eli5: What do people mean when they say “Newton invented calculus”?

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.

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.

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.

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.

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.

In the same way that nobody really creates anything - a statue carved from rock is already inside the rock waiting to be carved, the sculptor creates the form from the stone. Similarly Newton may not have created the concepts he described, but he gave them form in the minds of people.

Like Michaelangelo crafting David from the rock, Newton crafted calculus from the universe.

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.)

That’s the same with most inventions though, the properties already existed.

An Internal Combustion Engine was always possible, oil and iron have been readily available. Yet someone needed to refine the materials, design the parts and put them together to make the engine. The “ingredients” were always there, the invention is putting them all together.

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.

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.

Was the game of Monopoly discovered or invented? It's a list of rules, that someone made up. The consequences of those rules follow from the application of logic.

Mathematics is the same way. Someone says "hey, these are the rules I would like to study" and then we follow those rules to their inevitable consequences. We want, desperately for the rules we come up with to reflect something about reality, and we do a pretty good job of that, but ultimately the harsh truth in that reality is reality, and math is a language. At some point, someone must make a decision about HOW to translate between these two realms, and there is no Right or Wrong way about it. How well a rule actually reflects reality is simply a matter of opinion from person to person. A famous joke in mathematics is "The Axiom of Choice is obviously true, the Well–ordering theorem is obviously false; and who can tell about Zorn's Lemma?" But all three of these things are equivalent to each other!

There's a saying in science circles: all models are wrong, some models are useful. We choose the rules that give us useful results. Newton and Leibniz were able to codify the rules that lead to really, really, useful results. Their rules didn't come from on high. Their rules did not grow on trees. They made them up. If that's not inventing, then I don't know what is.

Fields of Mathematics require certain assumptions to be made, called axioms. When we're modeling things in the real world, we might choose axioms that give us a system that is similar enough to our reality as we understand it to make it useful for modeling, but that need not be the case.

Newton (and separately, Leibniz) came up with some axioms that allow us to have the field of Calculus. Newton also developed a set of symbols and norms that, along with already-accepted mathematical symbols and norms, allowed Newton to convey these ideas and perform useful operations for modeling things within this field he developed. He then practiced and studied this new field in order to find useful conclusions upon which more complicated ideas can be modeled and explored, and wrote about these extensively in ways that other mathematicians could use them as a foundation.

One could say that fields of mathematics are more discovered, rather than invented, as the math sort of naturally flows from a starting point of axioms, and everything else is just finding a good way to communicate it and write it down. Regardless of whether you view it as an invention or discovery, though, Newton certainly paved the way for very much of what we still use in Calculus today.

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.

Math is entirely a language, of sorts. Like other languages, it is completely a human invention. Unlike spoken languages, it only concerns numbers and their interactions. It is a very useful tool for modeling real world phenomena, but that doesn't mean math itself is somehow "natural." It doesn't exit independent of human society.

Depends on your definition of invented and discovered. Was the wheel invented? Or did we simply discover that round things roll? Either way, Newton was the first one to officially call it "the wheel" and use it in an academic setting. Before then yeah, anyone could have cut a round piece of wood or stone to roll down a hill, but they didn't.

Replace the wheel with calculus, and that's it.

Language and math are both technologies. Like, light waves really do make a harmonic oscillation, but someone invented that description for them

You would be happy to know that at the time of Newton they didn't describe achievements like this as inventions but discoveries. And others even still refer to them as "contributions" rather than attributing everything to one guy, after all a lot of mathematicians worked to build calculus.

Newton was working on a theory of "fluxions." In order to describe this he had to invent an entirely new mathematical notation and their mathematical associations (called derivatives).

We say Newton invented it because, he was the first one to do it. Leibniz (who is often given co-credit for discovering it independently)beat Newton to publishing. Newton wrote on it 50 years prior but didn't publish.

Often times in history many people invent things first around the same time (like first flight, the lightbulb etc) but typically when they have made the thing all others derive from or a final version.

In the case of Newton.... the derivatives we use today are the same Newton described when he was 23 years old.

not really, calculus is their way of manipulating numbers and representing phenomena; you can come up with your own way to do the same and name it whatever you want. But until then, it'll still remain that Newton is one of the people who invented calculus

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>

Newton was (arguably) the first to denote a manner of doing math to track change in a moving system.

Imagine a ball falling from a roof. You set up a camera to snag a picture every tenth of a second.

By plotting the ball's distance from the top at every picture on a graph, where the x-axis is tenths of a second and the y-axis is meters, you get a plot with an upward curve until it hits the ground, and then plateaus at the top of that graph. This is the position function, and you can calculate the curve to give a function for the position at any instant in time along the x-axis.

