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What is the axiomatic method?
Explain how to build a set of axioms
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The axiomatic method is a technique for studying math objects by formulating them as a type of math structure. Take some basic properties of the kind of structure you are interested in and set them down as axioms, then deduce other properties (that you may or may not have already known) as theorems. The point of doing this is to make your reasoning and all your assumptions completely explicit.
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voted helpful: davepamn
This site breaks it all down really well, in plain English... Hope this helps:
http://docs.google.com/gview?a=v&q=cache:sE0zeDS1kx0J:imps.mcmaster.ca/courses/CAS-701-02/slides/07-axiom-method.pdf+axiomatic+method&hl=en&gl=us&pid=bl&srcid=ADGEESgC0XWwhMt5OeI64vwVWOHxiVZ7QxcLpxfx35rQYVEyi_0WnCNkTq1tAMdwFemwD4pNifWwoeqSgzFPbkm_N_r7Uzyini_3HaxYD0OAfN3XTI2pCSNTabNhP_r-dQ-RPUfLoliu&sig=AFQjCNFEVTFpmxqL14TtRi8Qw-O-a9WCzw
http://docs.google.com/gview?a=v&q=cache:sE0zeDS1kx0J:imps.mcmaster.ca/courses/CAS-701-02/slides/07-axiom-method.pdf+axiomatic+method&hl=en&gl=us&pid=bl&srcid=ADGEESgC0XWwhMt5OeI64vwVWOHxiVZ7QxcLpxfx35rQYVEyi_0WnCNkTq1tAMdwFemwD4pNifWwoeqSgzFPbkm_N_r7Uzyini_3HaxYD0OAfN3XTI2pCSNTabNhP_r-dQ-RPUfLoliu&sig=AFQjCNFEVTFpmxqL14TtRi8Qw-O-a9WCzw
voted unhelpful: davepamn
It involves replacing a coherent body of propositions by a simpler collection of propositions.
-quote-
"The axiomatic method involves replacing a coherent body of propositions (i.e. a mathematical theory) by a simpler collection of propositions (i.e. axioms). The axioms are designed so that the original body of propositions can be deduced from the axioms.
The axiomatic method, brought to the extreme, results in logicism. In their book Principia Mathematica, Alfred North Whitehead and Bertrand Russell attempted to show that all mathematical theory could be reduced to some collection of axioms. More generally, the reduction of a body of propositions to a particular collection of axioms belies the mathematician's research program. This was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra.
The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that the mathematician would like to work with. For example, mathematicians opted that rings need not be commutative, which differed from Emmy Noether's original formulation. Mathematics decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff originally formulated.
The Zermelo-Fraenkel axioms, the result of the axiomatic method applied to set theory, allowed the proper formulation of set theory problems and helped to avoid the paradoxes of naïve set theory. One such problem was the Continuum hypothesis."
-end of quote-
-quote-
"The axiomatic method involves replacing a coherent body of propositions (i.e. a mathematical theory) by a simpler collection of propositions (i.e. axioms). The axioms are designed so that the original body of propositions can be deduced from the axioms.
The axiomatic method, brought to the extreme, results in logicism. In their book Principia Mathematica, Alfred North Whitehead and Bertrand Russell attempted to show that all mathematical theory could be reduced to some collection of axioms. More generally, the reduction of a body of propositions to a particular collection of axioms belies the mathematician's research program. This was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra.
The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that the mathematician would like to work with. For example, mathematicians opted that rings need not be commutative, which differed from Emmy Noether's original formulation. Mathematics decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff originally formulated.
The Zermelo-Fraenkel axioms, the result of the axiomatic method applied to set theory, allowed the proper formulation of set theory problems and helped to avoid the paradoxes of naïve set theory. One such problem was the Continuum hypothesis."
-end of quote-
voted helpful: davepamn
Please, give an example of an axiom and a proposition correlation
-quote-
"The Axiom of Causality is the proposition that everything in the universe has a cause and is thus an effect of that cause. This means that if a given event occurs, then this is the result of a previous, related event. If an object is in a certain state, then it is in that state as a result of another object interacting with it previously. For example, if a baseball is moving through the air, it must be moving this way because of a previous interaction with another object, such as being hit by a baseball bat."
-end of quote-
http://en.wikipedia.org/wiki/Axiom_of_Causality
"The Axiom of Causality is the proposition that everything in the universe has a cause and is thus an effect of that cause. This means that if a given event occurs, then this is the result of a previous, related event. If an object is in a certain state, then it is in that state as a result of another object interacting with it previously. For example, if a baseball is moving through the air, it must be moving this way because of a previous interaction with another object, such as being hit by a baseball bat."
-end of quote-
http://en.wikipedia.org/wiki/Axiom_of_Causality
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1. Axioms are true assertions made about abstractions of a concept.
2. The abstraction is generalized
3. A function can be deduced from the axiom assertion
4. Axioms help by making your assumptions explicit.