By Stephen Hewson
Even if larger arithmetic is gorgeous, ordinary and interconnected, to the uninitiated it could believe like an arbitrary mass of disconnected technical definitions, symbols, theorems and techniques. An highbrow gulf should be crossed earlier than a real, deep appreciation of arithmetic can advance. This booklet bridges this mathematical hole. It specializes in the method of discovery up to the content material, top the reader to a transparent, intuitive figuring out of the way and why arithmetic exists within the manner it does. The narrative doesn't evolve alongside conventional topic traces: every one subject develops from its easiest, intuitive place to begin; complexity develops obviously through questions and extensions. all through, the booklet comprises degrees of clarification, dialogue and keenness hardly visible in conventional textbooks. the alternative of fabric is in a similar way wealthy, starting from quantity concept and the character of mathematical idea to quantum mechanics and the heritage of arithmetic. It rounds off with a range of thought-provoking and stimulating routines for the reader.
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Additional resources for A Mathematical Bridge: An Intuitive Journey in Higher Mathematics
1 Negation As the simplest of examples, if we are given a single mathematical statement P then we can construct its negation which is false if P is true and true if P is false. We can use an obvious notation for the negation as NOT(P). Although negation appears to be very simple, in practice, negating a sentence can be somewhat involved. Furthermore there are passive ways of negating a sentence, in which we say what does not happen, and active ways to negate a sentence in which we say what does happen.
4Of course, nothing can ever be completely beyond doubt: we could keep probing each step in a proof endlessly. However, the m athem atical comm unity seems to be in agreement w ith th e nature of acceptable logical steps in a proof. We shall discuss these acceptable forms of logic and proof later. Note for now th e im portant point th a t proof is A Mathematical Bridge 10 argument which derives the theorem from the axioms. Deciding on the difference between a proof and a convincing argument takes some experience and often much thought, but the key idea is to be sure that every step in the proof is 100% certain to be true.
Definition of structure of a mathematical theory provides us with a series of statements which are not to be questioned: for the purposes of the theory in hand, they are simply facts or rules. Logical implication allows us to examine the consequences of these rules. It is assumed that in the context of the mathematical theory it is possible to make some true statements and some false statements. Put bluntly, mathematical endeavour is all about trying to find some true statements which are both interesting and not initially obvious from our definitions.