Newton determined the way to derive instantaneous velocity from this. It's one thing to know average velocity; total distance traveled over time, but to break out the position function for a related one that tells you the exact velocity at a moment in time is wild. That's the first derivative, by the way. It's technically the slope of a tangential line on the curve at that point of the x-axis, and could be generally approximated before, but Newton made it possible to find an exact measure. And he was able to provide the long-form proof of it.

And it worked again to derive acceleration from velocity. And changes in acceleration, called jerk. (Centuries later, the calculation of jerk's derivative, "snap", would be used to calculate optimal curve radii for train tracks.)

The natural relationships were always there, but Newton invented a series of new notations and rules in a subset of mathematics that didn't previously exist that allowed these calculations to be possible.

All math is just a language we can use to represent natural relationships. You're right, they obviously didn't create the underlying laws, they just discovered how they worked and figured out a way to represent them on paper. The relationships described by calculus are inherent properties of the universe.

What is a 'phenomena' to you? Math is used to describe other stuff (when useful at least). And it has rules that let you transform things into equivalent other things. Sometimes those other things are clearer or answer something for you.

Do you know what Calculus is, or is used for? And what happens with/in it? It's not just 1 + 1 = 2. It's about getting the area under a curve from the formula, and precisely. Or finding how much change is happening at a specific point (where a curve changes direction for instance). Or what happens between 0 and infinity along a curve. Or at least that's what I can recall from my long ago Calculus classes.

They're useful if you have questions they answer, and they're not obvious unless you already know them (to most people at least).

That's how a new math would be useful. It means we'd have a new way of answering something that stumps us today using symbols and numbers, maybe with new operations like sin/cos/tan did for triangles. It's the hope that some new way of looking at a problem will be much faster than existing methods, or answer new unknowns.

And as far as "Newton invented calculus" it's that he's credited with publishing the ideas first. Maybe he did, maybe he didn't... but it's not a big deal to me who specifically did something in a well known/public way. He worked out the rules and results of it, and shared it with other people. Who looked at it, and tried to see if it made sense to them or answered anything useful.

Maybe it’s more that he discovered it?

Does one invent a martial art, or discover it?

Did someone invent fishing, or discover it?

Did someone invent metalworking, or discover it?

Did someone invent the wheel? Or discover how to make a wheel?

Did someone invent the table, or discover how to make a table?

Even take a motor, or a lightbulb. Did someone invent those, or did they just discover a process by which to make them? The thing is, the process to create one would have worked before whoever first did it carried out or described that process. A lightbulb, or a table, or a blender are all amongst possible configurations of atoms. Was the inventor not just the first person to perform the process and represent it in "language" as you say?

In math and science, some people use the word “invented” but scientists usually say “discovered” Newton didn’t invent the fact increasing force increases acceleration, he discovered the correlation. Same thing for calculus or any type of math; they didn’t invent the phenomena, but they used they tested the phenomena and recorded results until they came up with the terms, variables, correlations, equations, etc.

If you tell someone directions to go to the store, are you actually going to the store? No, you're just explaining how to get to the store.

Math is the same. Math is how we explain how the universe works. It is not how the universe works, just an explanation. When someone invents something in math, they are inventing a new way of explaining how the universe works.

Now imagine you're used to giving directions in lefts and rights. In fact, you're very good at it, you even know all the angles down to the degree. This always worked when someone needed to get to the store. But suddenly you realize, helicopters exist. You discover this when a helicopter pilot comes up to you and asks you for directions to the store. The current way you tell directions no longer works. The helicopter can go straight there, but there's a mountain range in between, so altitude is very important. What do you do? You invent a new way of giving directions that makes sense in the context of helicopters.

That's new math, it's a new method that works within a context where the current math does not. It shares a lot of the same principles and, at its core, a lot will be the same. But you need new methods to explain problems that current math does not solve.

Can you see a continuum?

Math is discovered more than invented. It was there all along, we just had to name it's variables and so the calculations.

He discovered it while working on a problem during a plague.

Some people might say that they invented it... like some people say someone 'invented' gravity -- no they just figured out how things can be better described, or how numbers and mathematics can be used in new ways.

It's probably a bit off semantically. I think you're getting at the fact that math exists, and it can't really be created. It's probably better to say that newton is the one we give credit to for defining calculus and laying out the rules and symbols for others to use.

Calculus is a method for doing something.

The idea that "math exists and people just discover it, rather than invent it" is not usually a useful way to think, simply because we currently don't know all the things that are possible in this universe.

