Thread: Alternative mathematical notations

For many mathematicians I suspect the idea to sound out of track and superfluous: Has there been any serious attempts at finding alternative notations for mathematics? Sure, you can call any programming language an alternative notation, but rather they are often just adaptions for a limited environment. What I'm looking for is projects that try to take it further and try to take advantages of tools we didn't have for writing notation when it was first started to develop. Using visual automation, graphical enhancements etc... Not for plotting graphs but for drawing the actual language itself. I'm not looking for a replacement either, just an alternative.
Maybe notation doesn't have to be one dimensional, written on a line, contating letters, but shapes morphing for a particular meaning (expressing mathematical relations using shapes?).
While there is no reason going about creating new types of notation for no reason I think that much can be won in terms of usability, efficiency and clarity, for both the learner and the user, by complete revision. Especially if you rethink how technology can be used for enhancement.
What do you think?

Originally Posted by worldsayshi

For many mathematicians I suspect the idea to sound out of track and superfluous: Has there been any serious attempts at finding alternative notations for mathematics? Sure, you can call any programming language an alternative notation, but rather they are often just adaptions for a limited environment. What I'm looking for is projects that try to take it further and try to take advantages of tools we didn't have for writing notation when it was first started to develop. Using visual automation, graphical enhancements etc... Not for plotting graphs but for drawing the actual language itself. I'm not looking for a replacement either, just an alternative.
Maybe notation doesn't have to be one dimensional, written on a line, contating letters, but shapes morphing for a particular meaning (expressing mathematical relations using shapes?).
While there is no reason going about creating new types of notation for no reason I think that much can be won in terms of usability, efficiency and clarity, for both the learner and the user, by complete revision. Especially if you rethink how technology can be used for enhancement.
What do you think?

Mathematical notation is invented and revised constantly to suit the subject at hand.

It sounds as though you have been exposed to little or no modern mathematics. The typical undergraduate sees only a little that originated after about 1860.

What do you mean by "modern mathematics"? As for level of education, I'm currently going through a Master's degree in information technology related field. I've studied but not yet passed mathematics on a bachelor level. I'm talking about mathematical notation in general (or elementary), not so much mathematical notation in higher studies or for specific uses. (Elementary) algebra, first order logic. Managing and defining sets. I'm inclined to focus on fields related to programming.

Originally Posted by worldsayshi

What do you mean by "modern mathematics"? As for level of education, I'm currently going through a Master's degree in information technology related field. I've studied but not yet passed mathematics on a bachelor level. I'm talking about mathematical notation in general (or elementary), not so much mathematical notation in higher studies or for specific uses. (Elementary) algebra, first order logic. Managing and defining sets. I'm inclined to focus on fields related to programming.

By modern mathematics I mean mathematics dating from about 19o0 -- topology, functional analysis, modern abstract algebra, differential geometry, etc.

Given your education you are not likely to have seen much, and probably won't, thougfh you get a taste in some advanced undergraduate classes..

Relatively little is algorithmic. That which is related to computer science -- Turing machines, NP-completeness, etc. -- is abstract.

Ok. Let me rephrase then: Has there been any serious attempts at finding alternative notations for traditional mathematics (which is after all widely used in most practicalities)?

Originally Posted by worldsayshi

Ok. Let me rephrase then: Has there been any serious attempts at finding alternative notations for traditional mathematics (which is after all widely used in most practicalities)?

Sure, the current notation, developed by the Arabs, replaced Roman numerals. It has worked quite well ever since.

Babylonian children knew the quadratic formula.

Rather than worrying about notation, it would be nice if the mathematical competency of the average denizen of planet Earth were up to the standards of those Babylonians.

So mathematical competency has diminished? Have any theory why?

The current notation isn't an alternative to the current notation.

No kidding

Originally Posted by worldsayshi

So mathematical competency has diminished? Have any theory why?

Lack of motivation. Very poor understanding of mathematics and science by many primary and secondary teachers. over-reliance on canned software. .....

Anyone with a more progressive attitude in the matter?

Originally Posted by worldsayshi

Anyone with a more progressive attitude in the matter?

What do you mean by "progressive" ?

You have offered no objective for any notation change, nor any (even rough) description of any proposed change.

Progressive in that situation is indistinguishable from silly.

Change for the sake only of change is not progressive.

Coming from someone who has never done mathematical research the call for change is more than just naive.

Sure, let's change some things.

Let's start by rewriting the Pythagorean Theorem with some new notation.

where 3 is the big side of a perfect C-sider and 1 and 2 are the little sides. For example, .

Congratulations if you understood that. (Okay, so I left the operators and the name of the theorem alone, or it'd have been a lot worse.)

