Standard way of expressing 'no proof given'?

In summary: An extremely generous man, it would seem.In summary, in mathematical writing, it is not always necessary to give a proof for a sub-theorem if it is considered obvious or obtained by inspection. However, it is important to make sure that the statement is truly obvious and can be easily verified by the reader. Using phrases like "by inspection" or "by induction" can convey this idea without providing specific examples. It is also important to be mindful of casual sexism and ensure that any language used is inclusive and respectful.
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Jehannum
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In a proof of a theorem or in mathematical writing generally, if there is a statement of a sub-theorem, does a proof always need to be given if 'obvious' or if obtained by inspection? Is there a way of saying "I got this by trying some numbers in a calculator and the pattern was clear"?

The motivations are brevity, clarity and keeping to the main point.
 
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Jehannum said:
In a proof of a theorem or in mathematical writing generally, if there is a statement of a sub-theorem, does a proof always need to be given if 'obvious' or if obtained by inspection? Is there a way of saying "I got this by trying some numbers in a calculator and the pattern was clear"?
That wouldn't be a proof. For example, you cannot prove the Colaltz Conjecture just be trying a few starting numbers:

https://en.wikipedia.org/wiki/Collatz_conjecture

Or, more famously, Fermat's last theorem.
 
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You can find "it is obvious that..." but make sure it's really obvious. Looking at a few examples isn't enough.
 
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mfb said:
You can find "it is obvious that..." but make sure it's really obvious. Looking at a few examples isn't enough.
There is the old joke about the mathematics professor who goes to the blackboard writes down a few equations, and says "it is obvious that...". Then he pauses, scribbles furiously for twenty minutes and finally continues "yes, it is indeed obvious".
 
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I had a calculus instructor years ago who had these pat phrases:
"The proof is obvious to the most casual observer."
"Even my own mother could do this integral."
 
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Mark44 said:
I had a calculus instructor years ago who had these pat phrases:
"The proof is obvious to the most casual observer."
"Even my own mother could do this integral."
Hopefully, that sort of casual sexism is a thing of the past!
 
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This was in the age before rampant PC... To be honest, my mother was not able to do those integrals.
 
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Mark44 said:
This was in the age before rampant PC... To be honest, my mother was not able to do those integrals.
Nor is mine. But, the last time I visited her she was translating Immensee by Theodor Storm from a German edition written in gothic script!
 
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Jehannum said:
Is there a way of saying "I got this by trying some numbers in a calculator and the pattern was clear"?
Yes, if you have tried this 100 times you could say that "the table in Appendix A shows 100 (biased) sample inputs and the corresponding results; from these results the following relation was hypothesised" however this doesn't have anything to do with proof.
 
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We had a freakishly smart grad student grader once. The joke we told was that if you got lost in a proof just say ”trivially.” The grader would get to the comment, think a moment, say “yes, that is trivial” and give you credit.
 
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Jehannum said:
In a proof of a theorem or in mathematical writing generally, if there is a statement of a sub-theorem, does a proof always need to be given if 'obvious' or if obtained by inspection? Is there a way of saying "I got this by trying some numbers in a calculator and the pattern was clear"?

The motivations are brevity, clarity and keeping to the main point.

"By inspection, [...] is a solution" is always acceptable; the reader can easily check it, but it leaves open the possibility that there might be other solutions you didn't give.

"By induction we find that ..." is also acceptable, but don't bother giving the specific examples you looked at to deduce the general pattern. Again, the reader can easily check that the base case holds and verify the inductive step if it really is "obvious". Your reviewers will tell you if it isn't.
 
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jbriggs444 said:
There is the old joke about the mathematics professor who goes to the blackboard writes down a few equations, and says "it is obvious that...". Then he pauses, scribbles furiously for twenty minutes and finally continues "yes, it is indeed obvious".
I read that Euler used to write "it is obvious that ..." for stuff that took him months of labour.
 

1. What is the standard way of expressing 'no proof given'?

The standard way of expressing 'no proof given' is to use the phrase "no evidence to support" or "no evidence was found to support."

2. Why is it important to express 'no proof given' in a standard way?

Expressing 'no proof given' in a standard way helps to clearly communicate that there is no evidence to support a claim or hypothesis. It also helps to avoid confusion and misinterpretation of the results.

3. Are there any alternative ways to express 'no proof given'?

Yes, there are alternative ways to express 'no proof given' such as "inconclusive results," "insufficient evidence," or "no significant findings."

4. How do scientists determine if there is 'no proof given'?

Scientists determine if there is 'no proof given' by conducting thorough research and analysis of the available evidence. If there is no evidence to support a claim or hypothesis, then it is considered 'no proof given.'

5. Can 'no proof given' ever be considered a valid conclusion?

Yes, 'no proof given' can be a valid conclusion in certain cases. It simply means that there is currently no evidence to support a claim or hypothesis, but it does not necessarily mean that the claim is false. Further research and evidence may be needed to reach a definitive conclusion.

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