A response to a recent critique by Cem Bozşahin of the theory of syntactic semantics as it applies to Helen Keller, and some applications of the theory to the philosophy of computer science.

What psychological and philosophical significance should we attach to recent efforts at computer simulations of human cognitive capacities? In answering this question, I find it useful to distinguish what I will call "strong" AI from "weak" or "cautious" AI. According to weak AI, the principal value of the computer in the study of the mind is that it gives us a very powerful tool. For example, it enables us to formulate and test hypotheses in a more rigorous and precise fashion. (...) But according to strong AI, the computer is not merely a tool in the study of the mind; rather, the appropriately programmed computer really is a mind, in the sense that computers given the right programs can be literally said to. (shrink)

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...) with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

G. Priest's anti-consistency argument (Priest 1979, 1984, 1987) and J. R. Lucas's anti-mechanist argument (Lucas 1961, 1968, 1970, 1984) both appeal to Gödel incompleteness. By way of refuting them, this paper defends the thesis of quartet compatibility, viz., that the logic of the mind can simultaneously be Gödel incomplete, consistent, mechanical, and recursion complete (capable of all means of recursion). A representational approach is pursued, which owes its origin to works by, among others, J. Myhill (1964), P. Benacerraf (1967), J. (...) Webb (1980, 1983) and M. Arbib (1987). It is shown that the fallacy shared by the two arguments under discussion lies in misidentifying two systems, the one for which the Gödel sentence is constructable and to be proved, and the other in which the Gödel sentence in question is indeed provable. It follows that the logic of the mind can surpass its own Gödelian limitation not by being inconsistent or non-mechanistic, but by being capable of representing stronger systems in itself; and so can a proper machine. The concepts of representational provability, representational maximality, formal system capacity, etc., are discussed. (shrink)

Goal-directed problem solving as originally advocated by Herbert Simon’s means-ends analysis model has primarily shaped the course of design research on artificially intelligent systems for problem-solving. We contend that there is a definite disregard of a key phase within the overall design process that in fact logically precedes the actual problem solving phase. While systems designers have traditionally been obsessed with goal-directed problem solving, the basic determinants of the ultimate desired goal state still remain to be fully understood or categorically (...) defined. We propose a rational framework built on a set of logically inter-connected conjectures to specifically recognize this neglected phase in the overall design process of intelligent systems for practical problem-solving applications. (shrink)

Gödel argued that the incompleteness theorems entail that the mind is not a machine, or that certain arithmetical propositions are absolutely undecidable. His view was that the mind is not a machine, and that no arithmetical propositions are absolutely undecidable. I argue that his position presupposes that the idealized mathematician has an ability which I call the recursive-ordinal recognition ability. I show that we have this ability if, and only if, there are no absolutely undecidable arithmetical propositions. I argue that (...) there are such propositions, but that no recognizable example of one can be identified, even in principle. (shrink)

Queen Anne is dead, and it is a fallacy to substitute a definite description for another designator of the same object in stating the content of someone’s propositional attitudes. The fallacy can take subtle forms, as when Godel’s incompleteness theorems are used to argue against mechanistic views of mind. Some instances of the fallacy exemplify a more general logical phenomenon: the set of principles satisfied by one sentential operator can differ from, and even contradict, the set of principles satisfied by (...) another sentential operator coextensive with the first. These notions will be formally defined. The phenomenon will be explored through a series of example, with particular attention to the misapplication of Godel’s results. (shrink)

In a recent paper, Lyngzeidetson [1990] has claimed that a type of parallel computer called the ‘Connection Machine’ instantiates architectural principles which will ‘revolutionize which "functions" of the human mind can and cannot be modelled by (non-human) computational automata.’ In particular, he claims that the Connection Machine architecture shows the anti-mechanist argument from Gödel's theorem to be false for at least one kind of parallel computer. In the first part of this paper, I argue that Lyngzeidetson's claims are not supported (...) by his arguments; in the second part I consider some other aspects of parallel computation which may be of theoretical significance in cognitive science. (shrink)

This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly popular approach (...) to resolving indeterminacy, before offering some brief closing reflections. We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic. (shrink)

We argue that thinking of the man-machine comparison in terms of a contest involves, in a reasonable scenario, a criterion of success that is neutral. This is because we want to avoid a petitio principii. We submit, however, that, by looking at things this way, one makes the most essential human things invisible. Thus, in a sense, the contest approach is self-defeating.

