Not breaking news, but a bit of intellectual history is always a pleasant distraction: the BBC on the House of Wisdom, and the introduction of Arabic numberals to Europe.
How modern mathematics emerged from a lost Islamic library:
The House of Wisdom sounds a bit like make believe: no trace remains of this ancient library, destroyed in the 13th Century, so we cannot be sure exactly where it was located or what it looked like.
But this prestigious academy was in fact a major intellectual powerhouse in Baghdad during the Islamic Golden Age, and the birthplace of mathematical concepts as transformative as the common zero and our modern-day "Arabic" numerals.
Founded as a private collection for caliph Harun Al-Rashid in the late 8th Century then converted to a public academy some 30 years later, the House of Wisdom appears to have pulled scientists from all over the world towards Baghdad, drawn as they were by the city's vibrant intellectual curiosity and freedom of expression (Muslim, Jewish and Christian scholars were all allowed to study there).
An archive as formidable in size as the present-day British Library in London or the Bibliothèque Nationale of Paris, the House of Wisdom eventually became an unrivalled centre for the study of humanities and sciences, including mathematics, astronomy, medicine, chemistry, geography, philosophy, literature and the arts – as well as some more dubious subjects such as alchemy and astrology.
And, of course, centres of higher learning are commonly attacked by barbarians, so بيت الحكمة shared the fate of the Library of Alexandria, Taxila, and Nalanda:
The House of Wisdom was destroyed in the Mongol Siege of Baghdad in 1258 (according to legend, so many manuscripts were tossed into the River Tigris that its waters turned black from ink), but the discoveries made there introduced a powerful, abstract mathematical language that would later be adopted by the Islamic empire, Europe, and ultimately, the entire world.
And who was the instrument of this Muslim invasion of the West? That's right, Fibonacci!
Tracing the House of Wisdom's mathematical legacy involves a bit of time travel back to the future, as it were. For hundreds of years until the ebb of the Italian Renaissance, one name was synonymous with mathematics in Europe: Leonardo da Pisa, known posthumously as Fibonacci. Born in Pisa in 1170, the Italian mathematician received his primary instruction in Bugia, a trading enclave located on the Barbary coast of Africa (coastal North Africa). In his early 20s, Fibonacci traveled to the Middle East, captivated by ideas that had come west from India through Persia. When he returned to Italy, Fibonacci published Liber Abbaci, one of the first Western works to describe the Hindu-Arabic numeric system.
Some, however, still resist these new-fangled number systems from the East, but even so, Roman numerals are disappearing.
A vestige of sign-value notation, Roman numerals somehow persisted despite the introduction of Al-Khwarizmi's system, which relied on the position of digits to represent quantities. Like the towering monuments on which they were inscribed, Roman numerals outlived the empire that gave birth to them – whether by accident, sentiment or purpose, none can say for sure.
This year marks the 850th anniversary of Fibonacci's birth. It could also be the moment which threatens to undo the journeywork of Roman numerals. In the UK, traditional time-pieces have been replaced with easier-to-read digital clocks in school classrooms, for fear students can no longer tell analogue time properly. In some regions of the world, governments have dropped them from road signs and official documents, while Hollywood has moved away from using Roman numerals in sequel titles. The Superbowl famously ditched them for its 50th game, worried it was confusing fans.
But a global shift away from Roman numerals underscores a creeping innumeracy in other aspects of life. Perhaps more important, the disappearance of Roman numerals reveals the politics that govern any wider discussion about mathematics.
Innumerable people seem concerned about the large increase in innumeracy among the set of younger people today.
(Score: 4, Funny) by EJ on Thursday December 10 2020, @08:53AM (4 children)
"666 words in story"
(Score: 1, Interesting) by Anonymous Coward on Thursday December 10 2020, @10:39AM (1 child)
"DCLXVI words in story" FTFY
In itself interesting, because use all the first six symbols, in sequence ^_^;
https://en.wikipedia.org/wiki/Roman_numerals [wikipedia.org]
CYA
(Score: -1, Troll) by Anonymous Coward on Thursday December 10 2020, @01:08PM
deliberately childish exclamations vociferously intoned?
(Score: 0, Insightful) by Anonymous Coward on Thursday December 10 2020, @02:50PM (1 child)
Posted by Ari. Did you expect anything otherwise?
(Score: 0, Flamebait) by Anonymous Coward on Thursday December 10 2020, @05:20PM
Triggered by histoey bruh? Try Ignorall, now in fruity pebble flavors!
How predictable you bottom feeders are.
(Score: 4, Insightful) by stretch611 on Thursday December 10 2020, @09:08AM (15 children)
The difference today is that barbarians have been replaced by fundamentalist religions...
...and many politicians bribed to ignore science by those in real power to stay in power.
Now with 5 covid vaccine shots/boosters altering my DNA :P
(Score: 3, Interesting) by loonycyborg on Thursday December 10 2020, @10:05AM (3 children)
Mongols were progressive expansionist force in that time, but they believed that their own knowledge and culture are inherently superior to that of their enemies. Kinda like US nowadays.
(Score: 2) by crafoo on Thursday December 10 2020, @12:57PM (1 child)
Yes, we do go around the world shoving our progressive religion and culture down everyones' throats. I'd be pissed off too. Especially when it's such a destructive culture headed for such an obvious failure.
