# Is mathematics invented or discovered?

First of all, we need to notice differences between invention and discovery. “Invention” means that something that has never been made before or the process of creating something that has never been made before. “Discovery” means that the process of finding information, a place, or an object, especially for the first time, or the thing that is found. For example, Thomas Edison invented the lightbulb in December 31, 1879. The important thing for this sentence is there was a creation. The lightbulb was never been in the world until Thomas Edison created it. Then let us give an example for discovery. Clyde Tombaugh discovered Pluto in February 18, 1930. For that example the main idea should be that Clyde Tombaugh did not create Pluto. He just realized Pluto in our solar system for the first time.

If we notice that the differences between these two terms then we go on our main topic. For the people who interested in mathematics there is a huge conflict about mathematics is invented or discovered. In that article we give some examples about why people think their arguments are correct.

The most common example for people who think mathematics is discovered that Fibonacci Numbers. If we look at the nature then we obviously realized that there are brilliant patterns. Let us explain what Fibonacci Numbers is. Then give some examples in nature about it. Fibonacci Sequence defined as each number is the sum of the two preceding ones with F_0 = 1 and F_1 = 1. In mathematically;

If we put the values of in that sequence we get; 1,1,2,3,5,8,13,21,34,55,… Now let us take 9th and 10th term of Fibonacci sequence. 9th term is 34 and 10th term is 55. So if we calculate 55/34 we get 55/34≈1,6176470588. After that let’s look at 14th term =377 and 15th term =610. Then again calculate 610/377≈1,6180371352. Did you notice that if we take n sufficiently large then (n+1) th term / n th term converges “Golden Ratio”? Answer is yes! That approximation is alternately lower and higher than golden ratio and when n is sufficiently large it converges golden ratio. So mathematically;

Now you could ask that question. Is there a Golden Ratio or Fibonacci Sequence in nature or universe? Absolutely yes! We can see that sequence in number of petals in flowers. So many flowers satisfy the number of petals which is also Fibonacci Number. For example; iris has 3 petals, larkspur has 5 petals, delphiniums has 8 petals, cineraria has 13 petals, aster has 21 petals…

Iris has 3 petals.

Aster has 21 petals.

Sunflowers have 55 clockwise “spirals” and either 34 or 89 counter-clockwise spirals. The ratio of 89/55 or 55/34 is approximately equal to golden ratio.

The mathematics of nature is not limited with Fibonacci Sequences. If we look at honeycombs we notice that bee hives have “hexagonal shapes”. Additionally snowflakes have hexagonal shape. “Fractals” are also good observation in nature. Ferns are example of fractals because they are similar at any scale and repeat itself. Finally we can talk about “symmetries” which we learn in Algebra courses and it appears in nature. For instance starfish have radial symmetry and mammals have bilateral symmetry. These all examples show that mathematics is always has been. We just realized it and then formulated it.

Have you ever though about possibility of “another mathematics” ? Stephen Wolfram who invented Mathematica answer that question.

Mathematics as humans practice it is based on a handful of particular axioms system— each in effect defining a certain field of mathematics (say logic, or group theory, or geometry, or set theory). But in the abstract, there are an infinite number of possible axiom systems out there — in effect each defining a field of mathematics that could in principle be studied, even if we humans haven’t ever done it.

So why don’t we use another mathematical systems? Why we have been using same mathematical axioms thousands of years? Because we have historical background with our mathematics. We had built new structures , we had learnt purchase and sale. Because of requirement, we have to measure and count something. For recording quantities we have to symbolise something because numbers don’t appear in nature. We had assigned each quatities as numbers which we symbolise. We had reached an aggrement and it made facilitate our life. So that we had created a “new language” without noticing. After that we construct some axioms, some definations and then improving methods for solving physical problems. In summary we wanted to solve our real world problems then we invented mathematics accordingly.

Let us give an example. Collatz conjecture is a famous mathematical conjecture which is not proved yet. It means that our mathematical axioms or methods is not enough to prove it. Well, what happens if we change our axioms? Maybe it could be help to prove that conjecture. This example shows that sometimes our mathematical systems may not be enough. We can explain and prove so many things about universe but not all of them yet.

From these arguments we can notice that the people who defend “mathematics is invented” just seeing mathematics as a tool for explaining the physical laws and solving real world problems.