Re: Defining things in Mathematics and in Scheme

I do agree that few think that axiomatic method reveals the TRUTH in
capital letters. This traditional approach, in which axioms were
supposed to be indisputable is abandoned. However there are benefits
with the axiomatic methods.

1. It defines concepts so that you don't have to guess what the author
means.
2. It clarifies the components and relations between them and makes
them visible.

For instance group theory benefited when it was put on axiomatic basis.
A group is defined as a set that has the following 4 properties:
1. Closure: If A and B are two elements in G, then the product AB is
also in G.
2. Associativity: for all A,B,C in G, (AB)C==A(BC).
3. Identity: There is an identity element 1 such that 1A==A1==A for
every element A in G.
4. Inverse: There must be an inverse or reciprocal of each element.

Seeing this clearly let's me investigate variations of groups, for
instance Rings, Fields or Modules. Or I can specialize and investigate
the concepts of Abelian Groups, Finite Groups, Group Actions, Group
Automorphisms, Group Representation, Simple Group, Solvable Group and
so on.

Given an axiomatic framework you can
1. Accept my axioms and investigate the consequences.
2. Add to my axioms and investigate the consequences.
3. Change some of my axioms and investigate the consequences.

So the axiomatic method does help to clarify things and facilitates the
work of one researcher to build upon work of other researchers.

Bob

.

Relevant Pages

  • Re: Why will Cantor be a target for cranks?
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  • Re: Axioms of Boolean Algebra got from Concept Algebra
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    (sci.logic)
  • Re: Minimal logic valid?
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  • Re: Do mathematicians know their axioms?
    ... Anders Goeransson wrote:> ... How is "using the axiomatic method" normally conceived? ... You can use other propositions which you haven't ... to know what axioms you are assuming, ...
    (sci.logic)
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