Mathematical notation

[Amaton]

1) Why are there so many notations for vectors and related operations? v, [math]\vec{v}[/math], [math]v^j[/math], [math]v_j[/math], [math]|v\rangle[/math], etc. I understand that there are conventions endemic to certain fields of math/physics, but is there any practicality to get out if it?

 

2) As for mathematical notation in general, do you think it would be practical to establish a universal set of notation that could be used across all fields of mathematics and science?

Edited August 23, 2013 by Amaton

[ajb]

It can be confusing and different branches have their own notation. This can make it difficult to understand things.

 

The upside is that finding the "right" notation for you and your problem can make your thinking clearer.

 

 

The problem must be how would you establish a universal set of notation? What about clashes of notation? I don't want to have to use the same symbol for two different objects.

Edited August 23, 2013 by ajb

[Unity+]

It can be confusing and different branches have their own notation. This can make it difficult to understand things.

 

The upside is that finding the "right" notation for you and your problem can make your thinking clearer.

 

 

The problem must be how would you establish a universal set of notation? What about clashes of notation? I don't want to have to use the same symbol for two different objects.

Well, there are sometimes confusion especially with the description of a line. For example, when you begin learning about Geometry, you notate the line by . However, the next time you begin talking about the lines and points of a triangle you use capital letters instead of the other notation. Then, people begin notating points with capitals instead of the lines. The confusion is really with how different mathematicians teach notation.

[studiot]

1) It is impossible to avoid muplicity of notation. Take your example of vectors. Vectors are members of some set which we call a vector space V. Since there are lots of different types of vector there are lots of different vectors spaces V. V', V'''...... However there are plenty of other objects, that are not vectors, but are members of sets. So we could use general set notation for our vectors, at the cost of having other, non vecotr, objects with the same notation. Or we could introduce a second notation, specific to vectors, to maintain the distinction. But general set notation would still be correct.

 

2) Much notation has been determined by what can be easily presented. Presentation includes, hand written, printed, display screen. Each medium offers its own advantages and difficulties. I note in another thread here at SF there is discussion about a presentation that is not available through this forum's implementation of LaTex.

Edited August 23, 2013 by studiot

[Amaton]

It can be confusing and different branches have their own notation. This can make it difficult to understand things.

 

The upside is that finding the "right" notation for you and your problem can make your thinking clearer.

 

Yes, and so the matter returns to convention. I read that some fields use [math]v^j[/math] for vector notation, and I would have terrible frustration having to constantly remind myself that [math]x^2[/math] is not x squared!

 

I suspect that the field-endemic conventions of today are direct results of the personal preferences of very authors that contributed to the development of those areas.

 

 

The problem must be how would you establish a universal set of notation? What about clashes of notation? I don't want to have to use the same symbol for two different objects.

 

Then maybe this universal guidebook will have to be very comprehensive in its notational "alphabet". I'm thinking analogously to the International Phonetic Alphabet, which assigns a unique symbol or symbol pair to every practical sound the human mouth produces in speech. Of course, the 20 - 30 letters of modern Greco-Latin alphabets aren't enough to accomplish this, so many symbols exist beyond any individual one. Likewise, for universal notation to be comprehensive, many more forms will likely have to be introduced to avoid that non-uniqueness you brought up. We may also reserve multi-use notation for more "exotic" objects, which are less likely to be mixed up and confused. I'm just asking for us to weigh the pros and cons.

Well, there are sometimes confusion especially with the description of a line. For example, when you begin learning about Geometry, you notate the line by . However, the next time you begin talking about the lines and points of a triangle you use capital letters instead of the other notation. Then, people begin notating points with capitals instead of the lines. The confusion is really with how different mathematicians teach notation.

 

Yes, though these cases may not be so bad for certain reasons. Geometry helps one to visualize the scenario and the objects therein, whether by textual description or an actual diagram (where they are explicitly shown or stated). "Line segment AB intersects ray Q> at its point of tangency to circle P". One cannot do so this easily with symbolic analysis or similar approaches.