(Score: 0) by Anonymous Coward on Friday December 11 2020, @02:28PM
Don't worry there's plenty of locals desiring a return to a Stone Age society based on religious authority (or its equivalent in Ubermensches).
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @10:44PM
Except they were correct in their belief. Much like the US is now.
(Score: 1, Informative) by Anonymous Coward on Thursday December 10 2020, @12:30PM (10 children)
There's a group in the US tearing down statues and rewriting history that wants to change the rules of logic and the alphabet for being racist.
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @02:18PM
Pretty sure either "barbarians" or"fundamentalists" covered that.
(Score: 3, Insightful) by Tork on Thursday December 10 2020, @02:33PM (5 children)
🏳️🌈 Proud Ally 🏳️🌈
(Score: 0, Informative) by Anonymous Coward on Thursday December 10 2020, @03:44PM (1 child)
The Marxist Anglosphere changes our modern books (websites) INSTANTLY and CONSTANTLY whenever the information there contradicts or implicates their own causes. Even the dictionaries are updated. It's straight out of "1984".
(Score: 0) by Anonymous Coward on Friday December 11 2020, @02:36PM
Any examples...? Or just more waah waaah waaaah,
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @04:01PM
Live and learn.
https://fromthetrenchesworldreport.com/top-10-politically-incorrect-kids-books/176910 [fromthetrenchesworldreport.com]
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @09:52PM (1 child)
Altering, banning, and more recently, burning (search for Cameron “C-Grimey” Williams to learn more).
(Score: 0) by Anonymous Coward on Friday December 11 2020, @02:37PM
Are you referring to Trump's court cases ?
(Score: 3, Informative) by Immerman on Friday December 11 2020, @02:30AM (2 children)
Public statues don't record history - they celebrate it. If you want to record racist history, move it to a museum archive. If you want to celebrate it - don't be surprised when decent people fight against you.
(Score: -1, Troll) by Anonymous Coward on Friday December 11 2020, @06:02AM
"Decent" people. Stand and behold his virtue, oh world! He is more moral than any group of people who came before. Finally, man has achieved a righteous state. His accomplishments put to shame all those "old" people's. What did they ever accomplish, anyway?
(Score: 0) by Anonymous Coward on Friday December 11 2020, @02:38PM
Fortunately we have the Ark Encounter in Kentucky to safely preserve REAL history.
(Score: 1, Interesting) by Anonymous Coward on Thursday December 10 2020, @09:18AM (39 children)
I read somewhere that a lot of great math comes out of the fact that humans have ten fingers.
this has led us to use base 10 notation, which makes certain operations needlessly complicated --- using base 8 would be easier.
however, the mathematics of changing number bases requires a thorough understanding of number theory, hence having ten fingers has led us to understand integers better.
in this sense, maybe it would be best if we taught roman numerals in school for a couple of weeks --- the point is to understand numbers, not digits.
addition, in particular, is ridiculously simple with roman numerals...
it may have been in https://en.wikipedia.org/wiki/One_Two_Three...Infinity. [wikipedia.org]
if not, I won't apologize to anyone who reads that book looking for this particular paragraph, since you can't possibly regret reading it.
(Score: 3, Insightful) by RamiK on Thursday December 10 2020, @11:46AM (14 children)
Humans' short term memory is only good for 5-9 numbers and the "cache" for tables is slow so most binary & kin algorithms won't work well for humans. For handy algorithmic hacks that humans can pull comfortable, all you really need is a low enough prime in there. So, say, 2, 3 or 5 as multipliers for your base would work equally well.
compiling...
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @02:30PM (13 children)
It would definitely make the translation between human and byte representation much easier.
Good luck making such a change at this stage (was there ever a stage where it was easy?).
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @06:02PM (12 children)
Base 12, my friend, the most convenient of all bases.
Divides evenly by 2, 3, 4, 6, and 12. It is what is known as a "highly composite" number. That is why it is common in the pre-metric measurement systems. Metric-using people can't even divide evenly into thirds. So stupid for such a common thing.
(Score: 2) by Immerman on Friday December 11 2020, @02:41AM (4 children)
Base 60 is even better, as it adds 5 to the mix. Which is why the Babylonians used it, and it survives to this day in our geometry. Babylonians are why the angles in an equilateral triangle measure 60 degrees.
(Score: 0) by Anonymous Coward on Friday December 11 2020, @05:57AM (1 child)
60 is too high a number to be practical in a place value system.
It also as you pointed out only gives you one extra factor.
It is convenient to use orders of magnitude in many areas of science or engineering. Powers of 60 are too large for that to be convenient. Plus display scales and rulers with 60 divisions is not practical. Then you've got how to come up with written symbols for the digits 0 to 59. In base 10, it's 0-9; in base 12, you only need to add 2 new digits. Written base 60 digits would look a little like roman numerals or Maya numerals (combinations of sub digits that sum to a digit value) because 60 distinct digits is excessive. There is a reason some other bases besides 10 appear in the world, but as far as I know, 60 only did so once.
(Score: 2) by Immerman on Friday December 11 2020, @04:25PM
True it only gives you one extra prime factor, but adding 5, 10, 15, 20, and 30 is a lot of convenient splits, particularly for a species who has 5, 10, and 20 as "natural" digit-based numbers.
