Con-mathematical systems

[zelos]

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

[finlay]

Math concepts are universal. While there will be differences in notation and numeral systems, even the most exotic aliens will agree with us on the value of pi and the Pythagorean theorem.

You know, I read a website recently that suggested that we should be using 2π as our circle constant instead of π, because it's the ratio of the radius instead of the diameter. Apparently it makes certain things neater. They suggested calling it tau, mainly because it looks a bit like π, I think. But perhaps the aliens will come along and think we're mental for using "pi" instead of "tau", or something. Food for thought.

Here, it's a bit tongue in cheek and pseudo-philosophical and talking about consipiracies and pi-partisans, but it's worth a read, at least: http://tauday.com/ (warning: uses latex and very slow to load for me)

[bulbaquil]

Math concepts are universal. While there will be differences in notation and numeral systems, even the most exotic aliens will agree with us on the value of pi and the Pythagorean theorem.

You know, I read a website recently that suggested that we should be using 2π as our circle constant instead of π, because it's the ratio of the radius instead of the diameter. Apparently it makes certain things neater. They suggested calling it tau, mainly because it looks a bit like π, I think. But perhaps the aliens will come along and think we're mental for using "pi" instead of "tau", or something. Food for thought.

I actually do agree that we do tend to use 2π more than π (or in general kπ, where k is even), and it's certainly reasonable that an alien race might think the ratio of the circumference to the radius is more important than that to the diameter.

[pharazon]

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.

[stinja]

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.

[Anguipes]

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.

[pharazon]

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.

Yeah, my post above was pretty much getting at doing con-history instead. I guess numerology is one way you could get some genuine con-mathematics without having to do anything too difficult. There are plenty of little basic arithmetic statements that are not too hard to discover and even prove, and which are sort of silly from a mathematical point of view, but might look more impressive to someone silly like a numerologist: I mean things like "if N has an even number of digits, and Ñ means N written backwards, then N + Ñ is a multiple of 11". A few of these things are even useful, at least if you're doing arithmetic by hand: the rule of nines, say, or the divisibility tests for small primes.

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.

Does this statement include the ancient Greeks?

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.

Leibniz died in 1716... But the spirit of the statement is still true: I don't think many philosophers or mathematicians picked up on Leibniz' work in logic in the vicinity of his lifetime.

[Salmoneus]

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).

[Radius Solis]

The "problem" with con-mathematics is that, if you create and flesh out any novel system of mathematical thinking you've done actual mathematics.

This occurred to me a few days ago too, yeah. It does sorta explain why it's so hard to do - depending on the scope of what you want to do, it may be no less a task than real mathematicians face. Nevertheless I have hopes of getting somewhere with it, somehow, sometime.

I have had one little idea that may work out to be something, or may not. But lest I never post about it again, I'll mark it down here now. It seems reasonable that the basic notion of modular arithmetic or "clock math" could be arrived at quite early in a mathematical history, it being dead easy to get and potentially useful for dealing with weekday cycles or indeed clocks, and possibly many other things depending. It also doesn't seem implausible that this could be the chief avenue along which to pursue more advanced math, rather than geometry and ratios. For instance you can leap directly into trigonometry from there without aproaching it in terms of right triangles, as the heart of trig is really the circle anyway - especially if, instead of a number line, early mathematical thinkers conceive of the number continuum as a repeating cycle. And in a cycle you can set the "base" anywhere you want, in principle. Count by threes or count by tens, it works the same. So you could arrive at a Rennaisance-like historical period with a variable-base version of trigonometry that unifies the concepts of "quantity" and "angle" and that has its own representation system that forms the basis for much in later math, without necessarily involving the concept of pi. The main con-mathing task would then lie in creating that representational language and seeing what all could be expressed in it and how it would all come together. It would be nice if it were extendable to describing other sorts of curves too, or functions generally, but that might be too much to ask.

[finlay]

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..

[alice]

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?

