The dominant scientific and philosophical view of the mind – according to which, put starkly, cognition is computation – is refuted herein, via specification and defense of the following new argument: Computation is reversible; cognition isn't; ergo, cognition isn't computation. After presenting a sustained dialectic arising from this defense, we conclude with a brief preview of the view we would put in place of the cognition-is-computation doctrine.

What''s computation? The received answer is that computation is a computer at work, and a computer at work is that which can be modelled as a Turing machine at work. Unfortunately, as John Searle has recently argued, and as others have agreed, the received answer appears to imply that AI and Cog Sci are a royal waste of time. The argument here is alarmingly simple: AI and Cog Sci (of the Strong sort, anyway) are committed to the view that cognition (...) is computation (or brains are computers); butall processes are computations (orall physical things are computers); so AI and Cog Sci are positively silly.I refute this argument herein, in part by defining the locutions x is a computer and c is a computation in a way that blocks Searle''s argument but exploits the hard-to-deny link between What''s Computation? and the theory of computation. However, I also provide, at the end of this essay, an argument which, it seems to me, implies not that AI and Cog Sci are silly, but that they''re based on a form of computation that is well beneath human persons. (shrink)

Which function is computed by a Turing machine will depend on how the symbols it manipulates are interpreted. Further, by invoking bizarre systems of notation it is easy to define Turing machines that compute textbook examples of uncomputable functions, such as the solution to the decision problem for first-order logic. Thus, the distinction between computable and uncomputable functions depends on the system of notation used. This raises the question: which systems of notation are the relevant ones for determining whether a (...) function is computable? These are the acceptable notations. I argue for a use criterion of acceptability: the acceptable notations for a domain of objects D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document} are the notations that we can use for the usual D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document}-activities. Acceptable notations for natural numbers are ones that we can count with. Acceptable notations for logical formulas are the ones that we can use in inference and logical analysis. And so on. This criterion makes computability a relative notion—whether a function is computable depends on which notations are acceptable, which is relative to our interests and abilities. I argue that this is not a problem, however, since there is independent reason for taking the extension of computable function to be relative to contingent facts. Similarly, the fact that the use criterion makes a mathematical distinction depend on practical concerns is also not a problem because it dovetails with similar phenomena in other areas of computability theory, namely the roles of notation in computation over real numbers and of Extended Church’s Thesis in complexity theory. (shrink)

The property of smallness for Π0 1 classes is introduced and is investigated with respect to Medvedev and Muchnik degree. It is shown that the property of containing a small Π0 1 class depends only on the Muchnik degree of a Π0 1 class. A comparison is made with the idea of thinness for Π0 1 classesmsthm.

We consider some nonprincipal filters of the Medvedev lattice. We prove that the filter generated by the nonzero closed degrees of difficulty is not principal and we compare this filter, with respect to inclusion, with some other filters of the lattice. All the filters considered in this paper are disjoint from the prime ideal generated by the dense degrees of difficulty.

In Zwicker (1987) the hypergame paradox is introduced and studied. In this paper we continue this investigation, comparing the hypergame argument with the diagonal one, in order to find a proof schema. In particular, in Theorems 9 and 10 we discuss the complexity of the set of founded elements in a recursively enumerable relation on the set N of natural numbers, in the framework of reduction between relations. We also find an application in the theory of diagonalizable algebras and construct (...) an undecidable formula. (shrink)

We generalize a well-knownSmullyan's result, by showing that any two sets of the kindC a = {x/ xa} andC b = {x/ xb} are effectively inseparable (if I b). Then we investigate logical and recursive consequences of this fact (see Introduction).

The problem of how mathematics and physics are related at a foundational level is of interest. The approach taken here is to work towards a coherent theory of physics and mathematics together by examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, the complex numbers, (...) closer to what is actually done in experimental measurements and computations. The method replaces C by C n which is the set of pairs, R n,I n, of n figure rational numbers in some basis. The properties of these numbers are based on those of numerical measurement outcomes for continuous variables. A model of space and time based on R n is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n→∞ to obtain locally flat continuum-based space and time is outlined. Also R n based space is invariant under scale transformations. These correspond to expansion and contraction of space relative to a flat background. The location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics, based on C n and on R n space are briefly investigated. (shrink)

LetA be any nonrecursive set. We define a hierarchy of sets (and a corresponding hierarchy of degrees) that are reducible toA based on bounding the number of queries toA that an oracle machine can make. WhenA is the halting problemK our hierarchy of sets interleaves with the difference hierarchy on the r.e. sets in a logarithmic way; this follows from a tradeoff between the number of parallel queries and the number of serial queries needed to compute a function with oracleK.

