João Lucas

Sir Arthur Norrington deserved better of the world than to be known for his table. The Norrington Room, his presidency of Trinity, his long service to the University Press, deserve repeated coverage in the papers. But the only thing they say about him year after year is that he devised the table for comparing the academic prowess of the colleges in the Schools. It is not even true. Long before the Norrington table was first published, when I was an Assistant Tutor in a Cambridge college, I used to see the table which the Tutor drew up each year to show how our college was faring in tripos in comparison with other colleges. And, as a private tutorial check on Schools results, it is a legitimate and useful tool. But, published in the newspapers and made the subject of tea-time conversation in the country, it is a menace and counterproductive to the good things that Oxford should be chiefly concerned to foster.

Arguments have been much misunderstood. Not only has it been assumed that they must be deductive, but it has been assumed also that, although often expressed in loose and elliptical form, they must be capable, if they are valid at all, of being expressed with absolute precision, as a set of necessary and sufficient conditions for the action in question to be appropriate. Mathematical arguments are capable of being stated precisely, and it has long been a reproach to workers in other disciplines that they do not manage to achieve equal precision in their work. The assumption is made that absolute precision is in principle available, and it is only a lack of rigour by practitioners in the non-mathematical disciplines that prevents them casting their arguments in satisfactory form, and the philosopher should, therefore, reconstruct arguments to conform with these requirements, evaluating those that do, and rejecting any that do not.

The legend of the encounter between Wilberforce and Huxley is well established. Almost every scientist knows, and every viewer of the BBC's recent programme on Darwin was shown,* how Samuel Wilberforce, bishop of Oxford, attempted to pour scorn on Darwin's Origin of Species at a meeting of the British Association in Oxford on 30 June 1860, and had the tables turned on him by T. H. Huxley. In this memorable encounter Huxley's simple scientific sincerity humbled the prelatical insolence and clerical obscurantism of Soapy Sam; the pretension of the Church to dictate to scientists the conclusions they were allowed to reach were, for good and all, decisively defeated; the autonomy of science was established in Britain and the Western world; the claim of plain unvarnished truth on men's allegiance was vindicated, however unwelcome its implications for human vanity might be; and the flood tide of Victorian faith in all its fulsomeness was turned to an ebb, which has continued to our present day and will only end when religion and superstition have been finally eliminated from the minds of all enlightened men. Even churchmen concede that it was a disastrous defeat.1 Only Owen Chadwick strikes a note of caution, observing that the account given of the incident in Wilberforce's biography seems hardly consistent with an overwhelming defeat, and maintaining that the received account must be a largely legendary creation of a later date.

It would be improper for a President to play safe. After two years of curbing my tongue and not making all sorts of observations that have sprung to my mind, in order to let you have an opportunity of having your say, I am now off the leash. And whereas mostly in academic life it is appropriate to adopt a prudential strategy, and not say anything that might be wrong, I owe it to you on this occasion to play a maximax strategy, to speak out and say what I really think, being willing to run the risk of being wrong in order not to forgo the chance of actually being right in an area of the philosophy of science which must, I think for ever, be largely a matter of metaphysical speculation.

MANY thinkers deny the possibility of businessmen having responsibilities or ethical obligations. A businessman has no alternative, in view of the competition of the market-place, to do anything other than buy at the cheapest and sell at the dearest price he can. In any case, it would be irrational-if, indeed, it were possible-not to do so. Admittedly, there is a framework of law within which he has to operate, but that is all, and so long as he keeps the law he is free to maximise his profits without being constrained by any moral or social considerations, or any further sense of responsibility for what he does.

As the last College Meeting drew to its weary close, the Warden was moved to address the empty pews on the wickedness of not making attendance at meetings of the Governing Body a first call on one's time. Of course, it was waste of words to address them to absent auditors, but the sentiment was apposite. But it is inevitable with a Governing Body of fifty that each fellow feels on average only a 2% say in, and responsibility for, the affairs of the College. Large bodies are ineffective, and as the Fellowship has grown, papers have proliferated, committees have encroached on every available hour, and college life has degenerated into administration.

