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.

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.

Space, Time and Causality An Essay in Natural Philosophy.

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.

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)

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)

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.

It might be the case that absence of constraint is the relevant sense of ' freedom' when we are discussing the freedom of the will, but it needs arguing for. ...

Lucas introduces this paper by an account of how he began to be interested to questions about Materialism and Mechanism. Then he suggests a simple version of the Incompleteness theorem of Gödel, showing how this theorem proposes a version of the Epimenides’ paradox able to avoid the circularity of this paradox by means of the possibility to express meta-mathematics in terms of arithmetical propositions and by substituting questions concerning truth by questions concerning provability.

Two questions are distinguished: how to program a machine so that it behaves in a manner that would lead us to ascribe consciousness to it; and what is involved in saying that something is conscious. The distinction can be seen in cases where anaesthetics have failed to work on patients temporarily paralysed.

_The Conceptual Roots of Mathematics_ is a comprehensive study of the foundation of mathematics. J.R. 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.

John Stuart Mill’s Utilitarianism engages our interest and sympathy because it is flawed. It reflects the crisis in Mill’s life, when he lost his faith. He had been brought up by his father in the straitest tenets of utilitarianism, but had had nervous breakdown in early adult life from emotional ill-nourishment. Utilitarianism might work as a guide for the well-governing of India by James Mill and his colleagues, but gave little sustenance to the aspiring spirit of the Romantic Movement. It (...) treats people as units, not individuals. It takes no account of the “projects” that people pursue, as Bernard Williams.. (shrink)

These two articles are very interesting examples of how Lucas’ argument is not a direct proof but a dialectical argument depending from Mechanists’ first move. Good, starting from the Mentalists’ point of view, underlines that it is useless to argue that any program can be improved because the process for improving it can be programmed; he argues against Mentalism by denying that there are particular mental powers, because otherwise they could be described and so a computer could be programmed to (...) simulate them. The Lucas’ answer is constituted by a starting point’s change: his starting point is indeed the hypothesis that Mechanism is true and so that a complete specification of the mental mechanism of any human being can be given; therefore he argues that, once given, such a specification appears inadequate because it cannot produce as true its Gödelian formula, truth that a human being can see. (shrink)

David Lewis criticizes an argument I put forward against mechansim on the grounds that I fail to distinguish between OL, Lucas's ordinary potential arithmetic output, and OML, Lucas's arithmetical output when accused of being some particular machine M; and correspondingly, between OM the ordinary potential arithmetic output of the machine M, and ONM, the arithmetic output of the machine M when accused of being a particular machine N. For any given machine, M, N, O, P, Q, R,... etc., I can (...) in principle calculate a Godel sentence for that machine - indeed infinitely many, depending on the Godel numbering scheme adopted. The Godel sentence of a particular machine can, I claim, be seen to be true, if that machine is adequate for Elementary Peano Arithmetic. Hence, if I were accused of being M, I can on that supposition see that the Godel sentence of M is true, since I am capable of Elementary Peano Arithmetic and the machine M is said to be an adequate characterization of me. (shrink)

Coder’s argument is very similar to Lewis’ one: he maintains that some human beings are not able to follow Gödel’s theorem, so Lucas’ argument cannot show that their minds are not machines. The answer of Lucas is that one proposed against Lewis’ criticism, that is that Mechanism makes a universal claim and so a single counter-example – a single mind producing a singe truth not recognizable by any machine – is a disproof for it.

Benacerraf criticizes Lucas’ argument against Mechanism because, in his opinion, it depends too much on how the system we are talking about is presented and because the argument put in form of challenge reduces itself to a contest of wits between Lucas and the mechanists. In Benacerraf opinion, Lucas should clarify the sense of utilised notions and the argument would have to be reconstructed as formally as possible, in order to determine the involved philosophical premises. Moreover Benacerraf maintains that, instead (...) of abandoning the idea that human mind is a machine, we could assume that minds are machines for which it is not possible to prove the consistency or that they are inadequate for arithmetic; moreover minds could be machines whose characteristics we are not able to specify. However, Lucas answers that the requirement of reconstructing his argument in a formal way misunderstands his project: his argument is not a direct proof but a dialectical argument, a schema of disproof for any particular version of mechanist argument, and so the attempt to reconstruct it as a rigorous proof is a distortion of the original argument, that is essentially dialectical. What about the hypothesis suggested by Benacerraf, Lucas disputes that we are able to manage arithmetic and we don’t seem as inconsistent as an inconsistent system is, because we are selective while an inconsistent system is not; at the other hand, the idea that we are machine but we don’t know anything about what kind of machine we are evacuates Mechanism of all content. (shrink)

The views of Redhead ([2004]) are defended against the argument by Panu Raatikainen ([2005]). The importance of informal rigour is canvassed, and the argument for the a priori nature of induction is explained. The significance of Gödel's theorem is again rehearsed.

Hutton asserts that Lucas’ use of Gödel’s theorem against Mechanism is incorrect because of the impossibility to assume human minds’ consistency: he tries to show that there is a non-zero probability of a mind’s embracing mutually inconsistent propositions; moreover Hutton maintains that the request of human minds’ consistency is a request of infallibility. Lucas replies that the mistake of Hutton’s argument consists in his assigning probabilities to a mind’s accepting any proposition without considering what that mind has done hitherto, while (...) human beings are guided in their accepting a proposition as true by what they have already accepted. Besides, Lucas argues that his argument doesn’t require human infallibility but only that human beings have adequate backing for what they assert and that – in this sense – consistency is not something that can be established but a necessary assumption to begin any thinking at all. (shrink)

Introductions should introduce, but sometimes lead to engagements. That is our aim. We want to make Plato’s Republic more easily read by modern readers, but do not want to do only that. For philosophy is like poetry, and cannot be learned second-hand. I can learn all sorts of facts about a poem, but unless I have entered into the poet’s experience, if only in my imagination, it remains dead. Similarly, I shall not see the point of text-book analyses of philosophical (...) doctrines unless I have felt the force of the arguments that led the philosopher to propose them, and have some sense of the objections he encountered and the way he sought to surmount them. That is why we still need to read Plato and Aristotle, as we do Homer and Sophocles, in a way that we do not, save as a historical exercise, read ancient textbooks of medicine or mechanical construction. (shrink)