Randomness, said the Statistician,
is a concept essential to much of statistical theory, but few textbooks reveal that its validity is questionable. For example, to choose a random sample of people, you may not go onto the street and select people as the whim takes you. Although in common parlance that would be 'random', it would be subject to conscious and subconscious prejudices, and therefore not random in the statistical sense, which requires that every member of the population being sampled must stand an equal chance of being selected. Good statisticians do not make random selections haphazardly, but in a highly controlled manner, ignoring the paradox that this entails. Often they use tables of random numbers, which are constructed so that no digit or sequence of digits occurs more often than any other. The tables are so carefully impartial that if you were to delete a single digit, it would be possible to deduce which digit was missing. Can they then be regarded as random?
In any case, offered the Author,
chance may play a smaller part in real life than in statistical theory. For example, I have on my bookshelf a series of books about Hollywood film studios. In alphabetical order they are:
Notice that, in a list of seven titles, the second is already halfway through the alphabet. True, Allied Artists is missing, but then so are Monogram, Republic, and Twentieth Century Fox. Are the studio names distributed randomly throughout the alphabet? Any competent statistical test would find it highly improbable that they are. If they are not, what meeting of Hollywood moguls hatched the plot to favour the second half of the alphabet, and to what possible end?
It may have been a transatlantic agreement rather than a Tinseltown conspiracy, suggested the Historian.
It is surely no coincidence that British film companies fall mainly in the first half of the alphabet: Associated British Pathe, Amicus, Banner, British Lion, Ealing, Eros, Gainsborough, Gaumont, Hammer, and London Films, for example. It reminds me that towards the end of the fifteenth century, Pope Alexander VI drew a line down the Atlantic Ocean and decreed that all discoveries to the west belonged to Spain, and all to the east belonged to Portugal. Perhaps some pontiff of the movies made a similar division of the alphabet. If so, he was more even-handed than Alexander, who allotted the lion's share to Spain, so as to protect his personal and family interests, as you might expect of the father of Cesare and Lucrezia Borgia.
Talking of the church, there is a similar unequal distribution of the names of the great Christian philosophers, said the Author.
A list of the top ten would not feature Alexander, but it should include Anselm, Aquinas, and Augustine, and maybe even Abelard, which hardly seems fair to the rest of the alphabet.
A hush fell over the company at the mention of the first name in the Author's list. To our surprise, however, he did not pursue the matter. At last, the Philosopher remarked,
Perhaps it is time to discuss Anselm's proof. Apart from any other consideration, recent events in the Church of England have made it of topical relevance once again.
Before we start, let us recollect the relevance of this subject to our discussions. It was claimed by some among us that there can be no absolute logical proof of any proposition, because every argument starts with a premise, and therefore initially there must be an unproved premise. Consideration of this problem has long given rise to the hope of finding a self-proving premise. Descartes' Cogito ergo sum was an attempt at this; Anselm's ontological proof might be considered another. Our interest in it is philosophical rather than theological.
The Historian offered to set forth the relevant facts, so we made ourselves comfortable, and he began.
At the time of the Norman Conquest, before he was sanctified or appointed Archbishop of Canterbury, Anselm was the Prior of the Abbey of Bec in Normandy. While in that post, he wrote that he had discovered a wonderful proof of the existence of God. What was wonderful about the proof, he said, was that it was complete in itself, requiring no external support. In the nine centuries that have ensued, the argument over Anselm's proof has never ceased. There are two points at issue. First, is the proof logically valid? Most commentators have believed that it is not, but that raises the second and more thorny question: if it is not valid, at what point does it go wrong? Wherein lies its invalidity? The history of the debate is a study in itself. The proof has been reinvented more than once, and sundry rebuttals offered. Intriguingly, Anselm's proof seems to be accepted more often by atheists than by clerics. Those who worship reason feel compelled to accept any argument they cannot disprove, and rationalists are apt to find themselves simultaneously believing (a) that there is no God, and (b) that Anselm's proof of God's existence is irrefutable. Before we go any further, however, perhaps we should hear what Anselm's proof was. It must be remembered that Anselm wrote in Latin, so this account is not only a paraphrase, and a brutally abbreviated one at that, it is in a different tongue. I hope, however, not to do him too great an injustice.
According to the Psalms, 'The fool has said in his heart, There is no God.' Anselm's proof takes this as its starting point. He reasons that the fool must accept that even if God does not exist in fact (in re), He at least exists in the understanding (in conceptu). The fool must also agree what is meant by God, namely: that greater than which cannot be conceived. Now if the fool can conceive of a God that does not exist, he can also conceive of a God that does exist. But, says Anselm, a God that does exist would be greater than a God that does not exist. Therefore, God must exist, or else it would be possible to conceive of something greater, which is impossible by definition.
