The editor asked me, in my capacity as occasional reviewer, fellow traveller, and former ANPA president, to describe this organization for the readers of CYBERNETICS & HUMAN KNOWING. I will therefore begin by quoting from ANPA's `statement of purpose': "The primary purpose of the Association is to consider coherent models based on a minimal number of assumptions to bring together major areas of thought and experience within a natural philosophy alternative to prevailing scientific attitude. The combinatorial hierarchy, as such a model, will form an initial focus of our discussions". `Natural philosophy' being a (slightly archaic) synonym for what we now call `science', the interest of ANPA for this journal's readers is thus apparent.
A few historical and organizational facts must take temporary priority over an explanation of what `the combinatorial hierarchy' is. ANPA emerged - without name or other such formalities - in the 1960's in England as a loosely coordinated group of physicists and mathematicians who met roughly yearly to discuss fundamental issues. The key members of this original group were Frederick Parker-Rhodes, Ted Bastin and Clive Kilmister, and Ted and Clive are still active. ANPA came to exist formally in 1978 with the above statement of purpose, and in the course of the 1980's budded an American West Coast affiliate known as ANPA West at the initiative of H. Pierre Noyes (SLAC) and David McGoveran. The British branch, "ANPA" per se, meets yearly in September in Cambridge, whereas ANPA West meets in February at Stanford. See Box 1 for further details.
| Box 1: ANPA membership, newsletter, submissions, and proceedings: Dr. Faruq Abdullah, Physics Department, City University, Northampton Square, London EC1V 0HB ENGLAND; next ANPA meeting: Mid September Friday-Sunday 1994 in Cambridge. ANPA West membership, journal, and meeting info: Dr. Tom Etter, 409 Leland Avenue, Palo Alto, CA 94306 USA. All serious written contributions are considered, and ANPA strives to be a forum where such contributions, however `deviant' from conventional approaches, can receive qualified and considered feedback. |
The `combinatorial hierarchy' is a structure whose explication is the topic of the following section. At the most general level, it is a mathematical structure whose uniqueness and gene-rality invite multi-faceted interpretation. Given that the founders of ANPA were mostly interested in physics, it has traditionally attracted those interested in physical theory. However, we have increasingly been attracting those interested in biology, and hope this will continue. I note that semioticians should find the hierarchy of interest.
The combinatorial hierarchy
An understandable mathematical characterization cum description of the CH is difficult to fit into the present confines, so I will attempt it mostly in words. The hardy reader can try Box 2.
The combinatorial hierarchy (hereafter, `CH') is, first and foremost, combinatorial: that is, it is based on the combinatorics of arbitrary symbols, and contains in and of itself no a priori interpretation of said symbols. We say `The' because it turns out that the most general form - based on exactly two dichotomic symbols - is the sole exemplar. We say `hierarchy' because it consists of a series of `plateaus' or `levels', each of which is constructed on the basis of the preceding level. There are two operations on symbols: discrimination - denoted by `½', which compares symbols to see if they are the same (`0') or different (`1'); and what I will call aggregation - denoted by simple juxtaposition, which groups symbols together. I should perhaps note that the arithmetic properties of `1' and `0' play no particular role here.
The first step in constructing the CH is to postulate the existence some otherwise undifferentiated and totally structureless universe, f. [A ton of philosophical questions is herewith ignored!]
The second step is to compare this universe with itself, yielding f½f = 0, so we now have one symbol, `0'.
The third step generates yet another new symbol by noting that the universe is not the same thing as `0', that is, f½0 = 1, so now we have three symbols. Notice now that we cannot generate additional symbols from what we have in hand, since the as yet untried discriminations, f½1=0,1½0=1,1½1=0, yield nothing new. Thus at this stage we have exhausted the possibilities of generating new symbols via discrimination.
The fourth step therefore invokes our other operation, aggregation: we can simply replicate one or more of our existing symbols to produce some new, but more complex, symbols. For example, given a `0' and a `1' in our hand, and replicating `1' twice, we could produce `01' and `11'; or by replicating `1' and `0' once each, we could generate `10' and `01'.
[As an aside, the symbol consisting only of 0's is always implicitly present, so we consistently leave it out.]
Step five: Looking at these new symbols, we see that 01½11=10, or using the other example, 01½10=11. So given some set of symbols, we can generate others. There are, strictly speaking, three such sets here, {(10)}, {(01)}, {(10), (01), (11)}, since (e.g.) 01½01=00 which we always ignore, similarly for 10½10, and the 3-set, which as we saw above closes on itself too. Such sets are called `discriminately closed' sets, Dcs's. The `three' we have seen here is the `3' in level 1 of the hierarchy shown in Box 2, column (b).
We have again exhausted the possibilities of discrimination to expand the set of symbols, and so replicate and aggregate again. Notice that initial symbols of a new level are built out of combinations of the symbols at the current level. In particular, it turns out that the elements of the Dcs's contain exactly the ones to aggregate.
