That is the problem, non-mathematicians usually doesn't fully understand the math they use in their papers and thus the math becomes opaque. Papers written by real mathematicians are usually much easier to read, although of course the math in them is much much harder.
I remember going to lunch with one of my math professors in college. He was working on his PhD and was about to publish his thesis. As we sat down to eat he was very excited as he pulled out a sheet of paper from his pocket. It had been folded 3 or 4 times. You could tell he had been carrying this thing around, folding and unfolding it, for a long time because the folds showed wear.
This piece of paper was full of formulas, both sides, there was not a single blank area on the entire sheet.
He unfolded it and proceeded to give me a quick talk about what he was working on. He was very excited about it and I was happy for him. And yet that entire piece of paper looked like a language from another galaxy to me. I was on my third Calculus course. I had no clue what he was talking about.
Digressing a bit:
To this day I remember this when helping my kids with math, science and coding. As a matter of fact, I am currently working on an explanation of exponentiation and logarithms. In both cases everything looks great if things are even multiples of the base. The minute you do something like 2*2.1 or log_base_3(35.53) you hit the "and then a miracle occurs" problem, where you have to explain a thing by using the thing ("A white horse is a horse that is white").
I've spent the last couple of days working on cleaning-up an explanation of these things that makes sense without using a miracle to get to the answer. One of the problems is that there are natural explanations for things like square and cube (area and volume), but, what do powers of 2.1 and 3.25 mean? It is interesting how things completely break down. I don't think I have found a single mathematics text that bridges this gap.
If anyone has a sensible explanation of this I'd love to hear it!
When we start teaching math to students, we start with counting blocks: "You have 2 piles of blocks, one pile of 3 and another pile of 2. If you put them together, you get a pile of 5 blocks!"
That stops working as well when you deal with fractions. You can get away with 2.5 blocks, but 2.5 blocks is really 3 blocks, but one is a little smaller than the others. And at some point you can't use blocks to represent 2.3456 blocks. So you need different kinds of "natural" problems to represent those numbers.
But, as you point out. There are some things that aren't really representable as "natural problems". For a long time the idea of 0 wasn't natural. (People were actually killed for talking about the idea of 0) I mean, what does it mean to have 0 chickens? You either have some chickens, and you say "I have N chickens", or you don't have any chickens and you say nothing. Why would you need a number to represent nothing?
Maybe n^2.1 doesn't have a natural explanation. At least, not one you can hold in your hand. Can you imagine a shape with 2.1 dimensions to relate it to geometry? Probably not. But you can use geometry to prove that n^(a+b) = n^a * n^b and then you can apply those rules to "unnatural values" with an understanding of what is happening. The natural explanation of n^2 can be applied to the unnatural idea of n^2.1
Everything in math can't be understood with geometry or "natural examples", lots of math (most of math?) describes things that are not representable within the constraints of our physical world. That's what makes it so powerful!
Also, not everything in math can just be calculated (see: irrational numbers)
Yes! A long time ago I read a wonderful little book on just this bit of history:
https://www.amazon.com/Zero-Biography-Dangerous-Charles-Seif...
The square root of a number takes us from an area to the length of the side of the square corresponding to that area. The cube root is the same for a cube. What is the 10th root of x?
It's a number that, when multiplied by itself ten times equals x.
OK. How do you compute this number?
The best I can offer at this point is, for simplicity, a brute force search or, for faster results, a bisection search algorithm.
In other words, the "and then a miracle occurs" moment is right there. The fact that I can key these numbers into a calculator and get the answer isn't the kind of explanation I want to use for my kid. I don't want to say "once you get here you pick-up your calculator", because the legitimate question then might be "If it's magic, why don't I just pick it up at the start of the problem?"
To be clear, I don't mean "miracle" as anything other than "this shit is hard-to-impossible to explain or calculate by hand". That said, you could probably run through a quick bisection search by hand and likely converge on a low error answer in 2 to 5 cycles.
The meaning of of the e root of b explained with exponentiation and the exponentiation is explained with the root.
Moreover, mathematician seems driven by a frugal principle. They try to condense their though in the smallest number of symbols. To me it's like writing a Perl program with the shortest amount of text. Of course, the result is right, but it's super hard to understand.
Non-mathematicians discover something and then want the largest number of people to understand that thing as well, and they want to be the ones explaining it and see the sparkle in the eyes of the ones receiving the information and "getting it" for the first time.
Mathematicians want their peers to understand first and foremost in the quickest way possible, they then rely on 3rd parties to explain it to people who they consider "normies", they slap their name on the theorem or the demonstration and further use what they consider "subordinates" to explain it to the few "normies" who want to make an effort to understand it.