[finlay]

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.puw5nggjf8y4nfyoryfukso6ds803x1dwns1nadyo3uhakyxl4npuq1ep5n58b50hr7yna3qh3v4pkuo2f7xdvgsj3yxxfdjnstetqsd1e31lc0ouluxac6faomhzii3xmck8f75eh2tsbosr2581auok10zu5m8l5u7rnd0xtlljvq6vab8gzvdxzjd6dt5glxm96s75393j289fhknyjh7y3rltntppy43s1lb52ibt42263p5gn6msq0xyg9t0dvkldq4ahadih9gncsmkx2e694jdv88qklqj1wpul7l5kb3vgrwujtgg588ept9oqiinf1kp5m8yebizeyzm0cdmcqsxz7mdcxb1ea20v53fbudf3h0w2k08cejh9uyw30o55c7zb401vpnv6lnog3s6npxga3uoxnoohofcw5544bvytorcmb5kdswece44cq6hvzbgj4j4y5vn14d2jj09874mw1uxiihaqoofgs32wwhc81luu5rgjbq8drtnbmc0g8pnixrvxsnyksgmbv7te6s8x4i4mhndjs387bsklaiqi8nbxfvj6pzavr1oy7pw75fu6y0fq8yed1r7d18r3nq5w4zuvnt9iv1itsmy54ag0o7cg3r2n1v1nwy93tke5ekq9l8t01b082yoq8uilpdqlxedrrae57utn05oaed3qpbifzkt2rn0rg53hig2s2bh7qzrz0svyiurs4fg3aandhx6ed0jse5zons4e325g1urprbzzl12ealhtshejhk0vg9ajotepp8li5klimv04zswyyrjicg1s1zp3e1rxv4mtmitty3slaacz0x0eabou4gyijcga2bn2nv3b8uekub6ryoca4h4zz16ytzmv8ntzsh6qnmynw0b1peh2lxni9amfl1z6hmt0ewfrxdq8lr3nlj8lxltu7ulsdfoajtu4mtkoy0zp3vql8q9t5xaqr6cj02t74km6iciiwr0a53vlgc2z31wvtx1swwyfqccze3nc37v..._36

And phi:
1.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..._36

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

[alice]

I think we may just have invented a new method of generating new conwords...

[Radius Solis]

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?

Is that really true? Does having an infinite representation really imply that all possible sequences of digits must turn up somewhere in it? I'm not sure how it does. Obviously it doesn't with rational numbers that have infinite representations (in decimal) like 1/3 or 1/7, so the infiniteness alone can't require any such thing. Is there some other factor that does require this for irrational numbers?

[Pthagnar]

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?

[zelos]

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?

define "normal" number here

Math concepts are universal. While there will be differences in notation and numeral systems, even the most exotic aliens will agree with us on the value of pi and the Pythagorean theorem.

You know, I read a website recently that suggested that we should be using 2π as our circle constant instead of π, because it's the ratio of the radius instead of the diameter. Apparently it makes certain things neater. They suggested calling it tau, mainly because it looks a bit like π, I think. But perhaps the aliens will come along and think we're mental for using "pi" instead of "tau", or something. Food for thought.

Here, it's a bit tongue in cheek and pseudo-philosophical and talking about consipiracies and pi-partisans, but it's worth a read, at least: http://tauday.com/ (warning: uses latex and very slow to load for me)

I do believe the reason for Pi is simply that taking any decimal expansion of pi times an integer is alot easier than taking any decimal expanion of tau and divide it by any integer

[Aldwinkle]

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).

[pharazon]

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?

define "normal" number here

A normal number is one whose base b expansion "looks random" in the sense that it contains any finite string exactly as often as you'd expect for a random sequence of digits. To be precise, say S is a finite string of digits, and write D(x, S, d) for the frequency of occurrences of the string S in the first d digits of x; that is, D(x, S, d) = (# of appearances of S in first d digits) / (# of all strings of the same length as S in the first d digits). If S has length s, you'd expect a random sequence of s digits to match S with probability 1/b^s, since there are b choices for each digit. So x is normal if D(x, S, d) approaches 1/b^s as d approaches infinity.

This is the property Nancy Blackett was referring to, although as Pthag says, it is not actually known (or, I think, cared about terribly much) whether pi, e, or almost anything else, is normal. Statistical analyses of a lot of digits suggest that they are.

Math concepts are universal. While there will be differences in notation and numeral systems, even the most exotic aliens will agree with us on the value of pi and the Pythagorean theorem.

You know, I read a website recently that suggested that we should be using 2π as our circle constant instead of π, because it's the ratio of the radius instead of the diameter. Apparently it makes certain things neater. They suggested calling it tau, mainly because it looks a bit like π, I think. But perhaps the aliens will come along and think we're mental for using "pi" instead of "tau", or something. Food for thought.

