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