[GDC7]

To me it's the language.

You have to learn a whole new alphabet and signs.

This is done for the sake of quick communication between mathematicians, but it's necessary to make a study and see the pros and cons.

While it's true that it makes communication faster and straightforward it keeps so many people outside of the field.

Maybe the field would benefit to go more towards philosophy and logic, explaining it with words.

[_Microft]

The idea that unfamiliar symbols and alphabets are a huge problem for the accessibility of math is common. As physicist I do not agree. Math is hard. It's damned difficult. Symbols and alphabets are the least of your concerns when dealing with a math paper. I know a lot of these symbols by name, I sometimes understand the notation or could familiarize myself with it but the math itself? Nope, no chance, usually. If one cannot deal with the symbols, there is no chance in hell one could deal with the ideas.

[amcoastal]

I'll disagree. I read many papers with mathematics in them, and I get a lot of the concepts but the symbology used doesn't make sense to me so its hard for me to understand what is exactly going on. The sentence after the equation that explains each symbol is necessary for me, and many others as well. Not everyone has taken 8 math classes to know each kroniger delta by heart.

[Jensson]

> I read many papers with mathematics in them

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.

[robomartin]

Well, musical notation looks like gibberish to someone who did not learn it. That said, I do agree with you 100% on scientific papers. Without an explanation of the formulas to cater to a wider audience a lot of papers fall into the "and then a miracle occurs" fallacy. Not because that's what they actually do. Not at all. I say this because to a large set of readers the impenetrable math has to be taken as a divine act that moves you from step n to n+1.

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!

[hdctambien]

Following on from my other reply...

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)

[robomartin]

> For a long time the idea of 0 wasn't natural.

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...

[ectopod]

You can start explaining fractional powers with roots, e.g. x^0.5

[robomartin]

Right, the problem is that you quickly run into the "miracle occurs" territory.

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.

[hdctambien]

I think the magic/miracle of math is that you can go from "real world" into "math world" then back into "real world". If a rule is true for c and n and n+1, and you can physically represent the idea when n=2 and n=3, then you can apply that representation theoretically to n>3 to understand ideas that are not easily understandable.

The 10th root of x takes you from a measurement of an 10 dimensional object to the measurement of a 9 dimensional object. That's crazy, right? Without needing to "understand" what an 10 dimensional object is, you know something about it because you understand what roots mean with lower values...

Of course, that doesn't help you actually calculate the 10th root of x. Is there a better way than basically guess, check, and refine? The calculator is just really fast at doing that (and only needs to calculate a relatively small number of significant digits). Sometimes that's just how math is. The only magic there is that computers are very fast at computation compared to people.

[wiz21c]

except that mathematicians like to use shortcuts notation everywhere, shortcuts that only them understand... For example P(A|B,C) ?= P(A|B;C).

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.

[GDC7]

Yeah, it's like there is something in their brain which differs from non-mathematicians.

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.

[I_AM_A_SMURF]

Most mathematics paper will define the symbols they use beyond the basics (and sometimes even the basics). If you are thinking about extremely common symbols then... it's like complaining that somebody not trained in music cannot read a music sheet.

[andrewjl]

One can make the same argument for doing arithmetic using Roman numerals.

[GDC7]

Math will only emerge as a field when people will stop treating it as something that one only has to do at the frontier.

In other competitive fields such as banking, basketball and football you have the higher ups caring about the pyramid below them, if only as a place to recruit new talents.

Among the math higher ups, only Jim Simons cares about the "math pyramid" so to speak.

One has to be pragmatic, the goal of getting the population interested in math is GDP and median quality of life.

I know those things are very mundane for mathematicians who are absorbed in their world trying to be the ones cracking the Rienmann hypothesis, but even as that individual you have slightly better odds at making it if your surroundings look like Zurich or Cambridge vs. Baltimore or Mobile.

Matter of fact you have better odds if your country can extend the areas looking like Zurich and Cambridge and reduce the areas looking like Baltimore or Mobile.

[joeberon]

What are you talking about? Mathematics has well and truly "emerged as a field" lmao

[GDC7]

I think you'd be surprised if you gathered data on amount of cursing which goes on in colleges when students have to face math vs. English or history

Also cortisol levels spike before a math exam compared to again English or history.