The point is, the notation only exists by general agreement and really doesn't mean anything in and of itself. "Changing the notation" is a fairly meaningless exercise, since all that matters is that the symbols can be used to exchange ideas. "New tools" doesn't change that. (Besides, much of math is still done with pencil and paper, at least initially.)

I am 100% certain that there are advantages of a completely revised mathematical notation because of continual innovations and modifications in mathematics, as a result. Of course you cannot revise important theorems that contribute highly to certain surmises and conjectures, focus on unsolved problems and current problems that tend to arise. You can also spend time looking for holes (but why would you, unless you were getting paid a fortune to do so). Also, keep in mind that not everyone is banal or relies on dead Europeans to think for them. If that were the case, the human race would be screwed. So to answer your question, I would say yes.

Originally Posted by Sarah Afshar

I am 100% certain that there are advantages of a completely revised mathematical notation because of continual innovations and modifications in mathematics, as a result. Of course you cannot revise important theorems that contribute highly to certain surmises and conjectures, focus on unsolved problems and current problems that tend to arise. You can also spend time looking for holes (but why would you, unless you were getting paid a fortune to do so). Also, keep in mind that not everyone is banal or relies on dead Europeans to think for them. If that were the case, the human race would be screwed. So to answer your question, I would say yes.

What in the hell are you talking about ? Do you have any relevant example to bolster any of your purported points ?

Originally Posted by DrRocket

Originally Posted by worldsayshi

Anyone with a more progressive attitude in the matter?

What do you mean by "progressive" ?

By progressive I mean ideas that for example can further the concept of what can be accomplished by reinventing notation, rather than trying to find reasons for not having this discussion in the first place.

You have offered no objective for any notation change, nor any (even rough) description of any proposed change.

Originally Posted by worldsayshi

Maybe notation doesn't have to be one dimensional, written on a line, contating letters, but shapes morphing for a particular meaning (expressing mathematical relations using shapes?).

To be more specific:
Can formulas be described using node-charts? Can shapes be used for describing numbers (as suggested by the way some savants perceive them for example)? Can graphs be part of the notation instead of just a complement? Can interactivity and visualization, given by use of computers, reinforce the meaning of a formula or expression?

Change for the sake only of change is not progressive.

No, I very much agree to that, as I said.

Coming from someone who has never done mathematical research the call for change is more than just naive.

Most of the reason I'm bringing it up is that I'm planning to work on a visual programming language/computational mathematical notation for a design project. Trying to find some inspiration and similar projects I came here. Call it naive if you want. Its my time I'm wasting, not yours.

Interactivity and visualization are not the same as notation. Again, the only point to notation is that it conveys some meaning, and that only works by collective agreement.

Also, shapes won't let you write down 1,894,571 in any intuitive way, nor will they help with 3.141, much less .

Originally Posted by worldsayshi

For many mathematicians I suspect the idea to sound out of track and superfluous: Has there been any serious attempts at finding alternative notations for mathematics? Sure, you can call any programming language an alternative notation, but rather they are often just adaptions for a limited environment. What I'm looking for is projects that try to take it further and try to take advantages of tools we didn't have for writing notation when it was first started to develop. Using visual automation, graphical enhancements etc... Not for plotting graphs but for drawing the actual language itself. I'm not looking for a replacement either, just an alternative.
Maybe notation doesn't have to be one dimensional, written on a line, contating letters, but shapes morphing for a particular meaning (expressing mathematical relations using shapes?).
While there is no reason going about creating new types of notation for no reason I think that much can be won in terms of usability, efficiency and clarity, for both the learner and the user, by complete revision. Especially if you rethink how technology can be used for enhancement.
What do you think?

In terms of arithmetic, binomial arithmetic only uses the characters zero and one as operators. Arithmetic is very well understood, however.

Binary algebra only uses 0 and 1 as elements?

Originally Posted by MagiMaster

Binary algebra only uses 0 and 1 as elements?

I'm referring to the binomial numeral system, which uses only zero and one regardless of the operation carried out. The set A, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is equivalent to the following set, B, in the binomial numeral system {0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010}.

It's rather effective with regard to arithmetic and in fact I generally prefer it. However, the current arithmetic system is also rather good. Although, I am surprised to read that you are not particularly familiar with it considering you are a computer-scientist.

I am quite familiar with it, but, at least where I'm from, it's called binary numbers, not binomial numbers. And 0 and 1 would be elements, not operators. + and * are operators.

Originally Posted by MagiMaster

I am quite familiar with it, but, at least where I'm from, it's called binary numbers, not binomial numbers. And 0 and 1 would be elements, not operators. + and * are operators.

You, apparently, reside on planet Earth.

