A Clash of Symbols
Nov. 4th, 2007 09:15 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
rabidsamfanWhy did I spend a whole essay on place value? Because it's the most inescapable example of a simple truth -- the written language of math is not the same as the spoken language of math. The history of math is a history of symbols, and in order to keep the number of symbols to a manageable level, position and context have become as significant as shape. Compared to mathematical notation, even the jumble of written English is positively straightforward. We might have more than one way to write the sound /f/, but at least "ph", "f" and "gh" always do stand for a sound, and you always read from right to left. If writing English was like writing numbers, the "k" in "ark" would stand for a phoneme, the "k" in "eke" would stand for a syllable and the "k" in "key" would stand for a whole word. And that's without a whiff of grammar!
Take a "2" and put it under the line in a fraction, it means "halves". Put it in superscript on the left hand side of the "x" and now it's an exponent -- which is an abbreviation for multiplication. Leave it standing alone and you've still got the option of interpreting it as "two over one". And of course, it changes value by order of magnitude in 182, 821, and 218, and that's before we even kick in a decimal point.
Along with the number symbols we've got + and - for addition and subtraction, which are both (fairly) consistent. But multiplication and division can be indicated in (ahem!) multiple ways.
3 x 4 is the same as 3*4 is the same as 3(4).
3 ÷ 4 is the same as 3/4 is the same as 3:4 and I don't know how to put a long division symbol in, but you'll have to believe me...
Adding to the fun, there are a few exceptions to the general rules. Mixed numbers don't just confuse us when we're learning about fractions and we get told that putting the whole number next to the fraction means adding, they come back to haunt us in Algebra, when we're told that juxtaposition means multiplication. So is "2" set down right next to a fraction that reads "x/y" equal to "2 + x/y" or "2x/2y"?
Don't like Algebra? I know whose name you should be cursing!
François Viète. (He published under the Latinate version of his name, Franciscus Vieta, which I'll use because it's faster than adding the special characters.)
Vieta was an amateur mathematician, and rich enough to pay for publishing his ideas, and luckily for us, they were mostly good ideas. In 1591 he published a treatise called In artem analyticam isagoge. In it he suggested that a regular system of notation be used to distinguish the true unknowns in an equation from the "coefficients" -- the numbers which would be known in a specific equation, but are undetermined in a general equation.
Like this...
4(unknown) + 7 = 35, 9(unknown) + 5 = 50, and 220(unknown) + 100 = 300 are all specific equations, but they all have a similar pattern and are solved for the unknown in the same way, (by subtracting the second term from both sides and then dividing both sides by the coefficient.) Vieta suggested using vowels for the unknowns and consonants for the knowns. So all three of those equations would look like BA + C = D. Fifty years later, Descartes used the same principle, but different letters -- and we still use Descartes' convention today. Letters near the beginning of the alphabet stand for knowns, letters near the end stand for unknowns. The modern general form of the equation would be ax + b = c.
A modern mathematician, told to "solve" ax + b = c wouldn't even begin to think of asking if that meant solve for "a" or "b" -- the only unknown there is the "x" -- the other letters stand for things which are unspecified. It's obvious! Isn't it? Just like it's obvious that black is the color of mourning! Er... well, except in places where the color of mourning is red, or yellow, or purple... That's the trouble with conventions -- they're artificial, and they need to be explained to the newbies. (Who are usually busy trying to grasp fifteen other concepts at the same time...)
But the really dreadful thing about Vieta's contribution to the world of mathematics is the effect it had on what was considered possible and what was considered impossible. Paraphrasing from Dantzig, prior to Vieta, most medieval algebrists could divide equations into "possible" and "impossible" depending on what they knew about the numbers involved.
x + 4 = 6 is possible, x + 6 = 4 is impossible.
2x = 8 is possible, 2x = 5 is impossible.
x2 = 9 is possible, x2 = 15 is not.
That's because before Vieta, negative numbers weren't considered fair play by anyone who wasn't a banker, numbers could only be divided by numbers which went into them evenly, and only perfect squares had square roots. And sure enough, at first the general equations made in Vieta's literal notation had little qualifiers. x + a = b if b is greater than a, ax = b can only be solved if b is a multiple of a, that sort of thing.
