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 Post subject: Con-mathematical systems
PostPosted: Sat Apr 02, 2011 2:36 pm 
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I know that some of you have tried to create mathematical systems for your conworlds. To what extent did you succeed in making them different at some fundamental level from the algebra, geometry, calculus, and so on of our own mathematics? For example, would it be possible to express Euler's identity in a completely different form?

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PostPosted: Sat Apr 02, 2011 3:36 pm 
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PostPosted: Sat Apr 02, 2011 4:06 pm 
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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.

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PostPosted: Sat Apr 02, 2011 5:06 pm 
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lordofthestrings wrote:
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.


Amen.

And that's why mathematics is used in attempts to communicate with alien intelligences. However exotic they may be, at least the mathematics must be the same - everywhere in the Universe.

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PostPosted: Sat Apr 02, 2011 5:10 pm 
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So, we can assume that con-mathematics is doomed to be to Earth mathematics as conscript is to Tengwar?

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PostPosted: Sat Apr 02, 2011 5:42 pm 
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I did create a non-basal counting system about ten years ago, which Ive always been proud of. Im not sure I saved the details on a computer that still works, but I could reasonably well construct it from scratch if I had to. Essentially it's like counting on a checkerboard, with the corner square being 0, the next two being 1 and 2, then 3 and 4 and 5, and so on. But you spell the coordinates instead of the name, and it's an infinite checkerboard, so there is no limit to the size of each digit, but there will always be only two digits for any number.

The parts I dont remember are how to calculate the function without actually drawing a huge checkerboard, and how to extend it to more than two dimensions.

This system is not likely to be very useful to do math with, but I was creating it for an alien species which has trouble keeping more than one thought in their heads at the same time (indeed, their language only allows one morpheme per sentence, so they have a ridiculously huge vocabulary and no grammar.)

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PostPosted: Sat Apr 02, 2011 5:43 pm 
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Nancy Blackett wrote:
So, we can assume that con-mathematics is doomed to be to Earth mathematics as conscript is to Tengwar?


I wouldn't say that. Yes, the concepts will be the same since the laws of mathematics are fixed throughout the universe, but I'm sure your conpeople don't necessarily have to use base-10 like we do, and I can almost guarantee that their symbols for mathematical operators (+,-,/,x,=) will look different and work different. You might find this Wikipedia article on the History of Mathematics helpful.

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PostPosted: Sat Apr 02, 2011 5:56 pm 
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WeepingElf wrote:
lordofthestrings wrote:
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.

Amen.
And that's why mathematics is used in attempts to communicate with alien intelligences. However exotic they may be, at least the mathematics must be the same - everywhere in the Universe.

I've always heard this, and I guess I believe it... but why? I mean, for instance, we think in discrete quantities but, say, would a bug who only thinks in continuous develop math as we know it? Math is much about logic, but we can, and indeed have, developped different logic systems. or we could search for functional equivalents: there's a bunch of stuff algebra and arithmetics do for us, but maybe some weird silicon-based alien can only think in terms of geometry, or can't concieve the concept of number. I mean I'm talking seriously alien here, sure, and maybe this doesn't make sense, but still.

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PostPosted: Sat Apr 02, 2011 6:31 pm 
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i see.

so we humans are geniuses for being able to understand discrete sets and continua and logic and algebra and arithmetic and geometry and number and so on?

how lucky!!


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PostPosted: Sat Apr 02, 2011 7:10 pm 
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Pthug wrote:
i see.

so we humans are geniuses for being able to understand discrete sets and continua and logic and algebra and arithmetic and geometry and number and so on?

how lucky!!


the reverse case is by definition impossible to imagine

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PostPosted: Sat Apr 02, 2011 7:13 pm 
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I think any change in mathematics of a conculture would be either :

1) Mostly cosmetic. Different bases, symbols, computing algorithms, etc. If you want a handy reference, you can check "Mathematics across cultures : The History of non-western mathematics".