Originally Posted by MagiMaster

I am quite familiar with it, but, at least where I'm from, it's called binary numbers, not binomial numbers.

Nor is it called "binary algebra" (the appropriate terminology is binary arithmetic, with Boolean algebra being of relation). I'm not sure why I wrote "binomial;" This is indeed incorrect and so I ask of you to ignore of this mistake.

Originally Posted by MagiMaster

And 0 and 1 would be elements, not operators. + and * are operators.

Zero and one are not the only elements in the binary numeral system; however, they can be considered the subset of numbers that compose other numbers in the aforementioned system.

As far as operators are concerned, addition, subtraction, multiplication, division, and modulus are the operators of binary arithmetic, and in my introductory post I was referring to the only elements that the operators use (although I could see how the reason for this misunderstanding is on my behalf, and therefore apologize for my fault in this case).

I will try to weed out such mistakes in the future when posting; it is often the case that I only post before going to work (and I work nine hours a day and six days a week as an electrician) and am therefore in somewhat of a rush.

I also have a question Magimaster: would it be possible to implement Brownian motion as a solution to the random number generator?

Originally Posted by Ellatha

Originally Posted by MagiMaster

I am quite familiar with it, but, at least where I'm from, it's called binary numbers, not binomial numbers.

Nor is it called "binary algebra" (the appropriate terminology is binary arithmetic, with Boolean algebra being of relation). I'm not sure why I wrote "binomial;" This is indeed incorrect and so I ask of you to ignore of this mistake.

Originally Posted by MagiMaster

And 0 and 1 would be elements, not operators. + and * are operators.

Zero and one are not the only elements in the binary numeral system; however, they can be considered the subset of numbers that compose other numbers in the aforementioned system.

As far as operators are concerned, addition, subtraction, multiplication, division, and modulus are the operators of binary arithmetic, and in my introductory post I was referring to the only elements that the operators use (although I could see how the reason for this misunderstanding is on my behalf, and therefore apologize for my fault in this case).

I will try to weed out such mistakes in the future when posting; it is often the case that I only post before going to work (and I work nine hours a day and six days a week as an electrician) and am therefore in somewhat of a rush.

I also have a question Magimaster: would it be possible to implement Brownian motion as a solution to the random number generator?

Good advice -- When you have dug yourself a hole, quit digging.

Originally Posted by Ellatha

Zero and one are not the only elements in the binary numeral system; however, they can be considered the subset of numbers that compose other numbers in the aforementioned system.

As far as operators are concerned, addition, subtraction, multiplication, division, and modulus are the operators of binary arithmetic, and in my introductory post I was referring to the only elements that the operators use (although I could see how the reason for this misunderstanding is on my behalf, and therefore apologize for my fault in this case).

Well, {0, 1} would be the alphabet for an algebra over strings of those characters, but sometimes we work with just 0 and 1 (). Also, I didn't mean that + and - were the only possible operators, just that they were example operators. (In we usually only talk about + and * and sometimes ^.)

Anyway, what you're talking about is really just a relabelling of the integers in any case.

Originally Posted by Ellatha

I also have a question Magimaster: would it be possible to implement Brownian motion as a solution to the random number generator?

That question seems to be a bit of a non sequiter, and also a bit poorly worded. If you mean "can you use Brownian motion to implement a true random number generator" then, yes, but it might not be easy or practical.

Originally Posted by MagiMaster

That question seems to be a bit of a non sequiter, and also a bit poorly worded. If you mean "can you use Brownian motion to implement a true random number generator" then, yes, but it might not be easy or practical.

First of all, thank you for your answer. The reason I said "implement into a solution" is because I thought that Brownian motion may play a role into the solution rather than be entirely based around it. As for the randomness of my question: I thought it would be a good opportunity to ask a computer scientist a question relating to computer science that had long interested me (I'm not particularly well-read on this subject).

I am interested in learning from others when I get the chance, however the current education system is more based around memorization rather than understanding, and so I am an autodidact. Apparently some members seem to believe that I regard myself with some sort of supernatural neuro-kinesthetic cognitive aptitude, but I consider myself as no less and no more than than an individual interested in the truth of things, abstract or practical, or in a word a philosopher, with all the passion that I am, if that. (And so I thank you again for taking the time to help me).

Not to derail the thread, but yeah, any source of natural randomness can be used to generate true random numbers. All you need is something that returns a 0 or 1 with some truly random probability (as opposed to pseudo-random). Then there are standard ways of getting a fair random bit from that, and ways of getting any discrete distribution from that. (Continuous distributions can be estimated to any desired accuracy.) How often and how fair the original signal is will determine how fast you can generate other distributions though.