I've got to agree on the tedious aspects of Babylonian numbers - they basically used cuniform sub-digits for 10 and 1 so that digit 59 actually involved 5+9 marks to write. But there's no particular reason you couldn't use more concise sub-digits - e.g. if we wanted to adapt indo-arabic numerals as sub-digits we might do something like write it in two digits, with "5" being rotated sideways above or before 9 to indicate it was a count of tens, which would average only slightly more sub-digits than base 10 averages digits: (I'm using ' to indicate the previous digit is written sideways, with unecessary spaces inserted for extra clarity)
#of base60 sub-digits -- maximum b60 value = equivalent base10 value
2 -- 5'9 = 59
3 -- 9 5'9 = 599 (9*60+59)
4 -- 5'9 5'9 = 3,599 (59*60+59)
6 -- 5'9 5'9 5'9 = 215,999
8 -- 5'9 5'9 5'9 5'9 = 12,959,999
and so on.
In the end though I don't think it really makes a whole lot of difference what base you use - you still need to make the same number of loaves of bread to divide evenly X ways. It's the fact that there's 360 degrees in a circle that make it easy to subdivide, it doesn't really make a huge difference whether you write the number as 360 or 60.
(Score: 0) by Anonymous Coward on Friday December 11 2020, @02:47PM (1 child)
60 is the Esperanto of number bases. All it needs is for everyone to switch over, then our problems will be over
(Score: 0) by Anonymous Coward on Friday December 11 2020, @05:04PM
No, the right answer is 12, you damn Sexagesimalist!
Join the Duodecimal Revolution or be on the wrong side of history, like those blasted French Revolutionaries who falsely enshrined the number 10 as King of all Numbers! Decimal Oppression!
(Score: 2) by RamiK on Friday December 11 2020, @11:03AM (5 children)
Here's the thing about 2 & 3: You want one or the other but not both since they're just off-by-1 variants of each other but you don't really need 3 since it duplicates a lot of odd 1 tricks. From there you have 12 duplicating variants multiple times so you just end up with a lot of ways to get the same thing done at the same cost which leaves you with more cognitive load for larger tables and another IF...THEN...ELSE decision to choose what trick to use.
compiling...
(Score: 2) by Immerman on Friday December 11 2020, @04:29PM (4 children)
Really? So you've never wanted to divide something into thirds? Or is it halves you've never wanted to divide things into? 4ths? 6ths?
The whole point of a highly-divisible base is that it makes "simple" numbers like "10" and "100" easily divisible in many different ways. There's not really any other particular advantage of one base over any other.
(Score: 2) by RamiK on Friday December 11 2020, @05:42PM (3 children)
Personally, halves is common enough. Thirds mostly comes up for rough estimates so you can ball park it fine. 4th and 6th aren't worth the effort.
In general, when there's a real demand, people adopt an extra base system like how CS uses binary/hex/octal to indicate memory and the likes. So, I would argue, that while base-12 might looks useful on paper, in practice the problems it solves are just not significant enough.
compiling...
(Score: 2) by Immerman on Friday December 11 2020, @08:38PM (2 children)
I find I use thirds fairly frequently when building things, and exact thirds means you can use consistent pieces - e.g. a dresser with three columns of drawers all the same width, which makes construction much easier. Fourths come up a fair bit as well, though more often as a second halving than an up-front quartering. It's one of the few nice things about Imperial units, that something measured in feet will subdivide to exact whole-inch thirds and fourths. Sixths actually come up a fair bit as well with angular stuff since 6 equilateral triangles perfectly complete a circle.
Fifths though? I hardly ever use those, which makes base 10 a huge waste of potential - all those digits, and the only common way they divide neatly is in half.
I can't say it'd be worth the effort to change an established system for the benefits - but if you were choosing a base to start out with it's basically completely arbitrary, so you may as well use one that maximizes subdivision convenience. 12 has far more to recommend it than 10, precisely because it greatly simplifies thirds, fourths, and sixths, and only sacrifices the even-less-frequent fifths to get it. All that base 10 really has to recommend it is that the typical human can count to it in unary on their fingers.
(Score: 2) by RamiK on Friday December 11 2020, @09:54PM (1 child)
In the context of fabrication any point in favor of thirds will come at the expense of halves since you mostly just end up folding the piece or a reference rope instead of actively measuring anything. Then again, you get 0.35 halfway between 0.2 and 0.5 which is enough tolerance for just about anything that doesn't merit a calculator to begin with...
The fact 5 aren't too useful is to their merit since most people are already overloaded by just 2 in day-to-day mercantile context. And people being able to count to 10 on their fingers is actually very important since it lets younger kids learn to count earlier which hastens the rest of their mathematical education.
compiling...
(Score: 2) by Immerman on Saturday December 12 2020, @02:24AM
>In the context of fabrication any point in favor of thirds will come at the expense of halves since you mostly just end up folding the piece or a reference rope instead of actively measuring anything.
How so? Dividing things into 3rds in no way impairs your ability to divide into halves. The folded-rope trick even works for thirds (or fourths) with only slightly more difficulty. But if you're doing that you're avoiding numbers altogether, so your preferred number base is irrelevant.
>enough tolerance for just about anything that doesn't merit a calculator to begin with...
