I think it would be better to compare LISP to the Peano Postulates than to Euclidean Geometry.
I have constructed a complete Peano Natural Number Arithmetic in LISP, and can see a clear path to creating the Rational Arithmetic. A Natural corresponds to a List of Units, i.e. Unary Numerals. Cons corresponds to the Successor operation; An Integer is a pair of Naturals, the Difference between the two; a Rational is a pair of Integers, the Ratio of the Two. Rational arithmetic is closed under addition, subtraction, multiplication, and division, except division by zero. Addition and Multiplication are recursively defined as in Peano. The sums and products of Rationals are the standard definitions. Subtraction: change sign (reverse order of the pair of naturals) and Add; Division: invert (reverse order as in add) and Multiply.