the very tip of the iceberg of mathematical activity will be exposed. Whilst there is no substitute for actually working through mathematical problems, it can be helpful to understand that there are a variety of ways in which mathematics can be done.

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