# One plus one is equal to two - calculus in text is left as an excercise

May 24, 2014 8:00 AM Subscribe

I was surprised to learn that few people knew that almost all maths was written rhetorically before the 16th century, often in metered poetry. Even our wonderful symbol for equality – you know, those two parallel lines – was not used in print before 1575.

And they were all recited by parrots, hence the term "polymath." (Ducks shoe)

posted by dances_with_sneetches at 8:21 AM on May 24, 2014 [9 favorites]

posted by dances_with_sneetches at 8:21 AM on May 24, 2014 [9 favorites]

*I was surprised to learn....*

Oh please, you weren't born knowing this.

I expect he goes into it in greater detail in his book (which I shall put on the list). If renaissance mathematicians were not using symbols, they did use abbreviations for things such as plus and minus (p and m with lines over them).

Whestone of Witte is available in modern edition

posted by IndigoJones at 8:25 AM on May 24, 2014

It's interesting that moon4 +othermoon6 =2, rather than othermoon2.

posted by aniola at 8:27 AM on May 24, 2014 [2 favorites]

posted by aniola at 8:27 AM on May 24, 2014 [2 favorites]

I love the debate in the comments of the article about what that strange looking equation actually says. It looks like they decided on 14x+15=71 even though the article says it's 14x^2+15=71. I wouldn't have even guessed there was an x in it at all. Really interesting read and the comments link to a book that I may have to get my hands on.

Thanks!

posted by one4themoment at 8:29 AM on May 24, 2014

Thanks!

posted by one4themoment at 8:29 AM on May 24, 2014

1 + 1 = 10 in binary

posted by Orion Blastar at 8:43 AM on May 24, 2014

posted by Orion Blastar at 8:43 AM on May 24, 2014

Yeah, I have a set of the "Great Books" from my grandparents, and it includes a couple books by Newton. I remember trying to read Optics and just being totally fucking lost, even when I tried to keep track by symbols in my notes. Now, granted, I've forgotten a significant amount of my calculus, and so there's plenty of symbolic math that I just don't get either, but at least I feel like I could understand it in a lot less time than puzzling out the abstruse text of early modern mathematicians.

posted by klangklangston at 8:44 AM on May 24, 2014

posted by klangklangston at 8:44 AM on May 24, 2014

The standard history is Florian Cajori's A History of Mathematical Notations.

Jeff Miller also has a good website on the Earliest Uses of Various Mathematical Symbols.

posted by James Scott-Brown at 8:50 AM on May 24, 2014 [14 favorites]

Jeff Miller also has a good website on the Earliest Uses of Various Mathematical Symbols.

posted by James Scott-Brown at 8:50 AM on May 24, 2014 [14 favorites]

Those interested in the history of mathematical notation will find great pleasure in Florian Cajori's two-volume magnum opus on the subject. Published in 1928, it is now in the public domain and can be downloaded for free from the Internet archive.

posted by pguertin at 8:51 AM on May 24, 2014 [9 favorites]

posted by pguertin at 8:51 AM on May 24, 2014 [9 favorites]

I was recently introduced to Tartaglia's solution to a depressed cubic equation. I have a translation somewhere that rhymes, but I can't seem to find it. I like that x^3 + ax = b, x^3 = ax + b, and x^3 + b = ax are treated as three separate cases. The mindset of being perfectly okay with taking cube-roots but not entirely comfortable with negative numbers is strange to me, but such was the case for a long time.

posted by The Great Big Mulp at 8:56 AM on May 24, 2014 [5 favorites]

posted by The Great Big Mulp at 8:56 AM on May 24, 2014 [5 favorites]

pguertin, your link was a free download, so it was better. Maybe. The Internet Archive says the scan has a few problems, it says some pages are truncated on the left or right. It seems OK so far, other than the title page reading "A History of Athematical Notations." Actually, that is a book that should exist and I would read it.

