Glanville R. (1978) The nature of fundamentals, applied to the fundamentals of nature. In: Klir G. J. (ed.) Applied general systems research. Plenum, New York: 401–409. Available at http://cepa.info/2721

It is common nowadays to search for ’fundamentals.’ We assume there are such things and that they can be found, although this assumption may not be in any way justified.

I should like therefore to look at the whole notion of ’fundamentality.’ It is a concept of a non-reducible entity, from which other things (all other things) are made, or to which they refer. As it happens, most of those things we have assumed to have been fundamental have shown themselves not to be, (the sad history of nuclear physics) – and even those that we have set up and defined as being fundamental, such as the idea of a constant unit of length (which is fundamental to the activity of measuring) have an unpleasant habit of needing constant redefinition. So it seems that, from the point of view of past performance, fundamentals have an almost unique failure rate, Other things with similar rates of failure don’t usually survive as ideas worthy of continuing attention, but fundamentalism reigns regardless.

I take this survival as indicative of a particular significance in the idea, of a meaning which is vital even if elusive, of a quest for a reasonable basis which continues despite Gödel’s [Note 10] demonstration of the necessity of belief in axioms and Wittgenstein’s [20], [21] demonstration that what we can observe is, a priori, not the thing itself.

Work carried out in the last ten years has suggested reformulations of the problem such that, when re-stated, neither Gödel’s nor Wittgenstein’s discoveries are so alarming.

In this context, the notion of fundamentality merits attention. This paper hopes to show what its successful operation would entail, and hence the reformulation that would achieve this.

Something which is fundamental is non-reducible. Any attempt to derive it will involve self-reference, if it is truly fundamental, for, if it cannot be divided, it can only be described in terms that it, itself, has helped generate. While Maturana, [Note 12] Günther, [Note 11] von Foerster, [3], [4], [5], and particularly Varela [17], [18] have done much recently to liberate self-reference (and hence paradox) from the tyrannic domincance of the orthodox in logic, (and, in so doing, reduce the role of the fundamental to one of convention, not of formal necessity), their work is not directly relevant to our discussion, for they are discussing in extenso, not, as we are doing, in intenso[Note 1] .

By non-reducible, I mean that something cannot be divided in such a way that something else, from which it can be derived, and the means of deriving it, are, when taken together, distinctly smaller than that thing. This also means it has no structure.

As it happens, this is the definition of randomness: a random number string (or random number – there is no difference) is a string for which there is no shorter means of derivation.{@} In fact a random anything is something for which no pattern (i.e., no root) can be found. This is identical to fundamental: something, within any system or Universe, that is random, is also fundamental.

Normally, one considers that randomness is a non-provable characteristic. The elegant article by Chaitin (already referred to), discusses this most clearly. The problem is, that while non-randomness can be shown, it cannot be shown that something which has not been reduced yet, cannot be reduced. For the reduction of it only proves it was not random, not that the new root is random. Consequently, one is in the position of believing something is random, until it is proved not to be: the only proof lies in the negation. The resemblance to French law (believed guilty until shown to be innocent) is clear.

Given that fundamental is equivalent to random, and given a normal examination in intenso, it is no longer surprising that the fundamental particles of physics are continuously being shown not to be fundamental.[Note 2] , I attempt to show the logic and form of the actual description used, and hence to discuss the formulation of such descriptions. The ’Confinement of Quarks’ [Note 14] is an argument precisely about this.

In fact, it is in biology, alone among the natural sciences, that this fact has been accepted. In ’Chance and Necessity’ [Note 13] Monod argues convincingly that all biological activity is teleonomic (with typical French precision, he insists on a Greek-based technical term: we used to call it purposive, hence the ’Necessity’ of the title), and that it is constituted of teleonomic action on chance events/mutations, existing within an Environment that can successfully terminate an ’inappropriate’ existence. So that both organism and environment are teleonomic, both interact, and both are based on the chance occurrence of appropriate molecular configurations which become dynamically stable. ’Chance’ is our randomness, and it is, by definition, fundamental. That is, there is no external reason for the combinations. All Monod is saying is that the fundamentals, from which biological systems are built, are generated by chance. Logically, this is properly tautologous – a fundamental is random, the ’result’ of a chance, non-structured event, for it is in this that its fundamentalism lies.

