8. The Ten Strings

— Numbers As Rôle Models — 

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T E T R A C T Y S
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Gödel-Strings
Gödel-Numbers

Let us look at the Integers not only as “number” but also as performing rôles. Some numbers may perform different rôles depending upon context. The different ways in which a particular number relates to others would thus be helpful in defining it qua Gödel-string.

We could see ways to interpret the functions of different numbers.

Below some examples.


10 — 8 — 6 — 4 — 2

Eight can be arrived at within the Tetractys group via 4 times 2 or 4 plus 4. Naturally there would be other ways, such as 6 + 2, each contributing to an interpretation, but let us stay with the simplest possible.

Four plus itself seems to be the simplest expression of eight, involving the least number of terms and operations. It would thus be the most defining string for 8. It would be similar to the other even numbers in this, each string-wise relying upon one term only for its expression;

  • x + x = n

The Tetractys contains five different numbers that satisfy this string.


9 — 4 — 1

There is also the series of squares, which are expressions of the string

  • x × x = n

Four is unique among the ten in that it partakes in both sequences. Not even unity does that. This double partaking would be part of defining what is special about four, the rôle of four as string.


7

Seven, in contrast, can not be easily arrived at in a similar way. The only single term that it resolves to — within the Tetractys — is itself, and 1, if we allow division, but that a number divided by itself is unity is trivial and does not define in what ways it is unique.

Speaking of unique, it is also the only prime within the group to be like this. Three has six, five has ten. Should we count two or one as primes, the same for them.

It would thus represent that which cannot be computed with or within the means we have at our disposal. It is the prime of primes.

  • x = x

It is not surprising the Pythagoreans called 7 Monad. Unique among the ten, 7 is thus identity. Since 7 does not include any relation or operation, it would also be ‘stillness’; no-change; no-motion. The equal sign would belong to its species. As would, perhaps, the point.


10 — 6

These are interesting as they require dissimilarity for construction if we go by way of multiplication. Two is not included as it is less than obvious whether doubling unity is related to dissimilarity in any meaningful way.

  • x × y = n

or

  • y × x = n

1 — 2 — 3 — 4 — 5

These can be half with an equal other half.

  • 2n = x

This can be interpreted in a variety of ways, and perhaps we need not specify these ways right now. What we have started is to map the functions that are more or less self-evident if we require them to use a minimal number of operations and only interrelate with other Integers of the Tetractys.

This last equation — at least for now — and the other ones above would belong to the core of the Tetractys group. Others may be added, but not an unlimited amount if we are to call it the core.