2) Related to things only mathematicians care about. A lot of mathematical concepts tend to be used mostly for historical reasons, I suspect, but those are only on topics that concern mathematicians. The basis for mathematics used is very often set theory, but it could also work with logical combinators. The set theory used is mostly ZFC (Zermelo-Fraenkel with axiom of Choice), but there are many systems that can be used. Calculus is with the whole deltas and epsilons rigamarole, and so on.

Of course, the thing to remember is that mathematics in cultures is, at least at first, used to answer to needs relating to their life. But those things are, at least in part, related to the physical laws of the universe.

For instance, if there was a noticable curvature of space, or even if it wasn't at all similar to our old Euclidian space, the laws of geometry developped would reflect that. Sure, they could change later on to include more general spaces, like we did, but that's not what's taught to everyone. Calculus and such is quite a bit related to the apparently continuous space. And so on.


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PostPosted: Sat Apr 02, 2011 7:57 pm 
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I don't understand most of the responses so far. I mean, trivially, there is a huge amount of mathematics now that there wasn't 2000 years ago, or 1000, or 500, or 100. There's no reason at all to expect some culture we've never talked to before to have "the same concepts".


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PostPosted: Sat Apr 02, 2011 8:09 pm 
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Well how far technologically are they. A society with FTL tech would most defiantly have math. A civilization that lives in the desert hunting squirrels and tanning leather for shelters may not have math at all.


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PostPosted: Sat Apr 02, 2011 9:25 pm 
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My knowledge of math unfortunately isn't good enough to do this with the level of verisimilitude I'd like to have. But I rather like the idea of Saimi mages discovering public-key cryptography (the RSA algorithm or something like it), and treating it as an utmost sacred thing that makes the magical ideal of Perfect Secrecy possible. Partially this is because I've personally thought RSA to be cool as hell ever since I learned how it works like a year and a half ago. Unfortunately the Saimi mages, while they had a lot of things, did not have computers in our sense, and without computers it's hard to imagine people making widespread use of encoding and storing the written word as manipulable numbers.

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PostPosted: Sun Apr 03, 2011 7:57 am 
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I'm not entirely sure what the OP is asking about.
Could there be a conworld where the laws of mathematics are different? Could 1+1=3? No, because mathematics is also constructed - if you have invented the concept of addition, it will work the same as in our world, because otherwise it's not addition you've invented.
Could a conculture be interested in completely different concepts? Could they just not care about addition, and rather regard some completely different operations as fundamental? In theory, yes, but probably not in any universe which has dimensions like ours, so your whole conworld wold likely be nothing more than a peculiar number exercise. It's more a matter of conphysics, and that usually only leads to confusion.

On the other hand, you can certainly change the way they think of and handle numbers. Here are a few ideas:
- Different bases, obviously. My conpeople use base 16 for the most part, but sometimes 2 or 240.
- Entirely different ways of writing numbers. My conpeople used to use a completely additive system, where you could write the symbols in any order. Not unusual, come to think of it.
- Inventing zero. Our own culture still has a long way to go here - we still call the first day of the month "1", and plenty of other things.
- Inclusive/exclusive intervals. When we say "ages 10-12" we probably mean three years, which doesn't really make sense - we are basically saying "ages 10.0000 to 12.9999".
- Sticking to integers. In fact, our society went from inventing continuous numbers to realising that nature is quantised in only a few millennia - you could skip that step.
- Different angle measurements. It's a small thing, but the full circle isn't necessarily 360 or 2π. Some have also redefined π to be twice as big, which would mess up Euler's identity at least a little. My conpeople also treat angles a little different; they see a straight angle as 0, because it means there is no angle, and a full turn as 1/2. That means the sum of the angles is 1 for any geometrical figure, including a circle.
- Developing things in a different order. The Greeks developed geometry earlier than what was really useful, my conpeople developed symbolic logic instead.
- Relativistic maths. If you have some peculiar alien species which moves really fast, they would need different calculations. They might for example see rapidity as more fundamental than speed.
- Non-Euclidian space. I don't advise it, tho.
- Trivial arithmetic systems. Gödel's theorem says that a non-trivial arithmetic system can't be complete and coherent - what if your conpeople decide to go with one that's trivial?

