Differet Types of Mathematical Arguments and Proofs

[soopo]

Homework Statement

Are all types of mathematical arguments based on the following types of proofs?

Types of proofs
1. Direct proof, P -> Q
2. Proof by contradiction, \neg Q -> \neg P
3. ~Ad absodium, P and \neg Q -> false statement (such as 0 = 1)

I know the following types of arguments
1. Mathematical induction
2. Iterative argument
3. Least Criminals

The Attempt at a Solution

Mathematical induction seems to be a direct proof, similarly as the iterative
argument. In contrast, least criminal is apparently a combination of direct
proof, and the proof by contradiction, since least criminal argument is a
variant of Mathematical Induction.

 

[Focus]

I have no idea what you mean by iterative argument and least criminal. Your No.2 should be contra-position not contradiction. Reductio ad absurdum is proof by contradiction. If you are arguing by some sort of iteration, then you are referring to induction (be it natural or transfinite). There are no other iterative proof methods.

Induction is not a direct proof. You need an axiom of induction to use it. By the way here is the proper names of the "proofs",
1) Modus Ponens
2) Modus Tollens
3) Reductio ad absurdum

Here is a link to the rules of inference of natural deduction http://www.mathpath.org/proof/proof.inference.htm .

 

[soopo]

I have no idea what you mean by iterative argument and least criminal. Your No.2 should be contra-position not contradiction. Reductio ad absurdum is proof by contradiction. If you are arguing by some sort of iteration, then you are referring to induction (be it natural or transfinite). There are no other iterative proof methods.

Induction is not a direct proof. You need an axiom of induction to use it. By the way here is the proper names of the "proofs",
1) Modus Ponens
2) Modus Tollens
3) Reductio ad absurdum

Here is a link to the rules of inference of natural deduction http://www.mathpath.org/proof/proof.inference.htm .

Thank you a lot - I did not know that there are so many names for the same proof.

 

[Focus]

They are not really proofs, they are mainly rules of natural deduction. The methods of proof used in maths I come across are direct proof, contra-position, contradiction and induction. If you count counter example as a proof then that's in there too. Other than that I haven't yet come across any other methods.