Here is a test of your mathematical intuition. Can you answer the following question in plain English (it is a hard one)? The natural logarithm, e, is an irrational number that starts with the digits 2.71818… It is easily derived in a large number of different ways. But why is it that number and not some other number?
To elaborate, physical constants of interest have a value that is relative to their units. For instance, the speed of light is 186,000 if the units are miles per second, but 300,000,000 if the units are meters per second. The boiling point of water is 100 if the units are degrees celcius and 212 if the units are degrees fahrenheit, and so it goes with the gravitational constant, permeability of free space, Planck’s constant, etc.
Then there are pure constants that don’t have any units. For instance, π (pi), the ratio of the circumference to the diameter of a circle is a pure constant — just another point on the real number line. Why is it that point and not some other? One way of explaining why π is that particular point on the real number is that it is relative to Euclidean space. If we transform to hyperbolic or parabolic space then the ratio of the circumference to diameter is also transformed. So, the value of π is a property of the space we are operating in. Euclidean space is just one space, being the boundary between parabolic and hyperbolic space.
Returning to the natural logarithm. If the value of π is a property of the chosen space, then what is e the property of? What change in assumption, or reference point or axiom would give a different value of the natural logarithm? Try to explain it in terms that non mathematicians, like me, can understand. Obviously π and e are directly related in the equation exp(iπ) + 1 = 0 and in the central limit theorem and many other fundamental relationships. But in simple terms what would you have to change to change the natural logarithm?