The universe would have allowed lightbulbs to work during caveman times, but someone had to actually make one.

Perhaps it is possible to generate gravity, or create black holes, or time travel, or whatever. If anyone ever invents those methods, you could just as easily say that they simply discovered something that the universe was capable of doing the entire time. We use the word "invent" to describe the moment in time when someone passes the threshold from something being possible but unknown to them actually making it a reality.

Think about tools for a second. With the right tool you can reduce the time taken to do something or multiply your effort or both. For example, you can use a small blade to cut grass little by little or use a specialized tool like a scythe to do it much faster with less effort. If you want to join 2 things in such a way that you can unfasten them easily, you use a specialized tool like nuts and bolts.

Newton was trying to solve a problem. He was trying to figure out why an apple would fall from a tree but something much more massive like the moon wouldn't fall from the sky. (It's the 1600s people didn't understand much and were asking questions) What Newton did is that he invented mathematical tools i.e. Calculus to help him solve this problem.

Calculus ,like any other tool, when used in a certain way (rules like when and how) would let you solve complex problems by breaking them down into small chunks. That's what calculus is - a mathematical tool.

Newton built the framework for understanding the “Fundamental Theorem of Calculus”.

We had these two seemingly unrelated concepts.

• The slope of a line on a function, at a point. It was called “the derivative” of a function, and we knew about it, and studied it before Newton.

• The area underneath the curve of a function. It was called “the integral”, and we knew how to approximate it, and calculate it for special examples.

Newton proved that the two are inseparably linked. No one had any idea that they were. This was a novel discovery.

That most people are under the following false impression. “That the derivative and the integral were discovered by Newton, and he designed them, such that they are the same thing, but opposites.”

When in reality, “The derivative and the integral, were two completely separate mathematical objects. Newtown was the first person to figure out, and prove, that they are opposites of each other. We had no reason to believe, before Newton, that those two objects would be so incredibly similar.”

I'll suggest an analogy:

Multiplication has always worked: if you arrange items into rows and columns and make a rectangle, the length and height of that rectangle always gives you the number items without needing to count them.

But someone had to be the first person to write down times tables for the purpose of automating that multiplication. Times tables let you skip the visualization and counting steps.

math is totally made up and has no intrinsic corporeal truth. it's a fuzzy abstract way for us to model things using syntax and semantics, it's bounded by the limitations and architecture of our brains just like everything else we've defined

Whether math is “invented” or “discovered” is largely a matter of semantics and a topic of the philosophy of mathematics.

It is easy to take mathematics as some sort of thing that exists outside of people and embedded in the cosmos. However if you think about some of the symbols, they don’t really correspond to anything universally meaningful that isn’t defined by humans.

In the statement “2+2” what does “+” mean? Does it just mean that you’re taking 2 things and putting them physically closer to 2 other things? Does it mean you’re just manipulating symbols on a sheet of paper? It’s kind of just an abstract idea that corresponds to a vague and poorly defined pattern.

That brings us to the issue. When we really want to be rigorous, we need to have precise definitions. There are times where it makes sense to define “+” such that 6+7=13, but there are also (literally) times where it makes sense to define “+” such that 6+7=1 (like a clock).

You could probably get more precise and state Newton “invented” some of calculus (some of the symbols and definitions are uniquely his creation) and some of calculus was “discovered” by Newton (that Newton figured out some interesting facts that follow as consequence of reasonable base rules known as “axioms”).

Same as someone inventing English. The rock and trees exist whether you name them or not, but creating a structure in which they can be meaningfully discussed and understood is kind of the natural world equivalent of inventing something.

Like you said, it's a language that describes phenomena, meaning someone had to invent it.

Somewhere along the line somebody had to invent Arabic numerals, the number zero, negative numbers, and fractions, and so on.

If you go look at the history of math before the invention of modern algebraic notation, people would verbally describe how to solve things with words and it's really painful. Imagine how you would describe the Pythagoran theorem without equations.

Math is effectively just a language used to record, quantify, understand and communicate about natural phenomena. These events happen regardless of whether anyone is there to witness or record it. Newton just helped to plot out the dialect we use to understand these things. Saying he invented calculus is a bit of an over-simplification, I think.

Rather than invented think of it more like a discovery. Newton was the first one to discover the mathematical relationships that can be described by the formulas and terms used in Calculus. Before Newton had thought about this nobody knew the things he discovered or the formulas he created. But the universe still existed and numbers fundamentally still worked the same way before he was around he was just the first to work this out.