That's the point though - in base 10 practically any integer subdivision of 10 is a fraction that requires extra mental effort (or a calculator), the only whole-number subdivisions are 10/2 = 5 and 10/5 = 2. Whereas in base 12 most fractions of 10 are whole numbers: 10/2 = 6, 10/3 = 4, 10/4 = 3, and 10/6 = 2 - subdividing multiples of 10 will very rarely result in a fractional answer.
Also, it's pretty trivial to count to 16 on the fingers of one hand, and doing so is common in several cultures - use your thumb as an indicator to count finger tips and joints. That also makes it easy to count through all two-digit numbers using both hands in any base up to 16
(Score: 2) by hendrikboom on Saturday December 12 2020, @02:43AM
I once saw a book of math tables done to about ten of fifteen or so duodecimals. Yes, base 12. Logarithms, trig functions, etc. Done in early 1900's. Truly an awe-inspiring labour.
(Score: 1) by khallow on Thursday December 10 2020, @12:03PM (4 children)
It's even simpler with decimal numbers. 6+8=14 versus VI+VIII = XIV.
(Score: 2, Interesting) by Anonymous Coward on Thursday December 10 2020, @01:11PM (3 children)
you can shuffle roman numerals as long as you keep the largest at the right and you treat "small in front of large" as a single unit:
VI + VIII = VVIIII = XIIII ( = XIV if you insist).
see? all you need to know here is that VV = X
(Score: 1, Funny) by Anonymous Coward on Thursday December 10 2020, @08:17PM (1 child)
VV=X should be considered obvious, as X is composed of two V shapes.
(Score: 2) by Immerman on Friday December 11 2020, @04:32PM
So how does M fit in? It's also two V shapes. To say nothing of L, which is a rotated V shape. And then C throws everything out the window.
(Score: 1) by khallow on Sunday December 13 2020, @01:04PM
I didn't say you couldn't. Point is that's terrible math compared to the Arabic numerals. This gets more glaring as one goes to larger numbers, for an arbitrary example, 1996+518. In Arabic, one way to do it is add ones first 6+8 = 14. Then carry that 1. 9+1+1 = 11. Carry that one. 9+5+1= 15. Carry that one. 1+1=2. Then you have the digits for the thousands, hundreds, tens, and ones places - 2514.
Now do it in Roman numerals: MDCDLXLVI + DXVIII = MDD(CD)L(XL)XVIVIII = MM(CD)LLVVIIII = MM(CD)CXIV = MMDXIV.
So maybe it's easy in some sense to add Roman numerals. Well, it's even easier with Arabic numerals. And what happens when you're adding numbers that are billions or trillions or on the order of 10^80? The Roman numeral approach breaks quickly with each order of magnitude introducing two new symbols. Similarly, decimal fractions would also require their own symbols. After all, how are you going to represent 0.23 in Roman numerals? XXIII/C?
(Score: 4, Insightful) by GlennC on Thursday December 10 2020, @01:37PM (3 children)
When I was in elementary school in the 1970's, that's what we did in 3rd or 4th grade. I was surprised that my kids didn't have it.
I agree that being exposed to Roman numerals, along with maybe other notations, would help increase numeracy.
Sorry folks...the world is bigger and more varied than you want it to be. Deal with it.
(Score: 3, Interesting) by istartedi on Thursday December 10 2020, @05:34PM
Me too. The further back you go in the USA, the more our education system was based in the classics. Latin was still taught in our high school, but only as an elective that tended to be taken by college-bound students.
Any educated person was expected to know Roman numerals. When I took an intro programming course, one of our assignments was to write a program that took in a string and detected whether or not it was a valid Roman numeral, printing out the decimal equivalent if it were, and an error if it were not.
For some reason, that little exercise sticks out in my mind as a fond memory, perhaps because I was happy to get it working and have it print out:
Appended to the end of comments you post. Max: 120 chars.
(Score: 2) by Immerman on Friday December 11 2020, @09:21PM (1 child)
I really don't see how learning a hopelessly obsolete and convoluted number system improves numeracy (aka the ability to read and work with numbers) at all. There is literally nothing Roman Numerals do better than positional notation, the only reason to teach it is as a historical curiosity. Positional notation was a huge technological leap forward, rivaled only by the concept of zero - something that actually grew out of placeholder symbols positional notation.
Now, if you instead wanted to teach some math in a unary/tally system (e.g. I II III IIII
IIII) I could get behind that - tally marks can actually be quite useful in lots of scenarios, and since all math grew out of counting, it expose many of the "implementation details" of arithmetic far more clearly: e.g. multiplication is just shorthand for repeated additionIIII
IIII
IIII =
IIIIIIIIIIor count four groups of three (commutative property of multiplication)
III * IIII = count three groups of four
III
III
III
III =
IIIIIIIIIIWhich is also very short distance to understanding the relationship between multiplication and area, and WHY multiplication is commutative
And of course, division is utterly simple in unary as well
26/4 = deal 25 into 4 even piles:
IIIII .. IIIIII .. IIIIIIIIIII = 6 per even pile with 2 left overWhich also makes quite clear why division is NOT commutative
(Score: 2) by Immerman on Friday December 11 2020, @09:25PM
Bah, of course I don't notice until hitting submit that I bodged the most important line while going for increased clarity
That first example should start with the line found further down:
III * IIII = count three groups of four
IIII
IIII....