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PostPosted: Sun Apr 03, 2011 8:49 am 
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Chuma wrote:
I'm not entirely sure what the OP is asking about.


Essentially, "is it possible to invent a con-mathematical system which differs from our Earth one in more than just cosmetics like different bases, different numbers of degrees per circle, different names for sine and cosine, and so on?". Put this way, the answer seems to be "no", but if anyone has managed it, I'd love to know. A complete redesign of calculus, for example, would be very interesting.

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PostPosted: Sun Apr 03, 2011 9:58 am 
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con quesa wrote:
My knowledge of math unfortunately isn't good enough to do this with the level of verisimilitude I'd like to have. But I rather like the idea of Saimi mages discovering public-key cryptography (the RSA algorithm or something like it), and treating it as an utmost sacred thing that makes the magical ideal of Perfect Secrecy possible. Partially this is because I've personally thought RSA to be cool as hell ever since I learned how it works like a year and a half ago. Unfortunately the Saimi mages, while they had a lot of things, did not have computers in our sense, and without computers it's hard to imagine people making widespread use of encoding and storing the written word as manipulable numbers.


Not really. Gematria is a very old idea. Ciphers and codes are likewise old.

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PostPosted: Sun Apr 03, 2011 12:38 pm 
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It seems to me there are two quite different questions here...

#1, the fundamentals of nature are what they are regardless of who's doing the describing, yes.

But #2, what different species, or alternative human cultural histories, will care about while investigating those relationships may be subject to wild variation, as may their approach to various topics.

For instance the discrete/continuous issue Torco brought up is much more realistic than you guys gave him credit for - if a species starts out with a mainly continuous view of quantity, and here I'm talking about their roots, not what they are "capable of conceiving of" which is different and stupid to assume a limit on, then whatever mathematical structures they work out may be very different in flavor from ours.

An even deeper point of potential variation is what the purpose of math is conceived to be by the society/species. For instance I could easily imagine aliens that are not highly tolerant of purely theoretical musings, with only practical concerns considered legitimate. They'll get the same value for pi that we do, but they may not give a shit that ei*pi = -1 and might, for example, not bother to work out something like complex numbers until such time as working with electricity forces it. Or they might not care about proving things absolutely, the way we do. Proof is what our mathematicians are usually after when they investigate a behavior, as this is part of our conception of what understanding consists of: the immense value (as we see it of) of demonstrations that something must be true. Another culture may find it sufficient that something only seems to hold... and perhaps get some wrong answers that way, too.

Another point of variation: what the operations are. The hierarchy of operations we use goes like this: first there's counting, i.e. incrementing by one; then there's addition, which is a way to count incrementings; then there's multiplication, which is a way to count additions; then there's exponents, which are a way to count multiplications; and though we have little use for it there is also "tetratation", a way of counting exponentiation. So there's room for variation: a species/culture could just stop at multiplication, say, the way we usually stop at exponentiation, while another may go several levels further. It's not necessarily even arbitrary, there are reasons one might stop at multiplication - for example the fact that while addition and multiplication are commutative, exponents aren't. It might be held as a value that that makes them untenable, say. Perhaps another species might work out a different extension of multiplication that is commutative. (That may be impossible, but if anybody wants to argue that it is, I demand proof!). Also, for alien species, there's no reason an operations system even has to be based on the notion of counting (but I think anything human would have to be). What else might it be based on? Maybe operations could be based on the properties of things rather than their quantities, for example shape deformations, or cycles of change. Exactly how to make that work I couldn't say, but it seems plausible that ten thousand years of a species' smartest minds might be able to make something of it. Whether a conworlder could make something of it is less likely.


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PostPosted: Sun Apr 03, 2011 12:56 pm 
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Quote:
- Inventing zero. Our own culture still has a long way to go here - we still call the first day of the month "1", and plenty of other things.


That doesn't have much to do with the invention of 0.

Quote:
- Sticking to integers. In fact, our society went from inventing continuous numbers to realising that nature is quantised in only a few millennia - you could skip that step.


We didn't, really. As far as we know, space time is still continuous. It could not be, but so far, nothing indicates it.

