By Stephen Hewson
Even if greater arithmetic is gorgeous, usual and interconnected, to the uninitiated it could possibly suppose 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 boost. This booklet bridges this mathematical hole. It makes a speciality of the method of discovery up to the content material, major the reader to a transparent, intuitive knowing of the way and why arithmetic exists within the manner it does. The narrative doesn't evolve alongside conventional topic traces: each one subject develops from its least difficult, intuitive place to begin; complexity develops certainly through questions and extensions. all through, the booklet comprises degrees of rationalization, dialogue and keenness infrequently obvious in conventional textbooks. the alternative of fabric is in a similar fashion wealthy, starting from quantity idea and the character of mathematical concept 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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Extra info for A Mathematical Bridge: An Intuitive Journey in Higher Mathematics
Doing mathematics is not simply about making a step-by-step series of logical deductions until a problem is solved. It also involves guesswork, trial and error, following hunches, leaps of intuition, trying out a variety of approaches, making lots of mistakes and taking lots of false turns. However, once a solution to a problem is found, mathematicians like to tidy up their work into a neat, compact, clean, logical solution or proof. 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.
If so, then this axiom is not needed: it is a theorem, a derived result. We could make do with a smaller set of axioms. Consider, for example, a set N of whole numbers and an operation S which acts on whole numbers as follows: (1 ) For any whole number n, S(n) is also a whole number with S(ri) = n + 1 (2 ) For every number n in JV, S(n) is also in N 6T he concept of i will be covered in detail in C hapter 2. Its m ain property is th a t it squares to —1. 14 A Mathematical Bridge (3 ) 1 is in AT (4) 1 does not equal S(m) for any m in N (5) 1 is the smallest member of N In this list of axioms, the fifth point is not independent of the first four: it can be deduced as a logical consequence of the other axioms.
Since P and Q are either true or false and we require that we can determine the truth or falsehood of R directly from that of P and Q there are only actually four combinations of possibility for the truth Mathematics 21 or falsehood of R. We can tabulate these in a truth table: p T T F F Q T F T F R ?? ?? Since R can take one of two possible values for each row of this truth table, there are 16 distinct completions of the truth table: PQ T T T F FT FF 3 4 5 6 7 8 9 10 11 12 13 14 15 16 rpy r j l F F F F F F F F T T T F T F F F F F F T F T T T TTFT FTFF FFT FT F T T T F T T F F T F FT F F T T F T 1 2 r j "1 rj~l rj~] r j1 Of these values it is possible to extract a set of particularly key distinct logical possibilities.