Which branch of math does that?

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In summary, the conversation revolves around the topic of axiomatic mathematics and its relationship to abstract algebra. Peano's axioms are mentioned as a way of proving mathematical principles in abstract algebra. The conversation also touches on the deductive nature of mathematics and the use of assumptions and premises in constructing formal logic systems. Various resources are suggested for further study in this area.
  • #1
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Which branch of math deals with "proofs" of the axioms in arithmetic and such? I'd be interested to look into it.
 
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  • #2
You may want to investigate abstract algebra.
 
  • #3
I'm not sure exactly what you mean, as obviously nobody examines proofs of axioms!
 
  • #4
Well the Peano axioms are certainly provable... Radou, are the Peano axioms explored in abstract algebra?
 
  • #5
Werg22 said:
Well the Peano axioms are certainly provable...

Provable from what? If they are truly axioms then they can't really be proved true or false from other statements.
 
  • #6
Axioms can be found logically... this is what I mean by "proof".
 
  • #7
Werg22 said:
Axioms can be found logically... this is what I mean by "proof".

A logical assumption is very different from a proof.
 
  • #8
If you are interested in a proof of the Peano axioms, you might want to look at an introductory set theory book, such as Halmos'.
 
  • #9
Herstein's, Topics in Algebra, and then there are a few unknown books that I found which surveyed selections of abstract algebra, to give you an idea of how broad the spectrum is and provide you with an illustration so that you can select areas that you enjoy. If you are curious, I can give you those as well (although, they are REALLY old and seriously, no one has heard of them but I enjoyed pieces of them). I think this is the area that you are referring to? It's beautifully elegant, you will enjoy it.

As for axiomatic mathematics, I am under the impression that mathematicians construct axioms/assumptions/postulates to govern a particular formal logic system (such as an algebraic system) and from there, interesting properties emerge. If the system produces interesting properties, the system is studied further.

Mathematics are deductive because if the premises are true then the conclusion must be true because the conclusion is contained in the premises. However, the premises must be assumed true.
 
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  • #10
Given any system of axioms, there always exist a simpler system in which those axioms can be proved as theorems.

Peano's axioms assert that there exist a set of objects, N, called "natural numbers, and a function s (the "successor function"), from N to N such that
Axiom 1: There exist a unique member of N, call "0", such that s is a one to one function from N to N\{0}.
Axiom 2: If a subset, X, of N contains 0 and, whenever it contains n, it also contains s(n), then X= N.

You can, however, define the natural numbers in terms of sets: 0 is the empty set, 1 is the set containing only 1 (only the empty set), 2 is the set containing only 0 and 1, and, in general, given any n s(n) is the set containing n and all of its members. From that one can show that Peano's axioms are true.
Note: Historically, Peano's axioms included the number 0 as I have here. Nowadays, however, most people start with the number 1.

Werg22, you might find this interesting:
http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/$file/NUMBERS.pdf [Broken]
 
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  • #11
Ok thanks all. Complex, I will try to familiarize myself with the subject before considering any book purchase, so I must put you on hold. Halls, I will check the link you gave as soon as I can.
 

1. Which branch of math does algebra fall under?

Algebra is a branch of math that deals with symbols and the rules for manipulating those symbols to solve equations and understand relationships between quantities.

2. Is geometry considered a branch of math?

Yes, geometry is a branch of math that deals with the study of shape, size, relative position of figures, and the properties of space.

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Number theory is a branch of math that deals with the study of integers and their properties, such as prime numbers, divisibility, and arithmetic operations.

4. Does calculus belong to a specific branch of math?

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5. Which branch of math is used in statistics?

Statistics is a branch of math that deals with the collection, analysis, interpretation, presentation, and organization of data. It uses concepts from probability and combinatorics to make predictions and decisions based on data.

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