Bayesian reverse-engineering is a research strategy for developing three-level explanations of behavior and cognition. Starting from a computational-level analysis of behavior and cognition as optimal probabilistic inference, Bayesian reverse-engineers apply numerous tweaks and heuristics to formulate testable hypotheses at the algorithmic and implementational levels. In so doing, they exploit recent technological advances in Bayesian artificial intelligence, machine learning, and statistics, but also consider established principles from cognitive psychology and neuroscience. Although these tweaks and heuristics are highly pragmatic in (...) character and are often deployed unsystematically, Bayesian reverse-engineering avoids several important worries that have been raised about the explanatory credentials of Bayesian cognitive science: the worry that the lower levels of analysis are being ignored altogether; the challenge that the mathematical models being developed are unfalsifiable; and the charge that the terms ‘optimal’ and ‘rational’ have lost their customary normative force. But while Bayesian reverse-engineering is therefore a viable and productive research strategy, it is also no fool-proof recipe for explanatory success. (shrink)

Reverse mathematics is a program in mathematical logic that seeks to give precise answers to the question of which axioms are necessary in order to prove theorems of "ordinary mathematics": roughly speaking, those concerning structures that are either themselves countable, or which can be represented by countable "codes". This includes many fundamental theorems of real, complex, and functional analysis, countable algebra, countable infinitary combinatorics, descriptive set theory, and mathematical logic. This entry aims to give the reader a broad introduction (...) to the history, methodology, and results of the reverse mathematics program, with a particular focus on its connections to foundational programs such as finitism, constructivism, and predicativism. (shrink)

Why do we spend so much of our time passing on amusing anecdotes, making wisecracks,watching The Simpsons? In Inside Jokes, Matthew Hurley, DanielDennett, and Reginald Adams offer an evolutionary and cognitive perspective.

Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$ sentence of a certain form is provable using E-HA ${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the Dialectica interpretation. (...) These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis. (shrink)

In Beyond Neutrality, George Sher criticises the idea that state neutrality between competing conceptions of the good helps protect society from oppression. While he is correct that some governments are non-neutral without being oppressive, I argue that those governments may be neutral at the core of their foundations. The possibility of non-neutrality leading to oppression is further explored; some conceptions of the good would favour oppression while others would not. While it is possible that a non-neutral state may avoid oppression, (...) it is argued that the risks are so great that it is better to bet on government being neutral, thereby minimizing the possibility of oppression. (shrink)

Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied (...) in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert’s program due to Simpson [1988], and predicativism in the extended form due to Feferman and Schütte. -/- Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only omega-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a Pi-1-1 complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than Pi-1-1-CA0. (shrink)

We use methods of reverse mathematics to analyze the proof theoretic strength of a theorem involving the notion of coloring number. Classically, the coloring number of a graph $G=$ is the least cardinal $\kappa$ such that there is a well-ordering of $V$ for which below any vertex in $V$ there are fewer than $\kappa$ many vertices connected to it by $E$. We will study a theorem due to Komjáth and Milner, stating that if a graph is the union of (...) $n$ forests, then the coloring number of the graph is at most $2n$. We focus on the case when $n=1$. (shrink)

The AGM (Alchourrón-Gärdenfors-Makinson) model of belief change is extended to cover changes on sets of beliefs that are not closed under logical consequence (belief bases). Three major types of change operations, namely contraction, internal revision, and external revision are axiomatically characterized, and their interrelations are studied. In external revision, the Levi identity is reversed in the sense that one first adds the new belief to the belief base, and afterwards contracts its negation. It is argued that external revision represents an (...) intuitively plausible way of revising one's beliefs. Since it typically involves the temporary acceptance of an inconsistent set of beliefs, it can only be used in belief representations that distinguish between different inconsistent sets of belief. (shrink)

The all too common sunk cost effect is apparent when an investor influenced by what has been spent already persists in a venture, committing further resources or foregoing more profitable opportunities, when the economically rational action is to quit. Less common but arguably just as much a sunk cost effect is the mistake of giving up on a failed or failing venture too readily, sometimes out of nothing but pique at what has been lost, or perhaps through the more subtle (...) psychological forces posited by Kahneman, Tversky, Thaler and others within prospect theory and related work on ``mental budgeting''. Two case examples are considered, wherein decision makers dissatisfied with the results of their investments, and having lost money, appear to compound their losses by selling out at prices less than their own estimates of the remaining financial worth of the failed assets. These decisions are evaluated from the perspectives of both behavioral and prescriptive economics, and are found to have possible explanations in both. Their prescriptive rationale assumes a portfolio theory of investment decisions, and is demonstrated within both expected utility (economics) and mean-variance (finance) frameworks. (shrink)