Here, it's a bit tongue in cheek and pseudo-philosophical and talking about consipiracies and pi-partisans, but it's worth a read, at least: http://tauday.com/ (warning: uses latex and very slow to load for me)

I do believe the reason for Pi is simply that taking any decimal expansion of pi times an integer is alot easier than taking any decimal expanion of tau and divide it by any integer

I don't know why that would be relevant. Division is not hard, and what's so important about integer multiples of pi compared to 2pi / integers that anyone would care anyway?

[pharazon]

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).

Yeah, even though it's not known for almost any particular number whether it's normal, it is known that almost all real numbers are normal (where "almost all" has a precise meaning which you might take to be, if you pick a real number at random, it will be normal with probability 1).

I have had one little idea that may work out to be something, or may not. But lest I never post about it again, I'll mark it down here now. It seems reasonable that the basic notion of modular arithmetic or "clock math" could be arrived at quite early in a mathematical history, it being dead easy to get and potentially useful for dealing with weekday cycles or indeed clocks, and possibly many other things depending. It also doesn't seem implausible that this could be the chief avenue along which to pursue more advanced math, rather than geometry and ratios. For instance you can leap directly into trigonometry from there without approaching it in terms of right triangles, as the heart of trig is really the circle anyway - especially if, instead of a number line, early mathematical thinkers conceive of the number continuum as a repeating cycle. And in a cycle you can set the "base" anywhere you want, in principle. Count by threes or count by tens, it works the same. So you could arrive at a Renaissance-like historical period with a variable-base version of trigonometry that unifies the concepts of "quantity" and "angle" and that has its own representation system that forms the basis for much in later math, without necessarily involving the concept of pi. The main con-mathing task would then lie in creating that representational language and seeing what all could be expressed in it and how it would all come together. It would be nice if it were extendable to describing other sorts of curves too, or functions generally, but that might be too much to ask.

Are you imagining a discrete thing? Like, starting at a point on a circle and moving by 36 degrees each step, so there are 10 different positions? Otherwise I'm not sure how the "base" comes in.

[Salmoneus]

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.]

[pharazon]

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)

[Salmoneus]

Congratulations, phar, you've invented a new branch of theology: weighting the opinions of popes by how often they appear encoded in particular numbers...

[finlay]

This is the maximum expansion that wolfram alpha is capable of:
3.53i5ab8p5fsa[...]swg6buyw[...]rs1xclck34ji4[...]m197rgfyg09uqi3v1pa..._36

Zompist, you haven't been smuggling advertising into the fundamentals of mathematics, have you??

[pharazon]

This is the maximum expansion that wolfram alpha is capable of:
3.53i5ab8p5fsa[...]swg6buyw[...]rs1xclck34ji4[...]m197rgfyg09uqi3v1pa..._36

Zompist, you haven't been smuggling advertising into the fundamentals of mathematics, have you??

No, it is his nemesis, although this only becomes apparent after looking at more digits:

3.53i5ab8p5fsa...swg6buyw...rs1xclck34ji4...qi16not

[Radius Solis]

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.

That was the idea, yeah. Although I hadn't thought of modeling it with a helix, I was just imagining a clock and some manner of keeping track of revolutions that can be used or not used as needed. A helix could provide a handy mental handle for that. On the other hand, in a conworld context, I rather like the idea of at least a timekeeping device that was a globe instead of a disc, incrementing a unit of longitude for every full cycle of latitude. That would extend the counting on circles instead of lines to its logical second dimension. Designed right, e.g. using the same multi-hand system we already use on clocks, you could get a full year out of it. (Hell, why not invent one such for Earth? It might even sell a little, as a novelty item for timekeeping enthusiasts and random nerd sorts like me.) Or a different sort of device: use actual wires for the longitude/latitude lines and stick beads on them and we might have the makings of an interesting abacus. Or possibly a pointless one... I don't know enough to say.

[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.]

Well, anything I might do with this would have to be understandable to the layman, because I am one, and now there's an actual con-math idea to work on I can see that I'm ill-equipped to make anything workable out of it. Your comments here are already ahead of what I'd gotten to. I'll get nowhere interesting without some help, be that from pharazon or from anyone here who cares to add their two cents... because there's the central idea but I don't know what I should be doing to make it real.