A discovered invention

2) Can we know if mathematics is invented or discovered?

To understand this question, first we have to define the terms “discover” and “invent”. One of the many meanings of the word “discover” is given by the dictionary as “be the first to find or observe something”. And the meaning of “invent” given in a dictionary is “to create or design something for the first time”. In my opinion, from the way I see mathematics it must be a combination of both invention and discovery. By that I mean, math seems like it was discovered at first but it is an ongoing invention. We cannot be sure about any conclusions we make about the topic though; we could not be more inside of the box than we already are. Math has been our base to most things we do in our everyday life and also to technological and scientific discoveries for thousands of years now. And we still do not know when or how it all started. Did math exist before we did?

Invented?

The traces of invention in math are much more clear to me than the discovery ones. The schema I have of math is what I have gotten from school. We learn the concepts and then we apply them to real life situations. Really seems like something we would create to make our lives easier. But going out of the box with the simple concepts we learn and create new theories from it, well that’s a whole new level. That is when math is influenced by the creativity and imagination of the human brain. At this point it all becomes inventions based on other inventions that were done in the past. The connection between these inventions is called logic. In Andrew Wiles’s film, he explains how he got to the answer to Fermat’s last theorem by connecting invented theories from other mathematicians to create his own method. Nothing has to follow a pattern in math if the creator of the system it’s based on says so. So, for example, even though nature seems much more random than the math system we follow, it can be a still be a system.

 

Fibonacci and Nature

Discovered?

It is hard to understand how math could be in nature before us as there was no one to build a system/pattern to work from at that time. Well, there are very clear connections between the math world and nature. For example the Fibonacci sequence and the relationship it has with plants. The Fibonacci sequence works like this: the next number of the sequence is the sum of the two previous numbers. Surprisingly, branches and leaves in trees and petals in flowers follow the same pattern.

A quite more complicated example is the relationship between sea slugs/corals with hyperbolic geometry. The shape of these creatures is the answer to a mathematical theory that could not be modeled until recently.

Finally, as Steven Wolfram explains in Closer to Truth, what have been created and applied to our world based on our mathematical system is much less than what has not been created and applied yet. Many things like these are still out there to be discovered. And these discoveries in the world around us together with the logic we use from our mathematical system and our imagination a lot can and will be invented. Like I have said before, it feels to me that math is a bit discovered and invented. As long as we keep on moving forward the way we are, I believe we can use our mathematical inventions to discover the meaning of our existence.

Here is the coral reef math video.
http://www.ted.com/talks/margaret_wertheim_crochets_the_coral_reef#t-656834