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Is Math Truth?

A few weeks ago as we were on our way back from a weekend camping retreat, one of my friends asked me "Is math real?" This was inspired by a video from the PBS Idea Channel (Shown Below). The basic question being explored here is "What actually is math?" Is it something we made up, or is it something fundamental to the universe, like bacon? As the discussion progressed, we talked about whether there was a universal mathematical truth that describes everything.

The two sides of the debate presented in the video are that math is the underlying reason things work in the universe, and that math is a thing created by humans. On the one hand, the universe clearly works somehow (unless we're the illusion of some guy with his brain in a jar, sci-fi style). Things happen, and we can clearly see that things happen the same way in the same situation. Therefore there is something governing how things happen. On the other hand math as we know it is made up. Based on a set of axioms presented in Principia Mathematica I, II, and III, all of the results in math are derived. When you change the axioms or the presentation, you can rederive all of these results into something that is the same, similar, or completely different to what you had before.

But why math in the first place? Math is used to describe things. How many strips of bacon I have? How many do I have left when I eat all of them (0 is not a trivial thing!)? Where is this ball going to fall when I throw it? How do I build this building? Math has an origin in these kind of physical questions about magnitude, and numbers, and forms/geometry. From this the language of mathematics developed over human history. Fast forward a few thousand years to the Principia Mathematica and people were asking harder and harder questions about both the nature of the physical world and the nature of the language we're using to describe it. As old theories are shown to break down (ex. Newton's laws for really big or really small stuff), new theories arise.

To me, this constant updating is the main argument that math is not "true". None of our models perfectly describe reality. We always seek to improve our estimations. But that doesn't mean that math isn't real. Even though it's made up, Math is as real as language or art or any abstract human creation. Terry Pratchett argues a lot about this type of idea. In the Discworld series, things that are made up are very real, and ideas and belief shape the fold of reality. Through these physical manifestations of abstract ideas he argues that abstract objects are real things, and that reality is a more complicated notion than physically measuring something.

So our math is a real, but wrong attempt to describe the universe. Is there a "meta-math" that is truly right? I don't care. If we can predict reality well enough, we can take advantage of it even if it's completely false. We made huge strides through civilization with the power of math that was fundamentally flawed. From an engineers perspective, I want to improve math in the sense of improving our predictions and models of the world. It doesn't ever have to be right. Even if we developed a theory that perfectly described reality, there is no guarantee that the universe doesn't work through some completely different mechanism. What matters is picking a framework and using it to create something meaningful. And I believe that is the "truth" about math.

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