But the Guardian article itself is kind of thin. It kind of misses the point, the development of mathematics is tightly bound to the development of symbology to express it. And there were early mathematical expressions that did not involve writing or anything other than the raw numbers. Cajori depicts cuneiform tablets that are just lists of numbers, which look surprising like columnar spreadsheets. He also shows a quipu, which are just bundles of knotted cords, each representing a number. This book is well worth exploring.

posted by charlie don't surf at 9:10 AM on May 24, 2014

But the Guardian article itself is kind of thin. It kind of misses the point, the development of mathematics is tightly bound to the development of symbology to express it. And there were early mathematical expressions that did not involve writing or anything other than the raw numbers. Cajori depicts cuneiform tablets that are just lists of numbers, which look surprising like columnar spreadsheets. He also shows a quipu, which are just bundles of knotted cords, each representing a number. This book is well worth exploring.

posted by charlie don't surf at 9:10 AM on May 24, 2014

"

Kind of? I've seen football articles 3 times as long: he could have written a long and interesting article but this is almost nerd clickbait. And I love how the comments descend into an argument on the nature of magnetism. Oh grauniad.

posted by marienbad at 9:29 AM on May 24, 2014 [1 favorite]

*But the Guardian article itself is kind of thin.*"Kind of? I've seen football articles 3 times as long: he could have written a long and interesting article but this is almost nerd clickbait. And I love how the comments descend into an argument on the nature of magnetism. Oh grauniad.

posted by marienbad at 9:29 AM on May 24, 2014 [1 favorite]

[derail]

A common problem for autodidacts is not being able to remember what they picked up on their own vs. what was part of a normal public education. In addition there are pieces of lore (the SAIT joke for example) which are so endemic to one social group that you get confused when they get zero recognition in another.

In short I don't think he was being snotty and claiming that somehow people should have known this, just that he was caught off guard when they didn't.

posted by Tell Me No Lies at 9:45 AM on May 24, 2014 [5 favorites]

*>I was surprised to learn....*

Oh please, you weren't born knowing this.Oh please, you weren't born knowing this.

A common problem for autodidacts is not being able to remember what they picked up on their own vs. what was part of a normal public education. In addition there are pieces of lore (the SAIT joke for example) which are so endemic to one social group that you get confused when they get zero recognition in another.

In short I don't think he was being snotty and claiming that somehow people should have known this, just that he was caught off guard when they didn't.

posted by Tell Me No Lies at 9:45 AM on May 24, 2014 [5 favorites]

One of my pet theories is that a "worse is better" dynamic is why abstract mathematical notation developed in Europe rather than the Arab World.

Having learned Arabic and written at it, I can say from experience that not only is the writing system amenable to calligraphic flourishes, it's also the most comfortable writing system to be using with a quill and paper. You can really go at it for hours and hours. So between the invention of wood paper and the invention of moveable type, the health of a literate culture depended the most on the comfort and efficiency of the scrivener class, and the Muslim World had the clear advantage.

That's why there was more impetus to develop shorthand and movable type in the Latin World rather than the Arab World.

posted by ocschwar at 9:54 AM on May 24, 2014 [5 favorites]

Having learned Arabic and written at it, I can say from experience that not only is the writing system amenable to calligraphic flourishes, it's also the most comfortable writing system to be using with a quill and paper. You can really go at it for hours and hours. So between the invention of wood paper and the invention of moveable type, the health of a literate culture depended the most on the comfort and efficiency of the scrivener class, and the Muslim World had the clear advantage.

That's why there was more impetus to develop shorthand and movable type in the Latin World rather than the Arab World.

posted by ocschwar at 9:54 AM on May 24, 2014 [5 favorites]

Hey, my last AskMe was about the Pythagorean Theorem in the original Greek. And not only does it not use symbols, it doesn't even say "plus", "added", or "combined" anywhere.

posted by benito.strauss at 10:00 AM on May 24, 2014 [2 favorites]

posted by benito.strauss at 10:00 AM on May 24, 2014 [2 favorites]

I know someone who, in the wake of brain trauma, lost the ability to write language at all, either print or cursive, but whose ability to write numbers and numerical expressions was unimpaired; I only wish I knew how he handled algebraic expressions containing letters as unknowns, but I suspect that would have been very difficult for him, and I've often wondered whether the separated existence of numbers and letters in his brain which this implies is reflected in the blank and occasionally almost incorrigible incomprehension many kids seem to experience when they first encounter numbers and letters mixed together in algebraic equations.

