There is a difference between the language of math and the calculating of math.
Calculations prove knowledge of lower levels.
Language of math proves knowledge of higher levels.
It is the difference between reading topology and being able to reduce it.
It is the difference of reading a differential equation and solving one.
Reading is connected to concepts while calculations are based on pushing buttons.
"Stop teaching calculating.  Start teaching math." ~Conrad Wolfram
He says I should think about posing questions in the real world (or the holographic one).
University still hasn't become up to date.

Every day in math I face the halting problem:
whether I can finish a proof,
or whether I will go on trying to prove forever.
At any given point
all problems become an undecidable problem.
Time for a break.

I truly must shut down my right-brain from seeking more meaning from my partial left-brain understanding,
or else I'll lose sight of any lesson except the utmost whole.

...is like solving a crime that has left microbial evidence all over the city and in the adjoining ocean.

If it doesn't matter spacially whether there is a 1st or 2nd, etc., connection, then this matter shouldn't matter within reason.
As long as a connection is made or not made, that is all that will stand out.  
The kind of connection made cannot be spacially signified except via overlap or by tangency, so therefore, within reason the kind of connection made should not stand out either.
Therefore, this subject of topology is really a matter of 'connect the dots'.