The thinking behind "4 times 5 is 12, and 4 times 6 is 13, and 4 times 7 is-oh dear! I shall never get to 20 at that rate!" from Alice in Wonderland? Excerpt from Lewis Carroll's Alice in Wonderland:

"Let me see: four times five is twelve, and four times six is thirteen, and four times seven is-oh dear! I shall never get to twenty at that rate!"

My questions:

What mathematical machinery might the writer have had in mind when writing down the sentences above? Could one reengineer Carroll's thoughts behind? Is there any explanatory hint/link/allusion in Carroll's work or somewhere else?

I know that Carroll was a mathematician. So, there must be something out there...
 A: We have $4\cdot 5=20,\ldots ,4\cdot 10=40$. Assuming that one "has to stop" there (because one does stop there in elementary school), it would correspond to $12,13,14,15,16$ and $17$ in the "wrong" counting. So we will never reach $20$. Of course, there are many other possibilities.
A: In theory there are other possibilities...
Imagine that 
$$x\circ y=5x-8^{6-y}7^{y-5}$$ 
and it reads as $x$ times $y$.
So, 
$$4\circ y=20-8^{6-y}7^{y-5}.$$
In concreto
$$4\circ 5=20-8\cdot7^{0}=12 \text{ and } 4\circ 6=20-8^{0}7^{1}=13$$
(and $4\circ 7=20-8^{-1}7^{2}=20-\frac{49}8...)$

PS: It can be shown easily that $x\circ y$ $(y=5,6,7,...)$ will never reach $4x$. 
A: This blog has a suggestion
The idea is that she is doing calculations in base 10 but the answers are coming out in different bases ...
$4 \times 5 = 12 $ ( in base 18 )
$4 \times 6 = 13 $ ( in base 21 )
$4 \times 7 = 14 $ ( in base 24 )
So she is expressing $4n$ in base $3+3n$
indeed she can't get to 20 that way - if she could, $n$ would be a solution to $4n=6+6n$ which has solution $n=-3$  
A: There is a much easier one.
Take this.
$4\times n=4+n+(4-1)=n+7$
This way, $$4 \times 5=5+7=12$$ $$4 \times 6=6+7=13$$ $$4 \times 7=7+7=14$$
Now, note that a multiplication table usually continued up to $12$ only.
Ref.
In that case, we would have
$$4 \times 5=5+7=12$$ $$4 \times 6=6+7=13$$ $$4 \times 7=7+7=14$$ $$4 \times 8=8+7=15$$ $$4 \times 9=9+7=16$$ $$4 \times 10=10+7=17$$ $$4 \times 11=11+7=18$$ $$4 \times 12=12+7=19$$
And it would not reach $20$.
