Perceptual Ambiguities,

Decideability,

And

Category Errors

Introduction

The human brain is a very wonderful and mysterious organ. It must be the most valuable and yet most fragile part of the human body, 'housing' as it does, our mind.

Incoming data, can, sometimes, present as an ambiguous interpretation. This happens particularly in the visual field.

In his encyclopaedic volume (1), "The Oxford Companion to the Mind", Richard Gregory illustrates this effect with graphic examples on page 340. Illustration 4a is especially interesting as the duck is only seen as a rabbit when the image is rotated clockwise through 90 degrees. 4b gives two ambiguous interpretations without need for rotation. Figure 5, the Necker Cube, is an extremely good example of such ambiguity.

When the brain is confronted with ambiguous data, the interpretation will suddenly switch between the two interpretations in a rather disconcerting way!

It is the contention, in this paper, that perceptual ambiguities are not restricted to the visual field. Indeed, our experience, as human beings, is often rich with such impressions. When an acquaintenceship is developing into a friendship, we often hesitate before a firm friendship is possible. "Can I trust this person?". This frequently happens when we are young and inexperienced especially when getting to know a person of the opposite sex. I might think "I am 'in love' with this person" ."Are they serious about me or am I being taken for a 'ride'? ". There are two possibilities: the other person is trustworthy or they are not. We may have a 'sixth sense', an intuition about the future intentions but in the final analysis only the passage of time will reveal the answer.

Decideability and Completeness

However, the issue of ambiguity is even more fundamental, enigmatic and illusive then we might imagine. It bedevils the very fabric of logic and reason themselves. Back in the 1920's, the Czech born, Austrian mathematician, Kurt Godel, developed metamathemtics and published a paper which shattered the cosy world of logic established by Robert Boole, Lewis Carroll, Bertrand Russell and many other logicians.

Godel had discovered a 'defect' in logic, which was quite different, in principle, from the mechanism of mathematical induction. I am quoting, now, from (2) Norbert Wiener's 'Cybernetics', second edition, page 125 where he explains Induction.

        1. Pn is a proposition involving the number 1
        2. Pn has been proved for n=1.
        3. If Pn is true, then Pn+1 is true.
        4. Therefore Pn is true for every positive value of integer n.

Wiener then goes on to explain how…'some very interesting situations arise, in which we may be able to prove every case of a theorem Pn but if there is no systematic way of subsuming these proofs under a single argument, independent of n, such as we find in mathematical induction, it may be impossible to prove Pn for all n.

Godel's paper on decideability and completeness did just that.

His method was very ingenious and consisted of 'tagging' all known propositions, in turn with the unique series of prime numbers. E.g. One cannot multiply or divide each side of an algebraic equality expression by zero and obtain any valid result. This proposition could be labelled '1'. The next would be '2' , then '3', then '5,7,11…' and so on.

In the course of two millennia, no mathematician has been able to devise a formula, which can evaluate all of the primes, yet it is known that there are an infinite number of such entities (3). The consequence of metamathematics is that there are some propositions or 'truths' which may be true but which cannot be proved to be so (or proved to be false). Also no test can be constructed to show that a subject or discourse or study is complete. There may be further 'chapters' of knowledge waiting to unfold in the fullness of time.

The Theist - Atheist Debate

Hence ambiguities of perception and a basic 'flaw' in the fabric of logic make the ongoing debate between theism and atheism particularly troublesome.

Alister McGrath (4) in his critique of some of the writings of Richard Dawkins, has been scrupulously fair, in that he has only really seriously challenged (5) Richard's book: 'The God Delusion'. However it is perfectly possible to raise objections to the arguments put forward in Richard's earlier book (6): 'The Blind Watchmaker'

In this book, Dawkins recalls how William Paley (1802), walking across a heath stumbles across a watch (dropped by someone to the ground) and compares the design of the watch with the (apparent) design of the human eye. Dawkins goes on to reason that a conscious person i.e. the watchmaker designed the watch but that the human eye came about by a series of 'blind' processes, in which at each stage a working eye, (of some sort) resulted. (My parentheses).

Before making any comments on Dawkins's approach, it is considered necessary to explore some further underlying issues relevant to the processes under discussion.

Categories and Category Errors

Any process, whether it be design or a series of evolutionary steps can be thought of as a mathematical function in time and the kernel of this function is certainly more elaborate than a simple scalar function. It is far more likely to be a time dependent matrix or tensor function. This so because the data culled represent an orthogonal (statistically independent) set of components i.e. details of the objects concerned, the watch and the eye, as objects will have colour, materials of construction etc.etc.

Thus in order to make a comparison between two entities, we need to check carefully that the category or type of each is the same. ‘typing’ is a serious consideration in the field of computer programming and it was the handling of this area that Bjarne Stroustrup improved so markedly, in 1985 when he devised (7) the C++ language: a language especially designed to deal with objects.

To summarise, we cannot compare objects of a different category or type.

Admittedly, I am extending the definition of ‘type’ to a mode of thinking in the human brain, but this category error is at the root of what often seems to occur in arguments in contentious domains. There is a very old saying that you can not compare chalk with cheese. They are both a shade of white but there the similarity ends. In his book, ‘The Blind Watch Maker’, Dawkins makes a category error when he tries to compare the processes of evolution and human design. We are told that evolution is a ‘blind process’. However, we are now to believe that the skilled watchmaker (a person to be sure) purposefully designs the watch. Surely though, if we are to be strictly rationalist and reductionist in both cases, we must think along the following lines. In a thought experiment, it must be imagined that the skull of the watchmaker is to be removed during the design process. All that would be seen by an observer would be a 'blind' process of evolution of neural synaptic connections taking place in the brain cortex. 'Ah', Dawkins would say, 'the watchmaker is a conscious being.' But no one seems to know how to study or analyse consciousness. Some American philosophers believe that consciousness itself is a delusion and does not really exist. In short, the issue of consciousness is a non-problem! It is possible that the observer in a quantum phenomenon experiment needs to be a conscious being, but again this matter is highly controversial.

I suppose that what has been constructed in this commentary is a reduction ad absurdum argument. It demonstrates, at least, that the issues involved are far more subtle than might be imagined.

In conclusion, it is worth noting that, several thousand years ago, the Buddha declared that 'God both exists and does not exist'. This may be another way of stating Godel's hypothesis that there are some propositions that can not be proved or falsified.

The Theists know what the truth is for them, through personal revelation. The Atheists don't even bother to consider!

 References

(1) Richard. L. Gregory, The Oxford Companion to The Mind, Oxford University Press, 1987, p340.

(2) Norbert Wiener, Cybernetics, 2nd Edition, M I T Press, 1948, p125 - 126.

(3) Marcus du Sautoy, Music of the Primes, 2004.

(4) Alister McGrath, The Dawkins Delusion, SPCK, 2007.

(5) Richard Dawkins, The God Delusion, London, Bantam, 2006.

(6) Richard Dawkins, The Blind Watchmaker, Penguin Books, 1988, p4

(7) Bjarne Stroustrup, The C++ Programming Language, Addison Wesley, 3rd Edition, 1997.