Maybe our much greater facility with numbers and arithmetical thinking compared to the average literate person of the 16th century is a result of learning-- in the developmental process under the influence of education-- how to hive off a part of our brains and devote it to them.

Language per se just doesn't seem to be up to the job.

posted by jamjam at 10:18 AM on May 24, 2014 [3 favorites]

Maybe our much greater facility with numbers and arithmetical thinking compared to the average literate person of the 16th century is a result of learning-- in the developmental process under the influence of education-- how to hive off a part of our brains and devote it to them.

Language per se just doesn't seem to be up to the job.

posted by jamjam at 10:18 AM on May 24, 2014 [3 favorites]

*In short I don't think he was being snotty and claiming that somehow people should have known this, just that he was caught off guard when they didn't.*

posted by Tell Me No Lies at 12:45 PM on May 24 [+] [!]

posted by Tell Me No Lies at 12:45 PM on May 24 [+] [!]

That was my take on it, too. And I'm one of the ones who had no idea that math wasn't always equations. This concept blows my mind.

posted by FirstMateKate at 10:23 AM on May 24, 2014

After many years away from math, I started taking Calculus again about 9 months ago. I knew something had changed when I picked up a copy of Marx's Capital, Vol. 3 yesterday. When I got to a part with some discussion of the rate of surplus value compared to the rate of profit, I was able to sit parse through it with a pen and paper, and then notice that Marx had chosen genuinely terrible notation to express what he was trying to say. At the same time I took a look at a commentary with a somewhat turgid description of the falling rate of profit and the relationship between two variables (surplus value and constant capital) to variable capital. Realized that he was trying to narratively explain something that actually called for multivariable calculus.

I'm just starting to see it now. Pretty fascinating. The specialized symbolic logic of math is pretty powerful-- I'm in awe.

posted by wuwei at 10:55 AM on May 24, 2014 [4 favorites]

I'm just starting to see it now. Pretty fascinating. The specialized symbolic logic of math is pretty powerful-- I'm in awe.

posted by wuwei at 10:55 AM on May 24, 2014 [4 favorites]

As Hawking said, if your book has equations you lose half your readers. For me, I struggle through books with equations having a smattering of math education, but in the back of my mind I keep thinking that someone somewhere could write a guide to reading equations for us non-mathematicians so we could at least parse them into something more intelligible. But mathematicians make this hard because their own symbology is inconsistent. What looks like an operator might be a variable while that variable might be an operator and that equation there isn't. It's a name for something. I asked Roger Penrose once about this and he said that it just requires years of study to understand all this. I said that sounds like Masonic hermeticism....

posted by njohnson23 at 11:00 AM on May 24, 2014 [3 favorites]

posted by njohnson23 at 11:00 AM on May 24, 2014 [3 favorites]

I can't wait to tell every kid I help with their math homework that all problems used to be

The moon example reminds of my my childhood. As a kid, my younger brother wanted to prove that he was smarter than me (probably because I was a jerk and simply had a few years of math education he did not yet have) so he invented all sorts of retorts/word problems like "what's apple plus apply? ... orange. Duh!".

@IndigoJones:

Sometimes you learn things and they're fairly intuitive. Other times you learn something and it's surprising in that it isn't what you would have assumed or expected. This one of those latter instances for OP (and it seems, many others).

posted by Brian Puccio at 11:03 AM on May 24, 2014 [1 favorite]

*word problems*. I am so excited.The moon example reminds of my my childhood. As a kid, my younger brother wanted to prove that he was smarter than me (probably because I was a jerk and simply had a few years of math education he did not yet have) so he invented all sorts of retorts/word problems like "what's apple plus apply? ... orange. Duh!".

@IndigoJones:

No one is saying that.I was surprised to learn....

Oh please, you weren't born knowing this.

Sometimes you learn things and they're fairly intuitive. Other times you learn something and it's surprising in that it isn't what you would have assumed or expected. This one of those latter instances for OP (and it seems, many others).

posted by Brian Puccio at 11:03 AM on May 24, 2014 [1 favorite]

*Can an Equation be a Poem?*

Bernoulli would have been content to die,

Had he but known such a^2 cos(2phi)!

