I.4 Proving a More Abstract Result, Identifying Characteristic Properties, Generalization and Reformulation, Higher Dimensions and Several Variables, Discovering Patterns: Pages 56 - 58

A more abstract result, Lagrange’s theorem,  can be used to prove a more specific result, Fermat’s little theorem.  Fermat’s little theorem states that if p is a prime and if a is not a multiple of p, then a^p-1 leaves a remainder of 1 when you divide by p.  Lagrange’s theorem states that the size of a group is always divisible by the size of any of its subgroups.  An aurgument shows that Fermat’s little theorem is just a special case of Lagrange’s theorem.

This process of abstraction has multiple benefits: existence of a more general theorem, revealing other interesting specific cases, one only needs to prove this more general theorem once and it  connects otherwise unconnected results.

Characteristic properties are the defining properties of some objects rather than the object itself.   Examples, are the imaginary number, i, the square root of two and x raised to the power a for x and a real numbers.  Abstraction is also portrayed as the opposit of classification.

It is shown how reformulation of the concept of dimension allows one to generalize the concept of dimension to non-integer dimensions.  Nocommutative geometry uses this same process.  A manifold X is the set, C(X), of all complex valued continuous funtions defined on X.  C(X) is a vector space.  C(X) is an algebra and even a C*-algebra.  This algebra is commutative in the operation of multiplication.  If one now considers non-commutative algebras then this provides a reformulation with an inferred noncommutative geometry.  A third example is given that generalizes the fundamental theorem of arithmetic that uses primes and multiplication to form all integers in exactly one way.  Rings of numbers such as a + i b√5 with a and b integers may have multiple factorizations.  Ideal numbers are formed and defined.  We asssociate the set of all multiples of a number with the number.  A subset of a ring with the closure property is called an ideal.  One can use this ideal to determine a unique factorization. 

A single linear eauations in one variable is much easier to analyze than are polynomial systems of equations in several variables.  Partial differential equations using multiple variables and are much more difficult to analyze than differential equations in just one variable.  It is a fruitful method to generalize an idea to higher dimensions and/or to several variables.

One can discover patterns to solve packing problems in low dimensions.  For example, packing the 2-dimensional plane with circles or 3-dimensions with spheres.  In high dimensions this is an unsolved problem.  One should not just give up then but it may be fruitful to reformulate the problem so that it can be solved.  For example, one can find the most dense known packing in a higher dimension.

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