Gödei's Theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines. So also has it seemed to many other people: almost every mathematical logician I have put the matter to has confessed to similar thoughts, but has felt reluctant to commit himself definitely until he could see the whole argument set out, with all objections fully stated and properly met. This I attempt to do.

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. (shrink)

Responsibility is a key concept in our moral, social, and political thinking, but it is not itself properly understood. J.R. Lucas here presents a lively, broad, and accessible discussion of responsibility in various areas of human life, from personal and sexual relations to politics.

That space and time should be integrated into a single entity, spacetime, is the great insight of Einstein's special theory of relativity, and leads us to regard spacetime as a fundamental context in which to make sense of the world around us. But it is not the only one. Causality is equally important and at least as far as the special theory goes, it cannot be subsumed under a fundamentally geometrical form of explanation. In fact, the agent of propagation of (...) causal influence is electromagnetic radiation. In this examination, the authors find support for a rationalist approach to physics, never neglecting experimentation, but rejecting a simple empiricist or positivist view of science. (shrink)

The application of Gödel’s theorem to the problem of minds and machines is difficult. Paul Benacerraf makes the entirely valid ‘Duhemian’ point that the argument is not, and cannot be, a purely mathematical one, but needs some philosophical premisses to be able to yield any philosophical conclusions. Moreover, the philosophical premisses are of very different kinds. Some are concerned with what is essential to being a machine—these are typically intricate, but definite, easily formalised by the mathematician, but unintelligible to the (...) layman: others attempt to capture what is essential to being a mind, a person or a self—these are typically intuitive, but vague; resistant to exact definition by the logician, but, none the less, widely used and well understood. Gödel’s theorem itself, like many other truths, can be taken either way: it can be taken as a formal proof sequence yielding certain syntactical results about a certain class of formal systems, but it can also be taken as giving us a certain type or style of argument, which we can understand, and, once having got the hang of it, adapt and apply in innumerable different circumstances. In my dispute with the mechanist, I take Gödel’s argument both ways: I first take it as an argument which the mechanist, even according to his own mechanist principles, must accept as scoring some point against his favourite machine; and then I hope that the mechanist, as a man, will see that he can do better and that this sort of argument will always apply against any form of mechanism he espouses. (shrink)

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. (shrink)

Space, Time and Causality An Essay in Natural Philosophy.

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. (shrink)

The issue is obscured by the fact that the word `space' can be used in four different ways. It can be used, first, as a term of pure mathematics, as when mathematicians talk of an `n-dimensional phase-space', an `n-dimensional vector-space', a `three-dimensional projective space' or a `twodimensional Riemannian space'. In this sense the word `space' means the totality of the abstract entities-the `points'-implicitly defined by the axioms. There is no doubt that there exist, iii this sense, non-Euclidean spaces, because all (...) that is claimed by such an assertion is that sets of non-Euclidean axioms constitute possible implicit definitions of abstract entities, that is to say that some sets of non-Euclidean axioms are consistent. If Kant or any other philosopher had denied this, he would have been wrong; but Kant himself took care not to deny it, 2 and there is little reason to suppose that any philosopher concerned about space has been using the word in this, the pure mathematician's, sense. (shrink)

Equality is the great political issue of our time. Liberty is forgotten: Fraternity never did engage our passions: the maintenance of Law and Order is at a discount: Natural Rights and Natural Justice are outmoded shibboleths. But Equality—there men have something to die for, kill for, agitate about, be miserable about. The demand for Equality obsesses all our political thought. We are not sure what it is—indeed, as I shall show later, we are necessarily not sure what it is—but we (...) are sure that whatever it is, we want it: and while we are prepared to look on frustration, injustice or violence with tolerance, as part of the natural order of things, we will work ourselves up into paroxysms of righteous indignation at the bare mention of Inequality. (shrink)

The application of Gödel’s theorem to the problem of minds and machines is difficult. Paul Benacerraf makes the entirely valid ‘Duhemian’ point that the argument is not, and cannot be, a purely mathematical one, but needs some philosophical premisses to be able to yield any philosophical conclusions. Moreover, the philosophical premisses are of very different kinds. Some are concerned with what is essential to being a machine—these are typically intricate, but definite, easily formalised by the mathematician, but unintelligible to the (...) layman: others attempt to capture what is essential to being a mind, a person or a self—these are typically intuitive, but vague; resistant to exact definition by the logician, but, none the less, widely used and well understood. Gödel’s theorem itself, like many other truths, can be taken either way: it can be taken as a formal proof sequence yielding certain syntactical results about a certain class of formal systems, but it can also be taken as giving us a certain type or style of argument, which we can understand, and, once having got the hang of it, adapt and apply in innumerable different circumstances. In my dispute with the mechanist, I take Gödel’s argument both ways: I first take it as an argument which the mechanist, even according to his own mechanist principles, must accept as scoring some point against his favourite machine; and then I hope that the mechanist, as a man, will see that he can do better and that this sort of argument will always apply against any form of mechanism he espouses. (shrink)

The argument is a dialectical one. It is not a direct proof that the mind is something more than a machine, but a schema of disproof for any particular version of mechanism that may be put forward. If the mechanist maintains any specific thesis, I show that [146] a contradiction ensues. But only if. It depends on the mechanist making the first move and putting forward his claim for inspection. I do not think Benacerraf has quite taken the point. He (...) criticizes me both for "failing to notice" that my ability to show that the Gödel sentence of a formal system is true "depends very much on how he is given. (shrink)

PROFESSOR LEWIS 1 and Professor Coder 2 criticize my use of Gödel's theorem to refute Mechanism. 3 Their criticisms are valuable. In order to meet them I need to show more clearly both what the tactic of my argument is at one crucial point and the general aim of the whole manoeuvre.