We obtain a large class of significant examples of n-random reals (i.e., Martin-Löf random in oracle $\varphi ^{(n - 1)} $ ) à la Chaitin. Any such real is defined as the probability that a universal monotone Turing machine performing possibly infinite computations on infinite (resp. finite large enough, resp. finite self-delimited) inputs produces an output in a given set O ⊆(ℕ). In particular, we develop methods to transfer $\Sigma _n^0 $ or $\Pi _n^0 $ or many-one completeness results of (...) index sets to n-randomness of associated probabilities. (shrink)

One of the main problems in effective model theory is to find an appropriate information complexity measure of the algebraic structures in the sense of computability. Unlike the commonly used degrees of structures, the structure degree measure is total. We introduce and study the jump operation for structure degrees. We prove that it has all natural jump properties (including jump inversion theorem, theorem of Ash), which show that our definition is relevant. We study the relation between the structure degree jump (...) (in the sense of Soskov) and the jump degrees of a structure (in the sense of Jockusch) and give necessary and sufficient conditions for their existence in the terms of structure degrees. We show some properties, distinguishing the structure degrees from the enumeration degrees. (shrink)

Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a, so that for every \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, i.e., two sets A, B, with A properly contained in B, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of ω; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On the (...) other hand, we also give a sufficient condition on a, that yields that there is a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal ω + ω; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n ∈ ω, or n = ω, and every notation a of a nonzero ordinal there exists a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of cardinality n, whose Rogers semilattice consists of exactly one element. (shrink)

We show that for every ordinal notation ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\xi}$$\end{document} of a nonzero computable ordinal, there exists a Σξ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{-1}_\xi}$$\end{document}—computable family which up to equivalence has exactly one Friedberg numbering, which does not induce the least element in the corresponding Rogers semilattice.

Steve Awodey and Audrej Bauer. Sheaf Toposes for Realizability.

Ash and Nerode [2] gave natural definability conditions under which a relation is intrinsically r. e. Here we generalize this to arbitrary levels in Ershov's hierarchy of Δmath image sets, giving conditions under which a relation is intrinsically α-r. e.

In this paper we argue that Revision Rules, introduced by Anil Gupta and Nuel Belnap as a tool for the analysis of the concept of truth, also provide a useful tool for defining computable functions. This also makes good on Gupta's and Belnap's claim that Revision Rules provide a general theory of definition, a claim for which they supply only the example of truth. In particular we show how Revision Rules arise naturally from relaxing and generalizing a classical construction due (...) to Kleene, and indicate how they can be employed to reconstruct the class of the general recursive functions. We also point at how Revision Rules can be employed to access non-minimal fixed points of partially defined computing procedures. (shrink)

The appropriate methodology for psychological research depends on whether one is studying mental algorithms or their implementation. Mental algorithms are abstract specifications of the steps taken by procedures that run in the mind. Implementational issues concern the speed and reliability of these procedures. The algorithmic level can be explored only by studying across-task variation. This contrasts with psychology's dominant methodology of looking for within-task generalities, which is appropriate only for studying implementational issues.The implementation-algorithm distinction is related to a number of (...) other “levels” considered in cognitive science. Its realization in Anderson's ACT theory of cognition is discussed. Research at the algorithmic level is more promising because it is hard to make further fundamental scientific progress at the implementational level with the methodologies available. Protocol data, which are appropriate only for algorithm-level theories, provide a richer source than data at the implementational level. Research at the algorithmic level will also yield more insight into fundamental properties of human knowledge because it is the level at which significant learning transitions are defined.The best way to study the algorithmic level is to look for differential learning outcomes in pedagogical experiments that manipulate instructional experience. This provides control and prediction in realistically complex learning situations. The intelligent tutoring paradigm provides a particularly fruitful way to implement such experiments.The implications of this analysis for the issue of modularity of mind, the status of language, research on human/computer interaction, and connectionist models are also examined. (shrink)