Voices were raised against the proof as soon as it was published. Count Gaunilo, a monk of another abbey, made the telling point that the same structure of argument could be used to prove that almost anything really exists. For example, one could imagine an island more excellent than all other lands. But an island which does exist is obviously more excellent than one which does not, therefore the imagined island must exist in fact. Anselm shrugged off this argument, denying that it was truly parallel.
Many churchmen ignored Anselm's argument, feeling instinctively that a proof of God's existence was incompatible with belief as a matter of faith. In the 13th century, St Thomas Aquinas formally denounced the proof as fallacious. In the 17th century, Descartes re-invented it, with an argument that may be summarised as:
In the 18th century, Anselm's proof was refuted by Immanuel Kant, who was the first to call it the ontological proof. Kant's view, which many subsequent commentators have been content to endorse, is often summarised as Existence is not a predicate.
And what, pray, does that mean? asked the Physicist.
The Philosopher undertook to explain the point.
Every meaningful statement consists of a subject and a predicate; the subject is that which the statement is about, and the predicate is what is said of it. Kant's point is that a statement such as The cat exists is different in kind to statements like The cat eats, or The cat is five years old, because as soon as we offer the subject The cat, we are already implying that it has some sort of existence. In fact, it is difficult to imagine circumstances in which one would utter such a statement. Can you imagine asking Does that cat exist?
Perhaps not, said the Author,
but I can imagine making statements in which existence is the predicate. For example, I might say, Zsa Zsa Gabor really exists, but Peter Pan does not.
True, said the Philosopher,
but of course there is a sense in which Peter Pan does exist, and a sense in which Zsa Zsa Gabor is an invention. Your statement is about different sorts of existence rather than the existence or otherwise of the ostensible subjects. This is a question central to the ontological argument, and one to which we must return, but for the meantime perhaps I can persuade you of the peculiar nature of existence as a predicate in this way. Suppose that you are inventing a character for a story. You may predicate whatever you please about the character, may you not? You may say that he is eight feet tall, that he was reared by apes, that he is 300 years old, that he is wise and handsome, that he turns into a wolf when the moon is full, that he has X-ray vision, that he shot JFK, that he comes from the planet Vulcan, and that he can foresee the future and walk through walls. Is there anything that you cannot predicate of him? Yes, one thing and one thing only: you cannot say that he exists. Your readers already know that they are expected to suspend disbelief and suppose that he exists. If you say He exists, you are adding nothing to your previous presentation of the character.
The Author objected.
In a certain type of novel, fictitious and historical characters are involved. The admirable Flashman novels of George MacDonald Fraser contain informative notes making it clear which are which. It seems to me that the author can meaningfully say, This character existed; that character did not exist. However, I do take the point that, as a predicate, existence raises questions that other predicates do not.
Are we to take it, then, the Physicist asked the Philosopher,
that you concur with Kant in rejecting Anselm's argument?
Yes, but on rather different grounds, or perhaps on the same grounds expressed in different terms. Any proof like Anselm's has three parts: it starts with a premise which is either agreed to be axiomatic or has already been proved, it ends with the conclusion or demonstrandum, that which was to be proved, and in between is the logic, which purports to lead inescapably from the former to the latter. The logic connects the premise and the conclusion like a ladder leading from the floor to the ceiling. It is a mistake, I think, to spend much time examining Anselm's ladder - whether the rungs will bear any weight, are they too far apart, and so on - because both his floor and his ceiling are of dubious soundness.
With regard to the premise, the fool should have interrupted Anselm as soon as he started to speak of God, and should have asked him: this word God that you use, is it a proper noun, that is, a name that denotes, or is it a common noun, that is, a term applicable generally to all members of a certain class? If it is a common noun, then there is nothing in the proof that prevents there being numerous equally great gods. Presumably, that is not Anselm's intention. If however God is a proper noun, then any statements made about God are descriptive rather than definitive. The use of a name implies that there is already agreement between the speaker and the audience as to the person or thing denoted. In that case, if a series of statements leads to contradiction, it shows that one or more of the statements is untrue, not that the person or thing denoted does not exist. For example, if I were to say:
my argument would clearly be fallacious.
