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StephenPrivitera
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In general, what is the conventional method of proving a theorem? What makes a proof valid? I hope that question is clear enough.
Originally posted by loop quantum gravity
i think (and i might be wrong) but a proof should be prooved by Deductive reasoning first you have the premesis which is the data you have in hand in order to proove the theorem after that you conclude from the data the conclusion (theorem).
i hope the explanation is ok.
edit:
here is link to an article about the origins of proof there you might find the answer you were looking:http://plus.maths.org/issue7/features/proof1/
Originally posted by loop quantum gravity
now isn't this definition paradoxical?
Originally posted by StephenPrivitera
Prove that addition is not distributive over multiplication (domain=natural numbers).
Well that was much too easy!Originally posted by lethe
to show that something is not true, it is sufficient, and usually easier, to simply provide a counterexample.
if addition were distributive over multiplication, then 1+1*1 would equal (1+1)*(1+1). but 2 does not equal 4.
that is all one needs to do.
never mind my idea was a wrong one.Originally posted by lethe
no. why would you say it is paradoxical?
Stephen, maybe it might help to make a list of common types of proofs. Here's some examples that I remember from scratch:Originally posted by StephenPrivitera
In general, what is the conventional method of proving a theorem? What makes a proof valid? I hope that question is clear enough.
A theorem is a mathematical statement that has been proven to be true using a logical sequence of steps. Proving the validity of a theorem is important because it provides evidence and support for the statement, allowing it to be accepted as true and applicable in various mathematical contexts.
A valid theorem is one that has been proven to be true using a logical sequence of steps. It follows the rules of logic and can be applied in various mathematical contexts. An invalid theorem, on the other hand, is one that has not been proven to be true using a logical sequence of steps. It may contain errors or assumptions that make it unreliable or incorrect.
The most common methods used in proving the validity of a theorem are direct proof, proof by contradiction, proof by induction, and proof by contrapositive. Direct proof involves using logical steps to show that the theorem is true. Proof by contradiction involves assuming the opposite of the theorem and showing that it leads to a contradiction. Proof by induction involves showing that the theorem is true for a base case and then proving that it holds for all subsequent cases. Proof by contrapositive involves proving the contrapositive of the theorem, which is a statement equivalent to the original theorem but with the antecedent and consequent switched.
There are several potential limitations of a theorem's proof, such as relying on incorrect axioms or assumptions, using flawed logic, or overlooking counterexamples. Additionally, a proof may be valid within a specific mathematical context, but may not hold true in other contexts. It is important for mathematicians to carefully examine and critique the validity of a proof to ensure its reliability.
Peer review plays a crucial role in validating a theorem's proof. It involves having other experts in the field review the proof and provide feedback and critiques. This helps to identify any potential errors or flaws in the proof and ensures that it is reliable and valid. Peer review also helps to improve the clarity and presentation of the proof, making it easier for others to understand and replicate.