Generalization in mathematics

by Neil Rickert

Generalization is an important part of mathematics, and I shall discuss that here.  My discussion will mostly consist of examples with commentary on those examples.

I’m planning a future post on generalization in science.

Numbers

The simplest example has to do with numbers.  And our use of numbers presumably started with counting.  By assigning names, from a fixed sequence (1,2,3, …), we could count objects.  And then we could compare the results from counts of different collections of objects. This turned out to be useful for keeping track of quantities.

Rules were developed to deal with counting of groups of objects.  If we knew the counts of each of two groups, we could combine those with addition rules, to get the combined count.  And if we has several groups of the same count, we could combine those with multiplication rules.

And then people started using names from that sequence as if they were themselves objects.  And that gave us the natural numbers. The rules for addition and multiplication were used as operations on numbers.  This allowed theorizing about counting.

And then people tried with fractions, or rational numbers.  These were also useful for keeping track of quantities, and they allowed dividing into smaller sized quantities.  Then they discovered that they could define addition and multiplication for rational numbers in a way that allowed the same rules of addition properties (the commutative, associative and distributive laws).

Thus we see rational numbers as a generalization of natural numbers.  The concept of number was made more general, and ways were found to apply the rules for addition and multiplication to these more general numbers.

With further generalization, we get the negative numbers.

At this stage, the result was not completely satisfactory.  The equation x^2 = 2 was known to not have a solution within the rational number system.  And then study of infinite series, and the use of limits in the calculus, revealed further problems.  These were solved by generalizing to the real numbers.  That allowed for a square root of two and also solved the problems with limits.  But the equation x^2 = -1 still could not solved.  So mathematicians experimented with adding a square root of minus 1.  And that gave us the complex numbers.

What we see here, is that the concept of number has been made more general.  And the operations of addition and multiplication have been made more general.

No doubt there was some guessing and some trial and error experimentation to see what might work and what might be useful.  But once mathematicians had settled on what they wanted for these generalized number systems, they could deductively prove that everything worked the way that they wanted.  They did not have to rely on guessing or on trial and error to be sure that it would all work.

Fields, rings, groups

Another way of generalizing number systems is to go with the idea of a field.  A field is expected to have elements and operations of addition and multiplication, which satisfy the usual associative, commutative and distributive rules.  But, apart from that, it is made completely general.  The elements don’t have to be numbers.  A field can be any system that satisfies the usual rules for those operations.

The idea here is that if we can prove a theorem for fields, then that theorem applies to numbers systems.  But it also applies to anything that satisfies the axioms of a field.

A ring is more general still.  We drop some of the requirements for a field.  For example, we drop the requirement that there can be an inverse to multiplication (i.e. division).  The integers are then a ring, but not a field.  And we might also drop the requirement that multiplication be commutative.  That is, we might allow that ab \ne ba.

Instead of requiring two operations, addition and multiplication, we can instead require only a single operation.  And that gives us the idea of a group.  The operation is sometimes called addition, and sometimes it is called multiplication.  This depends on which terminology is more convenient for the particular use of group theory.

Other generalizations

Non-Euclidean geometries arises from dropping the parallel postulate from Euclidean geometry.  One example of this is spherical geometry, useful in navigation systems.  And Einstein’s general relativity uses non-Euclidean geometry.

Topology arises from the ideas of limits and continuity that come out of the calculus.  A topological space is a space (or set of points) where we can have some notion that is an analog of continuity.

Summary

In mathematics, generalization is usually something like the generalization of a concept.  That is, a version of the concept is provided that is less restrictive than the original from which it is derived.  It typically becomes more general (or less restrictive) by dropping or weakening some of the original requirements or some of the original axioms.

What is true for the original system might have a true analog for the generalized system.  We still require deductive proof of what we will hold to be true in the generalized system.  There isn’t any jumping to conclusions.  There might be some initial guessing on what we should try to prove.  But we still need the proof.

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