Abstract Failure

Who remembers being in grade school and being introduced to the mathematics concept of 'variables'?  For some reason, if you took a standard 2 + 2 = 4 formula and put in a blank ( 2 + _ = 4 ) then most of them were alright, but if you put in a variable (2 + X = 4 ) some of my classmates were brought tears.  They could not handle the undefined nature of the variable before the solution was found.  They needed X to have a concrete value, and will be even more confused if we try to explain that the "X" itself is arbitrary, and it could have been a "Y" or "Z". They simply never grasped the power of the algebraic concepts they were being taught.  The fact that they could very easily subtract 2 from 4 to find the value of the blank in their workbook did not magically imbue them with the understanding that they could use any symbol instead of a blank for any problem and perform the same operations. They failed to abstract the general principle, despite many concrete examples, and never became comfortable with dealing with abstracts, nevertheless sequences of abstract operations.

I've seen many examples of Abstract Failure, in all arenas of life. The funny thing is that some people will be completely capable of abstracting in their professional capacity, or in one particular subject, but then lose all ability to abstract in other domains. This can make it difficult to have conversations about important intellectual issues. Let's take a common example:

"Consider some statement," I might say, "whatever that statement might be. And let's say that statement has a truth value; it is either true or false. Okay, so then ..."

"Wait," the other person interrupts, "what is the statement?"

"That's not the point," I continue, "we are trying to discuss what it means for a statement to be true or false. All that matters now is that we agree that we have a statement that is either true or false."

"But," it comes again, but with more urgency, "What Is The Statement!?!"

At this point, if it seems that my conversation partner is being earnest (as opposed to just playing a jerk), I am honestly at a loss about what to do next.  The conversation we are currently engaged in is doomed; we will be talking past each other.

From time to time I'll explore this further.  Those posts will be labeled Abstract Failure.