"The use of God as a name," said the Historian, "reminds me of a curious story in the book of Exodus. After God had spoken to Moses from a burning bush, and had commanded him to bring the children of Israel out of Egypt, Moses asked, 'Behold, when I am come unto the children of Israel, and shall say unto them, The God of your fathers hath sent me unto you; and they shall say unto me, What is his name? what shall I say unto them?' God made a remarkable answer:
'And God said unto Moses, I AM THAT I AM: and he said, Thus shalt thou say unto the children of Israel, I AM hath sent me unto you.'According to this account, God's name is a claim to exist. Logically, of course, this is nonsense: a name cannot be a statement. There are, however, many examples in the temporal world of the use of names to suggest qualities, especially for those in positions of power. Thus judges must be addressed as Your Honour and Members of Parliament as Honourable Members however dishonourable they may be, churchmen are Reverends whether they command reverence or not, and members of royal families are Highnesses however low they sink. In The Golden Bough, Frazer relates an extreme example. There is a special language, he tells us, devoted to the person and attributes of the King of Siam: every part of his person has its own particular name, and every action he performs has its own word which signifies that the action is being performed by the sovereign; these words cannot be applied to the acts of any other person. There is no word in the Siamese language, Frazer says, by which any creature of higher rank or greater dignity than a monarch can be described; and the missionaries, when they speak of God, are forced to use the native word for king. In the event, I AM, (YAH-WE, or JEHOVAH) was not much used by the Israelites as a name for God. Perhaps they felt that it was too sacred."
The Philosopher was plainly fearful of losing the thread of his thoughts.
There is another objection to Anselm's premise. When Anselm defines God as that greater than which cannot be conceived, what does he mean by greatness? The only clue we have is that he believes that a God who exists is greater than a God who does not exist. Suppose the fool had countered that he could conceive of a God who was arbitrarily destructive, and that such a God was greater than a God who was constrained by mercy, love, and forgiveness. How would Anselm have reacted? There have been many instances of cultures that worshipped awesome gods of destruction, but there is no doubt that Anselm would have refused to accept such qualities as an indication of greatness. His view on this topic would surely have derived from his religious beliefs. In other words, Anselm would define godliness in terms of greatness, and greatness in terms of godliness.
Now as to Anselm's conclusion. What is it that he is seeking to prove? He says he will prove that God really exists, but what are we to understand by that? Let us start by disposing of the simplest explanation. Does he mean that God has corporeal existence? If he does, there are few who would give his argument any further attention. The primary meaning of the Latin word res is 'thing', implying physical substance, but it was also used figuratively. (The word real is similarly used by us in both senses.) Perhaps in the 11th century many were prepared to believe in a bearded old gentleman above the clouds, but Anselm was no peasant, and I think we can assume that he would no more propound such a thesis than any modern cleric - perhaps less so than some TV evangelists. So Anselm presumably means something other than physical existence. His argument depends on his meaning something more than existence as an idea, however: the fool has already granted that. Some sort of existence more substantial than conceptual existence, but less substantial than corporeal existence, is required. Anselm is implying that there are at least three kinds of existence. Are any of them provable?
It seems self-evident that something which exists only in the mind, as an idea, does not exist independently of the person conceiving it, nor can any other person know that it exists - he only knows what the conceiver relates about the concept. According to Locke, Berkeley, and Hume, even objects commonly understood to have material existence cannot be proved to exist independently of the perceiver. Our knowledge of them is limited to what our senses perceive; it makes no sense to ask What do they look like when nobody is looking at them? Scientific analysis itself now makes material existence problematical. The search for the building blocks of matter led to atoms, then to subatomic particles, and then to quarks, charm, and zones of probability of electrical charges. The solid stuff of which reality was supposed to be composed seems to have evaporated with the advance of science. What about the third kind of existence? Can God's existence be provable when other sorts of existence are not? Opinions on the matter are apt to be polarised: either God's existence is so fundamental that he who does not know it knows nothing, or else it is so transcendental that it is unknowable.
It may be useful at this point to remember that many modern philosophers believe that the certainty we can have in a proposition is in inverse ratio to its information content. We can be absolutely certain only of analytic propositions, which give us no information about the external world. We cannot be certain about synthetic propositions, and the more information they contain, the less certain we can be.
The Statistician interrupted. "The point is nicely illustrated by the rules of binomial sampling. If I take a random sample of 100 balls from a bag which contains a mixture of red balls and white balls, and my sample contains 63 red balls and 37 white, I may be
|90% certain that the actual percentage of red balls in the bag is between||55% and 71%|
|95% certain that it is between||53% and 73%|
|99% certain that it is between||50% and 76%|
|99.9% certain that it is between||47% and 79%|
Quite so, the Philosopher continued.
