A good notation reflects the underlying mathematics. Early mathematical problems were written as text. See for example the Moscow Mathematical Papyrus.. A problem is given as...

A number taken 1 and 1/2 times and added to 4 to make 10. What is the number?

The method of solution is given as...

Calculate the excess of this 10 over 4. The result is 6. You operate on 1 and 1/2 to find 1. The result is 2/3. You take 2/3 of this 6. The result is 4.

If we represent the unknown as X then we have the equation...

X + X/2 + 4 = 10

The power of using the equals sign is that the left and right arguments are treated equally. When we first learned how to simplify equations, we would have subtracted 4 from both sides. We then learned a typographical rule that lets us take the 4 from the left, change its sign, and put it on the right...

X + X/2 = 6

We can then multiply everything by two to eliminate the fraction...

2X + X = 12

...and so on. We cannot just reply on typographical rules. If X was the divisor, then we could have to remember to treat the special case where X is zero. But it makes things a lot easier.

There are lots of ways of writing two times X. We can write 2X, and some mathematicians do, but we cannot write two times two this way. A multiplication sign (dot, cross, or star) represents the multiplication by a symbol, and makes the notation the same for numbers and symbols.

Sometimes we get stuck with the second-best choice. The ancients took the pi symbol to represent the ratio of a circle's circumference to its diameter. It might have been better to choose the ratio of the circumference to the radius, but it is probably too late to change things now.

Let's pretend we are inventing a new notation. I never liked the notation for numeric intervals where...

[A,B)

...represents the numeric range between A and B, including A, but not including B. The unbalanced brackets bugs me, and will bug most text editors. I would like to write this as...

[A:]B

I have replaced the comma with a colon, which is used in some programming notations as a vertical ellipsis. [A:B] is used in some computer languages to represent the range from A to B. I have taken B out of the range by putting it outside the square brackets denoting the range.

If I wrote this in a paper, people would reject it as non-standard notation. However, if that had been suggested early in the evolution of set theory, it might have stuck. I think this is what happens with notation in general: there is a short period when everything is new, and you can rework the notation. There may be special fields where alternative notations flourish, such as suffix notation for matrices and tensors. But the opportunity for radical change evaporates once the notation is in general use.

A good notation reflects the underlying mathematics. Early mathematical problems were written as text. See for example the Moscow Mathematical Papyrus.. A problem is given as...

A number taken 1 and 1/2 times and added to 4 to make 10. What is the number?

The method of solution is given as...

Calculate the excess of this 10 over 4. The result is 6. You operate on 1 and 1/2 to find 1. The result is 2/3. You take 2/3 of this 6. The result is 4.

If we represent the unknown as X then we have the equation...

X + X/2 + 4 = 10

The power of using the equals sign is that the left and right arguments are treated equally. When we first learned how to simplify equations, we would have subtracted 4 from both sides. We then learned a typographical rule that lets us take the 4 from the left, change its sign, and put it on the right...

X + X/2 = 6

We can then multiply everything by two to eliminate the fraction...

2X + X = 12

...and so on. We cannot just rely on typographical rules. If X was the divisor, then we could have to remember to treat the special case where X is zero. But it makes things a lot easier.

There are lots of ways of writing two times X. We can write 2X, and some mathematicians do, but we cannot write two times two this way. A multiplication sign (dot, cross, or star) represents the multiplication by a symbol, and makes the notation the same for numbers and symbols.

Sometimes we get stuck with the second-best choice. The ancients took the pi symbol to represent the ratio of a circle's circumference to its diameter. It might have been better to choose the ratio of the circumference to the radius, but it is probably too late to change things now.

Let's pretend we are inventing a new notation. I never liked the notation for numeric intervals where...

[A,B)

...represents the numeric range between A and B, including A, but not including B. The unbalanced brackets bugs me, and will bug most text editors. I would like to write this as...

[A:]B

I have replaced the comma with a colon, which is used in some programming notations as a vertical ellipsis. [A:B] is used in some computer languages to represent the range from A to B. I have taken B out of the range by putting it outside the square brackets denoting the range.

If I wrote this in a paper, people would reject it as non-standard notation. However, if that had been suggested early in the evolution of set theory, it might have stuck. I think this is what happens with notation in general: there is a short period when everything is new, and you can rework the notation. There may be special fields where alternative notations flourish, such as suffix notation for matrices and tensors. But the opportunity for radical change evaporates once the notation is in general use.

