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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

#111

Post by hodgehog11 »

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 always tell my students that if they don't understand why things are done a certain way, that they should try to do it in the way most natural to them and then iterate to improve it. Eventually they will settle on something very similar to most common practice.

(2) Higher-level proofs are using so many ideas simultaneously that doing this would be tantamount to writing Lean code from scratch: painful.

Re: Mathematics is hard for mathematicians to understand too

#112

Post by [deleted] »

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.

[deleted]

Re: Mathematics is hard for mathematicians to understand too

#113

Post by wakawaka28 »

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…

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 has settled on a terse notation (the other reason being that you need to write stuff over and over). This helps up to a point, but mainly IF you are writing the same things again and again. If you are not exercising your memory in such a way, it is often easier to try to make sense of more verbose names. But at all times there is tension between convenience, visual space consumed, and memory consumption.

Re: Mathematics is hard for mathematicians to understand too

#114

Post by kragen »

bmacho wrote: post #109
kragen wrote: post #104

Earlier quoted context omitted.

When you render it for proper typesetting, do the parentheses around dy/dx disappear? (Oh, I guess you've removed them in your edit.) If they do, it seems like an error-prone way to write your math. If they don't, it seems like it will make your math look terrible. Supposing that the parentheses aren't necessary, as implied by your edit: how does AsciiMath determine that e^y isn't in the numerator in "e^y dy/dx", or…

Here is an online renderer and the description: https://asciimath.org/ The rules are basically the same as LaTeX, with saner symbol names, support for fractions, \ is not needed before symbols and () can be used instead of {}. > Supposing that the parentheses aren't necessary, as implied by your edit: how does AsciiMath determine that e^y isn't in the numerator in "e^y dy/dx" It seems to me that dx,dy,dz,dt behave li…

Thanks! It does better than I expected on tricky input like [0, 1/2). It seems like there are a lot of special cases, though. It does indeed remove parentheses from the output in some cases but not others.

Probably figuring out how to write things in AsciiMath is more trouble than copying and pasting them from Wikipedia though. (The alt text on equation images is the LaTeX source preceded with \displaystyle.)

How do you do \bigg(\big((4x + 2)x + 1\big)x - 3\bigg)x + 5 in AsciiMath? (((4x + 2)x + 1)x - 3)x + 5 makes all the parens the same size.

Re: Mathematics is hard for mathematicians to understand too

#115

Post by assemblyman »

wakawaka28 wrote: post #113
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…

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 what historically motivated the development of this system (the Indian system). Why did the Romans not think of it? What problems were the Indians solving? What was the evolution of ideas that led to the final system that still endures today?

I don't mean to underplay the importance of notation. But good notation is backed by a meaningfully different way of looking at things.

Re: Mathematics is hard for mathematicians to understand too

#116

Post by voidhorse »

matheme wrote: post #87
voidhorse wrote: post #60

Earlier quoted context omitted.

Precisely. Think of mathematics like a game. Players of magic the gathering will say a creature "has flying" by which they mean "it can only be blocked by other creatures with reach or flying". Newcomers obviously need to learn this jargon, but once they do, communication is greatly facilitated by not having to spell out the definition. Just like games, the definitions in mathematics are ethereal and purely formal as…

My entire being is anthithetical to this type of gatekeeping. > You expect the players of the game to learn the rules before they play. TFA is literally from a 'player' who has 'learned the rules' complaining that the papers remain indecipherable. > You expect the players of the game to learn the rules before they play. Actually, I expect to have to teach rules to new players before they play. We are different.

Many mathematicians do in fact teach the rules of the game in numerous introductory texts. However, you don't expect to have to explain the rules every time you play the game with people who you've established know the game. Any session would take ages if so, and in many cases the game only become more fun the more fluent the players are.

I'm not fully convinced the article makes the claim that jargon, per se, is what needs to change nor that the use of jargon causes gatekeeping. I read more about being about the inherent challenges of presenting more complicated ideas, with or without jargon and the pursuit of better methods, which themselves might actually depend on more jargon in some cases (to abstract away and offload the cognitive costs of constantly spelling out definitions). Giving a good name to something is often a really powerful way to lower the cognitive costs of arguments employing the names concept. Theoretics in large part is the hunt for good names for things and the relationships between them.

You'd be hard pressed to find a single human endeavor that does not employ jargon in some fashion. Half the point of my example was to show that you cannot escape jargon and "gatekeeping" even in something as silly and fun as a card game.

Re: Mathematics is hard for mathematicians to understand too

#117

Post by UltraSane »

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()?

Re: Mathematics is hard for mathematicians to understand too

#118

Post by carlCarlCarlCar »

Applied math is little more than semantics compression.

This fundamental truth is embedded in the common symbols of arithmetic...

+ ... one line combined with another ...linear...line wee

- ...opposite of + one line removed

x ...eXponential addition, combining groups

•/• ... exponential breaking into groups ...also hints at inherent ratio

From there it's symbols that describe different objects and how to apply the fundamental arithmetic operations; like playing over a chord in music

The interesting work is in physical science not the notation. Math is used to capture physics that would be too verbose to describe in English or some other "human" language. Which IMO should be reserved for capturing emotional context anyway as that's where they originate from.

Programming languages have senselessly obscured the simple and elegant reality of computation, which is really just a subset of math; the term computer originated to describe humans that manually computed. Typescript, Python, etc don't exist[1]. They are leaky abstractions that waste a lot of resources to run some electromagnetic geometry state changes.

Whether it's politics, religion or engineering, "blue" language, humans seem obsessed with notation fetishes. Imo it's all rather prosaic and boring

[1] at best they exist as ethno objects of momentary social value to those who discuss them

Re: Mathematics is hard for mathematicians to understand too

#119

Post by wasabi991011 »

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()?

You think changing sigma to sum() would make it easier to understand the 5 paper, 1000 page proof of the geometric Langlands conjecture?

Re: Mathematics is hard for mathematicians to understand too

#120

Post by mcmoor »

wakawaka28 wrote: post #113
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…

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…

Considering that post-arithmetic math rarely use numbers at all, and even ancient Greeks use lots of lines and angles instead of numbers, I don't think Roman numerals would really hold math that much.
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