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