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Mathematics is hard for mathematicians to understand too

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Re: Mathematics is hard for mathematicians to understand too

#131

Post by Chinjut »

matheme wrote: post #88
karmakurtisaani wrote: post #72

A lot of people here suggesting they'd be great mathematicians if only it wasn't for the pesky notation. What they are missing is that the notation is the easy part..

> What they are missing is that the notation is the easy part. This is so wrong it can only come from a place of inexperience and ignorance. Mathematics is flush with inconsistent, abbreviated, and overloaded notation. Show a child a matrix numerically and they can understand it, show them Ax+s=b, and watch the confusion.

The fact that there is a precise analogy between how Ax + s = b works when A is a matrix and the other quantities are vectors, and how this works when everything is scalars or what have you, is a fundamental insight which is useful to notationally encode. It's good to be able to readily reason that in either case, x = A^(-1) (b - s) if A is invertible, and so on.

It's good to be able to think and talk in terms of abstractions that do not force viewing analogous situations in very different terms. This is much of what math is about.

Re: Mathematics is hard for mathematicians to understand too

#132

Post by karmakurtisaani »

matheme wrote: post #88
karmakurtisaani wrote: post #72

A lot of people here suggesting they'd be great mathematicians if only it wasn't for the pesky notation. What they are missing is that the notation is the easy part..

> What they are missing is that the notation is the easy part. This is so wrong it can only come from a place of inexperience and ignorance. Mathematics is flush with inconsistent, abbreviated, and overloaded notation. Show a child a matrix numerically and they can understand it, show them Ax+s=b, and watch the confusion.

> This is so wrong it can only come from a place of inexperience and ignorance.

Thanks for the laughs :D

> Show a child a matrix numerically and they can understand it, show them Ax+s=b, and watch the confusion.

Show a HN misunderstood genius Riemann Zeta function as a Zeta() and they think they can figure out it's zeros. Show it as a Greek letter and they'll lament how impossible it is to understand.

Re: Mathematics is hard for mathematicians to understand too

#133

Post by karmakurtisaani »

UltraSane wrote: post #117
karmakurtisaani wrote: post #72

A lot of people here suggesting they'd be great mathematicians if only it wasn't for the pesky notation. What they are missing is that the notation is the easy part..

Not at all. Over and over I find really intimidating math notation actually represents pretty simple concepts. Sigma notation is a good example of this. Hmm, giant sigma or sum()?

Wait until you learn about integration. Measures, limits and the quirks of uncountable spaces don't become simpler once you call the operation integrate().

Re: Mathematics is hard for mathematicians to understand too

#134

Post by mzl »

xigoi wrote: post #125
mzl wrote: post #21

Earlier quoted context omitted.

I wrote about overlapping intervals a while ago, and used what I thought was the standard math notation for closed and half-open intervals. From comments, I learned that half-open intervals are written differently in french mathematics: https://lobste.rs/s/cireck/how_check_for_overlapping_interva...

We don’t talk about the French notation for intervals. Let it stay in France.

Yep, I agree on that. But still, interesting to see that such a "standard" thing can be so different in different dialects of mathematical notation.

Re: Mathematics is hard for mathematicians to understand too

#135

Post by zwnow »

gjulianm wrote: post #75
abraxas wrote: post #64

Earlier quoted context omitted.

There has to be a happy medium between the tersness of the current notation systems and the verbosity of code-like expressions. We just need to rethink this so more people can learn it. Math still stands a bit like writing did in ancient culture. It's a domain reserved for a few high priests inducted into the craft and completely inaccessible to everyone else.

I wonder why so many people are under the impression that the notation is what is keeping them away and if only the notation was easier then the underlying concepts would be clear. For example, if you don't know what the pullback of a differential form is, it doesn't matter if I write it in clear text or if I write the common notation φ^* ω. > It's a domain reserved for a few high priests inducted into the craft and…

φ^* ω is just 2 weird symbols. There are tons of symbols to learn. I gave up trying to learn the Russian alphabet after a few days so why do u think I am capable of memorizing the Greek one?

The theories are learnable, making sense of all the weird symbols is what's breaking my brain. I tried to get into set theory thrice now, not happening with all the math lingo, hieroglyphs and dry ass content. Learning can be incredibly fun if it was designed fun. Math is a dry and slow process. Make it fun, make it readable and people will be capable to learn it easier.

Re: Mathematics is hard for mathematicians to understand too

#136

Post by gjulianm »

zwnow wrote: post #135
gjulianm wrote: post #75

Earlier quoted context omitted.

I wonder why so many people are under the impression that the notation is what is keeping them away and if only the notation was easier then the underlying concepts would be clear. For example, if you don't know what the pullback of a differential form is, it doesn't matter if I write it in clear text or if I write the common notation φ^* ω. > It's a domain reserved for a few high priests inducted into the craft and…

φ^* ω is just 2 weird symbols. There are tons of symbols to learn. I gave up trying to learn the Russian alphabet after a few days so why do u think I am capable of memorizing the Greek one? The theories are learnable, making sense of all the weird symbols is what's breaking my brain. I tried to get into set theory thrice now, not happening with all the math lingo, hieroglyphs and dry ass content. Learning can be inc…

> why do u think I am capable of memorizing the Greek one

No one memorizes the Greek alphabet. We just learn it as we go because it’s useful to have different types of letters to refer to different types of objects. That’s it.

> I tried to get into set theory thrice now, not happening with all the math lingo, hieroglyphs and dry ass content.

That sounds like you’re trying to learn a specific field without actually having any of the prerequisites to learn it. I don’t know what you’re specifically referring to when you say “set theory” as that’s an incredibly wide field, and depending on what you’re trying to learn it can be quite technical.

