Stop 5: Planet of Continuity

Charlie Ahrendts (Image: John Burns)

Having learned about many fascinating axiomatic systems on our voyage so far, we finally make our way to the last stop on our journey. Upon landing on the planet one particular observation immediately strikes us. There are no edges on this planet. Everything seems to be smooth…

As you may imagine, this also has an impact on the mathematics of this alien civilization. We learn that they are very familiar with ‘continuous maths’. What this means can be best understood in terms of functions. There are lots of situations where we want to find the area under the curve of a function, and we do so using a technique called integration. The general idea is estimate the area using better and better approximations until eventually we find the exact formula, function, or number these values converge to. Sadly, this only works for some cases, for others we often need to divide our function into small pieces and then look at each one individually. Every time we do this, it falls under the category of discrete modelling. Opposed to this is continuous modelling. Any function is continuous if it has no gaps, jumps or sharp edges – it is completely smooth. Here is a simple example with the graphs of two functions x2 and |x|:

As we can see, the parabola is smooth at every point, while the other function has a sharp edge at x=0. If we try to calculate the gradient (or derivative) of each function, we are going to run into problems at x=0 for the modulus function. If we approach zero from the right (decreasing from a small positive number), then function is a straight line with gradient +1. However, if we approach zero from the left (increasing from a small negative number), then the function is a straight line with gradient -1. Since these do not agree at x=0 we say the derivative is ‘discontinuous’ here.

Now that we have a better grasp of continuity, we may start to wonder how these aliens intuitively think about functions. They might perceive them as a single object, rather than our notion of a set of inputs that is mapped onto a set of outputs. With this fundamentally different intuition, it is likely the aliens are able to perform integrations and derivatives much easier than we are. This extends to a better understanding of natural processes, since calculus is our most powerful tool when modelling change. We usually approximate time dependent functions in a discrete way, often because working with continuous models is very complicated, but this would not be an issue for our alien friends.

Here we can see again how our environment shapes the way we build our mathematics. When your only way to describe the objects around you is through continuous functions, they will naturally be more central to your axioms and thus to your understanding of the universe.

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