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PostPosted: Tue Apr 05, 2011 11:43 am 
Sanci
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All previous ones are correct, under the same initial assumptions of mathematics (Axioms) all societies reach the same conclutions ultimately no matter how isolated they are.

But certain parts of mathematics are more or less arbitrarily choosen, Sin/Cos are two functions that can be exchanged and render new things but Sin/Cos are still a preferable choice to for example Chord and Cochord.

Though having worked with Crd/ccd I can say in such a system the euler identity goes from e^(pi*i)+1=0 to e^(pi*i)-1=0


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PostPosted: Tue Apr 05, 2011 1:14 pm 
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PostPosted: Tue Apr 05, 2011 4:44 pm 
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PostPosted: Wed Apr 06, 2011 12:19 am 
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Creating genuinely new mathematics is, of course, non-trivial, but you can surely get away with doing less than that while also doing something more creative than just shuffling notation. Maybe your people have some ideas that are important to them but seem weird to us now, like wanting to construct everything by ruler and compass (in fairness to the Greeks, this seems pretty reasonable as a rigorous basis for synthetic geometry--but now that we have the power of algebra and analytic geometry, it seems quaint). The Greek emphasis on geometry also led to centuries of people feeling weird about writing things like "x + y^2": after all, if y^2 represents an area and x a line segment, it doesn't make much geometric sense to add them. And so on. Being stuck with theorems we already know doesn't seem like a terrible problem to me, since what's really interesting is how they are arrived at, conceptualized, and assembled into a coherent whole. Thinking that you can't do anything interesting because you'll still end up with pi = 3.1415926535... seems like worrying that you can't create an interesting language because you'll still end up being able to say "the dog bites the man".

You might also try to imagine things like: what if the Greeks had thought of probability? What would the next 1000 years of mathematics look like? Basic probability / combinatorics is simple enough that I think this is entirely reasonable, and really it's kind of a surprise to me that it took so long for such ideas to come about! Didn't people ever roll dice? Of course this also leads you back to the idea of expressing old facts in new ways. Eudoxus and Archimedes and so on anticipated the integral calculus with the geometric "method of exhaustion" for computing areas, but it's easy to imagine also approaching these ideas via, say, probability. The law of large numbers is pointing towards limits, for example, or you might think of measurements in a probabilistic way: the area of a shape inside a 1x1 square is the same as the probability that a dart thrown at the square hits the shape.

Also, do not forget that the things you know now were almost certainly not arrived at in their final, nicely polished form. Give your conpeople glimpses of some bigger ideas, different glimpses than we had historically. There are lots of ways to arrive at the same idea, not all of them obvious. Nowadays (at least in America), kids tend to first be exposed to complex numbers in the context of solving quadratic equations: sometimes the quadratic formula tells you to take the square root of a negative number. But no one cared about that 500 years ago: a solution of a quadratic equation likely represented some geometric thing, and there was no reason to care about ???square roots of negative numbers??? (indeed, you probably threw out negative solutions too). The first place it became apparent that complex numbers really mattered was after Tartaglia/Cardano/etc. worked out how to solve cubics. For example, the cubic x^3 - 7x^2 + 14x - 8 = (x-1)(x-2)(x-4) has roots 1, 2, 4. But the cubic formula gave Cardano, for one of the roots, something like:



This turns out to be equal to 1. Cardano may not have been interested in non-real solutions to his equations, but he certainly would be interested in getting "1, 2, 4" here, and so for the first time someone was forced to care about complex numbers.


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PostPosted: Wed Apr 06, 2011 2:48 pm 
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The "problem" with con-mathematics is that, if you create and flesh out any novel system of mathematical thinking you've done actual mathematics. It's the same problem that con-philosophy, con-programming or the like—it's actual, real-world work and the fact that it's supposed to be based in a conworld doesn't deny the fact that you are doing mathematics (or philosophy/computer science/etc) research in the real world.