According to the legend, Bishop Wilberforce at a meeting of the British Association for the Advancement of Science in Oxford on Saturday, June 30th, 1860, turned to Thomas Huxley, and asked him ``Is it on your grandfather's or your grandmother's side that you claim descent from a monkey''; whereupon Huxley delivered a devastating rebuke, thereby establishing the primacy of scientific truth over ecclesiastical obscurantism. Although the legend is historically untrue in almost every detail, its persistence suggests that it may nonetheless be true in some deeper, mythical, sense. To explore this possibility the British Academy has invited Dr Janet Browne to be a neo-Huxley confronting Mr J.R. Lucas, as a neo-Wilberforce, with each reconsidering their earlier arguments.

The Unity of Science is often thought to be reductionist, but this is because we fail to distinguish questions from answers. The questions asked by different sciences are different---the biologist is interested in different topics from the physicist, and seeks different explanations---but the answers are not peculiar to each particular science, and can range over the whole of scientific knowledge. The biologist is interested in organisms--- concept unknown to physics---but explains physiological processes in terms of chemistry, not a mysterious vital force. The replacement of Laplacian determinism by quantum mechanics further erodes the tendency towards reductionism. The answers given in different explanations are not subsumed under one complete theory; and quantum mechanics does not have a concept of haecceitas, "this-i-ness" which would make its entities the fundamental constituents of everything.

The ontological argument has run for a long time, regularly refuted, regularly re-appearing in a new form. Something can be learnt from its longevity. Its proponents must be on to something, or it would not have survived its many refutations. But equally, it must have been much misformulated, or it would not have seemed evidently fallacious to its many critics. Perhaps it does express a deep philosophical intimation. Certainly it has been taken to prove more than it really can establish. Like many other philosophical arguments it has suffered by being made out to be more rigorous than in the nature of the case it can be. For some philosophers addressing some questions it may have been decisive in leading them to adopt one of the few options open to them: but it is quite inconclusive for others, with different presuppositions or different problems, and cannot be reduced into a valid proof cogent for all comers and compelling them to accept the conclusion it claims to demonstrate.

IT is ungenerous to pick holes in The Concept of Law. It is a great work. Its clarity is luminous, and its argument sustained and convincing. Hart is eminently successful in rescuing the concept of law from the Legal Realists, the Positivists, and the Formalists, who attempt to straitjacket it within schemata which are too narrow or too vague to give an adequate elucidation of it. But sometimes Hart is not carried along by his arguments as far as he should. He makes too many concessions to his opponents, and his own account of the law is, in consequence, too formalist, in spite of having himself adduced cogent considerations elsewhere for rejecting the purely formalist line of argument. The rule of recognition, although important, is not fundamental. We should, rather, see law as a social phenomenon, to be distinguished from other social phenomena, but intelligible only in a social context, and not-as lawyers are too ready to suppose-an autonomous discipline which can be explained and understood entirely in its own terms.

Thus far the logic out of which mathematics has developed has been First-order Predicate Calculus with Identity, that is the logic of the sentential functors, ¬, →, ∧, ∨, etc., together with identity and the existential and universal quotifiers restricted to quotify- ing only over individuals, and not anything else, such as qualities or quotities themselves. Some philosophers—among them Quine— have held that this, First-order Logic, as it is often called, con- stitutes the whole of logic. But that is a mistake. It leaves out Second-order Logic, which we need if we are to characterize the natural numbers precisely, and pays scant attention to the logic of relations, especially transitive relations, which is the key to much of modern mathematics. Quine’s argument for restricting logic to First-order Logice was based on the grounds that only First- order logical theories display “Law and Order” and himself regards modal logic as belonging with witchcraft and superstition.1 Pred- icates are ontologically more suspect than individuals, and have a different logic, which is liable to give rise to paradox and inconsis- tency. Moreover, Second-order Logic lacks the completeness that First-order Logice has, which provides a pleasing parallel between syntactic and semantic notions, and argues for the analyticity of deductive logic.

