By Sean La

When I was in high school, I was a scientismist. Not a scientist (I was quite the mediocre student in Science 10). No, I was a scientismist, I believed in scientism. Scientism is the belief that the only sure way to understand the world is through the scientific method [1].

To me, science was the answer. Every phenomenon could be reduced. Culture is herd behaviour. War is natural selection. Love is chemistry. And religion is adaptation. To me, science was the word and to suggest otherwise was blasphemy.

And physics and math were the most powerful because of the power of their predictions.

But as I grew a little bit older and a little bit wiser, I made two great observations about science: there are things you can’t know, and there are things you can’t know to know.

Things you can’t know.

The most fundamental field of physics is quantum mechanics. It studies the behaviour of the tiniest particles known to man such as electrons, quarks and gluons. For those who are well-read in physics, I don’t need to tell you that when you zoom in onto this scale, the universe doesn’t behave like you think it would. Photons acting like waves and particles at the same time, particles linked through time and space, things get weird at this scale of the universe. But the phenomenon in quantum mechanics that surprised me the most as a wee lad reading physics was Heisenberg’s Uncertainty Principle.

Heisenberg’s Uncertainty Principle states that the more you know about where a particle is going, the less you know where it is and vice versa [2]. This is quite abstract, so I’ll illustrate this to you with a joke I modified from a post I found on Reddit [3].

Heisenberg gets pulled over for speeding. The cop asks Heisenberg “Do you know how fast you were going?” Heisenberg replies, “No, but I know exactly where I am!” The officer looks at him confused and says “you were going 108 miles per hour!” Heisenberg throws up his arms and cries, “Great! Now I’m lost!”

So I know this is kind of confusing, but what this means is there is a fundamental limit to how much information we can know about our universe. No matter how accurate our instruments or advanced our mathematics, we will never be able to tell the exact behaviour of a simple particle.

Things you can’t know to know.

Alright, so physics is inherently limited, how about mathematics? Surely those symbols math teachers shove down our throats must be correct, because why else would they make us learn it? Well, no. Mathematics shows us that there are things you can’t know to know.

So the first question I will ask before moving on is what is mathematics? This question doesn’t have a definitive answer, but we can view mathematics as a sort of language. For example, take the equation x + 2 = 12. This equation is saying there exists a number x, such that if you add 2 to it, it is equal to 12. Really, what this is conveying is knowledge about a particular abstract object. And mathematics aims to be a business of only true knowledge. What use would an equation be if it gave you the wrong answer? But there are answers to questions in mathematics that can’t be known. That is the essence of the final theorem I will be presenting to you, Godel’s Incompleteness Theorem.

Godel’s Incompleteness Theorem states that any consistent mathematical system that can do arithmetic is incomplete [4]. Here, “consistent” means that are no contradictions in our mathematical system, and “complete” means that every statement in the system can be proved true or false.

To give you a quick sketch of what the theorem means, we know we can view mathematics as a language, and like our own language English, we can construct a statement that refers to itself. Take, for example the liar paradox, “this sentence is false”. Is this sentence true? Well no, because if it is true, then it is false. Is this sentence false? No, because then this sentence is true. Either way we can’t know the truth value of this sentence, it’s a paradox.

What Godel’s Incompleteness Theorem says then, in fancy-shmancy math talk, is that every mathematical system must necessarily contain the statement “this statement cannot be proven”. Can we prove it to be true? No, because then it can’t be proven. Thus we can’t prove it, and the sentence is true. Weird right? We have just shown that there are things that we can’t prove to be true in mathematics. There are things we can’t know to know.

Let’s sit back and contemplate this for a second. What we just showed here is that both physics and mathematics are inherently limited; that math and physics can’t explain everything about this world, because they can’t even explain themselves! Then really, when we study math and physics, we are in a way accepting that it is true without question. In other words, we’re placing faith in these subjects, just like a religion. And this is kind of humbling. No matter how intelligent we become, there will always be questions for which we can never have an answer for.

Sean studies Mathematics at Simon Fraser University and is an editor for SFU SURJ. His interests include writing, programming and debating.

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[1] http://www.thenewatlantis.com/publications/the-folly-of-scientism
[2] https://www.theguardian.com/science/2013/nov/10/what-is-heisenbergs-uncertainty-principle
[3] https://www.reddit.com/r/Jokes/comments/2tm2ub/heisenberg_schrodinger_and_ohm_are_in_a_car/
[4] http://plato.stanford.edu/entries/goedel-incompleteness/