The article discusses the V.V. Tselishchev’s original and unique systematic study of the specific and extremely complicated problems of Gödel results regarding the question of artificial intelligence essence. Tselishchev argues that the reflexive property should be considered not only as an advantage of human reasoning, but also as an objective internal limitation that appears in case of adding Gödel sentence to a theory to build a new theory. The article analyzes so-called mentalistic Gödel’s argument for fundamental superiority of human intelligence (...) over machine one and the non-algorithmic nature of natural thinking. The discussion about the Gödel argument is not entirely speculative, but contains new knowledge. An example of such knowledge are the results of R. Smullyan levels of computers “awareness,” which are may be interpreted in a psychophysical sense. The concept of “zero level of intelligence” is proposed for such a reflexive property as “awareness of selfconsciousness.” Reflexive ranks below the awareness of self-consciousness can be considered negative levels of thinking in the sense that the intelligence, being reduced to them, significantly loses its completeness. Even self-consciousness turns out to be a negative level of thinking, since, according to Smullyan, the subject of self-consciousness is unaware of the type of thought to which he belongs. A thought experiment is proposed that allows us to establish the distribution of the properties of Smullyan stability and normality and to answer the question “Does an intuitive belief in the truth of a formal proof affect the truth of a proposition being proved?” According to intuitionism, the most unpleasant epistemic property is instability: beliefs that are not based on deep intuitions have no value. According to the constructivist philosophy of mathematics, instability is a less negative property than abnormality: the fact that high-ranking beliefs cannot be immersed to the very foundations is not significant because violation of truth due to lowering the rank of reflection is not critical. (shrink)

The aim of this paper is an attempt to give an answer to the question what does it mean that a computational system is intelligent. We base on some theses that though debatable are commonly accepted. Intelligence is conceived as the ability of tractable solving of some problems that in general are not solvable by deterministic Turing Machine.

The thinkers who have reflected on the problem of a coming superintelligence have generally seen the issue as a technological problem, a problem of how to control what the superintelligence will do. I argue that this approach is probably mistaken because it is based on questionable assumptions about the behavior of intelligent agents and, moreover, potentially counterproductive because it might, in the end, bring about the existential catastrophe that it is meant to prevent. I contend that the problem posed by (...) a future superintelligence will likely be a political problem, that is, one of establishing a peaceful form of coexistence with other intelligent agents in a situation of mutual vulnerability, and not a technological problem of control. (shrink)

In this paper I argue that it is more difficult to see how Godel's incompleteness theorems and related consistency proofs for formal systems are consistent with the views of formalists, mechanists and traditional intuitionists than it is to see how they are consistent with a particular form of mathematical realism. If the incompleteness theorems and consistency proofs are better explained by this form of realism then we can also see how there is room for skepticism about Church's Thesis and the (...) claim that minds are machines. (shrink)

The psychological orientation treats semantics as a matter of idealized computation over symbolic structures, and semantic relations like denotation as relations between linguistic expressions and these structures. I argue that results similar to Gödel's incompleteness theorems and Tarski's theorem on truth create foundational difficulties for this view of semantics.

Representationalist accounts of mental content face the threat of the homunculus fallacy. In collapsing the distinction between the conscious state and the conscious subject, self-representational accounts of consciousness possess the means to deal with this objection. We analyze a particular sort of self-representational theory, built on the work of John von Neumann on self-reproduction, using tools from mathematical logic. We provide an explicit theory of the emergence of referential beliefs by means of modal fixed points, grounded in intrinsic properties yielding (...) the subjective aspects of experience. Furthermore, we study complications introduced by allowing for the modification of such symbolic content by environmental influences. (shrink)

This piece continues the tradition of arguments by John Lucas, Roger Penrose and others to the effect that the human mind is not a machine. Kurt Gödel thought that the intensional paradoxes stand in the way of proving that the mind is not a machine. According to Gödel, a successful proof that the mind is not a machine would require a solution to the intensional paradoxes. We provide what might seem to be a partial vindication of Gödel and show that (...) if a particular solution to the intensional paradoxes is adopted, one can indeed give an argument to the effect that the mind is not a machine. (shrink)