A good notation reflects the underlying mathematics. Early mathematical problems were written as text. See for example the Moscow Mathematical Papyrusyrus.. A problem is given as...

A number taken 1 and 1/2 times and added to 4 to make 10. What is the number?

The method of solution is given as...

Calculate the excess of this 10 over 4. The result is 6. You operate on 1 and 1/2 to find 1. The result is 2/3. You take 2/3 of this 6. The result is 4.

If we represent the unknown as X then we have the equation...

X + X/2 + 4 = 10

The power of using the equals sign is that the left and right arguments are treated equally. When we first learned how to simplify equations, we would have subtracted 4 from both sides. We then learned a typographical rule that lets us take the 4 from the left, change its sign, and put it on the right...

X + X/2 = 6

We can then multiply everything by two to eliminate the fraction...

2X + X = 12

...and so on. We cannot just reply on typographical rules. If X was the divisor, then we could have to remember to treat the special case where X is zero. But it makes things a lot easier.

There are lots of ways of writing two times X. We can write 2X, and some mathematicians do, but we cannot write two times two this way. A multiplication sign (dot, cross, or star) represents the multiplication by a symbol, and makes the notation the same for numbers and symbols.

Sometimes we get stuck with the second-best choice. The ancients took the pi symbol to represent the ratio of a circle's circumference to its diameter. It might have been better to choose the ratio of the circumference to the radius, but it is probably too late to change things now.

Let's pretend we are inventing a new notation. I never liked the notation for numeric intervals where...

[A,B)

...represents the numeric range between A and B, including A, but not including B. The unbalanced brackets bugs me, and will bug most text editors. I would like to write this as...

[A:]B

I have replaced the comma with a colon, which is used in some programming notations as a vertical ellipsis. [A:B] is used in some computer languages to represent the range from A to B. I have taken B out of the range by putting it outside the square brackets noting the range.

If I wrote this in a paper, people would reject it as non-standard notation. However, if that had been suggested early in the evolution of set theory, it might have stuck. I think this is what happens with notation in general: there is a short period when everything is new, and you can rework the notation. There may be special fields where alternative notations flourish, such as suffix notation for matrices and tensors. But the opportunity for radical change evaporates once the notation is in general use.

A good notation reflects the underlying mathematics. Early mathematical problems were written as text. See for example the Moscow Mathematical Papyrus.. A problem is given as...

A number taken 1 and 1/2 times and added to 4 to make 10. What is the number?

The method of solution is given as...

Calculate the excess of this 10 over 4. The result is 6. You operate on 1 and 1/2 to find 1. The result is 2/3. You take 2/3 of this 6. The result is 4.

If we represent the unknown as X then we have the equation...

X + X/2 + 4 = 10

The power of using the equals sign is that the left and right arguments are treated equally. When we first learned how to simplify equations, we would have subtracted 4 from both sides. We then learned a typographical rule that lets us take the 4 from the left, change its sign, and put it on the right...

X + X/2 = 6

We can then multiply everything by two to eliminate the fraction...

2X + X = 12

...and so on. We cannot just reply on typographical rules. If X was the divisor, then we could have to remember to treat the special case where X is zero. But it makes things a lot easier.

There are lots of ways of writing two times X. We can write 2X, and some mathematicians do, but we cannot write two times two this way. A multiplication sign (dot, cross, or star) represents the multiplication by a symbol, and makes the notation the same for numbers and symbols.

Sometimes we get stuck with the second-best choice. The ancients took the pi symbol to represent the ratio of a circle's circumference to its diameter. It might have been better to choose the ratio of the circumference to the radius, but it is probably too late to change things now.

Let's pretend we are inventing a new notation. I never liked the notation for numeric intervals where...

[A,B)

...represents the numeric range between A and B, including A, but not including B. The unbalanced brackets bugs me, and will bug most text editors. I would like to write this as...

[A:]B

I have replaced the comma with a colon, which is used in some programming notations as a vertical ellipsis. [A:B] is used in some computer languages to represent the range from A to B. I have taken B out of the range by putting it outside the square brackets denoting the range.

If I wrote this in a paper, people would reject it as non-standard notation. However, if that had been suggested early in the evolution of set theory, it might have stuck. I think this is what happens with notation in general: there is a short period when everything is new, and you can rework the notation. There may be special fields where alternative notations flourish, such as suffix notation for matrices and tensors. But the opportunity for radical change evaporates once the notation is in general use.