Disagreement among experts is a central topic in social epistemology. What should an expert do when confronted with the different opinion of an epistemic peer? Possible answers include the steadfast view (holding to one’s belief), the abstemious view (suspending one’s judgment), and moderate conciliatory views, which specify criteria for belief change when a peer’s different opinion is encountered. The practice of Delphi techniques in healthcare, medicine, and social sciences provides a real-life case study of expert disagreement, where disagreement is gradually (...) transformed into consensus. An analysis of Delphi shows that moderate conciliatory views are descriptively more adequate than rival views. However, it also casts doubt on whether the debate in social epistemology is explanatory relevant vis-à-vis real life cases of expert disagreement, where consensus replaces truth, and acceptance is more explanatorily relevant than belief. (shrink)

Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach’s higher-order RM. As to splittings, there are some examples in RM of theorems A, B, C such that (...) A ↔, that is, A can be split into two independent parts B and C, and the aforementioned topological notions give rise to a number of splittings involving highly natural A, B, C. Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of, for example, the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory RCA 0. We obtain similarly “exceptional” results for the Urysohn identity, the Lindelöf lemma, and partitions of unity. We show that our results exhibit a certain robustness, in that they do not depend on the exact definition of cover, even in the absence of the axiom of choice. (shrink)

If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR 0. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of ℝ.

This paper contains a corrected proof that the statement “every non-empty closed subset of a compact complete separable metric space is separably closed” implies the arithmetical comprehension axiom of reverse mathematics.

Arguments for extended cognition and the extended mind are typically directed at human-centred forms of cognitive extension—forms of cognitive extension in which the cognitive/mental states/processes of a given human individual are subject to a form of extended or wide realization. The same is true of debates and discussions pertaining to the possibility of Web-extended minds and Internet-based forms of cognitive extension. In this case, the focus of attention concerns the extent to which the informational and technological elements of the online (...) environment form part of the machinery of the (individual) human mind. In this paper, we direct attention to a somewhat different form of cognitive extension. In particular, we suggest that the Web allows human individuals to be incorporated into the computational/cognitive routines of online systems. These forms of computational/cognitive extension highlight the potential of the Web and Internet to support bidirectional forms of computational/cognitive incorporation. The analysis of such bidirectional forms of incorporation broadens the scope of philosophical debates in this area, with potentially important implications for our understanding of the foundational notions of extended cognition and the extended mind. (shrink)

${\mathsf {CAC\ for\ trees}}$ is the statement asserting that any infinite subtree of $\mathbb {N}^{<\mathbb {N}}$ has an infinite path or an infinite antichain. In this paper, we study the computational strength of this theorem from a reverse mathematical viewpoint. We prove that ${\mathsf {CAC\ for\ trees}}$ is robust, that is, there exist several characterizations, some of which already appear in the literature, namely, the statement $\mathsf {SHER}$ introduced by Dorais et al. [8], and the statement $\mathsf {TAC}+\mathsf {B}\Sigma (...) ^0_2$ where $\mathsf {TAC}$ is the tree antichain theorem introduced by Conidis [6]. We show that ${\mathsf {CAC\ for\ trees}}$ is computationally very weak, in that it admits probabilistic solutions. (shrink)

Failure to take note of distinctive attributes in the distal stimulus leads to an inadequate proximal encoding. Representation of similarities in Chorus suffers in this regard. Distinctive qualities may require additional complex representation (e.g., reference to linguistic terms) in order to facilitate discrimination. Additional semantic information, which configures proximal attributes, permits accurate identification of true veridical stimuli.

We initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π2 1 comprehension. An MF space is defined to be a topological space of the form MF(P) with the topology generated by $\lbrace N_p \mid p \in P \rbrace$ . Here P is a poset, MF(P) is the set of maximal filters on P, and $N_p = \lbrace F \in MF(P) \mid p \in F \rbrace$ . If the poset P is (...) countable, the space MF(P) is said to be countably based. The class of countably based MF spaces can be defined and discussed within the subsystem ACA0 of second order arithmetic. One can prove within ACA0 that every complete separable metric space is homeomorphic to a countably based MF space which is regular. We show that the converse statement, "every countably based MF space which is regular is homeomorphic to a complete separable metric space," is equivalent to Π2 1-CA0. The equivalence is proved in the weaker system Π1 1-CA0. This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies Π2 1 comprehension. (shrink)