This is a defense and extension of Stephen Yablo's claim that self-reference is completely inessential to the liar paradox. An infinite sequence of sentences of the form 'None of these subsequent sentences are true' generates the same instability in assigning truth values. I argue Yablo's technique of substituting infinity for self-reference applies to all so-called 'self-referential' paradoxes. A representative sample is provided which includes counterparts of the preface paradox, Pseudo-Scotus's validity paradox, the Knower, and other enigmas of the genre. I (...) rebut objections that Yablo's paradox is not a genuine liar by constructing a sequence of liars that blend into Yablo's paradox. I rebut objections that Yablo's liar has hidden self-reference with a distinction between attributive and referential self-reference and appeals to Gregory Chaitin's algorithmic information theory. The paper concludes with comments on the mystique of self-reference. (shrink)

. The question of whether or not our universe is deterministic remains of interest to both scientists and theologians. In this essay I argue that this question can be solved only by metaphysical decision and that no scientific evidence for either determinism or indeterminism will ever be conclusive. No finite being, no matter how powerful its cognitive abilities, will ever be able to establish the deterministic nature of the universe. The only being that would be capable of doing so would (...) be one that is at once transcendent and immanent. Such a being is compatible with the God of the Christian tradition, which yields that a deterministic worldview is compatible with a deterministic worldview. A more important point is that because science is never able to establish the determinism of our universe, it can never definitely rule out divine action except on metaphysical grounds. (shrink)

"The Emperor's New Mind" by Roger Penrose has received a great deal of both praise and criticism. This review discusses philosophical aspects of the book that form an attack on the "strong" AI thesis. Eight different versions of this thesis are distinguished, and sources of ambiguity diagnosed, including different requirements for relationships between program and behaviour. Excessively strong versions attacked by Penrose (and Searle) are not worth defending or attacking, whereas weaker versions remain problematic. Penrose (like Searle) regards the notion (...) of an algorithm as central to AI, whereas it is argued here that for the purpose of explaining mental capabilities the architecture of an intelligent system is more important than the concept of an algorithm, using the premise that what makes something intelligent is not what it does but how it does it. What needs to be explained is also unclear: Penrose thinks we all know what consciousness is and claims that the ability to judge Go "del's formula to be true depends on it. He also suggests that quantum phenomena underly consciousness. This is rebutted by arguing that our existing concept of "consciousness" is too vague and muddled to be of use in science. This and related concepts will gradually be replaced by a more powerful theory-based taxonomy of types of mental states and processes. The central argument offered by Penrose against the strong AI thesis depends on a tempting but unjustified interpretation of Goedel's incompleteness theorem. Some critics are shown to have missed the point of his argument. A stronger criticism is mounted, and the relevance of mathematical Platonism analysed. Architectural requirements for intelligence are discussed and differences between serial and parallel implementations analysed. (shrink)

SummaryJ.R. Lucas has argued that it follows from Godel's Theorem that the mind cannot be a machine or represented by any formal system. Although this notorious argument against the mechanism thesis has received considerable attention in the literature, it has not been decisively rebutted, even though mechanism is generally thought to be the only plausible view of the mind. In this paper I offer an analysis of Lucas's argument which shows that it derives its persuasiveness from a subtle confusion. In (...) particular, I examine Benacerraf's reconstruction of Lucas's argument, thereby permitting a precise re‐statement of my own analysis and reinforcement of my conclusions regarding Lucas's mistake. This examination of Benacerraf's reconstruction permits us to evaluate certain radical and highly paradoxical conclusions concerning empirical psychology which Benacerraf draws, and this, in turn, permits us to consider certain suggestive implications for the mind which may, after all, follow from Godel's Theorem.RésuméJ.R. Lucas a dérivé du théorème de Gödel la conclusion que ľesprit ne peut pas être une machine ou être représenté par un systeme formel. Bien que cet argument célèbre contre la thése du mécanisme ait éveillé un interêt considérable dans la littérature, il ňa pas été réfuté de façon décisive, même si ľopinion générate prévaut que le mécanisme est la seule conception possible de ľesprit. Dans cet article, je propose une analyse du raisonnement de Lucas qui montre que sa force persuasive provient ?on;une subtile confusion. En particulier, j'examine la reconstruction, par Benacerraf, de la démonstration de Lucas, ce qui me pérmet de préciser ma propre analyse et de confirmer mes conclusions quant àľerreur de Lucas. Cet examen de la reconstruction de Benacerraf nous fournit ľoccasion de discuter certaines conclusions radicales et hautement paradoxals qu'il en tire concernant la psychologie expérimentale et nous conduit à certaines implications suggestives concernant ľesprit et qui pourraient finalement decouler du théorème de Gödel.ZusammenfassungJ.R. Lucas hat argumentiert, aus Gödels Theorem mUsse folgen, dass der menschliche Geist keine Maschine sein oder durch ein formales System dargestellt werden kann. Obschon dieses bekannte Argument gegen die mechanistische These in der Literatur beträchtliche Beachtung gefunden hat, ist es bisher noch nie eigentlich widerlegt worden, obgleich eine mechanistische Betrachtungsweise allgemein als die einzig plausible gilt. In dieser Arbeit biete ich eine Analyse von Lucas Argument an, die zeigen soil, dass seine Überzeugungskraft auf einer subtilen Konfusion beruht. Im besonderen untersuche ich Benacerrafs Rekonstruktion des Arguments, was rnirerlaubt, meine eigene Analyse in genauer Weise zu formulieren und meine Schlussfolgerung bezüglich des von Lucas gemachten Fehlers zu erharten. Diese Untersuchung von Benacerrafs Rekonstruktion ermöglicht es, gewisse radikale und im höchsten Grad paradoxe Schlüsse, die Benacerraf hinsichtlich der empirischen Psychologie zieht, zu beurteilen. Das wiederum erlaubt uns, gewisse den Geist betreffende Implikationen zu erörtern, die tatsächlich aus Gödels Theorem folgen kUnnten. (shrink)