Short of con-mathematics is con-history-of-math, in which there's a lot of room for variation. It's important to remember that pure mathematics (anything too far removed from counting, basic probability, and geometry) is something which took a long time and a substantial food surplus to get going. Higher math is a leisure profession, like the arts, but unlike the arts, it is appreciated pretty much exclusively by the people who do it. The connection between science and mathematics took a long time and a lot of inspiration to get going, and even then math basically rides along on the coat-tails of science. Computers brought math further into the mainstream—all this pure-mathy nonsense about the sorts of things it's possible to calculate suddenly had an actual application.

Remember that, while math has its fingers in pretty much all of the pies of human development, it has historically taken a pressing need before money gets tossed to the pure mathematicians. Without the information warfare that went on in WWII, it's possible that modern cryptography (maybe even computer science) wouldn't exist yet because there wouldn't have been a pressing need for it.

For your own con-historical mathematics, I recommend that you look at the ancient mesopotamian and egyptian mathematicians, as well as the mayans (who worked everything out from scratch) and the arabs and persians (who picked up where the greeks, romans, and egyptians left off and formulated much of algebra). In general, look for math that people use first. Mathematical logic is one of the most recent developments. Liebniz's work was less than two hundred years ago, Frege and Russell's about a hundred years ago, Gödel about eighty, and Kripke less than forty. Logic is the runt of the litter in more ways than one—its practical usefulness today applies mostly to mathematicians and computer scientists. In a pre-computerized society, it is useful pretty much only to philosophers.

A pretty canonical order of the pre-industrial development of mathematics is: counting systems, natural number arithmetic, Euclidian geometry & possibly early combinatorics, algebra/trigonometry & likely early combinatorics/probability.

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PostPosted: Wed Apr 06, 2011 3:57 pm 
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Thanks guys, now I'm doing conmaths instead of anything useful. Well, at least my conpeople will be able to trisect angles. They just won't be able to talk or write about it.

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PostPosted: Wed Apr 06, 2011 4:17 pm 
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PostPosted: Wed Apr 06, 2011 6:09 pm 
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Although logic never really died - in fact, it was one of the most vibrant branches of medieval philosophy. Rather than it being a late development, it was something developed early* and then abandoned as useless and sophistical, only to be RE-developed later on. This happened with most philosophy, but in the case of logic the abandoners were the scholars of the rennaissance and the enlightenment, who usually are the heroes.

*VERY early. Logic underlies Greek thought (scientific, mathematical and philosophical) in a way that is not true of earlier, middle-eastern thought. Logical principles and analysis were discussed from the earliest days; formal propositional logic (including modal logic) dates to at least the fourth century BC (eg Diodorus Cronus puts forward the same objections to material conditionals that moderns do, and proposes a modal alternative).

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PostPosted: Wed Apr 06, 2011 7:32 pm 
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PostPosted: Thu Apr 07, 2011 5:02 am 
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My con-maths system stretches to 'they use base 20'. I don't know what the implications of this would be, however. I'm not bad at maths but I don't really have the inclination to try and develop an alternative system that would go above and beyond what we have at the moment.

Just because it's a neat idea, I might throw in that they use 2π as the circle constant instead of π. Kinda makes sense I guess. Then I can type that into wolfram alpha and get 6.5d59cjd3fij643fhf279d15206c39253b2heg6igf793ea2g15b1cgh654c4f..._20 spat out at me. Useful I guess.

What's quite fun with base 20 is that you can spell a reasonable number of words with the first 10 letters of the alphabet. But wolfram alpha only converts 2π up to a couple of hundred places, and the only words that I can find are "beg" and "bee". π itself has "fag" on the first line, though, which makes me giggle, and "bed" and "age" somewhere else in the amount that it converts..


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PostPosted: Thu Apr 07, 2011 5:32 am 
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Use base-36 and you'll get the entire alphabet. No doubt some interesting words turn up early on in the expansions of well-known consonants.