Edited August 24, 2013 by Amaton

[Unity+]

 

Yes, and so the matter returns to convention. I read that some fields use [math]v^j[/math] for vector notation, and I would have terrible frustration having to constantly remind myself that [math]x^2[/math] is not x squared!

 

I suspect that the field-endemic conventions of today are direct results of the personal preferences of very authors that contributed to the development of those areas.

 

 

Then maybe this universal guidebook will have to be very comprehensive in its notational "alphabet". I'm thinking analogously to the International Phonetic Alphabet, which assigns a unique symbol or symbol pair to every practical sound the human mouth produces in speech. Of course, the 20 - 30 letters of modern Greco-Latin alphabets aren't enough to accomplish this, so many symbols exist beyond any individual one. Likewise, for universal notation to be comprehensive, many more forms will likely have to be introduced to avoid that non-uniqueness you brought up. We may also reserve multi-use notation for more "exotic" objects, which are less likely to be mixed up and confused. I'm just asking for us to weigh the pros and cons.

 

Yes, though these cases may not be so bad for certain reasons. Geometry helps one to visualize the scenario and the objects therein, whether by textual description or an actual diagram (where they are explicitly shown or stated). "Line segment AB intersects ray Q> at its point of tangency to circle P". One cannot do so this easily with symbolic analysis or similar approaches.

Well, for example it is is simpler to write capital letters instead of using the other notation(which is good). However, it is still confusing for people still learning Geometry or any other field of mathematics.

[ajb]

The other problem is that new concepts are introduced all the time and these need notation. You could not write a book that sets all the notation ever needed. Though, you could try to standardise "basic mathematics".

 

This reminds me of the question of naming things in physics. No one person sets the nomenclature and notation in mathematics (or physics) the community does that by its use.

[Myuncle]

The number of symbols to remember should be practical. Our numbers symbols for example are only ten: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. And that's practical, universally accepted, but in theory you can do the same calculations using 20 symbols, or only 5 symbols. For example: 0, 1, 2, 3, 4. So we would have

0

1

2

3

4

10 (which means our 5)

11 (which stands for our 6)

12 (our 7)

13 (our 8)

14 (our 9)

20 (our 10)

21 (our 11)

etc

etc.

[ajb]

That is just changing the base of the number system you are using. We have lots of other things in mathematics that need notation to represent them.

[Amaton]

The number of symbols to remember should be practical. Our numbers symbols for example are only ten: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. And that's practical, universally accepted, but in theory you can do the same calculations using 20 symbols, or only 5 symbols.

 

As ajb noted, that's simply using a different base system, base-5 in your case. Notation is a different matter, though I don't doubt the benefits of using certain bases.

This reminds me of the question of naming things in physics. No one person sets the nomenclature and notation in mathematics (or physics) the community does that by its use.

 

That's true. I guess a comprehensive standardization would either be impossible or impractical. In a way, I think notation (and nomenclature) in elementary mathematics is already rather standardized.

[John Cuthber]

Reminded me of this

http://xkcd.com/927/

[ajb]

In a way, I think notation (and nomenclature) in elementary mathematics is already rather standardized.

For elementary arithmatic this is true.

[Danijel Gorupec]

The math notation seems notoriously difficult to type on computer. If in the near future more mathematicians choose to type its math on computer (instaead of pencil-writing it on paper) some re-standardization will probably happen. If this will be the case, there is a chance that some diversity will be reduced among various math fields.

 

As I always say, math wants to be written - if you change the writing tool, the notation will adapt.

[Unity+]

The math notation seems notoriously difficult to type on computer. If in the near future more mathematicians choose to type its math on computer (instaead of pencil-writing it on paper) some re-standardization will probably happen. If this will be the case, there is a chance that some diversity will be reduced among various math fields.

