[assemblyman]

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 phenomenon under study. This requires looking at examples, making a hypothesis more specific or sometimes more general, using formal arguments, geometrical arguments, studying algebraic structures, basically anything that leads to better understanding. Ideally, one understands a subject so well that notation basically doesn't matter. In a sense, the really key ingredient are the definitions because the objects are chosen carefully to be interesting but workable.

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.

Someone quoted von Neumann about getting used to mathematics. My interpretation always was that once is immersed in a topic, slowly it becomes natural enough that one can think about it without getting thrown off by relatively superficial strangeness. As a very simple example, someone might get thrown off the first time they learn about point-set topology. It might feel very abstract coming from analysis but after a standard semester course, almost everyone gets comfortable enough with the basic notions of topological spaces and homeomorphisms.

One thing mathematics education is really bad at is motivating the definitions. This is often done because progress is meandering and chaotic and exposing the full lineage of ideas would just take way too long. Physics education is generally far better at this. I don't know of a general solution except to pick up appropriate books that go over history (e.g. https://www.amazon.com/Genesis-Abstract-Group-Concept-Contri...)

[auggierose]

Understanding new math is hard, and a lot of people don't have a deep understanding of the math they use. Good notation has a lot of understanding already built-in, and that makes math easier to use in certain ways, but maybe harder to understand in other ways. If you understand something well enough, you are either not troubled by the notation, because you are translating it automatically into your internal representation, or you might adapt the notation to something that better suits your particular use case.

[wakawaka28]

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.

[assemblyman]

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.

[wakawaka28]

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.

[mcmoor]

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.

[xg15]

> 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.

[matheme]

> 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.

[aleph_minus_one]

> 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 do.

The purpose of papers is to teach working mathematicians who are already deeply into the subject something novel. So of course only novel or uncommon notation is introduced in papers.

Systematic textbooks, on the other hand, nearly always introduce a lot of notation and background knowledge that is necessary for the respective audience. As every reader of such textbooks knows, this can easily be dozens or often even hundreds of pages (the (in)famous Introduction chapter).

[gjulianm]

> 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.

[agumonkey]

Math rarely emphasize on this. You either have talent and you get intuition for free or you're average and you swim as much as you can until the next floater. It's sad because the internal and external value is immense