(Score: 1, Funny) by Anonymous Coward on Thursday December 10 2020, @03:52PM (1 child)
It also shows that early mathematics was dominated by women, because men would have naturally used a base-11 system.
(Score: 0) by Anonymous Coward on Friday December 11 2020, @02:49PM
It's 1 better.
(Score: 3, Insightful) by ledow on Thursday December 10 2020, @05:03PM (8 children)
Ten fingers is pretty much not the reason, it's just how it happened. The Incas used 12, as did a load of other systems. And they didn't have 12 toes. It was just more sensible. There's a few based on 16. Just look at Imperial measures - everything from 7's to 144's and all of them using different "justifications". Truth is, that's just how things happened when maths merged and evolved, sometimes not even for the better.
If you want to use fingers, finger-binary makes far more sense anyway, and allows you to count in 1024's.
It's probably more related to a largely-illiterate populous struggling with more than that number of symbols, whereas today there's no reason to suspect that everyone doesn't understand 10 number digits, 24-26 letter digits, and a variety of character from a selection of over 100.
10 is a really crappy number, to be honest, in terms of factorisation, translation to the lowest number base (binary, because it's not a power-of-two), and all kinds of other reasons. It's not even prime (which would give it nice properties given that it's not nicely factorisable anyway). It's about the worst you could choose, I think. But nobody sits and argues that there are 26 letters in our alphabet because of some physical reason. There isn't. In fact our alphabet has duplicates not present in other languages, so even 24 is optimistic in terms of only choosing unique phonemes, etc. There's a reason that we need two characters to express ch-, sh-, th, etc. sounds - we didn't use enough characters to express all sounds. Chinese ideograms number in their thousands.
And you can't use the argument that Arabic numerals were chosen to be used on fingers and then in the same breath say that their greatest innovation was positional. I can't remember the last time I carried my little finger two positions to the left.
If we were starting again, making a utopia, beginning anew: Binary. Or 12. Or 24. 10 wouldn't be under consideration. Hell, as you say 8 stands more of a chance, then you don't need to use your thumbs, and it's a power of 2 and nicely binary to fit in with computing (8 bits in a byte - again, a completely random correlation based on some spurious restriction in days gone by). Doesn't solve the divide-by-3 problem, but neither does base-ten or binary.
Roman numerals are horrible because they have pre-subtraction, where IX is one less than X but XI is one more than X and MIX is not in power order at all. It's both positional and symbolic, so it has all the same problems as Arabic numerals, but in a non-linear fashion. It's actually a programming exercise for kids to convert Roman numerals to decimal or binary, because it's difficult to do without referring behind / ahead in the array of characters. Imagine the number of dyslexic Romans who totally got screwed on their purchasing because of that kind of system.
Pretty much, our history of numerals is just a complete mish-mash. Not unlike the history of calculus notations. There are several older notations that never took off but, in certain circumstances, are far neater to express problems in. It's far more "the one that just happened to stick" than any kind of logic, or evolution, or improvement, or physical characteristic.
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @06:12PM (2 children)
Base 12 for the win.
It's a highly composite number meaning it has many factors: 2, 3, 4, 6, 12.
You can exactly do the common operations of dividing things into halves, thirds, quarters (and sixths and twelfths).
In base 10, you can't do thirds, so you get nonsense like 33% or 67% as approximations. So stupid! This is a problem with the metric system too.
(Score: 0) by Anonymous Coward on Friday December 11 2020, @02:52PM (1 child)
We just need to grow another finger on each hand and this will take off.
(Score: 0) by Anonymous Coward on Friday December 11 2020, @05:08PM
You still count on your fingers?
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @07:26PM
actually 12 comes from knuckles, of which we do have 12 on each hand.
they said so on QI, so it must be true.
but otherwise: yes, I also believe that many of the things we now consider "natural" are just accidents that we got used to.
(Score: 1, Interesting) by Anonymous Coward on Thursday December 10 2020, @07:27PM
> 10 is a really crappy number
Did you forget...
There are 10 kinds of people.
Those who think in binary, and those who don't.
---
Perhaps there are 10 genders after all.
(Score: 0) by Anonymous Coward on Friday December 11 2020, @05:09PM (1 child)
Binary is only a consideration for people who fail to understand that math is also about communication between humans. For communication between humans, base 10 is the simplest, because you can just hold up the amount of fingers, for values greater than 10, you flash them in sequence.
One of the reasons base 10 won out was because it was much easier to teach illiterate people at least basic arithmetics with it. Historical examples include counting such as 2 hands of wagons, 5 hands of chickens, 3 hands of days away, and so on.
Just because you, Ledow, choose to be a slave to your computer, does not mean everyone else should be enslaved like that.
(Score: 2) by ledow on Saturday December 12 2020, @08:26PM
And it's really not.
That's why the Incans ended up on 12 and the Romans settled on weird numerals.
Yes, there's an element of baby-talk about it, but that's not why it propagates to become the entire numerical system. Much like your goo-goos and ga-gas aren't used or rewarded past babyhood and why you can't call it "spekettsi" for the rest of your life.
