How does one “invent new maths”? Like Isaac Newton inventing Calculus, or John Napier logs. How does one answer a mathematical question that’s never been answered?
In: Mathematics
Here’s one way you might discover logarithms on your own. (From now on, log = log base 10.)
Suppose you’re back in the olden days before calculators and computers and your job sometimes involves multiplying numbers with a lot of digits. One day your boss asks you to find the area of a rectangle with dimensions 15432.54 by 827361.37. The area is 15432.54 * 827361.37. Calculators haven’t been invented yet, and doing this with pencil and paper would take ages. Is there a way to simplify this?
You could make a multiplication table, like the one you memorized in grade school. Then you could just look up the product in the table! Except that such a multiplication table would be HUGE if you wanted one which would be actually useful. Is there a way to shrink the amount of information required to make this reasonable?
Then you remember a neat fact about exponents: b^(x+y) = b^x b^y. In a way, exponents turn addition (an easy operation) into multiplication (a very hard operation). What if we could do the opposite? What if we could turn multiplication into addition?
Enter logarithms, which are just reverse exponents. These have the neat property that log(xy) = log(x) + log(y). i.e. they turn multiplication into addition.
So let’s use this to find 15432.54 * 827361.37.
First, we will find log(15432.54 * 827361.37). Then at the end we can take 10 to the power of whatever the answer is to get the final result.
In scientific notation, this becomes log(1.543254 * 10^(4) * 8.2736137 * 10^(5)).
Using what we found earlier, we know that this is equal to log(1.543254) + log(10^5) + log(8.2736137) + log(10^5).
This is log base 10, so we have now log(1.543254) + 4 + log(8.2736137) + 5.
We can look up those logarithms in a logarithm look-up table, which is what people used back in the days before computers. By using logs and scientific notation, we only need to know the values of logs for numbers between 1 and 10. Much more feasible than a giant multiplication table.
If you calculate this, you would find that log(15432.54 * 827361.37) = 10.10613. Therefore, 15432.54 * 827361.37 = 10^(10.10613265) = (10^(0.10613265)) * 10^(10) which you can solve using a different lookup table.
Another angle most people don’t consider is that the very basis of mathematics is that if you have A) a new idea, and B) a body of already-accepted ideas, then what you need to do is 1) assume your new idea is correct and then 2) find where it either contradicts itself or contradicts what you already know to be true. That is how you might say somebody “invents” new math (sorry, not British). If your new idea violates what you already know, then that is generally a good sign that it *might* be incorrect. But if you can invent a theory of Calculus, and 1 + 1 still equals to 2, then you might be on to something.
For example, something most people don’t know is that the reason that division by zero is considered an invalid mathematical operation is because, you can get the result that any number equals any other number. The easiest place to see this is the graph of 1/x, where when x=0, y seems to be = ∞ AND -∞. Thus ∞=-∞. Clearly, this is infinitely incorrect. In fact, you can algebraically manipulate a division by zero to say anything you want, like 1=2. 1=2 doesn’t line up with what we already know to be true, so clearly there is something wrong with our new idea (lest there be something wrong with the old ideas– though the cumulative effect of this whole process is that we tend to suss out the right ones)
Was mathematics invented or discovered?
Just rip off some old math and put your name on it. Right Pythogoras?
First, if you’ve ever solved a mathematical problem that was not one that you were specifically *taught* how to solve, then you experienced a very small taste of what mathematical research is like. Because even if that particular question has been answered before by other people, it has never been answered before *by you* so for you it is a discovery which is a small-scale version of discovering “new” math.
You might think that is nothing like creating calculus, but no mathematical discovery comes out of thin air. Studying math gives you access to a variety of tools and ideas, and when you hit a problem that your tools/ideas are unable to solve, you modify them slightly to create new ones. Big mathematical discoveries are just particularly large or creative modifications. In the paragraph above, whatever you did to solve the “new” problem can also be used to solve other similar problems. If you were to try to “formalize” whatever trick or idea you used, that’s where new math comes from.
Since you mentioned calculus, I’ll try to use it as an example (though obviously this will have to go a bit beyond age 5). Since the ancient Greek Archimedes, we’ve been able to think of the area A inside a circle as follows: It has to be larger than the area L of any polygon inside the circle and smaller than the area U of any polygon that encloses the circle. That is, L < A < U, so L is a lower bound and U is an upper bound. By calculating areas of polygons with more and more sides, you can get L and U to be closer together, which means you can effectively compute A to any level of precision you want. This leads to a proof for why the standard formula for the area of circle is correct.
This idea makes sense for any “curvy” shape, not just circles. One thing that Newton did was *formalize* these ideas and use the formalization to help him calculate the area A for an astounding variety of shapes. In particular, he came up with the idea that A is the “limit” of those lower estimates L as those estimates become more accurate. This means that computing areas of curvy shapes boils down to being able to compute these limits, which led him to develop lots of tricks for computing such limits. In retrospect, it’s such a natural outgrowth of ancient Greek math that imho the surprising thing should be that it took so *long* for calculus to be invented.
Largely it is about identifying patterns, and connecting what you see to what we already know.
Sometimes what you see doesn’t match up with what you know, so you analyze it. You study it. Sometimes this allows you to extend what we know to include the new information and other times it allows you to create something new that we know.
When we get into the real abstract or theoretical stuff, it may be things we can’t observe. In this case, maybe you ask “what if this changed?” or some similar question, and then you explore how that will connect to what we know.
I know that wasn’t really an ELI5 answer, and was very vague, but it is a difficult concept to explain.
Also, Newton is not the sole inventor of Calculus credit is also due to Gottfried Leibniz who discovered it independently from Newton at around the same time.
We often thing that math is a bunch of rules and we can only do the math that the rules dictate. But it develops the other way around. There is some kind of math that we want to be able to do, and we just need to create rules that allow us to do that math.
Newton wanted to do a specific kind of math related to physics where he needed to divide things into infinitely small chunks. You can’t just do that without well thought out rules without it going crazy and incomprehensible. So the rules to do that were not fully developed around that time (though, there was already a lot of work done on Calculus before Newton), and so Newton just found out the rules needed to do what he wanted to do.
Lets say that maths is just a big lego set.
Now a friend of yours wants to build a robo-mech from the parts, but it turns out there
there are no suitable parts for it.
So your friend modifies the existing parts and uses them to build the robo-mech.
Discover isn’t really the word. Maybe “invent.” They basically came up with extensions to the language of math so that they could describe new problems.
Usually just looking at a problem from a different angle, or taking a path nobody explored before.
“I know you can only take the square root of a positive number but what would happen if we create a hypothetical number, let’s call it ‘i’, and claim that squaring it equals -1”.
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