Quote:
- Developing things in a different order. The Greeks developed geometry earlier than what was really useful, my conpeople developed symbolic logic instead.


Geometry was mostly developed for practical reasons (it's not called "earth measuring" for no reason). And the ancient greeks did start up a bit on logic, which was also quite studied in medieval times.

Quote:
- Trivial arithmetic systems. Gödel's theorem says that a non-trivial arithmetic system can't be complete and coherent - what if your conpeople decide to go with one that's trivial?


They're gonna have a hard time doing it, because anything better than a caveman understanding of mathematics will be non-trivial, Gödel-wise (basically addition and multiplication is already too much). Also Gödel only matters if you do axiomatic mathematics, which is probably gonna come a bit after multiplication, I think.

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A complete redesign of calculus, for example, would be very interesting.


No need to look too far. There's literaly dozens of ways to define calculus in modern mathematics.

Quote:
They'll get the same value for pi that we do, but they may not give a shit that ei*pi = -1 and might, for example, not bother to work out something like complex numbers until such time as working with electricity forces it.


Hell, practical methods for what would later be complex numbers already exist in our culture. Can't remember the name, but back around the 16th-17th century, they just represented some physical quantities in R² to solve some optics problems.


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PostPosted: Sun Apr 03, 2011 3:38 pm 
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When you get right down to it, all mathematics is conmathematics.

One idea I've been playing around with is an extension of Riemannian geometry, which works as follows:

Let the distance between two points z and y (which are points on some manifold) be the integral of a function L from z to y, where

L=sqrt(gab(x)dxadxb) + ka(x)dxa, summation over repeated indices is implied.

If the second term in L were dropped we'd have standard Riemannian geometry. For simplicity I assume that gab(x) is a symmetric tensor, though that's not strictly necessary. My inspiration for this comes from examining the motion of a charged particle in an EM field moving in a curved spacetime, however this physical interpretation is not necessary.


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PostPosted: Sun Apr 03, 2011 3:40 pm 
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Aldwinkle wrote:
That doesn't have much to do with the invention of 0.

It doesn't? If a culture hasn't understood the concept of zero, surely they would be likely to start their months on 1?
Or perhaps you are saying that starting on 1 is the logical choice even if you do have a concept of zero? I would be happy to discuss that with you, but perhaps we shouldn't derail this thread just yet.

Aldwinkle wrote:
As far as we know, space time is still continuous. It could not be, but so far, nothing indicates it.

I'd say there are a few indications, but I haven't specialised in quantum mechanics. At least it's possible to conceive of a culture which believes in discrete space - discovering atoms should give them plenty of encouragement.

Aldwinkle wrote:
the ancient greeks did start up a bit on logic, which was also quite studied in medieval times.

True. I meant in a more mathematical sense.

Aldwinkle wrote:
anything better than a caveman understanding of mathematics will be non-trivial

But only if we allow infinite sets of numbers, as far as I understand. And several cultures far more advanced than cavemen didn't have that sort of numbers. What if we decide that there are no numbers bigger than 1000? I think it's possible to define an arithmetic system which is complete and coherent in that case. Do let me know if I'm wrong, but be sure to explain it thoroughly. :)

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PostPosted: Sun Apr 03, 2011 7:19 pm 
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well yeah, the naturals mod 1000 but why? 1000 is not even particularly big a number


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PostPosted: Sun Apr 03, 2011 10:55 pm 
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lordofthestrings wrote:
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.


I think I actually wrote up a system trigonometry once which used a fraction of pi as the base unit instead of pi itself. It still works out, obviously, but I remember struggling a bit to get "nice"-looking numbers for right angles and so forth. I lost the notes, though.

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PostPosted: Sun Apr 03, 2011 11:55 pm 
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faiuwle wrote:
lordofthestrings wrote:
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.


I think I actually wrote up a system trigonometry once which used a fraction of pi as the base unit instead of pi itself. It still works out, obviously, but I remember struggling a bit to get "nice"-looking numbers for right angles and so forth. I lost the notes, though.


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PostPosted: Mon Apr 04, 2011 9:49 am 
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...it wasn't like that.

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