This paper investigates, in a simple risk-sharing framework, the extent to which the incompleteness of contracts could be attributed to the complexity costs associated with the writing and the implementation of contracts. We show that, given any measure of complexity in a very general class, it is possible to find simple contracting problems such that, when complexity costs are explicitly taken into account, the contracting parties optimally choose an incomplete contract which coincides with the ‘default’ division of surplus. Optimal contracts (...) with complexity costs are constrained efficient in our model. We therefore interpret our results as saying that, in the absence of a strategic role for complexity costs, their effect is entirely determined by their size relative to the size of payoffs. (shrink)

We define the bounded jump of $A$ by $A^{b}=\{x\in \omega \mid \exists i\leq x[\varphi_{i}\downarrow \wedge\Phi_{x}^{A\upharpoonright \!\!\!\upharpoonright \varphi_{i}}\downarrow ]\}$ and let $A^{nb}$ denote the $n$th bounded jump. We demonstrate several properties of the bounded jump, including the fact that it is strictly increasing and order-preserving on the bounded Turing degrees. We show that the bounded jump is related to the Ershov hierarchy. Indeed, for $n\geq2$ we have $X\leq_{bT}\emptyset ^{nb}\iff X$ is $\omega^{n}$-c.e. $\iff X\leq_{1}\emptyset ^{nb}$, extending the classical result that $X\leq_{bT}\emptyset '\iff (...) X$ is $\omega$-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every $X$ such that $\emptyset ^{b}\leq_{bT}X\leq_{bT}\emptyset ^{2b}$, there is a $Y\leq_{bT}\emptyset ^{b}$ such that $Y^{b}\equiv_{bT}X$. (shrink)

Terry Horgan and John Tienson have suggested that connectionism might provide a framework within which to articulate a theory of cognition according to which there are mental representations without rules (RWR) (Horgan and Tienson 1988, 1989, 1991, 1992). In essence, RWR states that cognition involves representations in a language of thought, but that these representations are not manipulated by the sort of rules that have traditionally been posited. In the development of RWR, Horgan and Tienson attempt to forestall a particular (...) line of criticism, theSyntactic Argument, which would show RWR to be inconsistent with connectionism. In essence, the argument claims that the node-level rules of connectionist networks, along with the semantic interpretations assigned to patterns of activation, serve to determine a set of representation-level rules incompatible with the RWR conception of cognition. The present paper argues that the Syntactic Argument can be made to show that RWR is inconsistent with connectionism. (shrink)

It is shown that there are incomparable Σ2 e-degrees a, b such that every e-degree strictly less than a is also less than b.

In this paper I argue that Turing’s responses to the mathematical objection are straightforward, despite recent claims to the contrary. I then go on to show that by understanding the importance of learning machines for Turing as related not to the mathematical objection, but to Lady Lovelace’s objection, we can better understand Turing’s response to Lady Lovelace’s objection. Finally, I argue that by understanding Turing’s responses to these objections more clearly, we discover a hitherto unrecognized, substantive thesis in his philosophical (...) thinking about the nature of mind. (shrink)

Turing computation over a non-linguistic domain presupposes a notation for the domain. Accordingly, computability theory studies notations for various non-linguistic domains. It illuminates how different ways of representing a domain support different finite mechanical procedures over that domain. Formal definitions and theorems yield a principled classification of notations based upon their computational properties. To understand computability theory, we must recognize that representation is a key target of mathematical inquiry. We must also recognize that computability theory is an intensional enterprise: it (...) studies entities as represented in certain ways, rather than entities detached from any means of representing them. (shrink)

The class $$\mathsf{TPA}$$ TPA of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of $$\mathsf{TPA}$$ TPA nor the first order theory of $$\mathsf{TPA}$$ TPA are decidable. Moreover, we show that the set of all equations valid in $$\mathsf{TPA}$$ TPA is exactly on the $$\Pi ^1_1$$ Π 1 1 level. We consider the class $$\mathsf{TPA}^-$$ (...) TPA - of the relation algebra reducts of $$\mathsf{TPA}$$ TPA ’s, as well. We prove that the equational theory of $$\mathsf{TPA}^-$$ TPA - is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work. (shrink)

It is shown that for every Turing degree d there is an ω-categorical group G such that the isomorphism type of G is of degree d. We also find an ω-categorical group G such that the isomorphism type of G has no degree.

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.