But if you're looking at an equation which is all made up of symbols, there's no obvious reason why those limitations should remain. It's faster, easier and better to change your ideas about what numbers are allowed to "exist". Bonus points if changing your ideas about numbers mean that some intractable problems from antiquity are suddenly solvable.
x = a - b works fine if you allow for negative numbers.
x = b/a is no problem at all if you don't mind your answer being a fraction, improper or proper, rather than an integer.
And the square root of a number that isn't a perfect square is acceptable if you don't mind being a little irrational about the whole thing.
Vieta's idea -- using symbols to stand for undetermined numbers as well as unknown quantities -- freed mathematicians from having to use words to describe what they were thinking about, and at the same time, elevated the symbols into things which could be manipulated in the same way that numbers were. You might want to define what the symbols stood for at the start of the problem, but once you had, you could gleefully ignore those definitions until you'd finished manipulating the symbols around.
Of course, the same economy of symbols which led to place value also led to mathematicians preferring to use old symbols rather than create new ones. Some new ones got created, of course -- the radical for square roots, which I can't construct in html, was one -- but for anything which indicated a sequence the number symbols were already around. (Like exponents!) And since they looked like numbers, some bright bunny started to think about whether they could be manipulated and... well, pretty soon you've got a page of symbols that ordinary mortals look at and think "I don't even know how to begin to read that out loud!"
(Not that reading equations out loud sounds much like what they mean anyway. Who on earth looks at a2 + b2 = c2 and says, "The square of the hypotenuse is equal to the sum of the squares of the other two sides" without a lot of prompting?)
Manipulating symbols around lets mathematicians to leave the concrete entirely and play in the abstract world. Ironically, it's the ability to play in the abstract world which has created the most useful applications of math back in the real world. Vieta freed math from the tyranny of words, but in the process things got more than a bit confusing.
If it's consolation, even people who like math get that feeling when the waters are deep. Dantzig took a look at what Bertrand Russell and Alfred North Whitehead were doing with symbols and said " ... my repeated efforts to master their involved symbolism have invariably resulted in helpless confusion and despair."
Next time, Math Words
Take a "2" and put it under the line in a fraction, it means "halves". Put it in superscript on the left hand side of the "x" and now it's an exponent -- which is an abbreviation for multiplication. Leave it standing alone and you've still got the option of interpreting it as "two over one". And of course, it changes value by order of magnitude in 182, 821, and 218, and that's before we even kick in a decimal point.
Along with the number symbols we've got + and - for addition and subtraction, which are both (fairly) consistent. But multiplication and division can be indicated in (ahem!) multiple ways.
3 x 4 is the same as 3*4 is the same as 3(4).
3 ÷ 4 is the same as 3/4 is the same as 3:4 and I don't know how to put a long division symbol in, but you'll have to believe me...
Adding to the fun, there are a few exceptions to the general rules. Mixed numbers don't just confuse us when we're learning about fractions and we get told that putting the whole number next to the fraction means adding, they come back to haunt us in Algebra, when we're told that juxtaposition means multiplication. So is "2" set down right next to a fraction that reads "x/y" equal to "2 + x/y" or "2x/2y"?
Don't like Algebra? I know whose name you should be cursing!
François Viète. (He published under the Latinate version of his name, Franciscus Vieta, which I'll use because it's faster than adding the special characters.)
Vieta was an amateur mathematician, and rich enough to pay for publishing his ideas, and luckily for us, they were mostly good ideas. In 1591 he published a treatise called In artem analyticam isagoge. In it he suggested that a regular system of notation be used to distinguish the true unknowns in an equation from the "coefficients" -- the numbers which would be known in a specific equation, but are undetermined in a general equation.
Like this...
4(unknown) + 7 = 35, 9(unknown) + 5 = 50, and 220(unknown) + 100 = 300 are all specific equations, but they all have a similar pattern and are solved for the unknown in the same way, (by subtracting the second term from both sides and then dividing both sides by the coefficient.) Vieta suggested using vowels for the unknowns and consonants for the knowns. So all three of those equations would look like BA + C = D. Fifty years later, Descartes used the same principle, but different letters -- and we still use Descartes' convention today. Letters near the beginning of the alphabet stand for knowns, letters near the end stand for unknowns. The modern general form of the equation would be ax + b = c.