In the light of this, a proof that God exists is possible only if the statement God exists is uninformative. This perhaps is another way of expressing the view of many churchmen that proof of God's existence is incompatible with faith.
We must also remember that in Anselm's day, certain questions raised by Plato and Aristotle concerning the existence of forms and universals were still in the forefront of philosophical enquiry. The classical philosophers had suggested that the objects that we perceive are but imperfect copies of the ideal objects which they represent. For instance, there are many examples of tables built by carpenters, and the word table applies to all of them. The true meaning of table, therefore, must be an incorporeal ideal table that embodies the essence of tableness that all man-made tables share. There can be only one ideal table, because if there were more, they would all be tables only because they shared certain attributes, and what they shared would constitute the ideal table.
As with objects, so with qualities. When we describe an object as red we mean that it exhibits the quality of redness. Redness, therefore, can be said to exist. The argument over the existence of universals, such as redness, was still being keenly debated in Anselm's time. Those who believed that universals really existed independently of the objects which exhibited them were known as realists, as opposed to the nominalists who argued that universals were nothing but the names we give to certain attributes perceived to be held in common by a number of objects. Needless to say, Anselm was a committed realist, and this should be borne in mind when considering what he means when he says that God really exists.
It is strange that you should choose colour as your example of a universal, the Physicist suggested,
because this is one quality for which science can be invoked to support the view that it does have independent existence. Suppose, for example, that no object emitted or reflected yellow light. We would observe no yellow objects - see no instances of yellowness. Yet physicists would know that electromagnetic radiation within a certain range of wavelengths produced visible light. They would know that blue light had a wavelength of about 400 millimicrons, green light about 500, and red light about 700. They would find it odd that there had been no observations of a wavelength around 600 millimicrons, but would infer that such a wavelength, if it could be produced, would have its own colour. Yellowness could be deduced to exist, even though nothing exhibited it.
At first, that sounds a very persuasive argument, said the Mathematician,
but there is a serious flaw in it. The human eye contains only three colour receptors, sensitive to red, green, and blue light respectively. Yellow light (that is, light with a wavelength of about 600 millimicrons) stimulates the red and the green receptors, but not the blue, and the signals that this condition sends to the brain are interpreted as the perception of yellow. But the same interpretation would be given to a mixture of red light and green light which stimulated the receptors to the same extent. The human visual system has no way of distinguishing what mixture of wavelengths is producing the impression of colour. Does the universal yellowness refer to yellowness as it is perceived or to yellowness as a particular wavelength of light? If the former, then universals are subjective and cannot have existence independent of the objects exhibiting them and the person perceiving them. If the latter, then there would have to be another universal of compound yellowness in addition to the universal of pure yellowness. Many perceived colours can be produced by several combinations of wavelengths. Each combination would have to have its own universal. There would be more universals relating to colour than there were colours. Is this reasonable?
There arises another difficulty also. It is known that some creatures can see light beyond the wavelengths visible to man. Are these also colours? Do they each have an associated universal? We refer to certain wavebands as ultraviolet and infrared, implying that they are colours of a sort, although invisible to us. At what point do electromagnetic radiations cease to be colours? We commonly divide the full spectrum into radio waves, infrared, visible light, ultraviolet, X-rays, gamma rays, and cosmic rays, but these are arbitrary divisions, relating to the manner in which we detect and use the radiation. Why would it be considered perverse to refer to radio waves as invisible colours? Because, I suggest, the very concept of colour is meaningful only so long as colours are actually perceived. If colour in general is subjective, then particular hues must also be subjective, a fortiori.
It is time we returned to our sheep. The Philosopher expressed a general view that the thread of the discussion was fast unravelling.
What we have established, I think, is that, from whichever direction one approaches Anselm's proof, one encounters the problem of existence in general. Is it a proper use of language to say, as Anselm does, that if we understand what is meant by a word, then that thing exists in our mind? Is he not rather too cavalier in using the word as freely as this?
I would like to raise another aspect, said the Author.
Anselm's proof is an example of the circularity that exists in almost all logic. You said that an argument consists of three stages: the premise, the logic, and the conclusion. Formally, that is true, but in reality there is a fourth element that precedes the premise, namely the speaker's prejudgment of the question. That is the true premise, and it usually takes the form of a blind faith in the truth of the conclusion which is sought. No amount of logical disproof will shake this prejudice, which is why beleaguered arguments are so often shored up with the exceptions and special cases against which William of Ockham inveighed.