The second you get into numbers past-10, addition, place value, and anything even VAGUELY useful, you end up with other number systems in preference. It's like saying that music is just "1, 2 ,3 ,4, 5, 6, 7, 8" with a colour for each number/key. Maybe when you're a toddler, not past that when you actually have to describe, use and process numbers (which every adult does).
Additionally, written base-10 is, as I said, the point at which you'd change because those same illiterate people who can't grasp numbers they can't see "fingers-of" wouldn't know what the ten mysterious symbols meant without a lot of training.
The reason we use base-10, the reason we have 10 digit symbols, the reason we wrote numbers down, has nothing to do with how many fingers we have. Any more than our clocks having 24 hours divided into 60ths and further 60ths, our years having 12 months, etc.
It's a bunch of random historical reasons, of which "we have 10 fingers" is the lowest consideration. Because, as I said, counting in 8's would be just as sensible (no thumbs which you can then use as pointers).
Fingers may be one reason that a few person started to work with numbers up to 10. It has almost no correlation whatsoever to why we have place-value base-10 counting, any more than having 24 ribs is "the reason" that our clocks have 24 hours.
It's just a spurious correlation.
And, as a mathematician who's worked their entire career in schools, let me tell me the biggest problem with teaching maths: Telling them that it's all arithmetic and that using their fingers will help. Fingers get in the way of maths, because you just have just as much to remember whenever you use them, while at the same time offloading the mental critical processing step to a physical object which acts only as a memory store, that you then have to picture for every calculation for years afterwards if you were told to use fingers whenever you struggled. It literally affects your mental processing abilities as you just rely on your fingers to be 100% correct memory storage / recalll devices, and leads to kids who just trust whatever answer is given back to them (which bodes very badly for later years when they have to know whether they are in the right ball-park for their answers).
The base used is actually immaterial. I could teach a bunch of innumerate rainforest tribe children in base-3. It would make no difference at all. It's about promoting an understanding, which focusing on "which finger do I fold down" is bad for... and incidentally doesn't progress you even past Year 1 / Year 2 arithmetic. Propagating that into adult life is an absolute educational mistake, which is why our use of base-10 is arbitrary and may even be counter-productive.
(Score: 2) by Immerman on Friday December 11 2020, @09:35PM
Base 10 is what you get when you invent a number system built around counting on your fingers. It has basically nothing else to recommend it.
Base 12 (or 60) is what you get when you actually think about how numbers are commonly used when inventing a positional number system.
Base 2 is exhaustively long-winded, and basically pointless unless you're working with digital computation machines. And base 2^N really have nothing to recommend them except that they're a convenient shorthand for working with base 2.
(Score: 2) by driverless on Friday December 11 2020, @01:08AM (1 child)
Good thing we didn't get our maths from Tennessee or Arkansas then...
(Score: 0) by Anonymous Coward on Friday December 11 2020, @02:54PM
I we define pi = 3, like it says in the Bible, then the proper base should be obvious.
(Score: 2) by Immerman on Friday December 11 2020, @03:11AM
>addition, in particular, is ridiculously simple with roman numerals...
But everything else is massively more difficult. It's hard to overstate just what a huge revolution positional notation was for mathematics, probably only the concept of zero as a number in its own right comes anywhere close - and zero got its start as a positional placeholder. Roman numerals were scarcely more than a shorthand form of unary (ticks and bundles)
Nothing particularly special about base 10 specifically, other than it makes it really intuitive to count on your fingers, but nothing particularly bad either. Base 12 is arguably better for integer division, but only if you're inclined to make things in batches of "10", regardless of base. And the widespread use of "dozens" shows people are quite willing to ease more easily divided values even when they aren't neatly represented in the number system.
I can't say I've heard of anything base 8 is better for, unless you're using it as shorthand for binary (in which case I'd argue base 16 is better - you can even easily count it one-handed by assigning a value to each finger tip and joint, and using your thumb as an indicator). But while binary has some advantage as the simplest possible positional notation system, those mostly aren't relevant unless you're building calculating machines from extremely simple components.
(Score: 2) by Immerman on Friday December 11 2020, @04:44PM
>addition, in particular, is ridiculously simple with roman numerals...
Only for very small numbers, unless you care to explain how XXXIII + XI = XLIV is simpler than 33+11 = 44?
The biggest problem with the supposed simplicity of Roman numerals is that they involve both addition and subtraction to interpret - unless you've memorized what a number means you may have to perform a rather involved calculation to figure it out.
Positional notation in contrast only involves addition (multiplication being a shorthand form of addition), and it scales very well to large numbers, and with the addition of a radix point, also to small fractional components.
23 = 10+10+1+1+1
(Score: 4, Insightful) by wisnoskij on Thursday December 10 2020, @01:58PM (11 children)
Arabic numbers were developed by Indians who practiced Hinduism. The Muslims just get the credit because they conquered them for a while a thousand years latter and helped spread the numbers around during their crusades.
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @02:24PM
I just got redpilled.
(Score: 1, Insightful) by Anonymous Coward on Thursday December 10 2020, @03:59PM (6 children)
The muslims get a lot of deserving credit because it is due to them that we even know the science and maths of antiquity. The Dark Ages tore all of that down and most of the writings of Ptolemy, etc., were preserved and recreated from the Arabic copies of those books. We were able to enter the Age of Enlightenment because not only of the advances they made themselves, but also the knowledge they valued and preserved. It is pretty misleading to suggest that people like al-Khwarizmi were just spreading notation around they picked up elsewhere.