The Conceptual Roots of Mathematics is a comprehensive study of the foundation of mathematics. Lucas, one of the most distinguished Oxford scholars, covers a vast amount of ground in the philosophy of mathematics, showing us that it is actually at the heart of the study of epistemology and metaphysics.

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. (shrink)

In this paper Lucas suggests that many of his critics have not read carefully neither his exposition nor Penrose’s one, so they seek to refute arguments they never proposed. Therefore he offers a brief history of the Gödelian argument put forward by Gödel, Penrose and Lucas itself: Gödel argued indeed that either mathematics is incompletable – that is axioms can never be comprised in a finite rule and so human mind surpasses the power of any finite machine – or there (...) exist absolutely unsolvable diophantine problems, and he suggest that the second disjunct is untenable; on the other side, Penrose proposed an argument similar to Lucas’ one but making use of Turing’s theorem. Finally Lucas exposes again his argument and considers some of the most important objections to it. (shrink)

Justice has always been regarded as one of the fundamental political virtues. No association of human individuals could subsist, says Hume, “were no regard paid to the laws of equity and justice”, and nearly every thinker who has turned to consider human society, has reached the same conclusion. Yet we are not at all clear what justice is, nor why it is so important. There are many other ideals which a society may cherish, and often reformers have felt impatient of (...) having to compromise their aims merely for the sake of justice, and have not seen why the claims of justice should in any way override other political goods. I shall try to show why justice is, along with liberty, a peculiarly fundamental ideal, which is not merely co-ordinate with other ideals a society may value; but also that this notwithstanding, considerations of justice are not all-sufficient, and do not, and should not, always override all other political ideals. (shrink)

There was once a leak from Hebdomadal Council. The Assessor told her husband, who told my wife, who told me that Monday afternoon had been spent discussing what Lucas would say if various courses of action were adopted, leading to the conclusion that it would be best to do nothing. I was flattered, but a bit surprised. The tide of philosophical scepticism had ebbed, and it was generally allowed that a reasonable way of discovering what someone would say was to (...) ask him. Dick Southwood did: he would quiz me in Common Room – sometimes ending "Thank you for letting me bounce these ideas off you" – and had reliable information about how one member of Congregation would react to various proposals. And not only me: he was a listening Vice-Chancellor, who used to bike from Wellington Square to Merton for lunch, greeting many as he passed them, and ready to stop if occasion warranted it. Of course, there are many other leaks. I remember once attending a meeting in the Town Hall to argue for cycle tracks, and someone coming up to me, and saying, "You’re having a tussle with Council, aren’t you? I think you ought to see the minutes of their latest meeting"; the next day there was a copy in my pigeon hole, giving me just the ammunition I needed. What members of Congregation tend the forget is the existence – the other side of the green baize door, so to speak – of a corps of bedells. (shrink)

This piece outlines concrete cases of benefit sharing that occur in relation to the sharing of human (biological) samples. For example, it surveys Indonesia’s decision, in 2006, to stop sharing virus samples of H5N1 (avian influenza) with the WHO Global Influenza Surveillance Network (GISN). It also outlines some of the ethical issues that arise in these cases.

My sights in this paper are trained on facts. Most people think that they know what facts are; that while their friends often, and themselves occasionally, are ignorant of the facts, at least they know what sort of things facts are---they can recognise a fact when they see it. Facts, in the popular philosophy of today, are good, simple souls; there is no guile in them, nor any room for subjective bias, and once we have made ourselves acquainted with them, (...) we have reached the beginning and summit of all wisdom. (shrink)

Abstract 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. (shrink)

[D. H. Mellor] Kant's claim that our knowledge of time is transcendental in his sense, while false of time itself, is true of tenses, i.e. of the locations of events and other temporal entities in McTaggart's A series. This fact can easily, and I think only, be explained by taking time itself to be real but tenseless. /// [J. R. Lucas] Mellor's argument from Kant fails. The difficulties in his first Antinomy are due to topological confusions, not the tensed nature (...) of time. Nor are McTaggart' s difficulties due to the tensed nature of time. The ego-centricity of tensed discourse is an essential feature of communication between selves, each of whom refers himself as 'I', and is required for talking about time as well as experience and agency. Arguments based on the Special Theory are misconceived. Some rest on a confused notion of 'topological simultaneity'. In the General Theory a cosmic time is defined, as also in quantum mechanics, where a natural present is defined by a unique hyperplane of collapse into eigen-ness. (shrink)

We can imagine a human operator playing a game of one-upmanship against a programmed computer. If the program is Fn, the human operator can print the theorem Gn, which the programmed computer, or, if you prefer, the program, would never print, if it is consistent. This is true for each whole number n, but the victory is a hollow one since a second computer, loaded with program C, could put the human operator out of a job.... It is useless for (...) the `mentalist' to argue that any given program can always be improves since the process for improving programs can presumably be programmed also; certainly this can be done if the mentalist describes how the improvement is to be made. If he does give such a description, then he has not made a case. (shrink)