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Comment by _Microft | Hacker News Reader

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.

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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.

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!

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

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.

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.

>The 10th root of x takes you from a measurement of an 10 dimensional object to the measurement of a 9 dimensional object.

Doesn't it take you from 10d to 1d? For instance, 10^10 is the hypervolume of a 10-cube with all side lengths = 10.

Imagine you are trying to explain this to a 15 year old.

If math is going to make sense to kids we can't resort to explanations that sound like "and then a miracle occurs".

BTW, I am not being critical of your answer. What I am saying is that there are these corners in seemingly simple math that have me scratching my head when it comes to explaining the concepts to a kid in a manner that makes sense and isn't circular. I have yet to find good answers to these questions.

Kid: What does the 10th. root of n mean?

Dad: It's the number, let's call it x, that, when raise to the 10th power is equal to n

Kid: So: n = x * x * x * x * x * x * x * x * x * x?

Dad: Yes! You got it!

Kid: How do you calculate it?

Dad: Well...

Kid: What if it is the 10.1 root of n?

Dad: Well, that's a little different...

Kid: How?

Dad: It's the number than when raised to the p-1 power times the base raised to the fractional portion of the power is equal to n

Kid: What's the fractional portion?

Dad: For the case of p = 10.1, it's 0.1

Kid: x * x * x * x * x * x * x * x * x * x^(p - int(p)) then?

Dad: Yeah.

Kid: How do I calculate x to the 0.1 power?

Dad: Well, you could use your calculator...(now starting to sweat)

Kid: How does the calculator do the math. You know, like when the math teacher says "Show your work"

Dad: Well, you could use logarithms...

Kid: What are logarithms?

Dad: A better method could be to use Newton's method. Here:

https://en.wikipedia.org/wiki/Newton%27s_method

Kid: It says: "start with an initial guess which is reasonably close to the true root, then to approximate the function by its tangent line using calculus, and finally to compute the x-intercept of this tangent line by elementary algebra"

Dad: Yes...

Kid: I don't know calculus. Is that the only way? I just wanted to understand how to calculate the 10th root of a number?

Dad: OK, let's try this. I just threw it together:

    # Calculate the exp root of n using a binary search
    #
    def root_binary_search(n, exp):
        # Return b, which is the exp root of n
        # b**exp should be equal to n
        #
        min = 0
        
        # For exponents < 1 the max needs to be sufficiently large
        max = n
        if exp < 1:
            while max**exp < n:
                max *= 2

        max_error = 0.00001

        while True:
            b = (max + min) / 2
            b_exp = b**exp
            error = abs(n - b_exp)
            # print(f"min: {min:15.4f}   max: {max:15.4f}   b: {b:15.4f}   b_exp: {b_exp:15.4f}   n: {n:15.4f}   error: {error:5.8f}")
            
            if error <= max_error:
                return b
            else:
                if b_exp > n:
                    max = b
                else:
                    min = b


    # Tests
    print(root_binary_search(4, 2), f"  result should be: {4**(1/2)}")
    print(root_binary_search(16, 2), f"  result should be: {16**(1/2)}")
    print(root_binary_search(5, 0.1), f"  result should be: {5**(1/0.1)}")
    print(root_binary_search(2, 10), f"  result should be: {2**(1/10)}")
    print(root_binary_search(4, 0.25), f"  result should be: {4**(1/0.25)}")

Kid: So...you are telling me to guess?

Dad: Yeah...? (looking embarrassed)

Kid: And to accept an error? 4-squared is 256, not 255.998046875?

Dad: Well, you have to understand that with a binary search...

Kid: And, did you see what happens if I run this case?

    print(root_binary_search(4, 1), f"  result should be: {4**1}")

Kid: Dad?

Dad: I have to get back to work. Why don't you ask your math teacher tomorrow?

I think you missed (or at least aren't building off of) the point of my comment.

I'm not questioning the pedagogy in the original comment, just the specific math. x^(1/10) takes a value of dimension [length^10] to a value of dimension [length].

Interestingly, I think you could take this in a few aesthetic directions. From a pure math perspective, this is where you can start talking about set theory, cardinality, etc. Irrational numbers are infinite sequences of digits we can only approximate. From a computer science perspective, you can talk about Newton's method, and also make the argument than an algorithm which converges to a number is a quite meaningful way to describe that number. Some would also add a caveat of 'efficiently' converging. And combining the two perspectives together, you can discuss that the set of computable numbers are of a lower cardinality than the set of reals -- aka 0% of real numbers are computable. You could also look at things from a geometrical perspective, and show how roots higher than square roots are tied to higher dimensions are are nonconstructible in the plane (this might be very hard to show!).

I understand what you are saying, believe me. I am trying to keep it simple because the objective is for the child to walk away with a useful non-scary answer that gives them a sense of proportion with which they can approach thinking about these things.