(Score: -1, Flamebait) by Anonymous Coward on Thursday December 10 2020, @05:58PM (4 children)
It is correcting a misconception spread by the article that this number notation was invented by the Arabs. It was not, and the very name Arabic numerals serves to reinforce the misconception. I have seen them referred to as "Hindu-Arabic numerals", and one would think in these PC times that name might have been used, but clearly the aim of this article was to pump up the Arabs, not spread the real truth.
As for Arabs "preserving" ancient Greek knowledge for Europeans: if the Arabs hadn't destroyed the Greek centers of learning, we wouldn't have needed the generous Arabs to "preserve" anything! Those Greek centers would still exist!
(Score: 5, Informative) by Anonymous Coward on Thursday December 10 2020, @07:34PM (3 children)
Like, in the fine Summary? Could you make it any more clear that you have not read even the summary, let alone the original article? And, Romans had much more of a role in destroying Greek centres of learning. When Muslims got around to burning the Library of Alexandria, it had been destroyed twice before, once by Julius Caesar, and once by the only thing worse than Fundie Muslims, Fundamentalist Christians.
(Score: 0) by Anonymous Coward on Thursday December 10 2020, @10:24PM (1 child)
And the Muslim Ottomans destroyed the Eastern Roman Empire, otherwise known as Byzantium. The Byzantines were keeping ancient knowledge alive and advanced it. Witness Hagia Sophia, the church which was the largest enclosed space by a building for 1,000 years. Poured concrete domes.
(Score: 0) by Anonymous Coward on Friday December 11 2020, @07:35AM
You like to think! Hagia is still there, now a Mosque, just to pique your Christian ass! Misbeliever! Blasphemer! Infidel? And his brother, RaulFidel!
(Score: 0) by Anonymous Coward on Friday December 11 2020, @02:59PM
Thankfully the enlightened Americans took great care of Baghdad to preserve its historical treasures.
(Score: 2) by Immerman on Friday December 11 2020, @05:02PM
Also because they were the center of the intellectual world for a long time - they passionately promoted their position as crossroads for intellectual (and economic) exchange between Europe, Asia, and Africa, from around 800AD to somewhere around 1200AD or so. Quite possibly aided by the fact that Islam was at the time interpreted to actively encourage intellectual pursuits, with many academics tracing the end of the Islamic golden age to the rise of religious philosophers who undermined that interpretation in favor of a more authoritarian interpretation that put ultimate intellectual authority in the hands of religious leaders.
(Score: 2, Funny) by Anonymous Coward on Thursday December 10 2020, @04:00PM (1 child)
The Muslims were in search of the fabled zero of Hindu maths. They found nothing.
(Score: 0) by Anonymous Coward on Friday December 11 2020, @07:06AM
As a set theorist, I strongly disagree!
(Score: -1, Troll) by Anonymous Coward on Thursday December 10 2020, @06:26PM
> It's a good story, But a Complete Fabrication
Not dissimilar to the hoax about Swedish meatballs allegedly being Turkish [sbs.com.au]. The hoax got coverage in most major papers and magazines, including Time. The correction got virtually no coverage.
You have to remember that the moslems have a strong social media presence and wrongly frame many narratives to their favor for political purposes.
(Score: 5, Informative) by Socrastotle on Thursday December 10 2020, @05:35PM (5 children)
This article is accurate that at one time the Mideast was a major hub of learning in the world. People of all different values and values congregated and shared different philosophies, worldviews, and challenged those of others. All was good and progress was rapid in light of constant intellectual 'competition' : The Islamic Golden age. But what most people don't seem to ask is, what happened? And a single man, Al-Ghazali [wikipedia.org], played an outsized role in what happened.
Al-Ghazali was a philosopher and theologian who, for many years in the late 11th century, did participate in these games of learning and sharing. But then at some point things changed. Near the turn of the century he wrote a new text, "The Incoherence of the Philosophers." [wikipedia.org] You can read it (with annotations) here [muslimphilosophy.com]. His worked aimed to strike down all logical challenges to Islam in one fell swoop. And how did he do this? He simply rejected the existence of logic. According to Al-Ghazali when a leaf burns after exposure to fire it is not caused by the fire, but by God choosing and deciding to set the leaf alight at that very moment. And so too as the leaf eventually turns to ash it is not driven by the interaction of the fire and the leaf but rather by God's decision, and only by God's decision, to morph the leaf.
It was a worldview that became immensely popular in an extremely short period of time. The reason was simple: it meant Islamic individuals (and even scholars) no longer needed concern themselves with the laws of the universe, let alone how those laws might interweave or run against the religion in various ways. By denying the very existence of any sort of natural logic, it all becomes a nonstarter. Nothing is as it seems. Everything happens only by the will of God, everything else is merely a social construct.
This belief remains with Islam to this very day which is a large part of the reason that in spite of Muslims being an enormous part of the entire world's population and having a history previously rich in the sciences (another example here is algebra - from al-jabr in a 9th century text by Al-Khwarizmi), they have played very little role in academic or technological progress of the world for the better part of a thousand years. And indeed the Islamic Golden Age went into rapid decline and would end up dying altogether shortly after Al Ghazali's new discovery.