 

As I always say, math wants to be written - if you change the writing tool, the notation will adapt.

Not really. Many mathematicians(like me) set up the keyboards on the computer to change certain keys to certain symbols within mathematics. Also, if you download the right software then you will have an easier time doing it.

[ajb]

As I always say, math wants to be written - if you change the writing tool, the notation will adapt.

That is the power of LaTex, which can be constantly updated with new packages.

[Amaton]

LaTeX is great and pretty comprehensive if you ask me. It is a bit tedious though for larger and/or more complex expressions. If mathematical notation is going to become standard on an electronic medium, I think it should go the route of Microsoft Word's equation editor. The interface is intuitive and everything's quick and easy. However, the version I'm familiar with isn't as complete as TeX.

[Danijel Gorupec]

That is the power of LaTex, which can be constantly updated with new packages.

But... there are two issues regarding math notation when math is used on computer... The first one, perfectly solved by the LaTeX, is concerned about rendering math. The second one is concerned about entering math into computer (either using keyboard or hand-writing recognition). The second one is the problematic one, IMO.

 

This is why I am a bit afraid of the LaTeX software - it is providing so much various symbols and those symbols are very appealing to mathematicians. But if mathematicians start using all those fancy symbols wildly, it will become problematic to find methods to type them effectively into computer... I am a developer of one rapid-math-typing software and when I look at those LaTeX symbols lists, I am having trouble sleeping

 

But now, we are probably getting too far from the original post.

[ajb]

It is a bit tedious though for larger and/or more complex expressions. If mathematical notation is going to become standard on an electronic medium, I think it should go the route of Microsoft Word's equation editor. The interface is intuitive and everything's quick and easy. However, the version I'm familiar with isn't as complete as TeX.

Latex gets quicker when you become more and more familiar with it. Equation editor however I find very slow, but then I have only used it about 2 times (never again!).

[Danijel Gorupec]

LaTeX is great and pretty comprehensive if you ask me. It is a bit tedious though for larger and/or more complex expressions. If mathematical notation is going to become standard on an electronic medium, I think it should go the route of Microsoft Word's equation editor. The interface is intuitive and everything's quick and easy. However, the version I'm familiar with isn't as complete as TeX.

FYI, you can try (free trial) DesignScience MathType software - this is a more powerful version of the MS Equation Editor.

You can also try sw of mine: Math-o-mir (better for rapid typing, but requires longer learning time - and, of course, renders equations more corase than the LaTeX)

[ajb]

The second one is concerned about entering math into computer (either using keyboard or hand-writing recognition). The second one is the problematic one, IMO.

 

That is going to be much harder.

 

 

Maybe mathematical notation will take this into account, but I doubt it.

 

 

 

They had all sorts of funky notations in there. In fact, I'm told--in a typical tale often heard of authors being ahead of their publishers--that Russell ended up having to get fonts made specially for some of the notation they used.

 

And, of course, in those days we're not talking about TrueType or Type 1 fonts; we're talking about pieces of lead. And I'm told that Russell could actually be seen sometimes wheeling wheelbarrows full of lead type over to the Cambridge University Press so his books could be appropriately typeset.

http://www.stephenwolfram.com/publications/recent/mathml/mathml2.html

 

[Amaton]

Latex gets quicker when you become more and more familiar with it. Equation editor however I find very slow, but then I have only used it about 2 times (never again!).

 

I see what you mean. It probably depends on whether one is more facile with "stream typing" or point-and-click. I can type fast, but I'm not very familiar with the TeX codes. It's like learning a language; one just has to practice to become more fluent.

But now, we are probably getting too far from the original post.

 

You're right. And don't think I didn't report you for derailing a perfectly good thread. (just kidding, the original poster does not mind, so discuss away!)

[ajb]

I think it is on topic, just an evolution of the opening question.

 

As computers are now the standard tool for writting mathematics, though pencil & paper or chalk & board are used to "do" mathemaics, notation will have to take this into account.