The aim of this paper is to present some philosophical considerations about the supposed AI emergence in the future. However, the predicted timeline of this process is uncertain. To avoid any kind of speculations on the proposed analysis from a scientific point of view, a metaphysical approach is undertaken as a modal context of the discussion. I argue that modal claim about possible AI emergence at a certain point of time in the future is justified from a temporal perspective. Therefore, (...) worldwide society must be prepared for possible AI emergence and the expected profound impact of such an event on the existential status of humanity. (shrink)

Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual (...) ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject. (shrink)

§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of forging (...) connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that Gödel's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine,M, and claims that the output ofMconsists of all and only the arithmetic truths that a human, or the totality of human mathematicians, will ever or can ever know. We assume that the output ofMis consistent. (shrink)

Computationalism, the notion that cognition is computation, is a working hypothesis of many AI researchers and Cognitive Scientists. Although it has not been proved, neither has it been disproved. In this paper, I give some refutations to some well-known alleged refutations of computationalism. My arguments have two themes: people are more limited than is often recognized in these debates; computer systems are more complicated than is often recognized in these debates. To underline the latter point, I sketch the design and (...) abilities of a possible embodied computer system. (shrink)

Turing's (1936) analysis of effective symbolic procedures is a model of conceptual clarity that plays an essential role in the philosophy of mathematics. Yet appeal is often made to the effectiveness of human procedures in other areas of philosophy. This paper addresses the question of whether Turing's analysis can be applied to a broader class of effective human procedures. We use Sieg's (1994) presentation of Turing's Thesis to argue against Cleland's (1995) objections to Turing machines and we evaluate her proposal (...) to understand the effectiveness of procedures in terms of their reliability and precision. A number of conditions for effectiveness are identified and these are used to provide a general argument against the possibility of a Leibnizian decision procedure. (shrink)

Granted that truth is valuable we must recognize that certifiable truth is hard to come by, for example in the natural and social sciences. This paper examines the case of mathematics. As a result of the work of Gödel and Tarski we know that truth does not equate with proof. This has been used by Lucas and Penrose to argue that human minds can do things which digital computers can't, viz to know the truth of unprovable arithmetical statements. The argument (...) is given a simple formulation in the context of sorites (Robinson) arithmetic, avoiding the complexities of formulating the Gödel sentence. The pros and cons of the argument are considered in relation to the conception of mathematical truth. (shrink)

I present two arguments that aim to establish logical limits on what we can know. More specifically, I argue for two results concerning what we can know about questions that we cannot answer. I also discuss a line of thought, found in the writings of Pierce and of Rescher, in support of the idea that we cannot identify specific scientific questions that will never be answered.

Certain selected issues around the Gödelian anti-mechanist arguments which have received less attention are discussed.

Austrian-born Kurt Gödel is widely considered the greatest logician of modern times. It is above all his celebrated incompleteness theorems—rigorous mathematical results about the necessary limits...