Anyone who has tried to teach a child math is familiar with just how hard it can to have them understand seemingly simple concepts. Simple example unrelated to powers/logs/roots. It took me about half an hour to explain how you can shift a parabola right and left by simply adding or subtracting a constant from x in the simplest form y = x^2. The fact that it moves in a direction opposite the sign caused even more confusion. It took telling the story in five different ways before the "aha!" moment happened.

The relationship between exponentiation and logarithms is another one that gets fun once things are not nice and even. Exponentiation is sequential multiplication and logs sequential division. Sounds good, until you can't multiply or divide by the base any more.

I find it interesting that in all of my searching I have not found a simple approach to explaining these things to children so they can build a tangible sense of what's in front of them.

That said, if the kid understands coding, yes, you can use programs to have them explore how things might work, create solutions, understand errors, estimation, etc. More the reasons to perhaps teach coding and math in parallel and to the same level of importance in schools.

>I understand what you are saying, believe me.

Do you? Almost nothing you've said has any relevance to my original comment.

Yes, I do. You can't approach kids with the kind of explanation you are proposing without their eyes glazing over. My kids are very comfortable with STEM and you still have to be careful. You cannot assume this to be the case with the general school population.

No, you still don't understand, because I'm not proposing an explanation!

I even clarified: "I'm not questioning the pedagogy in the original comment, just the specific math."

And yet, if you go back and look at this thread, what I am looking for is, in fact, an explanation that will work well for children. I thought I was explicit enough. I apologize if I was not.

Quote:

"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!"

In other words, I have no use for anything else as it quickly becomes an irrelevant time sink given the stated goal: Trying to explain this to children.

If you can translate what you wrote into something that can be taught to an average teenager (meaning, not a mathematically gifted or advanced student), you might just have the answer.

So far the only explanation I have found for how to solve these kinds of problems is successive approximation by guessing the answer. One level up from there is to use various algorithms to do the guessing, either on paper or through a computational solution (which requires a reasonable level of comfort writing code or using something like Excel).

I do want to thank you for taking the time to contribute to the conversation. Be well.

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)

> 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...

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.

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 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.

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.

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

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.

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

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

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".

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.

What has any of this got to do with "emerging as a field"? Mathematics is alive and well and, indeed, popular. It's not like Mathematics lectures are empty or that huge numbers are shying away from engineering or physics because of the Maths involved. Like, it's doing fine, and clearly "emerged as a field" centuries ago.

I don't give a fuck if people hate Maths, it's doing fine lol. You sound like someone who has literally no connection to the field and therefore can not perceive how it is actually doing, and instead you think "wow Maths must be in a bad state if everyone hates it". But no, that is just not true...

If there is any relevant criticism of Mathematics it is much more about going extremely deep into vastly theoretical domains without any reference to how practical such solutions might be than anything to do with whether or not the lay person "enjoys Maths".

All of this is irrelevant bullshitting. Maths emerged as a field centuries or even millenia ago. Please read the wiki page on "Mathematics".

Well, one of the GOAT mathematicians of our time, Jim Simons thinks math in America is in a bad shape.

And same goes for Terence Tao as he was unsatisfied with the low degree of collaboration between mathematicians and stepped up saying (politely) :"this sh-t has to stop"

Of course the rampant ego which is in the field prevents people from rallying around leaders like that or even arriving indipendently to the same conclusion because all these people see is the di-k measuring contest or slapping their name on a theorem.

You speak about practicality of math, what's more practical than turning Mobile, AL into a Cambridge or a Zurich.

If people in math cared about explaining their thoughts the way philosophers and rational thinkers do, then it could be possible.

Tbh, those people are the ones with huge egos. Mathematics is getting done on the raw ground regardless of if they are present or not. I hate this whining bullshit, if you want people to collaborate, make the tools to help people do so, rather than cringy pointless posturing.

> turning Mobile, AL into a Cambridge or a Zurich.

You are extremely naive if you think that this is something that would happen by raising general mathematics literacy. Nor is it something that we should necessarily even want to happen. Stop living in a dream land, not everyone needs to or should be a mathematician. We should aim for general literacy in statistics at most.

> If people in math cared about explaining their thoughts the way philosophers and rational thinkers do, then it could be possible.

Well, mathematics is in a much better state academically than philosophy, and literally in the UK we have ZERO philosophy education in the entirety of school. Philosophy hardly seems like an ideal to strive for, the average person know even less philosophy than mathematics and academically it is far smaller and less well funded.

> [philosophy and logic] don't get the same level of hatred that math gets

Because children aren't forced to study philosophy and logic for many hours each week.

And if you think logicians (even of the non-mathematical variety) aren't constantly inventing new notations...

> "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"

Those mathematicians aren't calling you "dumb and uninteresting", you're just not in their target audience.

> And your field will be kinda hated, which is what is happening.

Again, it's hated because children are forced to study it, because their parents think it's essential. Most people don't think mathematics is an evil cabal of men intentionally obfuscating their work with crazy symbols just for the heck of it.

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