TheoremProving: Difference between revisions
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I don't follow this. In my mind a theorem consists of a statement of the theorem followed by a proof of its truth. See, for example, the theorems in [[ElementaryGroupTheory]] |
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One may seek to prove a new theorem by hypothesis->investigation->conclusion, but that isn't the theorem. |
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There are many ways of proving a theorem correct, including: |
There are many ways of proving a theorem correct, including: |
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Revision as of 18:48, 12 March 2001
A mathematical theorem begins with a mathematical hypothesis, proceeds through mathematical reasoning to reach a mathematical conclusion.
Mathematicians seek to establish chains of reasoning that are convincing to other mathematicians. The main differences between mathematical argument and ordinary logical argument are in the topics of mathematical discourse.
The following diagram displays the relations among the terms:
- Theorem = Hypothesis--->Proof--->Conclusion
I don't follow this. In my mind a theorem consists of a statement of the theorem followed by a proof of its truth. See, for example, the theorems in ElementaryGroupTheory
One may seek to prove a new theorem by hypothesis->investigation->conclusion, but that isn't the theorem.
There are many ways of proving a theorem correct, including:
- Contradiction - If we can show that the assumption that our hypothesis is false leads to a logical contradiction, it follows that the hypothesis must be true. Also known as reductio ad absurdum.
By mathematical hypothesis, are we meaning the result to be proven or axioms?