This article explores reverse mission as practised by African Christians in Britain. The main research question is what crucial role does African identity play in African mission in Britain and how does that lead towards developing African British theology? It is argued that such a theology will help African Christians in Britain be affirmed in their cultural identity whilst at the same time reach beyond African communities in their mission engagement. African British theology is related to Black British theology (...) in that they both take the black experience seriously for theological reflection. However, African British theology is also distinct in that it seeks to understand African identity and mission in a postmodern multicultural British society. My research methods have been as an African Practical Theologian involving active participation as well participant observation. My approach has been interdisciplinary engaging the fields of practical theology, diaspora missiology, African theology and Black theology. (shrink)

Many have suggested that the transformation standardly referred to as `time reversal' in quantum theory is not deserving of the name. I argue on the contrary that the standard definition is perfectly appropriate, and is indeed forced by basic considerations about the nature of time in the quantum formalism.

Reverse inference in cognitive neuropsychology has been characterized as inference to ‘psychological processes’ from ‘patterns of activation’ revealed by functional magnetic resonance or other scanning techniques. Several arguments have been provided against the possibility. Focusing on Machery’s presentation, we attempt to clarify the issues, rebut the impossibility arguments, and propose and illustrate a strategy for reverse inference. 1 The Problem of Reverse Inference in Cognitive Neuropsychology2 The Arguments2.1 The anti-Bayesian argument3 Patterns of Activation4 Reverse Inference Practiced5 (...) Seek and Ye Shall Find, Maybe6 Conclusion. (shrink)

Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with two RM-phenomena, namely, splittings and disjunctions. As to splittings, there are some examples in RM of theorems A, B, C such that A↔, that is, A can be split into (...) two independent parts B and C. As to disjunctions, there are examples in RM of theorems D, E, F such that D↔, that is, D can be written as the disjunction of two independent parts E and F. By contrast, we show in this paper that there is a plethora of splittings and disjunctions in Kohlenbach’s higher-order RM. (shrink)

We investigate the reverse-mathematical status of several theorems to the effect that the natural number system is second-order categorical. One of our results is as follows. Define a system to be a triple A,i,f such that A is a set and i∈A and f:A→A. A subset X⊆A is said to be inductive if i∈X and ∀a ∈X). The system A,i,f is said to be inductive if the only inductive subset of A is A itself. Define a Peano system to (...) be an inductive system such that f is one-to-one and i∉the range of f. The standard example of a Peano system is N,0,S where N={0,1,2,…,n,…}=the set of natural numbers and S:N→N is given by S=n+1 for all n∈N. Consider the statement that all Peano systems are isomorphic to N,0,S. We prove that this statement is logically equivalent to WKL0 over RCA0⁎ source. From this and similar equivalences we draw some foundational/philosophical consequences. (shrink)

We examine the reverse mathematics and computability theory of a form of Ramsey's theorem in which the linear n-tuples of a binary tree are colored.

The counterfactual analysis of causation has focused on one particular counterfactual conditional, taking as its starting-point the suggestion that C causes E iff (C E). In this paper, some consequences are explored of reversing this counterfactual, and developing an account starting with the idea that C causes E iff (E C). This suggestion is discussed in relation to the problem of pre-emption. It is found that the 'reversed' counterfactual analysis can handle even the most difficult cases of pre-emption with only (...) minimal complications. The paper closes with a discussion of the wider philosophical implications of developing a reversed counterfactual analysis, especially concerning the differentiation of causes from causal conditions, causation by absences, and the extent to which causes suffice for their effects. (shrink)

Continuous recordings of brain electrical activity were obtained from a group of 176 patients throughout surgical procedures using general anesthesia. Artifact-free data from the 19 electrodes of the International 10/20 System were subjected to quantitative analysis of the electroencephalogram (QEEG). Induction was variously accomplished with etomidate, propofol or thiopental. Anesthesia was maintained throughout the procedures by isoflurane, desflurane or sevoflurane (N = 68), total intravenous anesthesia using propofol (N = 49), or nitrous oxide plus narcotics (N = 59). A set (...) of QEEG measures were found which reversibly displayed high heterogeneity of variance between four states as follows: (1) during induction; (2) just after loss of consciousness (LOC); (3) just before return of consciousness (ROC); (4) just after ROC. Homogeneity of variance across all agents within states was found. Topographic statistical probability images were compared between states. At LOC, power increased in all frequency bands in the power spectrum with the exception of a decrease in gamma activity, and there was a marked anteriorization of power. Additionally, a significant change occurred in hemispheric relationships, with prefrontal and frontal regions of each hemisphere becoming more closely coupled, and anterior and posterior regions on each hemisphere, as well as homologous regions between the two hemispheres, uncoupling. All of these changes reversed upon ROC. Variable resolution electromagnetic tomography (VARETA) was performed to localize salient features of power anteriorization in three dimensions. A common set of neuroanatomical regions appeared to be the locus of the most probable generators of the observed EEG changes. (shrink)