So here is the problem: the fundamentals of any system are random. Randomness can only be assumed (until it is disproved). If the fundamentals are not proved, the system that rests on them can only be assumed to correspond with the world it is describing. And assumed fundamentals seem to be continually destroyed.

We arrive at things we consider as fundamentals in two ways, which can loosely be termed ’description’ and ’measurement.’ In fact, one is a pared-down description while the other is a prescription. It is worth looking at these, for they reflect some basic differences in the ways we understand fundamentalism.

Without labouring the exact means by which descriptions can be made (which are covered in my other paper at this Conference), one can say that, when a description is made of something, the relevant qualities of descriptor and described are seen to be identical, although the process by which this happens (i.e., the actual distinctions drawn, in Spencer Brown’s terminology [Note 1] – or the actual states of affairs, in Wittgenstein’s [Note 20] ), depends on the observer’s choice.

In a scientific domain (and science is descriptive), an attempt is made to generate a system of unique, reductive descriptions by which meanings can be communicated unequivocally: a system, therefore, which defines a process for drawing the distinction and allocates a role to such distinctions; (this is in contrast to the essential ambiguity of the conversational paradigm of communication).[Note 3] These descriptions are constantly being refined and split down, so that a system of fundamentals can be found which, when put together, will give an accurate account of the behaviour of participants within the domain: in other words, an axiomatic system.

In contrast, a system of measurement – which is only a particular form of description – seems to comprise a more complicated arrangement. While a description is made in the same basic way described above, by drawing an analogy, the thing, which has the representative role seems to entail at least a defined standard, while the analogy is strictly defined as a comparison against this standard, and the description consequently and characteristically produces a number. The point here, of course, is that the standard is already defined – that is, the buck is passed.

However, this buck passing has a particular and valuable characteristic: the value placed in the definition of the standard, since it is a standard, means that it is no longer necessary for the standard to take an entirely rigid form itself. It is enough that it is defined as the standard, against which all else is measured. This means that, while the ’size’ of the standard (for instance, the standard meter) may change its medium of realization, while old realizations of the medium may be shown to be ’inaccurate,’ the actual idea of the standard does not change and, consequently, measurement retains its validity as an activity through the authority of the standard. When different standards are in use, however, communication can be very hard [Note 19] . Crudely put, then, a description finds an analogy between two things of roughly equal prestige, while a measurement, by giving one thing the authority to define itself and not to be further investigated, draws an analogy in which the thing described is definitely under the authority of the metric used. No wonder we get distressed when we perceive ’falling standards’ – which entirely undermine a constancy and predictability we equate with fundamentality.

I have introduced the comparison between a ’straight-forward’ description and a measurement because it demonstrates that there are, in fact, two ways in which we reason out fundamentals: by analysis of a series of analogies, in which a process of paring down is followed (i.e., reductive description); and by defining something as being fundamental and allowing its own act of continuous self-definition to be realised in different forms. Sometimes we change the fundamental to fit our description of the world (the U.K. has gone ’metric’). Sometimes we change our description of the world to fit the fundamental. (I do not believe men have freedom of choice, therefore they are S-R machines. S-R is the fundamental psychological mechanism: men have no freedom of choice.)

I would like now to return to the property of randomness. Something which is random is something that cannot be reduced (for reduction implies a structure and pattern). And this property of course, can never be proved. Or can it?

Let us consider the process of disproof. A number string is assumed to be random [Note 4] . After a considerable analysis, it is found to be non-random, for it can be split into a new number string or root (assumed to be random) and a process (algorithm) which produces the first number string from the newly derived root. This can be represented formally thus:

where R1 is the initially assumed random number string, R2 the new string, ? indicates assumed randomness, A1the reduction algorithm, X the operation of that algorithm on the number, ← the result of that operation, and the superscript ~ indicates demonstrated non-randomness.

Obviously there is a change in time state here (denoted by a subscripted #1) – the action of the algorithm causes a change in role in the assumed random number string to which it is applied (i.e., shows it to be non-random).

