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We know that if an argument is sound then it must be valid and its all premises must be true (as a result of this its conclusion is also true).

We prove mathematical statement by using logic rules. Then, can we say that all correct mathematical proofs are sound arguments?

I think that the answer is yes because a proof must be valid to reach the conclusion and if it is a correct proof that its premises and conclusion must also be true, but i am not sure about it? What is your notions?

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    $\begingroup$ What is your definition of a “correct mathematical proof” ? $\endgroup$
    – gandalf61
    Aug 6, 2020 at 15:47
  • $\begingroup$ @gandalf61 ordinary proofs $\endgroup$ Aug 6, 2020 at 16:19
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    $\begingroup$ You will need to give more detail than that. What do you mean by an “ordinary proof” ? Presumably it is not the same as a sound argument, otherwise you would not be asking your question. I doubt that anyone can help you unless you define your terms. $\endgroup$
    – gandalf61
    Aug 6, 2020 at 16:44
  • $\begingroup$ It sounds to me you are using to phrases to be defined to mean the exact same thing. Premises true and logic valid is a "sound argument". And correct mathematical proof is .... premises true and logic valid... $\endgroup$
    – fleablood
    Aug 7, 2020 at 0:46

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'True' is the sticky notion here. In real life, when we say that a statement is true, we mean true in our world. Thus, for example, any argument that starts out with: "Grass is purple ..." will not be a sound argument.

However, when mathematicians talk about 'truth' they mean that it describes some abstract mathematical world, e.g. the world of numbers, or the world of sets. And these kinds of mathematical worlds can really be anything the mathematician wants it to be. So in that context, a sound argument would be one whose premises are basically the axioms that the mathematician used to define that world.

For example: in the mathematical world of Euclidian geometry, a sound argument could start with "the angles in a triangle add up to the sum of two right angles", but any such argument is automatically not sound in the mathematical world of non-Euclidian geometry.

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  • $\begingroup$ thanks for this elegant clarification. $\endgroup$ Aug 7, 2020 at 8:56

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