One reason that mathematics exists is to represent complex systems in (relatively) simple forms. Another is to quantify things in a more precise manner than would be possible without the use of numbers and math.
In D&D, the core game mechanic is: roll a die, add a modifier, compare to a DC to determine success. (Armor Class is just a special type of Difficulty Class, after all.)
Now, the core mechanic is simply a way to determine success or failure.
We could use a bucket full of colored balls, and place the appropriate number of balls in the bucket such that when you pull one randomly, you have the correct percent chance to succeed. But that would be cumbersome, so we instead use dice and arithmetic to abstract this process.
Some people have advised that the player could simply state what she wants to do, and the DM (or other players) could translate it into a die roll. That's a fine idea, but at some point the player may want more precision in terms of her chance of success.
In the previously given example, the player says, "My character is scared. She wants to run far away."
The DM replies, "If you do that, you'll run past three orcs, each of whom will get a chance to attack you. Do you want to risk it?"
To which the player might respond, "How likely are the orcs to hit me?" or more abstractly, "How dangerous will that be?" or even, "Am I going to die?"
How do you answer those questions without using math? You can attempt an answer along the lines of "the orcs are sort of likely to hit you", but what does that mean?
Does it mean they hit you half the time? Less than half the time? How much less than half the time? Would you say it would be one-quarter of the time? Or maybe between a quarter of the time and half the time?
Note that even to ask these follow up questions, I had to use some mathematical terms.
So, I applaud the idea of bringing a non-math-loving player into the game. But I wonder if the game is even playable without using some mathematical concepts.