Utah Logic: Bracket Numbers (A.K.A. "Eidian Numbers")

A not uncommon observation is that in Science Fiction, most alien languages are recognisably Indo-European. (Albiet with different vocabulary and some features backwards.) Not only are they not very alien, but they are nowhere near as bizarre as some actual human languages. [Such as Kiliwa where numbers are verbs and adverbs]

And just as the language family is ours, alien numbers are recognisably our traditional base 10 notation with possibly different symbols. (Or in very rare cases, some other positive integer base. Even the familiar alternate notations (such as balanced ternary) don't appear, even though they have enough advantages that people on this planet have built comptuters based on them.

A common question (at least amoung the Utah Logic crowd) is how alien alien numbers could be, within the bounds of being something that could be reasonably justified mathematically.

LeRoy N. Eide once offered the following, with the given justification.

Derivation

Consider the (academically) reasonable idea of representing a number by its prime factorization. Since this is done for some types of mathematics, it is justifable in the sense above.

Since the primes are fixed, we need only write down the exponents of the primes, so instead of writing
2¹+3⁴+5⁰+7¹... we would have
(1 4 0 1 ...) This first step gets a counting list something like:

 1: (0)
 2: (1)
 3: (0 1)
 4: (2)
 5: (0 0 1)
 6: (1 1)
 7: (0 0 0 1)
 8: (3)
 9: (0 2)
10: (1 0 1)
11: (0 0 0 0 1)
12: (2 1)
13: (0 0 0 0 0 1)
14: (1 0 0 0 1)
15: (0 1 1)
16: (4)
17: (0 0 0 0 0 0 1)
...

But, we still have our old fashioned integers in the lists. We can get rid of these by replacing each such integer with its repreentation from the table. [Is it obvious why each number appears before it is needed?]

 1: (0)
 2: ((0))
 3: (0 (0))
 4: (((0)))
 5: (0 0 (0))
 6: ((0) (0))
 7: (0 0 0 (0))
 8: ((0 (0))))
 9: (0 ((0)))
10: ((0) 0 (0))
11: (0 0 0 0 (0))
12: (((0)) (0))
13: (0 0 0 0 0 (0))
14: ((0) 0 0 0 (0))
15: (0 (0) (0))
16: ((((0))))
17: (0 0 0 0 0 0 (0))
...

So now we are left with a representation that is only in terms of "(", ")", and "0" There is nothing magic about "0", other than it can be uniquely recognised, and isn't already in use. Since an empty set of prenthesis ("()") can't occur in the above, it is a suitable candidate for a representation of "0". Applying it yields us:

 0: ()
 1: (())
 2: ((()))
 3: (()(()))
 4: (((())))
 5: (()()(()))
 6: ((())(()))
 7: (()()()(()))
 8: ((()(()))))
 9: (()((())))
10: ((())()(()))
11: (()()()()(()))
12: (((()))(()))
13: (()()()()()(()))
14: ((())()()()(()))
15: (()(())(()))
16: ((((()))))
17: (()()()()()()(()))
...

Now THAT starts approaching a notation that is truely alien. But it is, with some work, just the familar prime factorization notation. And it is expressed only in raw grouping. Prentheses are a common notation in mathematics, but not the only way to show grouping even on this planet. Grouping by an overbar has at some times been seen in the mathematical and logic literature, so is justifiable in the sense above. With the use of overbar, every integer now has a unique symbol, but with an "obvious" pattern to an appropriate alien. And that, trimmed down to the minimim, gives us our current final version: (displayed as crude ASCII art... done properly it looks more like the I Ching.)

0: *

1: *
   *

2: *
   *
   *

3: ***
   * *
     *

4: *
   *
   *
   *

5: *****
   * * *
       *

6: ***  
   * *
   * *

7: *******
   * * * *
         *

8: ****
   ****
   ****  
   *  *
      *

9: ***
   ***
   * *
     *
     *

10: *****
    * * *
    *   *

11: *********
    * * * * *
            *

12: ***
    * *
    *
    *

13: ***********
    * * * * * *
              *

14: *********
    * * * * *
    *       *

15: *****
    * * *
      * *

16: *
    *
    *
    *
    *

17: *************
    * * * * * * *
                *

18: ***
    * *
    * *
      *

19: ***************
    * * * * * * * *
                  *

20: ***
    * *
    *
    *
    *
...

Of course, even if you adopt this underlying notation there are LOTS of other ways of showing grouping. (And the above assumes some sort of left to right ordering, or top to bottom, neither of which has any absolute justification.)

Go to ...

• This Page
• Misc. Topics Index
• Utah Logic home page
• Back to Back to the Philosophy Department
• University of Utah home page

This page is http://web.utah.edu/utahlogic/misc/index.html
This page was last modified on February 5th, 2007.