Mathematics - Both Discovered and Invented

There are two Scientific American links of interest to me.

One of the Scientific American articles is about the beauty of mathematical structures. There are some wonderful pictures in this slide show. The second article is in the August edition of Scientific American, which looks at 'Why does math work?' This goes to the age-old question: Is mathematics a natural entity that is there for us to discover, or is it invented by humans to satisfy our ability to quantify bits of our world? The author, Mario Livio, argues that it is both, and I tend to go along with this view.

There is a type of mathematics, pure math, that is analogous to pure science. This is where mathematicians investigate math for the sake of doing math, with no applications necessarily in mind. It is playing with curved surfaces, like Riemann investigated. Or when Galois was checking out the properties of various groups to solve polynomial problems. At their respective times, there were no real-world problems to which these mathematics were related. Perhaps one could say these topics were 'discovered.' Later on, years after the math was understood, scientists found new phenomena in Nature where the non-euclidean geometry (general relativity) and group theory (such as in particle physics) were important and necessary to describe.

Another type of math is applied math, where math is 'invented' to solve specific problems in the world. Newton developed calculus in order to solve gravitational problems, such as proving that a planet's mass can be reduced down to a single point, or the laws of motion, that required formal connections between displacement, velocity and acceleration. The rules of geometry were formulated in order to help quantify items being traded, or in designating property lines and areas. This is analogous to applied science, which places a focus on investigating real problems and finding solutions or producing a new tool or product that will be useful to humans.

An interesting thought experiment presented in the Livio article goes like this: If the intelligence of the world resided in a jellyfish that lives deep in the ocean, where it is generally isolated, would the concept of numbers exist? If there is nothing to count, and nothing discrete about the environment in which one lives, do numbers make any sense at all? So does this mean numbers are a natural concept of Nature, or that numbers are an invented entity because humans have a need to count things? Perhaps the jellyfish thought experiment leads to a conclusion that numbers are an invented concept. This is an interesting 'battle' to think about.