This paper reassesses the criticism of the Lucas-Penrose anti-mechanist argument, based on Gödel’s incompleteness theorems, as formulated by Krajewski : this argument only works with the additional extra-formal assumption that “the human mind is consistent”. Krajewski argues that this assumption cannot be formalized, and therefore that the anti-mechanist argument – which requires the formalization of the whole reasoning process – fails to establish that the human mind is not mechanistic. A similar situation occurs with a corollary to the argument, that (...) the human mind allegedly outperforms machines, because although there is no exhaustive formal definition of natural numbers, mathematicians can successfully work with natural numbers. Again, the corollary requires an extra-formal assumption: “PA is complete” or “the set of all natural numbers exists”. I agree that extra-formal assumptions are necessary in order to validate the anti-mechanist argument and its corollary, and that those assumptions are problematic. However, I argue that formalization is possible and the problem is instead the circularity of reasoning that they cause. The human mind does not prove its own consistency, and outperforms the machine, simply by making the assumption “I am consistent”. Starting from the analysis of circularity, I propose a way of thinking about the interplay between informal and formal in mathematics. (shrink)

In the 70 years since Alan Turing’s ‘Computing Machinery and Intelligence’ appeared in Mind, there have been two widely-accepted interpretations of the Turing test: the canonical behaviourist interpretation and the rival inductive or epistemic interpretation. These readings are based on Turing’s Mind paper; few seem aware that Turing described two other versions of the imitation game. I have argued that both readings are inconsistent with Turing’s 1948 and 1952 statements about intelligence, and fail to explain the design of his game. (...) I argue instead for a response-dependence interpretation. This interpretation has implications for Turing’s view of free will: I argue that Turing’s writings suggest a new form of free will compatibilism, which I call response-dependence compatibilism. The philosophical implications of rethinking Turing’s test go yet further. It is assumed by numerous theorists that Turing anticipated the computational theory of mind. On the contrary, I argue, his remarks on intelligence and free will lead to a new objection to computationalism. (shrink)

The Turing Test is one of the most disputed topics in artificial intelligence, philosophy of mind, and cognitive science. This paper is a review of the past 50 years of the Turing Test. Philosophical debates, practical developments and repercussions in related disciplines are all covered. We discuss Turing's ideas in detail and present the important comments that have been made on them. Within this context, behaviorism, consciousness, the 'other minds' problem, and similar topics in philosophy of mind are discussed. We (...) also cover the sociological and psychological aspects of the Turing Test. Finally, we look at the current situation and analyze programs that have been developed with the aim of passing the Turing Test. We conclude that the Turing Test has been, and will continue to be, an influential and controversial topic. -/- (This paper was reprinted in: The Turing Test: The Elusive Standard of Artificial Intelligence, J. H. Moor, ed., Kluwer Academic, pp. 23-78, 2003.). (shrink)

Roughly speaking, computationalism says that cognition is computation, or that cognitive phenomena are explained by the agent‘s computations. The cognitive processes and behavior of agents are the explanandum. The computations performed by the agents‘ cognitive systems are the proposed explanans. Since the cognitive systems of biological organisms are their nervous 1 systems (plus or minus a bit), we may say that according to computationalism, the cognitive processes and behavior of organisms are explained by neural computations. Some people might prefer to (...) say that cognitive systems are ―realized‖ by nervous systems, and thus that—according to computationalism—cognitive computations are ―realized‖ by neural processes. In this paper, nothing hinges on the nature of the relation between cognitive systems and nervous systems, or between computations and neural processes. For present purposes, if a neural process realizes a computation, then that neural process is a computation. Thus, I will couch much of my discussion in terms of nervous systems and neural computation.1 Before proceeding, we should dispense with a possible red herring. Contrary to a common assumption, computationalism does not stand in opposition to connectionism. Connectionism, in the most general and common sense of the term, is the claim that cognitive phenomena are explained (at some level and at least in part) by the processes of neural networks. This is a truism, supported by most neuroscientific evidence. Everybody ought to be a connectionist in this general sense. The relevant question is, are neural processes computations? More precisely, are the neural processes to be found in the nervous systems of organisms computations? Computationalists say ―yes‖, anti-computationalists say ―no‖. This paper investigates whether any of the arguments on offer against computationalism have a chance at knocking it off.2 Ever since Warren McCulloch and Walter Pitts (1943) first proposed it, computationalism has been subjected to a wide range of objections.. (shrink)

