Is Triviality the Same as Intuition?

[highmath]

If I say that something has triviality context, can I say that something is intuition?

Is Triviality = intuition?!

 

[MarkFL]

Generally speaking, I certainly would not say the two terms are interchangeable, or synonymous.

 

[I like Serena]

If I say that something has triviality context, can I say that something is intuition?

Is Triviality = intuition?!

In math we cannot prove anything with intuition. At best it can help us to find the way.

The word 'trivial' is actually a jargon word with a specific meaning.
It means the simplest possible mathematical structure.
As an example the 'trivial' real function is the function that maps all real numbers to zero.
In the same fashion, the 'trivial' solution of the differential equation $y'=y$ is $y=0$.

Long story short, triviality and intuition have nothing to do with each other.

 

[S.G. Janssens]

Generally speaking, I certainly would not say the two terms are interchangeable, or synonymous.

I very much second that.

To me, the Jordan Curve Theorem is a striking example of an intuitive result with a proof that I find non-trivial.

 

[I like Serena]

I very much second that.

To me, the Jordan Curve Theorem is a striking example of an intuitive result with a proof that I find non-trivial.

I think that merely means that it does not follow directly from the definition, which would otherwise be the simplest possible mathematical proof.
Put differently, if a proper mathematician writes that something follows trivially, it isn't meant to be condescending, it means that it follows from the definition. If it does not, it is not trivial, and is likely introduced by lack of understanding, laziness, or a mistaken sense of superiority. ;)

 

[Greg]

Perhaps the more one knows more things become trivial. Perhaps the more one knows the greater becomes one's intuition.

 

[highmath]

In math we cannot prove anything with intuition. At best it can help us to find the way.

The word 'trivial' is actually a jargon word with a specific meaning.
It means the simplest possible mathematical structure.
As an example the 'trivial' real function is the function that maps all real numbers to zero.
In the same fashion, the 'trivial' solution of the differential equation $y'=y$ is $y=0$.

Long story short, triviality and intuition have nothing to do with each other.

Can you exaplain to me the underline text?
Can you me more examples to more thing like the underline text? (If I can understand your exaplanation , Thanks...).

 

[Evgeny.Makarov]

I think that merely means that it does not follow directly from the definition, which would otherwise be the simplest possible mathematical proof.

I would say that all theorems follow from definition. The question is how directly. I think in most cases of trivial proofs the statement nevertheless does not follow from the definition using one or two inference rules of formal logic. It may take, I would guess, a couple dozen steps, such as breaking conjunctions, instantiating quantifiers, forming contrapositions and so on, and still many would call a proof "trivial". And the smarter a person is, the longer proofs he or she finds trivial.

 

[S.G. Janssens]

And the smarter a person is, the longer proofs he or she finds trivial.

This may also explain why the term "trivial" is abused by some (not those present in this topic): For the purpose of impression, they love to strongly state how they find something "trivial", even when it really isn't that trivial. (At least, not in the minds of their students, readers or colleagues.)

 

[I like Serena]

Can you exaplain to me the underline text?
Can you me more examples to more thing like the underline text? (If I can understand your exaplanation , Thanks...).

Not sure what you're looking for, so let me give a reference to an example:
https://proofwiki.org/wiki/Definition:Zero_Homomorphism