Minds, Machines and Gödel

J.R. Lucas

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 (indeed the procedure whereby a Gödelian formula is constructed is a standard procedure) 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.

God, the Devil, and Gödel

P. Benacerraf

Satan stultified: a rejoinder to Paul Benacerraf

J.R. Lucas

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.

Human and Machine Logic

I.J. Good

Human and Machine Logic: a Rejoinder

J.R. Lucas

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.

Lucas against Mechanism

D. Lewis

Lucas against Mechanism: a Rejoinder

J.R. Lucas

Lewis maintains that the problem of Lucas’ argument is that it has to require a certain arithmetic – ‘Lucas arithmetic’ – which is true and can be produced by human beings but that any machine can’t produce; nevertheless there is no reason to suppose that human beings are in general able to verify theoremhood of Lucas arithmetic. Lucas refutes this criticism by arguing that, in order to refute Mechanism, it is not necessary to show that any mind can produce all of Lucas arithmetic or that any mind can understand Gödel’s theorem, because a single counter-example is enough: that is a single mind producing a single truth not recognizable by any machine, exactly the Gödelian sentence of the machine itself.

Gödel’s Theorem and Mechanism

D. Coder

Lucas against Mechanism: a Rejoinder

J.R. Lucas

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.

This Gödel is killing Me

A. Hutton

This Gödel is killing Me: a Rejoinder

J.R. Lucas

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.

Minds, Machines and Gödel: a Retrospect

J.R. Lucas

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.

The Gödelian Argument: Turn over the Page

J.R. Lucas

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.

The Implications of Gödel Theorem

J.R. Lucas

After a brief and informal explanation of the Gödel’s theorem as a version of the Epimenides’ paradox applied to Elementary Number Theory formulated in first-order logic, Lucas shows some of the most relevant consequences of this theorem, such as the impossibility to define truth in terms of provability and so the failure of Verificationist and Intuitionist arguments. He shows moreover how Gödel’s theorem proves that first-order arithmetic admits non-standard models, that Hilbert’s programme is untenable and that second-order logic is not mechanical. There are furthermore some more general consequences: the difference between being reasonable and following a rule and the possibility that one man’s insight differs from another’s without being wrong. Finally some consequences concerning moral and political philosophy can arise from Gödel’s theorem, because it suggests that – instead of some fundamental principle from which all else follows deductively – we can seek for different arguments in different situations.

A simple Exposition of Gödel’s Theorem

J.R. Lucas

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.

Death, nature and technique

F. Ferrari

In this paper the author intends to examine, within the continental philosophical tradition, the developments of the idea of death, its transformation from a collective event into an individual one, its slow detachment from the religious ambit, and its introduction in the dimension of technique.

Value, pleasure, utility: an enquiry by A. Lambertino

P. Marrone

The article is a critical note of A. Lambertino's work, Valore e piacere. It focuses particularly on the chapters dedicated to Kant, Nietzsche, Freud and on the last chapter, which clearly expresses the author's point of view. The conclusion is that the ethical position presented by Lambertino corresponds to a moderate ethical intellectualism.

Mankind and the birth of society: anthropological and sociological myths within the dialogues of Plato

M. Mazzoni

Plato, in different dialogues such as Statesman, Protagoras, Republic, Timaeus-Critias and Laws, creates a number of myths that describe both the early condition of the human race and the efforts made by men to develop the first forms of social and political organisation. Myth - that ancient and unsurpassed mean of transmission of knowledge and moral precepts - seems indeed to be the only instrument able to reconstruct not only the anthropological characteristics and original state of man, but also the genesis of human aggregation and of the first technical and cultural realisations. Therefore, a determined group of dialogical characters – The Stranger of Elea, Protagoras, Socrates, Critias and the Athenian –, within the different platonic dialogues, presents a number of myths marked not only by their great literary fascination, but also by their independent philosophical value. Although each myth is linked to the others by many analogies both in structure and content, each one is an autonomous artistic and intellectual creation, but is however intrinsically bound to the dialogical context it is collocated in. As myths are introduced in order to give answers to precise questions and find solutions to certain problems (is the “golden age” lifestyle a desirable one? is it just an “animal paradise”? is man a selfish and aggressive being or is he a social and peace-loving one? is the political aggregation the product of an artificial and utilitarian contract or is it a natural result? is it possible to create a polis that guarantees collaboration and peace, without excluding cultural and aesthetical realisations?), it is essential not to sever the deep link between the mythical tale and the context in which it is collocated, but rather to examine carefully the function and meaning of the former in relationship to the latter and vice versa.