A modern mathematician, told to "solve" ax + b = c wouldn't even begin to think of asking if that meant solve for "a" or "b" -- the only unknown there is the "x" -- the other letters stand for things which are unspecified. It's obvious! Isn't it? Just like it's obvious that black is the color of mourning! Er... well, except in places where the color of mourning is red, or yellow, or purple... That's the trouble with conventions -- they're artificial, and they need to be explained to the newbies. (Who are usually busy trying to grasp fifteen other concepts at the same time...)
But the really dreadful thing about Vieta's contribution to the world of mathematics is the effect it had on what was considered possible and what was considered impossible. Paraphrasing from Dantzig, prior to Vieta, most medieval algebrists could divide equations into "possible" and "impossible" depending on what they knew about the numbers involved.
x + 4 = 6 is possible, x + 6 = 4 is impossible.
2x = 8 is possible, 2x = 5 is impossible.
x2 = 9 is possible, x2 = 15 is not.
That's because before Vieta, negative numbers weren't considered fair play by anyone who wasn't a banker, numbers could only be divided by numbers which went into them evenly, and only perfect squares had square roots. And sure enough, at first the general equations made in Vieta's literal notation had little qualifiers. x + a = b if b is greater than a, ax = b can only be solved if b is a multiple of a, that sort of thing.
But if you're looking at an equation which is all made up of symbols, there's no obvious reason why those limitations should remain. It's faster, easier and better to change your ideas about what numbers are allowed to "exist". Bonus points if changing your ideas about numbers mean that some intractable problems from antiquity are suddenly solvable.
x = a - b works fine if you allow for negative numbers.
x = b/a is no problem at all if you don't mind your answer being a fraction, improper or proper, rather than an integer.
And the square root of a number that isn't a perfect square is acceptable if you don't mind being a little irrational about the whole thing.
Vieta's idea -- using symbols to stand for undetermined numbers as well as unknown quantities -- freed mathematicians from having to use words to describe what they were thinking about, and at the same time, elevated the symbols into things which could be manipulated in the same way that numbers were. You might want to define what the symbols stood for at the start of the problem, but once you had, you could gleefully ignore those definitions until you'd finished manipulating the symbols around.
Of course, the same economy of symbols which led to place value also led to mathematicians preferring to use old symbols rather than create new ones. Some new ones got created, of course -- the radical for square roots, which I can't construct in html, was one -- but for anything which indicated a sequence the number symbols were already around. (Like exponents!) And since they looked like numbers, some bright bunny started to think about whether they could be manipulated and... well, pretty soon you've got a page of symbols that ordinary mortals look at and think "I don't even know how to begin to read that out loud!"
(Not that reading equations out loud sounds much like what they mean anyway. Who on earth looks at a2 + b2 = c2 and says, "The square of the hypotenuse is equal to the sum of the squares of the other two sides" without a lot of prompting?)
Manipulating symbols around lets mathematicians to leave the concrete entirely and play in the abstract world. Ironically, it's the ability to play in the abstract world which has created the most useful applications of math back in the real world. Vieta freed math from the tyranny of words, but in the process things got more than a bit confusing.
If it's consolation, even people who like math get that feeling when the waters are deep. Dantzig took a look at what Bertrand Russell and Alfred North Whitehead were doing with symbols and said " ... my repeated efforts to master their involved symbolism have invariably resulted in helpless confusion and despair."
Next time, Math Words
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
(no subject)
Date: 2007-11-05 02:48 am (UTC)(sorry, brain is too fried to actually read your entry right now, but these have been very informative! :-)))
(no subject)
Date: 2007-11-05 01:42 pm (UTC)Er, is it time for breakfast already?
(no subject)
Date: 2007-11-05 03:00 am (UTC)(no subject)
Date: 2007-11-05 01:42 pm (UTC)(no subject)
Date: 2007-11-05 04:05 am (UTC)The 'explanations' of arithmetic, I never understood. I did better memorizing. The explanations with algebra, etc - those, I got.
(no subject)
Date: 2007-11-05 01:41 pm (UTC)(no subject)
Date: 2007-11-05 11:18 am (UTC)What a perfect quote! I'll have to remember *that* one!
(no subject)
Date: 2007-11-05 01:40 pm (UTC)