Proof of mathematical theorems

In summary, the question posed is whether any theorem in mathematics can be proven with just a pen and paper or a super-computer. There is some debate over this, as some argue that not all theorems can be proven due to the limitations of the system. However, in general, a theorem is considered true if it has a proof, but the reverse is not necessarily true. It is important to verify the validity of claims of theorems and ask for evidence in the form of a proof.
  • #1
kent davidge
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My question is simple. Can one prove any theorem in mathematics by having only a pen and a paper, or a super-computer for that matter?

Since math is essentially all about theorems, and we usually take them as true. I guess someone went in and proved them at some point in our history. But some of them are rather misteryous and I don't think their proof reduces to writing down equations and axioms.
 
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  • #2
Are you the same @kent davidge , who's posted questions about general relativity? How is it that you don't understand the nature of mathematical proof?
 
  • #3
Stephen Tashi said:
Are you the same @kent davidge , who's posted questions about general relativity? How is it that you don't understand the nature of mathematical proof?
yes, but it is the other way. I should not stick my nose everywhere. I'm having my first classes of special relativity in the uni, but you know, general relativity and quantum mechanics are more interesting.
 
  • #4
kent davidge said:
Can one prove any theorem in mathematics by having only a pen and a paper, or a super-computer for that matter?
You don't need computation capacities, as these are only rarely used, mostly to cover a finite number of exceptional cases. In general, you cannot proof any theorem (cp. Gödel), but you don't know in advance and most common theorems can be proven or disproven by a counterexample. Whether this takes minutes or centuries is another question. And you probably should have access to a good library and many journals!
 
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  • #5
kent davidge said:
Can one prove any theorem in mathematics by having only a pen and a paper, or a super-computer for that matter?
By definition, a theorem is a well formed formula for which a proof exists.

Note that the notion of being "true" and the notion of being "provable" are different. Neither implies the other.
 
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  • #6
I would say "provable" implies "true".
 
  • #7
Vanadium 50 said:
I would say "provable" implies "true".
In an inconsistent system, one can prove things that are not true. In a system where the axioms are not true, one can prove things that are not true.
 
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  • #8
jbriggs444 said:
By definition, a theorem is a well formed formula for which a proof exists.
yes, the problem is that someone can make a false claim that a given assertion is a theorem
so we are left with no option but to believe what authors are claiming to be theorems
 
  • #9
kent davidge said:
yes, the problem is that someone can make a false claim that a given assertion is a theorem
so we are left with no option but to believe what authors are claiming to be theorems
You could ask that a proof (or an outline thereof) be supplied as evidence for the claim that the assertion is a theorem.
 
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1. What is the process of proving a mathematical theorem?

The process of proving a mathematical theorem involves using logical reasoning and mathematical concepts to demonstrate that a statement is true. This typically includes starting with known facts and building upon them to reach a conclusion.

2. What is the purpose of proving a mathematical theorem?

The purpose of proving a mathematical theorem is to provide a rigorous and logical justification for a mathematical statement. This helps to ensure that the statement is true and can be relied upon in further mathematical work.

3. How do mathematicians know when a theorem has been proven?

Mathematicians know when a theorem has been proven when all the steps of the proof have been logically and rigorously demonstrated. This often involves using previously established theorems and axioms to support the proof.

4. Are there different methods for proving mathematical theorems?

Yes, there are various methods for proving mathematical theorems. Some common methods include direct proof, proof by contradiction, and proof by induction. The specific method used depends on the statement being proven and the approach of the mathematician.

5. Can a mathematical theorem ever be proven wrong?

Yes, a mathematical theorem can be proven wrong if there is a flaw in the logic or if new evidence arises that contradicts the statement. However, the process of proving a theorem is designed to minimize the likelihood of errors and to ensure that the statement is as accurate as possible.

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