Some argue about mathematics and if it is discovered or invented. I’ve held an opinion for some time that mathematical things are there to be discovered, we humans invent language and notation to describe it.
When it comes to mathematics and to explore the idea of discovery and invention we need to look around use for evidence. Case one, back to the time of Newton and Leibniz and Calculus. Today we use the notation of Leibniz to do those calculations.
The notation that both used was different and the attempt was to achieve the same measurement albeit through slightly different paths. This was about measuring the area under a curve and tangents from that curve.
From this we could claim that the area under a curve was to give itself up and surrender to our tools even if the tool sets were different.
Let’s look back further in time, lets see what we can say about numbers and that most basic of skills, counting.
Different social groupings around the planet learned to count and created number systems to deal with their needs.
Some used base 60 such as the Babylonians and some groups in South America. The decimal system was used by many other groups, perhaps something to do with hands and fingers. What we know of zero and at least one version of the decimal system was that it was used for religious purposes, that story is from the Indus valley and surrounding regions.
Some people stop counting after a few single digits and then consider all other groups of numbers to be many. Some don’t stop and get to concepts like infinity representing it as a symbol.
We should pause a moment, think of the primes, 1, 2, 3, 5, 7, stopping there, there are infinitely many primes, a fact which is known since the time of Euclid circa 300 BCE.
However before we get bogged down in humans having the skills to count, how about the cicadas, those insects stay underground for for a prime number of years, seven years and then arise for six weeks and where they congregate and make so much noise that people tend to leave the areas they live in while they are around. It should be fair to say that cicadas know how to count in some way, even if they don’t use numerals as we do. They have evolved to do this thing.
Let us now head back to the numbers, people have ways to manipulate them, what people have done and some have become quite skilled with is to restate our discoveries and add to our understanding and abilities by refining the language we use.
This can be shown by the conversion of word problems into more symbolic conversations we have today, this http://jwilson.coe.uga.edu/EMAT6680Fa10/Shields/shields%20iu/BIT%20WP.htm gives a nice example of the conversion of words into symbols.
While this is no rigorous proof, it is the start of a story I’m interested in exploring at a later date.