I Standard way of expressing 'no proof given'?

[Jehannum]

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.

 

[PeroK]

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.

 

[mfb]

You can find "it is obvious that..." but make sure it's really obvious. Looking at a few examples isn't enough.

 

[jbriggs444]

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".

 

[Mark44]

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."

 

[PeroK]

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!

 

[Mark44]

This was in the age before rampant PC... To be honest, my mother was not able to do those integrals.

 

[PeroK]

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!

 

[pbuk]

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.

 

[Frabjous]

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.

 

[pasmith]

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.

 

[Jehannum]

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.