I can’t seem to wrap my head around the fact that math is invented? Maybe he came up with the symbols of integration and derivation, but these are phenomena, no? We’re just representing it in a “language” that makes sense. I’ve also heard people say that we may need “new math” to discover/explain new phenomena. What does that mean?
Edit: Thank you for all the responses. Making so much more sense now!
In: Mathematics
Think of it like this: ducks always existed. But humans invented the name to describe them.
In the same way, physics always existed. But Newton invented the way to describe them.
I suppose it depends on how you use the word “invented.” There is an argument that all math is naturally occurring – all we do is _discover_ it and create a notation to codify that discovery. It is a bit semantic, though – prior to Newton, we had no understanding of calculus and Newton is the one who discovered/invented it and gave that to the world.
To the second part of your question, calculus is about rate of change. You can have a function (equation) and know that the outputs change when you change the inputs. You can even plot that out on a graph and see the change with your eyes. However, that function alone doesn’t tell you anything about _how fast_ the outputs are changing as you move along the line.
Enter calculus. By taking the derivative of the function, you get a _new_ function that shows you the rate of change at any given point of the original function.
Newton created/discovered calculus to help explain how the planets in the solar system moved. Up until then, the functions we had to predict where planets would be at any given moment in time didn’t work – they were _kind of_ right, but still off. Calculus gave us the “new math” required to accurately predict how they would move. Specifically, he figured out that to understand position, you needed to first understand speed (the derivative function of position) and to understand speed you needed to first understand acceleration (the derivative function of speed).
Edit: Worth adding that Leibnitz also discovered calculus around the same time, though he is much less well known for it.
Newton (and Leibnitz) were the first people to realize that numbers could be manipulated in this way and describe the rules governing those manipulations and relationships, such as finding the area under a curve. They came up with the actual symbols we use and described the rules governing what those symbols mean and how they can be used. When we say things like “take the derivative of the function”, that is something that theoretically we could always have done, but Leibnitz and Newton were the first to recognize this truth and how it could be useful.
If you want to get into the philosophy of it, then it can be argued that all math just sort of already exists somewhere in the abstract sense, so no one ever really ‘invents’ or ‘creates’ math, but practically speaking if we don’t know about a certain mathematical principle or outlook then we can’t use it, so the distinction between ‘invention’ and ‘discovery’ is kind of academic.
The instantaneous rate of change of a function isn’t a thing that exists in nature and you can drop on your foot. It’s an abstract property about an abstract mathematical relationship.
It’s not even the same thing as a slope between two points on a line or curve. You need to apply the concept of limits, and you need to conceive of a function having a slope at a single point.
The ideas and techniques of differentiation and integration weren’t always known. Somebody had to invent the concepts for talking about them and the tools for computing them, and the notation for formalizing them.
If you write a certain equation and then solve it, and you could think of this as ‘discovering’ a mathematical ‘phenomena’. But you still have to first think of writing the equation in that specific way. If it’s a new way, then it’s an invention, right? It’s a new method for creating an equation, and people haven’t done it that way before, so you can say you’ve invented it.
To use a simpler example than Calculus, let’s go back to Archimedes, the ancient Greek. When he was alive, nobody in Greece could measure the circumference of circles easily, because they didn’t know about the number π that we use to calculate it. Archimedes said okay, put a hexagon around the circle. Now put a smaller hexagon inside the circle. We can calculate the circumferences of the these hexagons and the circumference of the circle must be somewhere between the two. Now, double the number of sides of both hexagons. The difference between their circumferences is now smaller, but the circle’s is still between them. So you keep doing that over and over again until you can calculate something you know is very close to the circumference of the circle.
You could say that by doing this, Archimedes discovered π. He figured out the ratio of the diameter of a circle to its circumference, a mathematical truth that had existed before him even though nobody knew about it. But, the *method* he used to find it was invented by him. Nobody had thought to do that before he did it, so it was his invention. (Probably.)
Calculus is a method of describing, calculating and predicting the results of a vast variety of physical and theoretical principles, along with all the associated proofs that that method is accurate.
Isaac “invented” that in thay he developed the methods and proofs and got them publicized. At least heavily from the derivative side Leibniz was the contemporary coming at it from the integral side.
This feels kind of like saying you couldn’t have invented a ruler because everything had a length already. Everything did, but the tools to measure and describe it reliably and with consistency across different observers still needed to be invented.
Math and physics didn’t exist and don’t exist. We watched how everything behaves and then came up with a language to describe it and predict it. That’s why we change/expand our math, our observations of the world around us don’t match what this language says. So we alter equations until they predict accurately again. IMO, we are inventing it, it doesn’t exist, the universe doesn’t care what our math says. Our math is just a different representation of what already is.
Calculus is a branch of mathematics that deals with how to calculate rates of change at a variety of time frames. Algebra and related mathematics had already been described and thoroughly studied by scholars for centuries before Newton. But Newton realized that these older branches of mathematics were insufficient to describe the phenomena he was studying. So he developed a new way of calculating rates of change at instantaneous intervals thanks to the core concept of calculus: limits. Now, he wasn’t the only scholar doing this. Other scholars, such as Gottfried Leibniz, were also doing similar work. But Newton’s contributions are the most well-known
And that’s what your hypothetical “new math” essentially means: sometimes researchers realize that the existing schools of mathematics are insufficient to mathematically describe what they’re observing, so you need to develop new methods. Entire branches of mathematics come from these practical considerations. Statistics, the branch of math where I personally did most of my studies, originated from insurance companies trying to quantify which clients were of greater or lesser risk of requiring payouts.
Math is a language that we can use to articulate ideas, and its specialty is logical arguments, and procedures, concerning numbers and functions.
Newton and Leibniz were the first to use math-language to describe how we could solve problems via calculus. You could argue that those concepts always existed, but its very, very possible that no earthly mind had ever had those thoughts before, so if you’re the first to ever express them, we usually credit you with “inventing that math”.
<opinion>It’s a stretch to say that every possible story, poem, computer program, article and mathematical structure already exists and they’re just floating in the ether, waiting to be noticed; ideas are ‘software’ that don’t meaningfully exist unless they have hardware to run on (or at the bare minimum, be stored in), and they take a very important step toward becoming ‘real’ the first time the meat-computer in somebody’s head runs them. You discover/invent an idea if you’re the first to ever think it. Ideas are not in the same class as e.g. mountains, which exist whether or not anyone’s seen them.</opinion>
It means he literally invented it.
He invented the mathematical processes for working with derivatives, limits, infinite series, integrals, and other things that define what calculus is. He exploited existing mathematics to formalize a new way of using mathematics. Geometry is not very good at working with the infinitely small, but it is arbitrarily easy for calculus and this allows you to do all sorts of cool things. Newton invented that.
Newton actually did not invent the notation. We use (at least largely) the notation preferred by Gottfried Wilhelm Leibniz, who is credited with inventing calculus independently of Newton at about the same time.
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