Mathematics is language to describe relationships. Humans observed the universe, recognized patterns, and invented language to describe those.

Science and mathematics have a cool relationship that is mutually beneficial to both. Mathematics is the primary language of science, and is used to analyze data, represent scientific phenomena, and understand scientific concepts.  Newton's basic system defined the framework and language for calculating and comparing the motion of objects (back when nobody knew anything about it.)

He didn’t invent calculus, he invented the formulas we use to describe calculus

I like to think of math as representing or simulating quantity. When I write an equation like ( v = 1 \times d ), the value of 1 can represent anything I want – an apple, the moon, or even the entire galaxy – and ( d ) is whatever value I choose to simulate, whether it's time, distance, circumference, weight, or luminosity.

The variable ( v ) can then be written to simulate an aspect, quality, or physical attribute of whatever entity I've chosen.

It's all about imagination. I pretend that the number 1 represents an entire moon, and ( v ) could represent the moon's speed, weight, or brightness.

Numbers are like sounds, symbols are like words, and calculus is no different from a language.

nobody is really answering your question, which i take to be philosophical in nature. the question is, as it appears to me: is math invented or discovered? there is no correct answer (that we know and agree on). both sides have solid arguments for and against. i think i side with you on math being discovered (you reference the existing ‘phenomena’). another way of asking it is: if math is possible, does it exist? Like, calculus was possible before newton ‘invented’ it- so did it exist before, in thought of some form?

Colours and "artistic things" exist in the real world, but in order to manipulate those colors as humans, we needed to develop paints, pigments, brushes, canvas, and 'theories' about why things look good.

It's not enough that "the thing exists" - in order to actually USE and understand it we need a framework and tools with which to analyze it and manipulate it.

Reality follows the "rules of reality" i.e. the laws of physics, which would exist with or without human existence.
In order to understand and manipulate reality, we needed to create a language / framework / tools to describe it. That's what math is. A language and a framework that represents, at each period, our best understanding of natural laws that allow us to explain natural phenomena, predict them, and manipulate them.

The invention of complex numbers (imaginary numbers) was a triumph of logic, because they seem to have no relation to reality at all but still show up in all sorts of natural phenomena.

I think "discovered" is more apt here, as evidenced by the fact that someone else (Leibniz) did it at almost the exact same time, several countries away.

Calculus was always true, we just didn't always know about it, like discovering a new land mass, or a new species, or that female Custodes exist.

Oh, math is absolutely invented. It's invented all the time.

I mean, just look at knot theory. Stuff like that is clearly made up.

You might say, "oh but it reflects real life so well!" Yeah, some of it. The rest of the field is just pure made-up abstract stuff.

And you know what? There's nothing wrong with that. That's what math is — playing around with abstractions.

We say that Newton invented calculus and discovered gravity. You can argue that the semantic difference here is arbitrary because in both cases what he did was figure out something about the world — one a property of physical objects and the other a property of numbers.

Newton wanted to make math do things that it couldn't do with what existed at the time. Algebra has serious limitations, especially about dividing by zero, that caused him to have a frustrating time calculating specific things like instantaneous velocity or acceleration without having velocity at 2 different points.

Newton developed ways to bend but not break the rules of algebra to be able to practically but not technically divide by zero. He came up with the idea of limits and rigorously proved that his bending of the rules did not break them. Using these limits, he developed a tool that divides by "almost but not quite exactly" zero. You can look up the Epsilon-Delta definition of a limit if you want to understand this more technically.

This allowed him to develop derivation and eventually integration. Derivation and integration allow us to create a function that represents (with conditions met) exactly all the values of an "instant" velocity or acceleration at a single point.

These operations can be performed, with some limitations/conditions, under conditions where traditional algebra either can't be used or where it is laborious and slow. This invention expanded what he could represent with math and paved the way for many other advancements.

Image 10 + 10 +10 and someone invents 3 x 10. Seems easy for us, but that was a leap.

Imagine a curve described by a formula, like a sin wave or part of a circle. Finding the slope of that curve at a specific spot can be approximated with a straightedge, but it takes calculus to find it exactly.

Math is not about the symbols you use to note it down, math is about logical concepts. And those logical concepts aren't just there, carved into fabric of reality or something, someone has to come up with them to begin with. Newton came up with the concept of calculus and figured out how to use it in practice. Leibniz also invented calculus at the same time, independently. And he is actually the one whose notations we use. Newton called it "Method of Fluxions" and for a while the term was actually used in English schools, but obviously it didn't stick.