BTW, this is the ending of a famous poem by Stanislaw Lem (with major credit to the translator Michael Kandel) and makes a lot more sense if you plot the equation and see the sinuous, sensuous, undulating form.

posted by charlie don't surf at 11:25 AM on May 24, 2014 [2 favorites]

*who, in the wake of brain trauma, lost the ability to write language at all, either print or cursive, but whose ability to write numbers and numerical expressions was unimpaired; I only wish I knew how he handled algebraic expressions containing letters as unknowns*

He almost certainly handled them without difficulty because in this usage letters aren't expressing language but are used as abstract symbolic number containers. While the "meaningful" variable names used in programming might cause him to scratch his head I'm sure symbols like x, y, and z would have been just as accessible as + and the integral sign.

posted by localroger at 12:30 PM on May 24, 2014 [1 favorite]

*No one is saying that.*

The author affected surprise that a piece of extremely arcane knowledge was not a commonplace, even among his sophisticated friends. I know lots of obscure crap too, and I'm aware of the phenom being caught up short when others do not. I was also raised to believe that it is a form of showing off to point the fact out.

He may not have meant to be condescending, but it all too easily comes off that way. A little alienating if you're trying to encourage interest in a pet subject (his book on which, note, I said was going on my to-read list). "Hey guys, check this out!" would have gone down smoother.

posted by IndigoJones at 12:46 PM on May 24, 2014

My vote is that we call this a derail and drop it, and that people who want to discuss it further take it to memail.

posted by benito.strauss at 1:19 PM on May 24, 2014 [1 favorite]

posted by benito.strauss at 1:19 PM on May 24, 2014 [1 favorite]

Wow - I really like that the reason we have equals is "gemini lines" because there can be nothing more equal... that is "twins"...

posted by symbioid at 2:05 PM on May 24, 2014

posted by symbioid at 2:05 PM on May 24, 2014

YOU WON'T BELIEVE WHAT I JUST LEARNED ABOUT MATHEMATICAL SYMBOLISM!

posted by symbioid at 2:06 PM on May 24, 2014 [2 favorites]

posted by symbioid at 2:06 PM on May 24, 2014 [2 favorites]

In the future all mathematics will be taught through interpretive dance.

posted by blue_beetle at 2:09 PM on May 24, 2014

posted by blue_beetle at 2:09 PM on May 24, 2014

Keep in mind that before the invention of arabic numerals you couldn't have things like long addition, subtraction, multiplication and division. Those Greek mathematicians did all their work on abaci and in their heads. There was no way to write it out, except in explanatory text.

posted by alms at 4:09 PM on May 24, 2014 [1 favorite]

posted by alms at 4:09 PM on May 24, 2014 [1 favorite]

*My vote is that we call this a derail and drop it, and that people who want to discuss it further take it to memail.*

Yeah and next thing you're going to bring semiotics into the argument, that sure won't go well.

posted by sammyo at 4:14 PM on May 24, 2014

HAHAHAHA...

And this is fucking hilarious, because... Plato's Meno expounds the Socratic theory of anamnesis. That is to say, anamnesis is the act of remembering that which you had forgotten upon your incarnation into the human state... In other words,

posted by symbioid at 4:32 PM on May 24, 2014 [6 favorites]

IndigoJones: "followed by the link...I was surprised to learn...."

Oh please,you weren't born knowing this.

benito.strauss: "Wherein, a comment links to this Socratic dialog from Meno.Hey, my last AskMe was about the Pythagorean Theorem in the original Greek."

And this is fucking hilarious, because... Plato's Meno expounds the Socratic theory of anamnesis. That is to say, anamnesis is the act of remembering that which you had forgotten upon your incarnation into the human state... In other words,

**you WERE born knowing this**, you just FORGOT you knew it.posted by symbioid at 4:32 PM on May 24, 2014 [6 favorites]

About 8 years ago, I wrote a short paper discussing a relationship a student had found (by luck, on an exam, doing the problem wrong) between cross-ratios and ratios of cross products. One of the interesting things we discovered is that the notation for the cross-product of two vectors is actually very recent---as near as we could tell, it appeared in the 1880s.

posted by leahwrenn at 5:49 PM on May 24, 2014 [2 favorites]

posted by leahwrenn at 5:49 PM on May 24, 2014 [2 favorites]

Warning: That Jeff Miller page that James Scott-Brown linked to? A person can end up spending a lot of time there.

posted by benito.strauss at 7:06 PM on May 24, 2014 [1 favorite]

posted by benito.strauss at 7:06 PM on May 24, 2014 [1 favorite]

I love the length of that equals sign.

