Sida Liu

There is a classical question that is quite entertaining: is mathematics invented or discovered?

Most people (non-mathematicians) tend to think that math is invented, that is the outcome of the work of mathematicians. But mathematicians themselves often express the feeling that they are not inventing math but discovering what is already there.

Recently, I read a paper titled “From Entropy to Epiplexity: Rethinking Information for Computationally Bounded Intelligence”. The paper points out that deterministic processes can generate new information for bounded observers. The author even mentions mathematics as an example:

“And yet, we use pseudorandom number generators to produce randomness, synthetic data improves model capabilities, mathematicians can derive new knowledge by reasoning from axioms without external information, …”

This sentence seems to give a very satisfying answer to the question of whether math is invented or discovered. Mathematicians usually do two things: they lay down axioms, and then they derive new theorems and corollaries. The former is invention, and the latter is discovery.

Inventions: the Axioms

If we write down some rules for computation, and if we have unlimited computational power, we can say that those rules immediately give rise to a new world. For example, each cellular automata rule gives rise to a world that is either repeating, chaotic, or “interestingly complex”, so to speak.

Another example is the universe itself: if we have the initial state of the universe along with all the laws of physics, and if we have infinite computational power, we could in principle derive the entire universe. If we slightly change the initial state or the laws of physics, we could have a different universe.

Similarly, in mathematics, if we write down the axioms of Euclidean geometry, we define the entire Euclidean world.

This is the incredible part, and we can call them inventions.

We can say that inventions of computational rules create new worlds. Anyone can create new worlds, but some worlds are more interesting than others.

Discoveries: the Theorems and Corollaries

After those worlds were created, curious people can explore them and discover patterns. Some of the patterns may be analogous to patterns in other worlds, and by understanding how those patterns evolve, we might be able to predict the outcomes in other worlds without actually carry out more expensive computation.

One of the worlds we care about most is the physical world we live in. When the patterns in a world we create are analogous enough to the physical world, we may call those patterns “useful”, because they allow us to solve “real-world” problems with less computation.

When mathematicians create new worlds that are “useful”, those worlds often become popular, like Euclidean geometry. People are more willing to explore popular and useful worlds rather than spend their time in apparently useless ones. However, some worlds are not immediately analogous to the real world, but may be analogous to other worlds. People often consider them “interesting” but not “immediately useful”.

The patterns of those worlds are captured by mathematical theorems and collaries, or under other names. But no matter what we call them, they are definitely patterns of those worlds that either “useful” or “interesting”.

Conclusion

In conclusion, the answer to the classic question “is math invented or discovered” is clear now:

Those who lay down the axioms are the inventors of the basic rules of those worlds. Deterministic computation alone populates or determines the worlds based on the rules. And others who generate useful theorems or corollaries are explorers of the worlds they consider useful or interesting, they discover.