We begin by distinguishing computationalism from a number of other theses that are sometimes conflated with it. We also distinguish between several important kinds of computation: computation in a generic sense, digital computation, and analog computation. Then, we defend a weak version of computationalism—neural processes are computations in the generic sense. After that, we reject on empirical grounds the common assimilation of neural computation to either analog or digital computation, concluding that neural computation is sui generis. Analog computation requires continuous (...) signals; digital computation requires strings of digits. But current neuroscientific evidence indicates that typical neural signals, such as spike trains, are graded like continuous signals but are constituted by discrete functional elements (spikes); thus, typical neural signals are neither continuous signals nor strings of digits. It follows that neural computation is sui generis. Finally, we highlight three important consequences of a proper understanding of neural computation for the theory of cognition. First, understanding neural computation requires a specially designed mathematical theory (or theories) rather than the mathematical theories of analog or digital computation. Second, several popular views about neural computation turn out to be incorrect. Third, computational theories of cognition that rely on non-neural notions of computation ought to be replaced or reinterpreted in terms of neural computation. (shrink)

A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call ‘internalism’, according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably (...) explains how we pick out unique mathematical structures, this does not suffice to account for the determinacy of mathematical discourse. (shrink)

This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for (...) Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical
objection” to his view that machines can think. Logico-mathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it
should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human
mathematicians presumably do. (shrink)

In his 1951 Gibbs Memorial Lecture, Kurt Gödel put forth his famous disjunction that either the power of the mind outstrips that of any machine or there are absolutely unsolvable problems. The view that there are no absolutely unsolvable problems is optimism, the view that there are such problems is pessimism. In his 1995—and, revised in 2013—Verificationism Then and Now, Per Martin-Löf presents an illustrative argument for a constructivist form of optimism. In response to that argument, Solomon Feferman points out (...) that Martin-Löf’s reasoning relies upon constructive understandings of key philosophical notions. In the vein of Feferman’s analysis, one might be object to Martin-Löf’s argument for either its reliance upon constructivist considerations, or for its appeal to non-unproblematically mathematical premises. We argue that both of these responses fall short. On one hand, to be critical of Martin-Löf’s reasoning for its constructiveness is to reject what would otherwise be a scientific advance on the basis of the assumption of constructivism’s falsehood or implausibility, which is of course uncharitable at best. On the other hand, to object to the argument for its use of non-unproblematically mathematical premises is to assume that there is some philosophically neutral mathematics, which is implausible. Martin-Löf’s argument relies upon his third law, the claim that from the impossibility of a proof of a proposition we can construct a proof of its negation. We close with a discussion of some ways in which this claim can be criticized from the constructive point of view. Specifically, we contend that Martin-Löf’s third law is incompatible with what has been called “Poincaré’s Principle of Epistemic Conservation”, the thesis that genuine increase in mathematical knowledge requires subject-specific insight. (shrink)

The paper undertakes to characterize Hao Wang's style, convictions, and method as a philosopher, centering on his most important philosophical work From Mathematics to Philosophy, 1974. The descriptive character of Wang's characteristic method is emphasized. Some specific achievements are discussed: his analyses of the concept of set, his discussion, in connection with setting forth Gödel's views, of minds and machines, and his concept of ‘analytic empiricism’ used to criticize Carnap and Quine. Wang's work as interpreter of Gödel's thought and the (...) importance of his writings as a source are also discussed. (shrink)

In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational (...) models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes. (shrink)

Although Artificial Intelligence (AI) and Machine Learning (ML) are attracting a lot of scientific and engineering attention nowadays, nothing up to now has been achieved to reach the level of building machines that possess human-like intelligence. Though, the engineering community continuously claims that several engineering problems are solved using AI or ML. Here, it is argued that engineers are not being able to build intelligent machines, implying that the systems claimed to have AI/ML belong to different engineering domains. The base (...) of the syllogism is the existence of four main obstacles on which extensive elucidation is performed. In addition, attempt to clear out the developed confusion, mis-use and ab-use of the phrases “Artificial Intelligence” and “Machine Learning” by scientists and engineers is carried out. Furthermore, mathematical, and philosophical approaches are also mentioned that strengthen the argument against AI implementability as part of the whole syllogism. Finally, an alternative approach (not being unique) is suggested and discussed for performing research on AI and ML by the engineers. It is based on complexity theory and non-linear adaptive systems and provides the benefit of eliminating the before mentioned pragmatic and philosophical obstacles that engineers are facing and ignoring, without creating confusion on this scientific endeavor. (shrink)