Now the chain can expand recursively:

Until it reaches a value of (n) in which one of two things happens: either (n) becomes infinite [Note 5] , in which case, as von Foerster demonstrates [Note 6] , the state before and the state after can be exchanged. But what happens if, before then, there is an algorithm which, acting on the (assumed) random string produces a string that is the same as itself or larger? Under normal circumstances, neither of these conditions is accepted as a demonstration of randomness, for it is tested by non-reducability, and what we have achieved is not reduction, but expansion. (Since the algorithm plus the same string is longer than the string itself, through the addition of the algorithm.)

Again, all appears lost. But there is one condition in which this is not actually so: that is, when the algorithm and the resulting string are the same. In this case, a reduction is produced, even when the algorithm and resulting string appear larger, together, than the original string, for they are no different. That is to say, that the one string can fulfil both roles (algorithm and resulting string), and the process of reduction can continue.

Does this lead anywhere, other than to the potentially continuing reduction or the larger answer?

It can, in the following way. If the string, assumed to be random, to which the algorithm is applied, and from which an identical answer to the algorithm is produced, is itself identical to the initial string, the string is no longer reducable, for it operates, via itself, on itself, to produce itself. This can be shown thus:

In this case, we have a very peculiar event, for the string by operating itself upon itself, produces itself. It is non-reducible, for the self-reduction it carries out is itself, and consequently, the doubt about its randomness is overcome by its recursive self- definition (as is shown at sn+1).

Until recently, such phenomena were considered entirely silly paradoxes without meaning, mediaeval serpents eating their own tails. But Varela’s [Note 17] work, in extenso, showed this not to be so. And so, paradox changes its role.

In the case of the idea of demonstrable randomness described above, there is a strong parallel with the recently developed ideas of Eigen-values and -functions. Von Foerster [Note 6] uses the infinite regress proof (which I used above to demonstrate the irrelevance of randomness after infinite operations on infinite strings) to demonstrate the sensible existence of such things – things which, through operation of the self on the self produce their own values (selves).

This concept of Eigen-value and -function giving rise to Eigen-behaviours, creates Eigen-systems which von Foerster equates with ’Objects.’ This excites me, for the Objects he refers to are precisely those Objects that I describe elsewhere – Objects with their own description built in, where a means of observation allows the Object to be built through the mutually complementary roles of observer and observed (which also allows for independent observations of other observers), and which are essentially unique, just as a random number is.

Monod’s insistence on chance as the basic building block of biology, of Random Molecular Happenings as providing the fundamentals of life, is thus not only formally sensible, but is demonstrably so; just as the self-defining nature of an Object (an Eigen-function and -value) is necessarily random and unique, and consequently fundamental.

But such formulations can only exist at the point where intenso and extenso meet. For a universe which consists of such fundamentals, such self-defining, random Objects, while being finite, allows for any predications of the Objects. Yet, this infinity of predications (or observations), always connects these Objects, in a massively complex, network which, from its own nature, re-enters: the universe’s inhabitants, self-defining as they are, nevertheless define each other. And, when this happens, extenso and intenso complement each other so well, that they meet, and lose their frightening significance: for to talk of all Objects as self-defining (i.e., random), talks of the universe which is also an Object. And so it is that a concept of the fundamental can be entertained, albeit that in the ’real world’ we sanitise fundamentals to fulfill the roles we crave of them.

There is one final twist to this argument. If I am correct, something which is fundamental is random, and randomness is an Eigen-property of Objects which describe themselves, and hence have a stability from continuous regeneration. Thus, the inevitability of Monod’s chance. But what is Necessity? In Monod’s terms, it is the teleonomic need to self-reproduce. And, in formal terms this means that things describe themselves, and through this self-description, gain a stability from continuous regeneration: that is to say, Monod’s Necessity is another way of denoting an Object, and Chance and Necessity are the essentially interrelated as the same.

There remains, of course, the question of whether all Eigensystems, all Objects, are random, as opposed to all random strings being Objects. But that is another discussion.

References

[1] Brown, G.S., Laws of Form, George Allen and Unwin, London, 1969.

[2] Chaitin, G., “Randomness and Mathematical Proof,” Scientific American, May 1975.

