In a previous blog post entitled WHAT ACTUALLY IS A NUMBER?, we looked at an argument from Plato that mathematical objects are not simply an invention by mankind but have instead some objective truth, maths is not a formalism we created but is something fundamental, something “out there” in the real world.
To illustrate this, we will look at a fun piece of mathematics, seemingly simple but with a tremendous amount of depth. The equation below is what we call a “power tower”. We take some variable, x, and raise it to the power of itself, and raise that to the power of itself, and so on and so on an infinity of times. We then form an equation by writing that this infinite tower of exponents equals some value, in the below example we set it equal to 2:
This certainly looks like a tricky equation to solve for x. The breakthrough comes however when we make an observation, namely that if we take off the bottom x and look at just the power tower of that bottom x, as in the red box below:
Now we ask ourselves what the infinite exponential in the red box is equal to. Well since it is infinite, we already know that it is equal to two by the first equation we wrote down. (NB this is a common trick to use when we are dealing with infinities in Maths, taking one term off an identically infinite function, still results in an infinite function). Replacing the tower in the red box with two, we therefore have:
And plugging this value of x into the original equation we have the mathematical statement that:
Now this is surprising, and a beautiful result, but this is not the focus of our endeavour. We may ask for example, instead of saying the power tower equals 2, can we set the power tower equal to any value we choose? To investigate we will now set our power tower equal to 4 instead of 2 and see what happens:
As before we proceed by considering the part of the equation boxed in red, noting that it is identical to the tower in the original equation, making a substitution and then solving for x:
Uh-oh! Once again we have arrived at x being the square root of 2, but this time from a different initial equation!. This implies that:
No doubt we have a glaring contradiction here, but what has gone wrong?
In fact, the mere assumption that there exists an x such that an infinite power tower x is equal to 4 is wrong. To solve an equation, you need to first know that the equation has a solution, and this just is not the case for the power tower equal to 4.
And this is the point: we are often inclined to think of Mathematics as invented by us. We have created the terms of Maths and we have created its rules, but this is not the case. Mathematical statements only have truth on the basis of whether they are true or not. We have not defined maths, we are exploring it. It is not something created by us, but has an empirical foundation outside of ourselves.
Written by William Brooke, Director of William Brooke