It's like 16th century scholars programmed in javascript.

posted by zoo at 10:42 AM on May 25, 2014 [2 favorites]

It's like 16th century scholars programmed in javascript.

posted by zoo at 10:42 AM on May 25, 2014 [2 favorites]

*I was surprised to learn....*

Thanks to the 'net, I'm "surprised to learn" damned-near every day. Ah well ... IIRC someone said that knowing the extent of our ignorance is the beginning of real knowledge. Given our limited time and the scope of the mystery, I guess ignorance ===== permanent situation.

posted by Twang at 11:35 AM on May 25, 2014

Relevant: woodcut of a smug Boethius showing up a dismayed Pythagoras with his fancy Arabic notation tech.

posted by Erasmouse at 12:38 PM on May 25, 2014 [3 favorites]

posted by Erasmouse at 12:38 PM on May 25, 2014 [3 favorites]

I literally have Heath's translation of Diophantus's

Diophantus played a role in the intermediate stage of mathematical notation where he used some limited notation for various things; Heath devotes Chapter III of his introduction,

All of the great Arab algebraists, from al-Khwarizmi through al-Qalasadi (1412–1486 CE) used some notation and each expanded it; by al-Qalasadi there was quite a bit of notation established.

To place the development of mathematical notation squarely and (almost exclusively) in Europe in the early modern period is to be eurocentric, just as it's the case that the western mathematical canon (which I studied in college) is quite annoyingly eurocentric when it just takes the development of algebra for granted with the introduction of cartesian geometry. There was a lot of groundwork laid for the symbolic abstraction of mathematics which proved so very powerful when it flourished — it was, indeed, an incredibly important and enabling development, but it didn't come from nowhere, fully-formed. No small part of it was already there in the work of the Arabs and, conversely, it was not immediately fully articulated, regularized, and established for a very long time in European mathematics, as any student of this subject is well aware of.

† It's been on my Amazon wish list and I got it this year for xmas.

posted by Ivan Fyodorovich at 5:00 PM on May 25, 2014 [9 favorites]

*Arithmetica*(about 250 CE) right here two steps away from me in my bedroom† — in this work, Diophantus did a lot of the ground-work for the development of algebra, and it was a primary influence for al-Khwarizmi's*The Compendious Book on Calculation by Completion and Balancing*(820 CE) (الكتاب المختصر في حساب الجبر والمقابلة), the title transliterated as*Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala*, which gives us the term*algebra*(and an older transliteration of Al-Khwarizmi's name,*Algoritmi*, gives us the term*algorithm*).Diophantus played a role in the intermediate stage of mathematical notation where he used some limited notation for various things; Heath devotes Chapter III of his introduction,

*Notations and Definitions of Diophantus*, to this. It's quite interesting and it shows that there was a fair amount of notation in*Arithmetica*.All of the great Arab algebraists, from al-Khwarizmi through al-Qalasadi (1412–1486 CE) used some notation and each expanded it; by al-Qalasadi there was quite a bit of notation established.

To place the development of mathematical notation squarely and (almost exclusively) in Europe in the early modern period is to be eurocentric, just as it's the case that the western mathematical canon (which I studied in college) is quite annoyingly eurocentric when it just takes the development of algebra for granted with the introduction of cartesian geometry. There was a lot of groundwork laid for the symbolic abstraction of mathematics which proved so very powerful when it flourished — it was, indeed, an incredibly important and enabling development, but it didn't come from nowhere, fully-formed. No small part of it was already there in the work of the Arabs and, conversely, it was not immediately fully articulated, regularized, and established for a very long time in European mathematics, as any student of this subject is well aware of.

† It's been on my Amazon wish list and I got it this year for xmas.

posted by Ivan Fyodorovich at 5:00 PM on May 25, 2014 [9 favorites]

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allword problems.On another note this is an absolutely fascinating article. Thanks for posting it.

posted by Tell Me No Lies at 8:05 AM on May 24, 2014