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Physics is clearly discovered, while languages are invented. The distinction here is clear: one observes nature and the other is a construct of humans. However, things become less evident when we get to Maths - is it observations of relationships in nature or is it methods made by humans to tackle our problems? This long standing debate hasn’t been resolved in centuries and I plan on shedding some light to it, and perhaps help you reach a conclusion.
I am, myself, a conventionalist which is a middle-ground of some sorts. Conventionalists argue that mathematics is a human-constructed language used to describe discovered truths. That may sound intimidating, but I will break it down so you can fully understand the meaning behind that sentence.
We must first start with a simple truth: the terms and methods we use in mathematics are conventions (agreements). For example, the coordinates system - which helps describe relationships between different points - is a man-made tool. The same goes for our number system; “1”, “2” and “3” are terms created to help us describe what we see.
A similar, second point we must recognise is that truths are context-dependent (strange, right?). In geometry, we could say that all angles in a triangle add up to 180°, and that this is a universal, eternal, and unchangeable truth - but this is not the case. In non-euclidean geometry (fancy word for geometry in curved surfaces), they can add up to 270°! Therefore, we reach the conclusion that our conventions must be context-dependent and may vary over time.
However, not everything is man-made. These truths we find in nature, and that we observe with mathematics are, as the word implies, discovered. The Pythagorean formula, a2+b2=c2, is true for all of the right angle triangles (even in curved surfaces!) we find in nature. No human created that to help us tackle a problem, but the terms “a”, “b”, and “c” were, and help us describe this relationship we find in nature. Hopefully, it now seems evident.
I would like to conclude that we have proven that maths are both created and discovered, and that they are formed by human conventions to describe patterns in nature. Sadly, this is not entirely the case. We have indeed found convincing evidence for the statement, but it is not that simple: conventionalists often struggle with the fact that, if mathematics are merely conventions, they would lack a deeper connection to the objective reality of the real world.
Javier G, Year 11