The "Fencepost" Problem (or so I just learned...)

I've felt for a while as if there is something fundamentally different in my logic and the logic of people who understand numbers. Maybe I have a hard time dealing with things that make sense at face value, I don't know. I'm sure that anyone who really grasps mathematical concepts can then whirl off into my fun happy land of theory and abstraction, but I, a humanities-head, just never hired the captain for that particular boat.

There was a wall I hit when I was young, regarding math. It was an obstacle to my understanding, a conceptual dilemma so deep that I still don't get it. It perplexed me so much that rather than dwell, than trying to wrap my third grade mind around it, I just pushed it away. But deep in the back of my mind, this Unknown instilled a deep mistrust in me regarding even the simplest of equations. To this day I check myself over and over again. I never relaxed, never relinquished my mind to the muscle-memory required to know instantly that six times seven is forty-two.

I'll explain the problem by example. Next week is the 22nd to the 26th. That's five days. But 26 minus 22 is 4. That makes NO SENSE to me. I guess it is because the day of the 22nd is also an entity that is being taken away, but I always felt like the function of subtraction was to reveal a quantity that has been parsed from a whole. So what's the point if it gives you the wrong number? I mean I must conclude here that when we subtract numbers from each other, we are really counting the spaces between them, the integers. The concept of integers helps me a little bit, but no one explained integers to me in third grade so I was pretty disheveled about the whole thing.

I see numbers as things that exists independently of the objects they quantify. I see them like ghost skins. If we have three apples, the first apple is wrapped in a ghost skin that is 1. The third apple is wrapped in a ghost skin that is 3. 3 minus 1 is 2. That makes sense because if you take one apple away you have two left.

But what does it mean when you say that the "difference" between 3 and 1 is 2? Seems to me that the difference between 3 and 1 is 1. Apple 2 is the difference, the solid object that lies between apple 1 and apple 3.

When you say the "difference" is 2, you must be saying that there are 2 spaces between apples 1, 2 and 3. "Difference" in math is supposed to be synonymous with "subtraction". But the only way you can say that the difference between 1 and 3 is 2 is by counting the spaces. But when you "subtract" 1 from 3 and get 2, you are counting the apples - the solid objects - not the spaces.

Way more complicated than it has to be. Well, welcome to my brain. This is something I've avoided trying to think about for 15 years.

The problem is not necessary in my ability to wrap abstract numbers around the physical universe. It's something to with my expectations of how this marriage works. I feel subtly betrayed by the 1 up or 1 down problem when trying to add or subtract real quantities, and most of the time I just give up and count on my fingers. Because fingers are real, physical, and won't trick you with silly things like "differences". I looked up math disabilities in grade schoolers and none of them really sound like what my issue was. So I'd just be curious to know if anyone else knows what I'm talking about.

Edit: I feel incredible satisfaction at finding this wiki page, thanks to the all-knowing Bjorn.

http://en.wikipedia.org/wiki/Fencepost_problem#Fencepost_error