[3] Von Foerster, H., “Introductory Comments to Francisco Varela’s Calculus for Self Reference,” Int. J. General Systems, 1975, Vol. 2.

[4] Von Foerster, H., “On Self-Organising Systems and their Environments,” in Yovits, M. and Cameron, S. (Eds.), Self-Organising Systems, Pergamon Press, London, 1960.

[5] Von Foerster, H., Notes on an Epistemology for Living Things, BCL, Illinois University, USA, 1972.

[6] Von Foerster, H., “Recursions, a Primer, Position Paper,” unpublished manuscript, 1976.

[7] Glanville, R.S., “The Logic of Descriptions – or, Why Physics Won’t Work,” paper delivered at 1st International Conference on Applied General System Research, 1977.

[8] Glanville, R.S., “The Object of Objects, the Point of Points – or Something about Things,” unpublished Ph.D. Thesis, Brunel University, 1974.

[9] Glanville, R.S., “What is Memory, that it can remember what it is?” Proc. 3rd E.M.C.S.R., Vienna, 1976.

[10] Gödel, K., ”Über formal unentscheidbare Sätze der Principia Mathematica und Verwandter Systems,” Monatshefte für Mathematik und Physik, Vol. 3, 1931.

[11] Günther, G., Formal Logic, Totality, and the Super-Additive Principle, BCL, Illinois University.

[12] Maturana, H. Varela, and F. Uribe, R., Autopoesis, University of Chile, Santiago, 1974.

[13] Monod, J., Chance and Necessity, Collins, London, 1972.

[14] Nambu, Y., “The Confinement of Quarks,” Scientific American, November 1976.

[15] Pask, G. and Scott, B.C.E., “Caste: A System for Exhibiting Learning Strategies and Regulating Uncertainties,” Int. J. Man-Machine Studies, 1973, Vol. 5.

[16] Pask, G., “Co-Evolution Quarterly, Position Paper,” unpublished manuscript 1976.

[17] Varela, F., “The Arithmetic of Closure,” Proc. 3rd E.9.C.S.R. Vienna, 1976.

[18] Varela, F., “A Calculus for Self-Reference,” Int. J. General Systems, Vol. 2, 1975.

[19] Wittgenstein, L., Remarks on the Foundations of Mathematics, Blackwell, Oxford, 1967.

[20] Wittgenstein, L., Tractatus Logico Philosophicus (2nd Ed), Routledge & Kegan Paul, London, 1971.

[21] Wittgenstein, L., Zettel, Blackwell, Oxford, 1967.

Endnotes

1

There is an apparent problem here, which will be resolved later. It is that to talk about a fundamental (i.e., non-reducible) in intenso is to reduce it. (The corresponding argument is that one cannot talk about a Universe in extenso.) The solution to this problem that will be discussed here follows the idea that, where Universe and fundamental meet, one can talk about them in these terms which are basically undistinguishable at this point, i.e., this is where intenso and extenso re-enter each other.

2

In another paper, ’The Logic of Descriptions – or, Why Physics Won’t Work’ [Note 7] , I attempt to show the logic and form of the actual description used, and hence to discuss the formulation of such descriptions.

3

One of the major differences between the so-called ’natural’ and ’social’ sciences lies in the communication paradigm assumed. Conversation Theory’s manner of description constantly refers back to the described Object, and does not develop the same sort of argument that science normally strives for. Pask [Note 15]

4

This means the number string as a whole, not the ’individual’ numbers. The notion is critical – we are not talking about a grouping in which the individual numbers are of concern, such as the collection of all Gödel numbers, or of primes or whatever, when the act of collection makes the total (but not, necessarily, the parts) non-random. In the case of a Gödel number collection, we might consider whether the sequence is random or not. Chaitin’s article explains this most clearly.

5

This assumes an infinite initial number string – and, consequently, only shows that, for the initial number, randomness is a quality with no meaning. This, in turn, implies that the idea of ’fundamental’ only has value in a finite Universe. Hence, if the Universe consists of self-observing Objects, as I argue [8], [9], and such objects are fundamental, then the Universe is finite (if indefinitely large). The fact of the experience of the Universe, for any Object (or processor) is its being infinite (Pask [15], [16]) is not contradicted by this.

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