"Anselm's argument exhibits this effect in two places, one covertly, the other overtly. The former occurs in his exchange with Gaunilo, who presented an argument along these lines:
Anselm, however, takes more or less the same elements and arranges them thus:
The overt instance is even more revealing. Having proved that God must exist, Anselm pursued the argument further and also proved that it was impossible to conceive of God as not existing. This gave him pause for thought, for he recognised that he was in danger of cutting off the branch whereon he sat. His whole line of reasoning had started with the fool who said in his heart There is no God, and Anselm admitted that this meant that the fool did conceive of God as not existing. One would think that the contradiction was fatal to Anselm's case, but he attempted to reconcile it by distinguishing between conceiving in the sense of knowing what a word means, and conceiving in the sense of understanding the entity signified by the word. According to Anselm, when the fool says There is no God, he knows what the word God means, but he does not understand what God is. That is why he can say There is no God. But for those who truly understand what God is, it is impossible to conceive of God as not existing.
By this argument, Anselm built existence into the definition of God, and thus rendered his proof of God's existence mere tautology, but he did not see it that way because of a curious ambivalence he had towards the purpose of his proof. He stated expressly that his proof was not intended to convert non-believers, but to help believers augment their religious faith with intellectual knowledge; indeed, one of the works in which he set forth his proof was entitled Faith in Search of Knowledge. This stance was at variance with his claim that his proof was self-sufficient, and also ignored the fact that the proof itself took the form of confuting a non-believer. However, it enabled him to imply that for those who start with a belief in the existence of God as a matter of faith, his proof would provide a certainty of the existence of God as a matter of fact.
This claim is quite untenable. To suppose that the conclusion of an argument can have any higher status than its premise is little short of ludicrous. There is no limit to what one might believe as a matter of faith: UFOs, astrology, or witchcraft, for example. Given a false premise, logic can lead to any false conclusion. Therefore, if Anselm's claim were valid, absolutely anything could be proved to be true. This, in effect, is the objection Gaunilo made.
The influence of prior faith is not peculiar to a belief in God - it applies to all fiercely held theories. A great deal of supposedly objective research is nothing but a determined attempt to concoct evidence in favour of some irrationally held belief.
The Physicist was not having this.
There is nothing wrong in a scientist having an inspired insight into a possible solution, and then allowing this to guide his enquiries.
You think not? said the Author.
But such prejudice can destroy objectivity and result in error even if the proposition is true. Galileo, for example, had an unreasoning belief in his version of the solar system which led him to defend it with invalid arguments. In the end, he relied solely on the argument of the tides, which was not only scientifically unsound but was based on a misconception of the actual tidal cycles. That was why the better scientists of his day rejected his thesis, not because of ignorant conservatism, as popular mythology suggests.
The Physicist thought better of contesting the point, and, turning to the Philosopher, changed the subject.
You mentioned that recent events in the Church of England had made the ontological proof topical once more?
That is so, the Philosopher replied.
A Church of England priest, the Rev Anthony Freeman, has been dismissed from his appointment as Priest-in-Charge of St Mark's in West Sussex because he has written a book in which he questions the existence of God. He wrote: There is nothing out there - or if there is, we can have no knowledge of it. His critics claim that this is tantamount to saying that he does not believe in God. In his farewell sermon, the Rev Freeman emphasised that he did not believe in God as a person, but as a figure whose mercy and grace are mediated through human beings. He denied that this made him an atheist.
There are two points of interest in this otherwise parochial dispute. Firstly, it raises the question: is believing that God exists either the same as or necessary to believing in God? The Rev Freeman's critics seem to think that it is. One suspects that Anselm would have agreed with them. Why else does he wish to prove God's existence? When he says God really exists, it is not a mere statement of fact, but a declaration of religious belief, a slogan representing a body of complex doctrine. The use of slogans is defensible in religion as in politics, but it is a fatal mistake to suppose that proving the literal truth of the slogan will justify the body of policies or doctrines that it stands for.
The second point of interest harks back to the matter of universals which we have already discussed. The Rev Freeman describes God as a figure. This is a positively inspired choice of vocabulary. It hints at substance without entailing it. It reminds us of the figures of Euclidean geometry - the most easily acceptable examples of Platonic ideal forms. No man-made circle, for example, is perfect, but we accept that the ideal circle exists - we know how to find its centre and compute its circumference and area. At the same time, etymologically figure is akin to fiction and figment.
This serves to remind us how often philosophical arguments purport to be about concepts, but turn out to be about words. Indeed, there is a school of thought that believes that the true subject of philosophy is language, its meanings and uses.
Anselm's proof is indeed truly wonderful, but not for the reason that he supposed. What is marvellous about it is that consideration of this simple little proof raises most of the great questions which have concerned philosophers throughout the ages.