x10.1 Locality Newton's Law of Universal Gravitation was always open to the complaint that it involved \Action at a Distance", contrary to the Principle of Locality. But it was very well established empirically, and had to be accepted. Similarly in contemporary quantum me- chanics we seem to have correlations between measurements that defy the Principle of Locality, but have to be accepted none the less.1 Although locality is a characteristic mark of causal con- nexion, it is not, as Hume supposed,2 an essential one. Nor is it merely a uniformity we have found to hold for the most part|else we should feel little compunction in accepting that in some cases it happened not to hold. It is clearly an a priori principle, though not an absolutely necessary one.

x2.1 Non-contradiction One can think wrong. The fact that after much thought one has reached a conclusion is no guarantee that the conclusion reached is right. Only a very opinionated man would refuse to concede the possibility of error, and once the admission of fallibility is made, the problem of justifying one's beliefs becomes acute. So we formulate our reasons as best we can. But even when formulated, they may fail to convince. Only if people are willing to be reasonable can they be reasoned with. None so obdurate as those who will not listen to reason, and with them at least it is better to save one's breath than to attempt to convince them. You just cannot argue with people who will not be argued with. With them we can only let them go their way, as did Socrates; E 1&# (ea chairein).

Equality is one of the great issues of our age, but few people stop to wonder at its being an issue in politics at all. Yet it is surprising that a concept which has its natural habitat in the mathematical sciences should have taken root in our thinking about how we should be governed. We do not naturally think of society in terms of group theory, or rings or fields, and have long been aware of the difficulties in establishing any over-arching social or political order. But we unthinkingly assume that we can meaningfully ask, and reliably tell, whether people are, or ought to be, equal to one another even while admitting how difficult it would be to say whether they were more or less than one another. Formal logic can help. Equality is an equivalence relation, that is to say one that is transitive, symmetric and reflexive. Equivalence classes pick out classes of people or things that are the same, or similar , in some respect or other. There are many such, and we need to specify in respect of what two things are or are not equivalent before we are saying, or asking, anything definite. I can be equivalent to you in respect of age, or height, or weight, and many equivalence classes— contemporaries, co-religionists, comrades—may be of great importance socially or politically. But equality is, in its original context, more than just an expression of sameness. It suggests also a possibility of being either more than or less than. I can be the same age as you, but if I were not, I should be either older than you, or younger. This is always the case in mathematics. The law of trichotomy holds, that if two things are not equal, then one is greater, in the relevant respect, than the other, and the difference can itself be measured. With human beings, however, there are rather few respects in which we can be properly measured. Age, height and weight apart, the ascription of most numerical measures is a dubious affair. At one time psychologists were confident that they could measure intelligence, and economists still purport to measure wealth, but the ascriptions they actually make do not seem to deserve the confidence called for..

It is meet and right that pride and humility should be the two human characteristics on which University sermons have to be preached. Left to myself, although I might have picked on my modesty as something I should share with you, I should have given the preeminence to other among my sins than pride. My greed, my sloth, my avarice or, in this salacious age my lust, are subjects on which I could tell you much that might interest you. Pride lacks immediate appeal. We are not sure what it is, or whether it is a bad thing, when we think of it in purely individual terms. But when we consider it collectively, we can see that it is, together with humility, something Oxford is peculiarly well qualified to preach on. We all of us are proud of our university. We were proud, and our schoolmasters were proud, when we first got our places here. We are, dons and undergraduates alike, proud of our colleges, each grateful that good fortune has brought him to the best college in Oxford, and anxious that everyone else should secretly acknowledge it to be the best. Our parents were proud when we took our degrees, and although we profess to be unconcerned with classes, we are deeply content to record our firsts when occasion requires us to do so, or have our contemporaries allude to them as opportunity offers. We are studious, as dons, not to pull rank, safe in the knowledge that others will do it for us, and that we shall receive the deference due to a fellow of an Oxford college. In an age that is egalitarian in theory but elitist at heart, Oxford men have benefited greatly, as other forms of social eminence have been eroded, leaving a clear field for our own claims to public esteem, which are, if not entirely unchallenged, still generally allowed. Oxford is, as we like to be told by outsiders, a centre of excellence, and a lot of the resplendence rubs off on us, not altogether undeservedly. It is, as we corporately admit on Commemoration Sunday, largely due to our having entered into other men's labours..

