Jean Lucas

Quantum mechanics has seemed to defy all attempts to construe it realistically, but antirealism, like the many‐worlds hypothesis, is even more difficult to accept. In order to give a realist construal of quantum mechanics, we need first to distinguish the objective and rational aspect of reality from the paradigmatic thing‐like aspects of having determinate physical properties: quantum‐mechanical entities may be real in the former sense though not in the latter. Anti‐realist arguments are based on the difficulty of giving an account of quantum‐mechanical collapse and the apparent superluminal velocities involved. Objections to superluminal velocities on the score of the special theory of relativity are found not to be conclusive, and the price—there being some preferred frame of reference—to be acceptable. A sketch of a probabilistic account of quantum‐mechanical collapse is offered, which makes the difference between the macro‐ and the micro‐world a matter of degree rather than kind. If that, or some other, account proved acceptable, we could be quantum‐mechanical realists, though quantum‐mechanical reality would be very different from that of material objects in hardware shops.

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.

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.

In this paper Lucas comes back to Gödelian argument against Mecanism to clarify some points. First of all, he explains his use of Gödel’s theorem instead of Turing’s theorem, showing how Gödel’ theorem, but not Turing’s theorem, raises questions concerning truth and reasoning that bear on the nature of mind and how Turing’s theorem suggests that there is something that cannot be done by any computers but not that it can be done by human minds. He considers moreover how Gödel’s theorem can be interpreted as a sophisticated form of the Cretan paradox, posed by Epimenides, able to escape the viciously self-referential nature of the Cretan paradox, and how it can be used against Mechanism as a schema of disproof. Finally, Lucas suggests some answers to the most recurrent criticisms against his argument: criticisms about the implicit idealisation in the way he set up the context between mind and machine; questions concerning modality and finitude, issues of transfinite arithmetic; questions concerning the need of formalizing rational inference and some questions about consistency

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.

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.

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.

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).

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.

In this article, Lucas maintains the falseness of Mechanism - the attempt to explain minds as machines - by means of Incompleteness Theorem of Gödel. Gödel’s theorem shows that in any system consistent and adequate for simple arithmetic there are formulae which cannot be proved in the system but that human minds can recognize as true; Lucas points out in his turn that Gödel’s theorem applies to machines because a machine is the concrete instantiation of a formal system: therefore, for every machine consistent and able of doing simple arithmetic, there is a formula that it can’t produce as true but that we can see to be true, and so human minds and machines have to be different. Lucas considers as well in this article some possible objections to his argument: for any Gödelian formula we could, for instance, construct a machine able to produce it or we could put the Gödelian formulae that we had proved as axioms of a further machine. However - as Lucas underlines - for every of such machines we could again formulate another Gödelian formula, the Gödelian formula of these machines, that they are not able to proof but that we can recognize as true. More general arguments, such as the possibility to escape Gödelian argument by suggesting that Gödel’s theorem applies to consistent systems while we could be inconsistent ones, are moreover refuted by Lucas by maintaining that our inconsistency corresponds to occasional malfunctioning of a machine and not to his normal inconsistency; indeed, a inconsistent machine is characterized by producing any statement, on the contrary human being are selective and not disposed to assert anything.