Of course,"squarerootofminus1", "ratioofradiustocircumference", "baseofnatuarallogarithms" will turn up in the values of e, phi, and pi respectively, along with the entire Babel text in all conlangs ever created. But that's just one of the LORD's merry jokes. Don't you love infinity?


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PostPosted: Thu Apr 07, 2011 6:15 am 
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This is the maximum expansion that wolfram alpha is capable of:
3.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..._36

To be honest, I'm just seeing a lot of w and z and not a lot of words.

Here's e:
2.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..._36

And phi:
1.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..._36

Oh shit, they've made the screen wide. Sorry about that....


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PostPosted: Thu Apr 07, 2011 7:34 am 
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I think we may just have invented a new method of generating new conwords...


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PostPosted: Thu Apr 07, 2011 9:46 am 
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PostPosted: Thu Apr 07, 2011 10:17 am 
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i do not think that there are any known proofs that pi and e are normal numbers -- i think their status as widely believed facts mostly comes from statistical analyses of known digits?


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PostPosted: Thu Apr 07, 2011 3:47 pm 
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PostPosted: Thu Apr 07, 2011 4:18 pm 
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Normal numbers are numbers that have the property that, in a given base, they will have every possible combination of digits (the actual definition requires that every sequence of digit is uniformely distributed). The most trivial normal number is the Champernowne constant, which is just 0,1234567891011121314... (though that one only works in base 10, but it can be generalized to a particular base n).

Fancier numbers can be normal in any base. This is never the case for rational numbers, for obvious reasons, but it is suspected of many of our best constants, like pi, e and such (it's also a pretty common with uncomputable numbers).


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PostPosted: Thu Apr 07, 2011 4:26 pm 
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PostPosted: Thu Apr 07, 2011 4:33 pm 
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PostPosted: Thu Apr 07, 2011 4:50 pm 
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I'm reading it as talking about a helix - so that the numbers return to the same point on the circle but at a 'higher level', so to speak. So in a base-3-helix, '4' would be at the same angle from 2 as 1 is.

I think it's a really clever idea.

[Polygons, for instance, could be described not with angles but with times of day. The area of a circle might be a basic unit, at least for mathematicians (because together with hours, it would define any polygon). Or, if the area of, say, a base-10 circle were given as constant, the size of other circles could be given in hours-per-day. It would be interesting how they might extend that to polyhedrons, though - spheres, skewed helices, spirals traced on a cone, or intersecting circles? There's a lot of possibilities for expansion, but understandable even to the layman. I'm a little envious.]

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PostPosted: Thu Apr 07, 2011 4:53 pm 
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Nancy Blackett: Here's a list of all the English words appearing in the first 100000 base-36 digits of pi in base 36, along with how often each word appears. Really it would make more sense to do this with base 26 and shift 0, ..., 9 to being a, ..., j, but it's too late now. The list is pretty boring. Since 36^3 = 46656 < 100000, there are loads of 3-letter words, and 100000/36^4 = 0.059, so we expect to see about 6% of 4-letter words showing up, so there's some selection there. But 5-letter words are few and far between. Here are some of the MOST EXCITING parts:

...9bdybgbogeyj992q3...
...81tacoj2hs...

...reijleo6u9...
...4dleo56g...
...uileoir20...
...p8vleoo09...
...k7tpiusj9t...

...ymca...

Also, djniymhvbbcnxsiugmbwusdbaoxdkhbmmtnf occurs twice.

(why yes, this WAS a colossal waste of time)


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PostPosted: Thu Apr 07, 2011 4:58 pm 
Sanno
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Congratulations, phar, you've invented a new branch of theology: weighting the opinions of popes by how often they appear encoded in particular numbers...

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PostPosted: Thu Apr 07, 2011 5:12 pm 
Sumerul
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PostPosted: Thu Apr 07, 2011 5:48 pm 
Lebom
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PostPosted: Thu Apr 07, 2011 7:29 pm 
Smeric
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