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What are patterns?

Are patterns ontic or epistemic?

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©  2009 GERRY NOLAN

 

What are patterns?

There are patterns everywhere in everything at every scale: spatial, temporal and statistical, and mathematics is generally regarded as the science of patterns (Devlin 1997, Resnik 1997). Devlin notes further that patterns can be ‘either real or imagined, visual or mental, static or dynamic, qualitative or quantitative’ (1997: 3). Because we are so familiar with patterns we take them for granted, with the result that what pattern is has attracted very little sustained inquiry; as Resnik remarks, ‘I know of no developed philosophical account of patterns’ (1997: 202). Consequently, if we want to learn what the ontology of pattern is per se, we must begin from a point where even the most elementary questions about pattern remain not only unanswered but unasked.

Although numerous, examples of patterns or types of patterns do not tell us what pattern is. Paraphrasing Socrates in his search with Theaetetus for an answer to the question ‘What is knowledge?’ we note that the question is not ‘What are the examples of pattern?’ nor ‘How many types of patterns are there?’ ‘We don’t ask the question because we want a catalogue, but because we want to know what pattern is.’ (1987: 146e)

Of course patterns must have elements, or something that we can coherently name ‘elements’; whether ‘real or imagined, visual or mental’, patterns must be patterns of something. Nevertheless, we talk about what knowledge is without being concerned with exactly of what it is knowledge. Analogously, it should be acceptable to ask what pattern is without being concerned with details of what is patterned.

If patterns were not organised in some way there would only be ‘randomness and absolute unpredictability’, as Dennett says (1991: 30), so it is apparent that all patterns must be organised and that some principle of organisation is required. Since it is this principle according to which patterns are organised, without specifying exactly what principle it may be, we will call it the ‘organising principle’. Because, without organisation there can be no pattern, an organising principle is a necessary attribute of pattern, but is it sufficient?

There are many things that are organised according to a principle which does not result in them having a pattern, for example, a novel may be organised according to a principle of dramatisation, where the very absence of pattern is essential. Again, an artist may organise a work according to some principle with the objective of creating surprise and unpredictability, in which case pattern would be detrimental. On the basis of these two examples we will say that an organising principle is not sufficient to be a pattern, that to be a pattern requires repetition either internal to an entity or event, or externally as in similar instances of an entity or event, that is there has to be more than one instance of an entity or event. The external condition will apply even if subsequent instances are only alike enough to be classified as being of the same pattern, for example individual lions vary but they are all recognisably the pattern of a lion; a person may have a consistent pattern of thought, being optimistic say, but it will vary from day to day; a certain pentagonal pattern may be repeated in many carpets, but may be a different set of colours in each.

Therefore, if p is pattern, we state the rule as follows:
p is pattern iff
(1) p has an organising principle
(2) the organising principle entails repetition

There does not seem to be any disagreement about the necessity of an organising principle but there may be a counter-example to the entailment of repetition; that is, in Champernowne’s constant (number)[1] the digits never literally repeat, as in some fixed block of digits recurring.
 

Neither, being an infinite sequence, can the whole number ever repeat. However, despite the lack of these two types of repetition, the first internal and the second external, the counter-example fails because it is obvious that the sequence is highly patterned in some sense, as in the same principle being applied over and over, namely, the digits of the next whole number are added at the end. In other words, there is an organising principle that results in one type of internal repetition, but there is no external repetition.

Part of the OED definition of ‘pattern’ is that it is regular and of course very many patterns are regular, in which case we will say that the regularity supervenes on the repetition. However an immediate objection to regularity as either a necessary attribute or one that is entailed by the organising principle, is that many events occur according to a power law—the law that small events occur exponentially more frequently than large ones—and many such events exhibit no regularity, hence no predictability. At first glance power laws do not appear to have an organising principle either, however, Per Bak maintains that power laws that apply to dynamic complex systems, including earthquakes, stock and commodity markets and avalanches on sand-piles, as well as many other systems, arise from what he has termed ‘self-organized criticality’ (1997, my italics). It is clear that power laws can be represented by histograms, many of which are obviously patterns, so without asking here how it works, the organizing principle for patterns of complex systems is self-organisation. Also, since power laws apply to statistical analyses there must be a repetition of events to be countable, so we can say that the organising principle of pattern entails repetition.

Therefore, we conclude that p is pattern iff (1) p has an organising principle, and (2) the organising principle entails repetition.

However a great deal more work is required.

In Plato’s Theaetetus, just raising the question of what knowledge is was the beginning of a philosophy of knowledge, on this basis, just raising the question of what pattern is suggests the beginning of a philosophy of pattern. It may be that, as said in Theaetetus (1997: Essay: 133), in the end our inquiry will be aporetic, however the fear of an aporetical conclusion has not deterred philosophers in the past.

References

Bak, Per. 1997. How Nature Works: The Science of Self-organized Criticality, Oxford University Press

Dennett, D. C. 1991. Real patterns, The Journal of Philosophy, Vol. 88, No.1: 27–51.

Devlin, K. 1997. Mathematics: The Science of Patterns, Scientific American Library, A division of HPHLP, New York.

Oxford English Dictionary. 1989. Second Edition, online version updated March 2009.

Plato. 1987. Theaetetus, translated by Robin A H Waterfield, Penguin Books, London.

Resnik, M. D. 1997. Mathematics as a Science of Patterns, Clarendon Press, Oxford.

1 Thanks to Jim Franklin for his suggestion in discussion that Champernowne’s number could be a counter-example to the entailment of repetition.