Maths is just a way to describe things and relationships between them, I think. That makes it another language so the terms used are probably as arbitrary as words in any language but the objects/phenomena described remain real. Languages change all the time to be able to describe new discoveries and new words (neologisms) are always being added when they help to simplify or clarify descriptions.

Edit: I'm linguistic, not mathematical but the above is my very basic understanding.

Math is invented in the sense that language is invented. Trees already existed, but humans had to make words to name and describe them.

Interestingly, the raw materials of calculus have been around almost as long as math itself. The Ancient Greek formula for the area of a circle (still used today) was derived through a primitive application of what we would know recognize as integral calculus. However, since the process involves either an infinite process or infinitesimally small values, mathematicians were unsure of the validity of the method and therefore tried to avoid it when they could. Newton and Leibniz, however, decided to set aside that question of validity and just went ahead and developed a system of integral calculus that depended on those infinitesimals. They couldn’t prove it true in the modern mathematical sense, but integral calculus had something better: it was useful. It could produce verifiably correct answers much more efficiently than previous methods. It wasn’t until the 19th century when infinitesimals were replaced by limits that calculus became “proven true” in the modern sense.

Math is invented because we define objects and we set which rules those objects should obey based on our comprehension of the world.

Math is a way of reasoning and understanding relationships between numbers. It is entirely a human invention, like language, music, or art. It would not exist without us.

But as with language, music or art, it helps us understand the world and each other.

The math that non-mathematicians learn and use everyday has practical applications because it happens to correlate with reality. But that's just a coincidence.

Pure math does not need to have any relationship with stuff we observe in the physical world. It is intrinsically and universally true and because of that it can help us understand things that we can't observe or experience directly.

Coming back to calculus, Newton invented it as a way of understanding and predicting the motion of planets, which for him were lights in the sky that showed up in a slightly different place every night. Leibnitz was looking at the properties of curves and tangents. It later turned out that they were describing the same thing, and that these relationships are useful for understanding more than just the motions of planets.

Yes there's a discussion about whether it's invented or discovered, but what is math?

My physics prof liked to say, "The fact that the structure of math matches the structure of nature is a miracle of the first order." I don't know enough to do a true deep dive on the philosophical questions there, but it's pretty cool that there's so much compatibility.

But math isn't just equations for describing physical reality. At its base, math is a set of axioms and the tools and rules all the things that follow. So we make a set of statements to define the framework that we are working in, define the rules that we will follow, and see what comes out. Axioms are kind of just a set of useful assertions. They're not necessarily derived from "first principles" back to some universal truth, but they are a foundation that we have found to have useful properties when we use them to as the basis of our math.

When mathematicians "do" math, they're poking at existing theory to find gaps in the logic and trying to figure out how to fill those gaps, so that any mathematical statement/operation can be explicitly tied back through some chain of proofs to the basic axioms. Those proofs show that a mathematical statement/operation is legitimate and consistent with our existing framework. Or they're working on ways to expand the math we have - to make new math like Newton did with calculus. Even such basic things as addition (which probably feel self evident to you) have proofs to show that they are legitimate operations that fit in our framework. And we want all of our math to be tied back to the source in that same way.

So when we say Newton invented calculus, we mean he came up with a new set of rules and a new set of operations, and showed how they fit into the existing framework of axioms etc.

So some of the basic ideas of calculus were known in Egypt and India many centuries before Isaac Newton.

Kerala school of astronomy and mathematics - Wikipedia

This basically comes down to semantics. Newton discovered Calculus in the same way that Columbus "discovered" North America. Sure, North America and Calculus could have been discovered before Columbus/Newton, but humanity as a whole did not know of it's existence before them. North America was actually discovered multiple times before Columbus: by the Vikings and the people who crossed the Bering Straight before them. It's possible someone discovered Calculus before Newton, but Newton gets the attribution (fairly or not) because he's the one who first widely disseminated the information and techniques.

The written symbols that make up math, THAT'S what humans have invented.  It's the language we've made to represent the real world as best we can, for science! 

Newton and Liebniz are both said to have invented calculus because each had their personally-invented notation for rates-of-change-math, and each were used to solve very big problems in mathematics.  They were each HUGELY influential with modern mathematical notation

I'm late to the party but I think for more insight into how mathematics can be "invented" it's worth looking into the history of other mathematical inventions, things like the number 0, or fractions, or integers vs real numbers. Some of that stuff is WILD.

Did you know there was a time where people argued whether or not 0 was real? Or that negative numbers could exist? We take both for granted now, but these were revolutionary discoveries of their day. Heck, the fact that we even use equations. Back in the day all proofs were made with geometry!