Notoriously math is the most hated subject/field. There is no point trying to hide it.

Unless one is a masochist, then it's desirable to be popular, and although the field cannot and should not be watered down for the sake of ease of access , I think there is lots of room before reaching that point, but nobody is interested in making such effort

[joeberon]

What's this got to do with "emerging as a field"? Yes Maths is hard, yes generally people don't like to do it, but that doesn't mean it hasn't "emerged as a field".

[GDC7]

It's self imposed because mathematicians love to use their own notation.

Math is at its core philosophy and logic, they are both hard too, but they don't get the same level of hatred that math gets because unlike mathematicians philosophers and rational thinkers take the time to explain what goes through their mind instead of condensing very complex thoughts in 20 characters strings.

Turns out there are some social rules which not even math can break, if your attitude is :

"I don't give a damn about people understanding what I am trying to say, they are all dumb and uninteresting because they don't even put the time in to learn my special notation"

You won't be very popular. And your field will be kinda hated, which is what is happening.

[robomartin]

> Maybe the field would benefit to go more towards philosophy and logic, explaining it with words.

Interesting perspective.

I studied Philosophy and Logic in university:

https://en.wikipedia.org/wiki/List_of_logic_symbols

Much of it was familiar to me because, earlier, I took a class my Physics professor insisted we should take, as he put it, "if you want to get out of the dark ages": Programming in APL.

AS it turns out many of the symbols used in APL come from Logic.

To this day I find it disturbing that Python uses "^" for bitwise XOR, because both in Logic and APL, this is the symbol for AND. Anyone who studied Logic instantly recognizes the APL logic operators.

I say "interesting perspective" because the reality of what you are asking is precisely opposite what you think the outcome would be.

[GDC7]

> https://en.wikipedia.org/wiki/List_of_logic_symbols

Among all those symbols only ">" and "<" are somewhat intuitive, all the others you have to learn what they mean.

Even "=" is derivative of "<" and ">" because by reasoning you can understand that you get to it by rotating the 2 lines about 30 degrees after realizing that you are dealing with 2 numbers which are in fact the same and not one being bigger than the other

[robomartin]

Yes, of course. BTW, I don't think my comment covered the fact that I agree with you 100% on the impenetrability of mathematical notation.

That said, all notations --including the written alphabets of many spoken languages-- are impenetrable until you learn them. As a personal example, for me, learning French and German was a million times easier than learning Chinese and Japanese. In the first two cases I could read and write the languages right away. In the case of the latter two the notation imposed both a significant time drain and a cognitive load that got in the way of learning. I did a lot better with Japanese than Chinese. And BTW, I would not dare say I know these two languages. I can rattle off a bunch of phrases in Japanese and understand them if spoken slowly. My brain has yet to synchronize to Chinese.

My point is that specialized notations have been a part of the human experience forever. From cuneiform to modern written languages. Our brains are pretty good at learning notation. I would not fault mathematics for anything other than, perhaps, practitioners assuming everyone reading a math-heavy text understands the notation as they do.

Personal example: One of my kids is going though an MIT CS class on edX. He got scared when he was presented a formula with a huge sigma "Σ" sign in front of it and numbers below and above it.

It took less than a minute to explain that this just means a sequence of sums, maybe ten seconds. I just wrote down something like: "(a0 * b0) + (a1 * b1) + ... + (an * bn)" and said: "This is what it means. Summation". Done.

The point is, notation doesn't have to be hard.

[GDC7]

> It took less than a minute to explain that this just means a sequence of sums, maybe ten seconds. I just wrote down something like: "(a0 * b0) + (a1 * b1) + ... + (an * bn)" and said: "This is what it means. Summation". Done.

I think the real world feedback is quite different, given that math can be explained textually with words , why should we not do it?

The burden of the proof is always on the institution trying to do something. In this case the US government trying to make the US population better at math.

The population is quite okay with the present day situation, it's the government's job to make stuff happen and change things around to obtain the desired result, that is an improvement compared to what we have today.

Math proficiency is in line with new notation foreign languages proficiency from your examples (Chinese, Japanese, Austrian and German to a certain extent), that's because as you said both math and those languages have a different notation.

Given that (unlike foreing languages) math can be explained WITHOUT having to teach a new notation, then why don't we do it?