> Learning can be incredibly fun if it was designed fun. Math is a dry and slow process.

This sounds like someone complaining that getting to run a marathon is tiresome and hard. Yes, teaching mathematics can always be improved and nothing is perfect, but it will still be hard work.

Re: Mathematics is hard for mathematicians to understand too

#137

Post by gjulianm »

matheme wrote: post #94
assemblyman wrote: post #89

I find software engineers spend too much time focused on notation. Maybe they are right to do so and notation definitely can be helpful or a hindrance, but the goal of any mathematical field is understanding. It's not even to prove theorems. Proving theorems is useful (a) because it identifies what is true and under what circumstances, and (b) the act of proving forces one to build a deep understanding of the phenome…

> If the idea is that the right notation will make getting insights easier, that's a futile path to go down on. I agree whole heartedly. What I want to see is mathematicians employ the same rigor of journalists using abbreviations: define (numerically) your notation, or terminology, the first time you use it, then feel free to use it as notation or jargon for the remainder of the paper.

> What I want to see is mathematicians employ the same rigor of journalists using abbreviations: define (numerically) your notation, or terminology, the first time you use it, then feel free to use it as notation or jargon for the remainder of the paper.

They already do this. That is how we all learn notation. Not sure what you mean by numerically though, a lot of concepts cannot be defined numerically.

Re: Mathematics is hard for mathematicians to understand too

#138

Post by xg15 »

assemblyman wrote: post #89

I find software engineers spend too much time focused on notation. Maybe they are right to do so and notation definitely can be helpful or a hindrance, but the goal of any mathematical field is understanding. It's not even to prove theorems. Proving theorems is useful (a) because it identifies what is true and under what circumstances, and (b) the act of proving forces one to build a deep understanding of the phenome…

> One thing mathematics education is really bad at is motivating the definitions.

I was annoyed by this in some introductory math lectures where the prof just skipped explaining the general idea and motivation of some lemmata and instead just went through the proofs line by line.

It felt a bit like being asked to use vi, without knowing what the program does, let alone knowing the key combinations - and instead of a manual, all you have is the source code.

Re: Mathematics is hard for mathematicians to understand too

#139

Post by vjk800 »

pathikrit wrote: post #51

I love math but the symbology and notations get in my way. 2 ideas: 1. Can we reinvent notation and symbology? No superscripts or subscripts or greek letters and weird symbols? Just functions with input and output? Verifiable by type systems AND human readable 2. Also, make the symbology hyperlinked i.e. if it uses a theorem or axiom that's not on the paper - hyperlink to its proof and so on..

1. I work in finance and here people sometimes write math using words as variable names. I can tell you it gets extremely cumbersome to do any significant amount of formula manipulation or writing with this notation. Keep in mind that pen and paper are still pretty much universally used in actual mathematical work and writing full words takes a lot of time compared to single Greek letters.

Large part of math notation is to compress the writing so that you can actually fit a full equation in your vision.

Also, something like what you want already exists, see e.g. Lean: https://lean-lang.org/doc/reference/latest/. It is used to write math for the purpose of automatically proving theorems. No-one wants to use this for actually studying math or manually proving theorems, because it looks horrible compared to conventional mathematics notation (as long as you are used to the conventional notation).

Re: Mathematics is hard for mathematicians to understand too

#140

Post by wakawaka28 »

assemblyman wrote: post #115
wakawaka28 wrote: post #113

Earlier quoted context omitted.

Notation makes a huge difference. I mean, have you TRIED to do arithmetic with Roman numerals? >If the idea is that the right notation will make getting insights easier, that's a futile path to go down on. What really helps is looking at objects and their relationships from multiple viewpoints. This is really what one does both in mathematics and physics. Seeing the relationships between objects is partly why math ha…

I haven't thought about or learned a systematic way to add roman numerals. But, I would argue that the difference is not notation but a fundamental conceptual advance of representing quantities by b (base) objects where each position advances by a power of b and the base objects let one increment by 1. The notation itself doesn't really make a difference. We could call X=1, M=2, C=3, V=4 and so on. I also don't know…

Adding and subtracting Roman numerals is pretty easy because it's all addition and subtraction. A lot of it is just repeating the symbols just like with tally marks. X+X is just XX for example. You do have to keep track of when another symbol is appropriate, but VIIII is technically equivalent to IX. It's all the other operations that get harder. If the Romans had negative numbers, then the digits of a numeral could be viewed as some kind of polynomial with some positive and negative coefficients. But they also didn't have that.

>The notation itself doesn't really make a difference. We could call X=1, M=2, C=3, V=4 and so on.

Technically, the positional representation is part of the notation as well as the symbols used. Symbols had to evolve to be more legible. For example, you don't want to mix up 1 and 7, or some other pairs that were once easily confused.

>Why did the Romans not think of it?

I don't know. I expect that not having a symbol for zero was part of it. Place value systems would be very cumbersome without that. I think that numbers have some religious significance to the Hindus, with their so-called Vedic math, but the West had Pythagoras. I'm sure that the West would have eventually figured it out, as they figured out many impressive things even without modern numerals.

>But good notation is backed by a meaningfully different way of looking at things.

That's just one aspect of good notation. Not every different way of looking at things is equally useful. Notation should facilitate or at least not get in the way of all the things we need to do the most. The actual symbols we use are important visually. A single letter might not be self-describing, but it is exactly the right kind of symbol to express long formulas and equations with a fairly small number of quantities. You can see more "objects" in front of you at once and can mechanically operate on them without silently reading their meaning. On the other hand, a single letter symbol in a large computer program can be confusing and also makes editing the code more complicated.

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