In Epiphany Term, 1942, C.S. Lewis delivered the Riddell Memorial Lectures in the Physics Lecture Theatre, King's College, Newcastle, which was then a constituent college of the University of Durham. The Riddell Memorial Lectures were founded in 1928 in memory of Sir John Buchanan Riddell of Hepple, onetime High Sheriff of Northumberland, who had died in 1924. His son, Sir Walter, was, like his father, a devout Christian, active throughout his life in public affairs. He was Fellow, and subsequently Principal, of Hertford College, Oxford, and Secretary, and subsequently Chairman, of the University Grants Committee---at a time when the interventions of the UGC in academic affairs were entirely benign. I myself have a special personal interest in the founder of the Riddell Lectures. Sir Walter and my father were contemporaries at Oxford and close friends. Owing to his untimely death I never knew him, though I am credibly assured that he must have viewed me when I was taken to Hepple in a Moses' basket at the age of six months. In being invited to give this lecture on the Riddell Lectures, I am not only honoured but enabled to discharge a debt of family..

I must start with an apologia. My original paper, ``Minds, Machines and Gödel'', was written in the wake of Turing's 1950 paper in Mind, and was intended to show that minds were not Turing machines. Why, then, didn't I couch the argument in terms of Turing's theorem, which is easyish to prove and applies directly to Turing machines, instead of Gödel's theorem, which is horrendously difficult to prove, and doesn't so naturally or obviously apply to machines? The reason was that Gödel's theorem gave me something more: it raises questions of truth which evidently bear on the nature of mind, whereas Turing's theorem does not; it shows not only that the Gödelian well-formed formula is unprovable-in-the-system, but that it is true. It shows something about reasoning, that it is not completely rule-bound, so that we, who are rational, can transcend the rules of any particular logistic system, and construe the Gödelian well-formed formula not just as a string of symbols but as a proposition which is true. Turing's theorem might well be applied to a computer which someone claimed to represent a human mind, but it is not so obvious that what the computer could not do, the mind could. But it is very obvious that we have a concept of truth. Even if, as was claimed in a previous paper, it is not the summum bonum, it is a bonum, and one it is characteristic of minds to value. A representation of the human mind which could take no account of truth would be inherently implausible. Turing's theorem, though making the same negative point as Gödel's theorem, that some things cannot be done by even idealised computers, does not make the further positive point that we, in as much as we are rational agents, can do that very thing that the computer cannot. I have however, sometimes wondered whether I could not construct a parallel argument based on Turing's theorem, and have toyed with the idea of a von Neumann machine. A von Neumann machine was a black box, inside which was housed John von Neumann..

Plato began it. After thinking about the nature of argument he concluded that the correct way of reasoning was the axiomatic way, and formulated the programme of axiomatization that Eudoxus and Euclid subsequently carried out. Since then the axiomatic method has been firmly established, not only as the method for mathematics, but as a paradigm to which all other disciplines should strive to be assimilated; and in this present century not only has axiomatization been carried through as completely as it can be, but the most determined efforts have been made to wish hypotheticodeductive schemata on to biology, economics, and even history.

viva was unmistakable; I had sat in when a friend was being done, to spot the form; it was the same room, which I had not been into since my own viva in Greats many years ago, the same table, the lonely candidate on one side, the sombre Inquisitors on the other, courteous, considerate, anxious that the candidate should acquit himself well, but sure to notice every fallacy or error. Others, too, had sensed the likeness. ``Yes, I think the candidate passed'' one tutor said meditatively of an ennobled Vice-Chancellor; ``I think the Examiners passed too; very fair, very fair---but very searching. You had better look out.'' A blur of people, perhaps friends and supporters. Kindly, encouraging questions, some expected, some not. Balls asking to be hit for six; dropped catches---why had not I gone armed with a list of a dozen different ways in which Oxford was more up to date than other places? missed cues; fumbled answers; moments of caution; moments of over-confidence-- -``I have heard tell of a women's college where...'', biting my tongue just too late.