The question then centers on whether or not such demonstrations are totally the result of an invented axiomatic foundation. Various geometrical theorems, algebraic concepts and formulae, differential integration, and calculus, logarithms, decimal point, graphical representation, geometrical instruments like the protractor or ruler, digital languages, and mathematical signs were invented, which helped mathematics scale new heights and evolve as a science. Twoless numbers are likely to be a barren area of study! People didnt have the same issues as we do today, such as calculating a sphere. The operations and elements are all by definition. This shouldn't even be a matter of dispute unless, of course, you have a view of knowledge acquisition that is as Plato subscribes in which we merely recall the truths from a prior life. Math also comes with sets of rules not created by people that set boundaries to what can be stated. I was referring to mathematics as a set of principles, not the objects which I take to be numbers, shapes etc - is that right? It is omnipresent, and its practical applications are found in many structures or patterns in the cosmos.
We can't one day wak outside and find every last number to Pi. However, if the content of all this can be distinguished from the language in which it is implemented, so to speak, reaching a level of abstraction that removes space and time altogether, one question arises, then, which is whether or not this abstract domain exists absent humans having these abstractions in their head, cloaked as they might be in spatial-temporal frameworks. Again, we may have such freedom in choosing the symbols that represent absolute truths but we can not invent the truths themselves. Now ask yourself, did human beings discover or invent monkeys? To quote the best definition of Independence concept that I found Mathematical objects are independent of intelligent agents and their language, thought, and practices. However, math has always been there.
Just think of my simple example. So in conclusion, I say that math was discovered, because it was always their we just didn't call it mat then until it was clearly defined. A pure mathematician working in, say, number theory is free to invent a new kind of number. A fourth, electric charge, was added in the nineteenth century. Someone had to create equations, no one discovered how to calculate the slope and y intercept in the pyramids. And another thing, if we don't start considering making an alternative to fossil fuels then in just a couple of years you are going to step outside and find yourself choking on pollution like the Chinese people! Both are concrete theories, both constructed both right and logically developed. So perhaps I should have asked; 'are the principles of mathematics, the laws and so on, out there waiting to be discovered? Or in the air as they fall.
The link between the two is unbreakable. However, I will try to give you some things to think about at least. He worked on it, he constructed the equations and so on. In many ways it is similar inventing a game like chess or go and discovering what makes for the best way to play. We may discover many ways to approach a single math problem, but we can't make up our own rules. Is mathematics discovered or invented? What about this: If you have one raindrop that falls to the ground.
Two raindrops hit the ground in the same spot and form a single larger raindrop. Math works because our universe supports it and operates like a giant computer. Newton, driven to model the mechanics of the physical world, introduced his fluxions. Both types of Geometry are based on very strict absolute rules that we most definitely had no choice in their invention. Of course, you don't seem to grasp the fact that our freedom of choice is in the set of rules we choose to follow and this is the difference between Euclidean and non-Euclidean geometries.
First of all, math occurs in nature and throughout our solar system in ways that are too complex and mind blowing for most to understand. I guess Mathematics was at first set by a set of rules and then developed according to them, and this first definition of these first rules is what baffles many. It expresses relations of abstract symbols according to well defined rules. The rate of increase of a sphere, as an example, that has to be a discovery, since that is how spheres behave in the real world. But let me pose a possible answer in the form of a conjecture: Conjecture: Definitions are invented. And this is so because mathematicians today assume a particular logical framework or foundation, either tacitly or explicitly in order to develop the challenges they wish to create for themselves. It's very unlikely that such a world would develop radio telescopes, so they aren't going to be among the worlds that get our strings of primes that we are trying to communicate our intelligence with.
Moreover, if the conjecture is proved, it becomes a theorem containing a self-reference, fertile ground for paradox. Stone age people such as the aborigines had no written language but they verbally communicated numerical data by pointing to fingers and toes and other body parts for larger numbers. . On the other hand, Euclidean geometry and non Euclidean geometry are both based as well on different but kind of related laws. And even saying there has to be a tie in to experimental results doesn't seem to be enough. Give our natural world a chance to thrive! Second of all, why are we worrying soooo much about this topic? Everyone would think of their own theory and code as concrete.
The definitions of derivative and integral having been made, the Fundamental Theorem of Calculus that relates them is an automatic, although far from obvious, consequence, awaiting discovery. But let me ask it a different way. Someone mentioned, well if there's 1 stone, and then there's another stone, you have 2 stones regardless of whether humans are around. Math must logically be proven with assumptions, theorems, and conclusions. The answers to specific questions in mathematics may be 100% determined, but the freedom in choosing which questions to ask is endless. That is absolutely worthless as a defense against invention. If I eat one vlorg, and then I eat another vlorg I have eaten two vlorgs.