We have now seen the construction of two levels - steps 2 & 3 and steps 4 & 5 - and this process can continue. But how far? One might think that one could go on indefinitely, and in one sense one can. But in another sense, it turns out that the sequence terminates with four levels. This termination comes about via the interplay between the number of Dcs's available at a given level and the number of symbols they can generate at the next level. In the language of vector spaces, the symbols can be viewed as vectors, and the cut-off occurs when the number of symbols available for building new vectors is insufficient to `span' the resulting space.
Thus there are two intertwined sequences, one for the number of symbols and one for the size of a `spanning set', between which one `alternates' in the construction. The first sequence's series is (1), 3, 7, 127, 2127 - 1 and the other is (1), 4, 16, 256, 2562. The cut-off arises because 4>3, 16>7, 256>127 but 2562 <2127. [See ref. 1]
This may all seem a little abstruse, but the point is that given some initial number of distinct symbols and operations for comparing and combining them in all possible ways, we find
* There is a unique hierarchy, based on two symbols, which has exactly four levels, and then stops! Starting with three or four symbols yields fewer levels before the cut-off, and five or more never get off the ground.
* However, intriguingly, the CH has an ourobouros1 property, in that once the fourth level is reached, and at any time thereafter, the set of symbols then available can be grouped into two sets, thus providing a `new' definition of `1' and `0', and the organic development thus proceeds self-similarly.
*This hierarchy has a precise mathematical characterization (see references).
*This same hierarchy turns up in other places - for example I have found it in the use I put Clifford algebras [see ref. 2] to in AI, and with the same interpretation. My aggregation operation is `can occur simultaneously' and my discrimination operation is `exclude each other', over computational events.
One last intriguing property of the CH, and the one which
prompted the most initial interest is the running sum of the number of
symbols: 3,3+7=10, 10+127=137,2127+136»1.7x1038,
and in particular the extent to which the last two in fact are more
than accidental approximations to the fine structure constant
and
, respectively2.
It can fairly be said that the ANPA program has been driven by the pursuit
of this question. In the course of this pursuit, a non-trivial start has
been made in the reformulation of modern physical theory on a purely combinatorial
basis.
Howsoever, the readers of the journal should I think take two basic thoughts from this brief introduction:
First, that there exists an abstract formulation of the space of symbols which, even before the symbols are given an interpretation, contains a great amount of structure; and
Second, that there exists an organization, ANPA, devoted to both this structure and the greater task of creating a viable alternative to contemporary scientific theory, which you are cordially invited to consider.
| Box 2: The CH is constructed from two inter-twined sequences (columns (b) and (d) below) and looks like this: | |||||
|
(a) level |
(b) # symbols per level |
(c) cumulative S (b) |
(d) map dimensionality |
(e) # of map elements |
(f) comment |
|
0 1 2 3 4 |
(1) 3 7 127 2127-1 |
(1) 3 10 137 2127+ 136 |
(1) 4 16 256 (256)2 |
(1x1) 4x4=16 16x16=256 256x256=65536 |
16>7 256>127 65536<2127-1 cut-off reached |
Column (b) is simply the full number of ways
a number of entities (symbols) can be combined - 1,2,3... at a time, which
is This
sequence thus counts the number of symbols that can be formed from some
given set of symbols by aggregation. The second sequence, column (d), is
related to the number of symbols from column (b) which can via discrimination
produce the remaining ones at the next level.
More precisely, we invoke the concept of discriminate closure on these `bit strings', and define a discriminately closed subset (Dcs) as a single non-null string, or as that set of non-null strings which, when any pair are discriminated yield another member of the set. If we start from linearly independent strings a, b, c,...(ie. a½b¹0, b½c¹0, a½c¹0, a½b½c¹0, ...we can form the Dcs's {a}, {b}, {c}, {a,b,a½b}, {b,c,b½c}, {a,c,a½c}, {a,b,c,a½b, b½c, a½c, a½b½c}. Starting with strings of 2 bits, we can form 22-1=3 Dcs's, for example {(10)}, {(01)}, {(10),(01),(11)}. To preserve this information about discriminate closure, we map these three sets by non-singular, linearly independent 2x2 matrices which have the members of these sets as eigen vectors. Rearranged as strings of four bits, these form a basis for 23-1=7 Dcs's. Mapping these by 4x4 matrices we get 7 strings of 16 bits which form a basis for 27-1=127 Dcs's. We have now organized the information content of 137 strings into three levels of complexity. We can repeat the process once more to obtain 2127-1»1.7x1038 Dcs's composed of strings with 256 bits, but cannot go further because there are only 256 x 256 linearly independent matrices available to map them, which is many too few. Thus after four stages, the information-carrying capacity of our information-preserving mapping scheme is exhausted. |
|||||
References
Noyes, H.P., and McGoveran, D.O. An Essay on Discrete Foundations for Physics. Physics Essays, v 2,1. 1989.
Hestenes, D. New Foundations for Classical Mechanics. Reidel (Holland) 1989. [Chapters 1 & 2 are an excellent introduction to Clifford algebras, including a very nice retrospective on classical Greek geometry which I can recommend.]
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