Bostrom and Ord’s reversal test has been appealed to by many philosophers to substantiate the charge that preferences for status quo options are motivated by status quo bias. I argue that their characterization of the reversal test needs to be modified, and that their description of the burden of proof it imposes needs to be clarified. I then argue that there is a way to meet that burden of proof which Bostrom and Ord fail to recognize. I also argue that (...) the range of circumstances in which the reversal test can be usefully applied is narrower than they recognize. (shrink)

This paper uses the framework of reverse mathematics to analyze the proof theoretic content of several statements concerning multiplication of countable well-orderings. In particular, a division algorithm for ordinal arithmetic is shown to be equivalent to the subsystem ATR0.

Perceptual learning can be defined as practice-induced improvement in the ability to perform specific perceptual tasks. We previously proposed the Reverse Hierarchy Theory as a unifying concept that links behavioral findings of visual learning with physiological and anatomical data. Essentially, it asserts that learning is a top-down guided process, which begins at high-level areas of the visual system, and when these do not suffice, progresses backwards to the input levels, which have a better signal-to-noise ratio. This simple concept has (...) proved powerful in explaining a broad range of findings, including seemingly contradicting data. We now extend this concept to describe the dynamics of skill acquisition and interpret recent behavioral and electrophysiological findings. (shrink)

Applying speech act theory to the phenomenon of hate speech, some philosophers seek to explain how even ordinary people can obtain the capacity, power, or authority to oppress, subordinate, or marginalise the targets of their verbal attacks. Such explanations are answers to what is called the authority problem. However, hitherto these philosophers have focused exclusively on standard examples of racist speech in which members of historically oppressor groups verbally attack members of oppressed groups. In this paper, I address the (or (...) an) authority problem in relation to examples of reverse hate speech, where members of historically oppressed groups verbally attack members of oppressor groups. I critically examine two conventional answers to the authority problem: the first is that even ordinary people can successfully perform prototypical hate speech acts by virtue of being participants in norm-governed social practices of racism; the second is that such people can do so by virtue of the silence of bystanders granting them licensed authority. I argue these conventional answers have lesser explanatory heft as applied to examples of reverse hate speech. I then introduce and defend a third answer to the authority problem, which I believe can adequately account for examples of both standard and reverse hate speech, namely the possession of informal social authority. (shrink)

In this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.

In this paper, we investigate the logical strength of completeness theorems for intuitionistic logic along the program of reverse mathematics. Among others we show that is equivalent over to the strong completeness theorem for intuitionistic logic: any countable theory of intuitionistic predicate logic can be characterized by a single Kripke model.

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. (...) In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics. (shrink)

'The arrow of time' refers to the curious asymmetry that distinguishes the future from the past. Reversing the Arrow of Time argues that there is an intimate link between the symmetries of 'time itself' and time reversal symmetry in physical theories, which has wide-ranging implications for both physics and its philosophy. This link helps to clarify how we can learn about the symmetries of our world, how to understand the relationship between symmetries and what is real, and how to overcome (...) pervasive illusions about the direction of time. Roberts explains the significance of time reversal in a way that intertwines physics and philosophy, to establish what the arrow of time means and how we can come to know it. This book is both mathematically and philosophically rigorous yet remains accessible to advanced undergraduates in physics and the philosophy of physics. This title is also available as Open Access on Cambridge Core. (shrink)

A theory is usually said to be time reversible if whenever a sequence of states S 1, S 2, S 3 is possible according to that theory, then the reverse sequence of time reversed states S 3 T, S 2 T, S 1 T is also possible according to that theory; i.e., one normally not only inverts the sequence of states, but also operates on the states with a time reversal operator T. David Albert and Paul Horwich have suggested (...) that one should not allow such time reversal operations T on states. I will argue that time reversal operations on fundamental states should be allowed. I will furthermore argue that the form that time reversal operations take is determined by the type of fundamental geometric quantities that occur in nature and that we have good reason to believe that the fundamental geometric quantities that occur in nature correspond to irreducible representations of the Lorentz transformations. Finally, I will argue that we have good reason to believe that space-time has a temporal orientation. (shrink)

We will overview the results in an informal approach to constructive reverse mathematics, that is reverse mathematics in Bishop’s constructive mathematics, especially focusing on compactness properties and continuous properties.