The reason I think this is so important is because it's something that not only can, but almost certainly *will*, happen to another civilization. From the height of progress to however one might want to describe the Mideast today. And all simply because it can be quite comforting to rejecting the world as we observe it, and replace those observations with ideology that is, as a consequence of its construction, unable to be challenged on anything like a logical level. For what does logic mean when you start with the assumption that it does not exist?
(Score: 0) by Anonymous Coward on Friday December 11 2020, @03:04PM
And their blaze was top notch too.
(Score: 1) by khallow on Sunday December 13 2020, @01:23PM (3 children)
I disagree on the reason. I think it's a combination of two factors. First, ignorance-coping. If God is responsible for every movement of reality rather than equally inscrutable physical law, then you have something to pray to. It's a more comfortable way to deal with a world one doesn't understand and far less work than actually trying to figure out those physical laws. And maybe God will choose to make those leaves burn differently, if you really need him to.
Second, such knowledge is a threat to power. If your power comes from ideology (which is a common situation with religious authorities), then science and knowledge are often threats (particularly, if you're lazy and don't bother to provide any logic for your ideological claims). The simplest way to deal with such threats ideologically is to deny their existence. That works until someone does something that you can't explain away with "God did it".
(Score: 1) by Socrastotle on Monday December 14 2020, @03:13PM (2 children)
It's probably a mix of many things but it does immediately solve many basic logical questions. The oldest and most basic being something like, 'Could God create a rock even he could not lift?'
It's something I think most people now a days, religious or not, would likely just see as a silly word game along the lines of "this sentence is false." But it actually poses a fairly substantial philosophical problem when looking to logically integrate religion and philosophy/science, as was the norm for the time. Whether the answer is yes or no, it suggests a limitation on God's power. And so it seems impossible to answer in a way that is theologically consistent. However, by rejecting physical laws altogether you can simply respond to such queries by altogether denying that the notion of lifting even exists. When you "lift" a rock, you do so only by the will of God. The stresses you feel are not from any sort of interaction with the rock itself, but solely at the leisure of God. So, with the notion of lifting being deemed a "social construct", the question becomes nonsensical.
(Score: 1) by khallow on Monday December 14 2020, @06:25PM (1 child)
It just makes the argument a little more difficult to make, say "Can God will a task he can't do?" or "Can God conceive of limits on his power?"
Nor does that refute the idea that physical law is important. After all, that peculiar consistency is the will of God. For example, grasping burning leaves hurts. Maybe that's God willing a sort of communication with us that we shouldn't grasp burning leaves? And once you have some reason to comprehend physical law (even if for some reason you don't care that God is willing such things), you have opened the door to accepting physical law.
(Score: 1) by khallow on Monday December 14 2020, @06:57PM
(Score: 0) by TomTheFighter on Thursday December 10 2020, @08:56PM (4 children)
Mathematics for the Nonmathematician
by Morris Kline
Puts Math in a historical context - definitely the coolest book on the subject I've read.
https://www.goodreads.com/book/show/281821.Mathematics_for_the_Nonmathematician [goodreads.com]
"Professor Kline begins with an overview, tracing the development of mathematics to the ancient Greeks, and following its evolution through the Middle Ages and the Renaissance to the present day. Subsequent chapters focus on specific subject areas, such as "Logic and Mathematics," "Number: The Fundamental Concept," "Parametric Equations and Curvilinear Motion," "The Differential Calculus," and "The Theory of Probability.""
Very readable to the barely Math literate such as myself ( did manage to make it through Stats for Psych majors and into to Trig for Engineers but that's a looong story ).
(Score: 1, Interesting) by Anonymous Coward on Thursday December 10 2020, @10:21PM (3 children)
Too bad he didn't start with the Babylonians.
The Greeks deservedly get a lot of respect, but there were much older civilizations.
(Score: 2) by hendrikboom on Saturday December 12 2020, @02:59AM (2 children)
People calculated and counted before the Greeks.
But the Greeks were the first to do what is now characteristic of mathematicians -- proving obvious results from obvious premisses.
-- hendrik
(Score: 0) by Anonymous Coward on Saturday December 12 2020, @05:58AM (1 child)
So... captains of the obvious, those Greeks!
(Score: 2) by hendrikboom on Saturday December 12 2020, @11:03AM
Yes, the beginning of axiomatic description.
This has progressed by now to a new attitude: A set of axioms is no longer considered to be an obvious truth, but instead is treated as a set of defining conditions. And there's a lot that can be done with that.
Even Euclid's axioms for geometry -- those are no longer considered sufficient. They were too vague. They never explained, for example, anything about a point being "between" two others on a line. Or what it means to "move" a triangle onto another when you prove they are congruent. In the last century a new set of axioms was formulated that are more precise. And more abstract. If you encounter any thing at all that satisfied the new axioms, you can call it a Euclidean geometry.
And category theory, for example, deals with 'objects' and 'arrows' between them. It is not specified what 'objects' and 'arrows' are -- there are just some specific conditions that they have to satisfy for you to have a category. Originally, the objects were mathematical entities such as groups and topological spaces and the arrows were mappings between groups and between topological spaces. But this rapidly got generalized and as a result there are an immense variety of different kinds of things that can be treated using category theory.
Sometimes even usefully.
-- hendrik