New notations are necessary for Chinese, but not for math, so why don't we remove this barrier to entry?

New notation is part of the human civilization but it has to be acquired early on to become like a second skin, which is what Latin letters are for us.

One has to be realistic . Mathematical notation will always take the backseat vis-a-vis literal notation. Kids just don't learn (and aren't taught) mathematical notation the same way they learn (and are taught) latin letters.

Instead of fighting against windmills we should take that as a given and try to influence what can be influenced.

As I said the institution trying to make a change in end results, must consider changes in the process...otherwise nothing happens.

[robomartin]

I think the notation is very much needed because it quickly becomes a tool for thought and communication. This is very much the case for every spoken language and other areas, such as music. Your point, which is quite correct, is that the math might not be explained well enough and internalized to the extent where the notation becomes a language for students beyond the simplest levels of mathematics.

A kid can learn the notation for whole, 1/4, 1/8, etc. musical notes and their positions on the staff very easily. An immediate relationship is created to the key on the piano or the fret on the guitar. I have been to math classes where the professor simply vomits formulas on the blackboard for one hour and you are left to figure out what they hell happened. That is a problem. Not the notation. The way math is taught.

[GDC7]

> A kid can learn the notation for whole, 1/4, 1/8, etc. musical notes and their positions on the staff very easily. An immediate relationship is created to the key on the piano or the fret on the guitar

I don't know about that. How many people can read music?

But also music, much like Chinese and Japanese is at a comparative disadvantage compared to math because there are no other tools to explain how high or low a note is.

You can use words to explain math, just like you did with your son.

Internalization is key, if you miss the window as a kid then it's gonna be an uphill battle and life being complicated as it is would mean people giving up on it.

And real world feedback is telling us that such window will be missed, that's why I had thought about math being always explained using the familiar English language which is almost always never missed as a kid.

[Jtsummers]

> Given that (unlike foreing languages) math can be explained WITHOUT having to teach a new notation, then why don't we do it?

Can you explain, say, orbital mechanics, without math notation? In a way where someone can determine where a satellite will be at a particular time given its position and velocity at a prior time taking into account disturbances to the ideal orbit caused by the Moon and Sun (we'll stick with just those 2 and pretend the Earth itself is a perfect sphere).

I don't mean explain in a pop-sci sense. That's actually feasible with very little math (though you will probably want some diagrams), I mean explain in a way that the audience can then apply this math-but-not-in-math-notation to solve real world problems.

[GDC7]

Again, you are assuming that math is only done at the frontier.

I don't care about the frontier, I care about improving standards of living and quality of life, and that you can do by moving the needle in a concrete manner for HS and college math proficiency.

Not to mention that the satellite operations you mention will benefit a lot thanks to a higher standards of living/quality of life which are synthetized in the GDP metric.

One can only imagine the GDP growth that would happen if math proficiency levels were to suddenly become on par with coastal China.

At that point the satellite operations you'd speak of would become much smoother without even needing to move the math frontier forward.

You'd see collapsing costs everywhere ranging from personnel, raw materials, building operations, security and so forth.

[umvi]

I think it would be better to just get away from ultra terseness. It's crazy to me how terse mathematics is compared to CS.

def velocity(time_ms): return ...

vs.

v(t) = ...

Like nearly every operation and variable is one character or symbol long (with the puzzling exception of trig where you get a whopping 3 characters - sin/cos/tan/etc.)

[whatshisface]

I really don't want to write a complete word hundreds of times when I am solving an equation on paper.

[Jtsummers]

At least that idea is better than the other way some people want to "improve" math notation: drop it and write everything in "plain English". Like anyone who does even basic algebra would really benefit from that.

"The position of a particle at some point in time is its original position added to the product of its initial velocity and the time added to half the acceleration times the square of the time. Now, if I tell you the initial position, initial velocity, current position, and acceleration, how much time has passed? Remember, you can't use algebra anymore because we banned it in favor of 'plain English', also good luck communicating your ideas to people who don't understand English."

[whatshisface]

That's actually quite close to how Newton's Principa was written. If you want a real challenge, go try and read it... and if you really want a challenge, try to read it without already understanding mechanics!

[umvi]

Use short hand with pencil and paper. But make it less terse inside textbooks, inside computers, etc